2002 Formulae and Tables

FORMULAE AND TABLES for EXAMINATIONS of THE FACULTY OF ACTUARIES and THE INSTITUTE OF ACTUARIES

2002

FORMULAE AND TABLES for EXAMINATIONS of THE FACULTY OF ACTUARIES and THE INSTITUTE OF ACTUARIES

2002

This Edition 2002 Ó The Faculty of Actuaries and The Institute of Actuaries No part of this publication may be reproduced in any material form, whether by publication, translation, storage in a retrieval system or transmission by electronic, mechanical, photocopying, recording or other means, without the prior permission of the owners of the copyright. Acknowledgments: The Faculty of Actuaries and The Institute of Actuaries would like to thank the following people who have helped in the preparation of this material: D Hopkins M Z Khorasanee W F Scott The Faculty of Actuaries and The Institute of Actuaries is licensed by FTSE International (“FTSE”) to publish the FTSE 100 indices. All information is provided for information purposes only. Every effort is made to ensure that all information given in this publication is accurate, but no responsibility or liability can be accepted by FTSE for any errors or omissions or for any loss arising from use of this publication. All copyright and database rights in the FTSE 100 indices belong to FTSE or its licensors. Redistribution of the data comprising the FTSE 100 indices is not permitted. The Faculty of Actuaries and The Institute of Actuaries gratefully acknowledge the permission of CRC Press to reproduce the diagram on page 20 adapted from the publication “CRC Standard Probability and Statistics Tables and Formulae” edited by Stephen Kokoska. The Faculty of Actuaries and The Institute of Actuaries acknowledge the permission to reproduce English Life Tables No. 15 (Males and Females). Crown Copyright material is reproduced with the permission of the Controller of HMSO and the Queen’s printer for Scotland. The Faculty of Actuaries and The Institute of Actuaries gratefully acknowledge the permission of Lindley & Scott New Cambridge Statistical Tables, 2nd Edition, 1995, Tables 4, 5, 7, 8, 9, 12, 12a, 12b, 12c and 12d — Cambridge University Press.

ISBN 0 901066 57 5

PREFACE This new edition of the Formulae and Tables represents a considerable overhaul of its predecessor “green book” first published in 1980. The contents have been updated to reflect more fully the evolving syllabus requirements of the profession, and also in the case of the Tables to reflect more contemporary experience and methods. Correspondingly, there has been some modest removal of material which has either become redundant with syllabus changes or obviated by the availability of pocket calculators. As in the predecessor book, it is important to note that these tables have been produced for the sole use of examination candidates. The profession does not express any opinion whatsoever as to the circumstances in which any of the tables may be suitable for other uses.

FORMULAE This section is intended to help candidates with formulae that may be hard to remember. Derivations of these formulae may still be required under the relevant syllabuses. Contents

Page

Mathematical Methods Statistical Distributions Statistical Methods Compound Interest Survival Models Annuities and Assurances Stochastic Processes Time Series Economic Models Financial Derivatives

2 6 22 31 32 36 38 40 43 45

Note. In these tables, log denotes logarithms to base e.

1

1

MATHEMATICAL METHODS

1.1

SERIES Exponential function x 2 x3 exp( x) = e = 1 + x + + +L 2! 3! x

Natural log function x 2 x3 log(1 + x) = ln(1 + x) = x + - L ( -1 < x £ 1 ) 2 3

Binomial expansion æ nö æ nö (a + b) n = a n + ç ÷ a n -1b + ç ÷ a n - 2b 2 + L + b n è 1ø è 2ø

where n is a positive integer (1 + x) p = 1 + px +

p ( p - 1) 2 p ( p - 1)( p - 2) 3 x + x +L 2! 3! ( -1 < x < 1)

2

1.2

CALCULUS Taylor series (one variable) h2 f ( x + h) = f ( x) + hf ¢( x) + f ¢¢( x) + L 2!

Taylor series (two variables) f ( x + h, y + k ) = f ( x, y ) + h f x¢( x, y ) + k f y¢ ( x, y ) +

(

)

1 2 h f xx ¢¢ ( x, y ) + 2hk f xy ¢¢ ( x, y ) + k 2 f yy ¢¢ ( x, y ) + L 2!

Integration by parts dv

b

b

b

du

ò a u dx dx = [uv]a - ò a v dx dx Double integrals (changing the order of integration) bæ

x

òa èòa b

òa

b b f ( x, y ) dy ö dx = ò æ ò f ( x, y )dx ö dy or ø ø aè y x

b

b

a

a

y

dx ò dy f ( x, y ) = ò dy ò dx f ( x, y )

The domain of integration here is the set of values ( x, y ) for which a £ y £ x £ b. Differentiating an integral b( y )

d f ( x, y )dx = b ¢( y ) f [b( y ), y ] - a ¢( y ) f [a ( y ), y ] dy a (òy ) b( y )

+

¶ f ( x, y )dx ¶ y a( y)

ò

3

1.3

SOLVING EQUATIONS Newton-Raphson method If x is a sufficiently good approximation to a root of the equation f ( x) = 0 then (provided convergence occurs) a better approximation is x* = x -

f ( x) . f ¢( x)

Integrating factors The integrating factor for solving the differential equation dy + P ( x) y = Q ( x) is: dx exp

(ò P( x)dx)

Second-order difference equations The general solution of the difference equation axn + 2 + bxn +1 + cxn = 0 is: if b 2 - 4ac > 0 : xn = Al1n + Bl n2 (distinct real roots, l1 ¹ l 2 ) if b 2 - 4ac = 0 : xn = ( A + Bn)l n (equal real roots, l1 = l 2 = l ) if b 2 - 4ac < 0 : xn = r n ( A cos nq + B sin nq) (complex roots, l1 = l 2 = reiq ) where l1 and l 2 are the roots of the quadratic equation al 2 + bl + c = 0 .

4

1.4

GAMMA FUNCTION Definition ¥

G ( x) = ò t x -1e -t dt , x > 0 0

Properties G ( x) = ( x - 1)G ( x - 1) G (n) = (n - 1)! , n = 1,2,3,K G (½) = p

1.5

BAYES’ FORMULA Let A1, A2 , ..., An be a collection of mutually exclusive and exhaustive events with P( Ai ) ¹ 0, i = 1, 2, …, n. For any event B such that P(B) ¹ 0: P( Ai | B ) =

n

P ( B | Ai ) P ( Ai )

, i = 1, 2, ..., n.

å P( B | A j ) P ( A j ) j =1

5

2

STATISTICAL DISTRIBUTIONS Notation PF PDF DF PGF MGF

= Probability function, p ( x) = Probability density function, f ( x) = Distribution function, F ( x) = Probability generating function, G ( s ) = Moment generating function, M (t )

Note. Where formulae have been omitted below, this indicates that (a) there is no simple formula or (b) the function does not have a finite value or (c) the function equals zero. 2.1

DISCRETE DISTRIBUTIONS Binomial distribution Parameters:

n , p ( n = positive integer, 0 < p < 1 with q = 1 - p )

PF:

æ nö p ( x) = ç ÷ p x q n - x , x = 0,1, 2,K, n è xø

DF:

The distribution function is tabulated in the statistical tables section.

PGF:

G ( s ) = (q + ps ) n

MGF:

M (t ) = (q + pet )n

Moments:

E ( X ) = np , var( X ) = npq

Coefficient of skewness:

6

q- p npq

Bernoulli distribution The Bernoulli distribution is the same as the binomial distribution with parameter n = 1 . Poisson distribution Parameter:

m (m > 0)

PF:

p ( x) =

DF:

The distribution function is tabulated in the statistical tables section.

PGF:

G ( s ) = em ( s -1)

MGF:

M (t ) = e m ( e

Moments:

E ( X ) = m , var( X ) = m

e -m m x , x = 0,1, 2,... x!

t

-1)

Coefficient of skewness:

1 m

7

Negative binomial distribution – Type 1 Parameters:

k , p ( k = positive integer, 0 < p < 1 with q = 1 - p )

PF:

æ x - 1ö k x - k , x = k , k + 1, k + 2,K p ( x) = ç p q è k - 1÷ø

PGF:

æ ps ö G (s) = ç è 1 - qs ÷ø

MGF:

æ pet ö M (t ) = ç ÷ è 1 - qet ø

Moments:

E( X ) =

Coefficient of skewness:

8

2- p kq

k

k

k kq , var( X ) = 2 p p

Negative binomial distribution – Type 2 Parameters:

k , p ( k > 0 , 0 < p < 1 with q = 1 - p )

PF:

p ( x) =

PGF:

æ p ö G (s) = ç è 1 - qs ÷ø

MGF:

æ p ö M (t ) = ç ÷ è 1 - qet ø

Moments:

E( X ) =

G (k + x) p k q x , x = 0,1, 2,K G ( x + 1)G (k ) k

k

kq kq , var( X ) = 2 p p

Coefficient of skewness:

2- p kq

Geometric distribution The geometric distribution is the same as the negative binomial distribution with parameter k = 1.

9

Uniform distribution (discrete) Parameters: a , b , h ( a < b , h > 0 , b - a is a multiple of h )

2.2

h , x = a, a + h, a + 2h,K, b - h, b b-a+h

PF:

p ( x) =

PGF:

æ sb + h - s a ö h G (s) = b - a + h çè s h - 1 ÷ø

MGF:

M (t ) =

Moments:

1 1 E ( X ) = (a + b) , var( X ) = (b - a )(b - a + 2h) 2 12

æ e(b + h)t - eat ö h b - a + h çè eht - 1 ÷ø

CONTINUOUS DISTRIBUTIONS Standard normal distribution – N (0,1) Parameters: none PDF:

1 - 12 x 2 f ( x) = e , -¥ < x < ¥ 2p

DF:

The distribution function is tabulated in the statistical tables section.

MGF:

M (t ) = e 2

Moments:

E ( X ) = 0 , var( X ) = 1

1t2

E( X r ) =

10

1 2

r 2

G (1 + r ) , r = 2,4,6,K æ rö G ç1 + ÷ è 2ø

Normal (Gaussian) distribution – N (m, s 2 ) Parameters:

m , s2 ( s > 0 )

PDF:

ìï 1 æ x - m ö 2 üï 1 f ( x) = exp í - ç ÷ø ý , -¥ < x < ¥ è s 2 s 2p îï þï m t + 12 s 2t 2

MGF:

M (t ) = e

Moments:

E ( X ) = m , var( X ) = s 2

Exponential distribution Parameter:

l (l > 0)

PDF:

f ( x) = le -lx , x > 0

DF:

F ( x) = 1 - e -lx

MGF:

tö æ M (t ) = ç 1 - ÷ è lø

Moments:

E( X ) =

-1

, t 0, l > 0)

PDF:

l a a-1 -lx f ( x) = x e , x>0 G (a )

DF:

When 2a is an integer, probabilities for the gamma distribution can be found using the relationship: 2lX ~ c 22a

MGF:

tö æ M (t ) = ç 1 - ÷ è lø

Moments:

E( X ) =

-a

, t 0)

PDF:

f ( x) =

Moments:

E( X ) =

G (a + b) a-1 x (1 - x)b-1 , 0 < x < 1 G (a)G (b) a ab , var( X ) = a +b (a + b) 2 (a + b + 1)

E( X r ) =

G (a + b) G (a + r ) , r = 1, 2,3,K G (a ) G (a + b + r )

Coefficient of skewness:

2(b - a) a + b + 1 (a + b + 2) ab

13

Lognormal distribution Parameters:

m , s2 ( s > 0 )

PDF:

ìï 1 æ log x - m ö 2 üï 1 1 exp í - ç f ( x) = ÷ø ý , x > 0 è 2 s s 2p x ïî ïþ

Moments:

E( X ) = e r

m+ 12 s 2

E( X ) = e

Coefficient 2 of skewness: æes + 2 è

)

)

2 2 , var( X ) = e2m+s æes - 1 è

r m+ 12 r 2 s 2

, r = 1, 2,3,K

2

es - 1

Pareto distribution (two parameter version) Parameters:

a , l (a > 0, l > 0)

al a

PDF:

f ( x) =

DF:

æ l ö F ( x) = 1 - ç è l + x ø÷

Moments:

E( X ) =

(l + x)a+1

, x>0 a

al 2 l (a > 1) , var( X ) = (a > 2) 2 a -1 (a - 1) (a - 2)

E( X r ) =

G (a - r )G (1 + r ) r l , r = 1, 2,3,K , r < a G (a )

Coefficient of skewness:

14

2(a + 1) a - 2 (a > 3) (a - 3) a

Pareto distribution (three parameter version) Parameters: a , l , k ( a > 0 , l > 0 , k > 0 ) G (a + k )l a x k -1

PDF:

f ( x) =

Moments:

kl k (k + a - 1)l 2 ( a > 1 ), var( X ) = (a > 2) E( X ) = a -1 (a - 1) 2 (a - 2)

G (a)G (k )(l + x)a+ k

E( X r ) =

, x>0

G (a - r ) G ( k + r ) r l , r = 1, 2,3,K , r < a G (a ) G ( k )

Weibull distribution Parameters: c , g ( c > 0 , g > 0 ) g

PDF:

f ( x) = c g x g -1e - cx , x > 0

DF:

F ( x) = 1 - e - cx

Moments:

æ rö 1 E ( X r ) = G ç1 + ÷ r g , r = 1, 2,3,K è gøc

g

Burr distribution Parameters: a , l , g ( a > 0 , l > 0 , g > 0 ) ag l a x g -1

PDF:

f ( x) =

DF:

æ l ö F ( x) = 1 - ç è l + x g ÷ø

Moments:

æ r ö æ r ö lr g E ( X ) = G ç a - ÷ G ç1 + ÷ , r = 1, 2,3,K , r < ag è g ø è g ø G (a )

(l + x g )a+1

, x>0 a

r

15

2.3

COMPOUND DISTRIBUTIONS Conditional expectation and variance E (Y ) = E[ E (Y | X )]

var(Y ) = var[ E (Y | X )] + E[var(Y | X )]

Moments of a compound distribution If X1, X 2 ,K are IID random variables with MGF M X (t ) and N is an independent nonnegative integer-valued random variable, then S = X1 + L + X N (with S = 0 when N = 0 ) has the following properties: Mean:

E (S ) = E ( N )E ( X )

Variance:

var( S ) = E ( N ) var( X ) + var( N )[ E ( X )]2

MGF:

M S (t ) = M N [log M X (t )]

Compound Poisson distribution Mean:

lm1

Variance:

lm2

Third central moment:

lm3

where l = E ( N ) and mr = E ( X r )

16

Recursive formulae for integer-valued distributions (a, b,0) class of distributions

Let g r = P( S = r ) , r = 0,1,2,K and f j = P( X = j ) , j = 1,2,3,K . bö æ If pr = P ( N = r ) , where pr = ç a + ÷ pr -1 , r = 1, 2,3,K , then è rø g 0 = p0 and g r =

r

æ

å çè a + j =1

bj ö ÷ f j g r - j , r = 1, 2,3,K rø

Compound Poisson distribution If N has a Poisson distribution with mean l , then a = 0 and b = l , and g0 = e

-l

l r and g r = å j f j g r - j , r = 1, 2,3,K r j =1

17

2.4

TRUNCATED MOMENTS Normal distribution If f ( x) is the PDF of the N (m, s 2 ) distribution, then U

òL

x f ( x)dx = m [F(U ¢) - F( L ¢)] - s [f(U ¢) - f( L ¢)]

where L ¢ =

L-m U -m and U ¢ = . s s

Lognormal distribution If f ( x) is the PDF of the lognormal distribution with parameters m and s 2 , then

ò

U k x L

where Lk =

18

f ( x)dx = ek m+½ k

2 2

s

[F(U k ) - F( Lk )]

log L - m log U - m - k s and U k = - ks . s s

19

20

X +k

X -k

Type 1 Type 2

Geometric p

min X i

Bernoulli p

Neg.Binomial k, p

kq m= p k ®¥

Poisson m

m = np n®¥

Binomial n, p

S X i (same p)

Uniform a, b, h

X +k

S X i Type 1 Type 2 (same p ) X -k

k =1

SXi

SXi (same p )

n =1

DISCRETE

Normal m, s²

log X

ab

m=

a a +b

S (ai + bi X i )

eX

h®0

X -a b-a Uniform a, b

Uniform 0, 1

a = b =1

Beta a, b

b ®¥

a ® ¥,

(a + b ) 2 (a + b + 1)

SXi (same p )

s =

2

s2 = m m®¥

n®¥

s 2 = np(1 - p)

m = np

SXi

-

SXi (same l )

a =1

X2 F n1, n2

min X i

Exponential l

l =½

n1 X , n 2 ® ¥

X1 n1 X 2 n2

t n

n2 n1

g =1

X1 g

l=

k = 12 n1

a = 12 n 2

CONTINUOUS

n®¥

n=2

1 ln X l

Gamma a, l

2l X

Chi-square n

X1 X1 + X 2

1 2

a + (b - a ) X

l =1

l=

Normal 0, 1

S Xi2

a = 12 n

m +sX

X -m s

SXi (same l )

Lognormal m, s²

P ai Xi

bi

RELATIONSHIPS BETWEEN STATISTICAL DISTRIBUTIONS

Weibull c, g

Pareto a , l, k

k =1

X1 g Pareto a, l

Burr a , l, g

g =1

21

For example, the arrow marked “ m = np , n ® ¥ ” connecting the binomial distribution to the Poisson distribution means: For large values of n , the binomial distribution with parameters n and p will approximate to the Poisson distribution with parameter m , where m = np .

Limiting cases (indicated by dotted lines)

Similarly, “ P X i ” and “ min X i ” denote the product and the minimum of a fixed set of independent random variables. Where a sum or product includes “ ai ” or “ bi ”, these denote arbitrary constants.

For example, the arrow marked “ S X i (same p )” connecting the binomial distribution to itself means: The sum of a fixed number of independent random variables, each having a binomial distribution with the same value for the parameter p , also has a binomial distribution.

Sums, products and minimum values

Note that the parameters of the transformed distributions may differ from those of the basic distributions shown.

Transformations For example, the arrow marked “ e X ” connecting the normal distribution to the lognormal distribution means: If X has a normal distribution, the function e X will have a lognormal distribution.

For example, the arrow marked “ n = 1 ” connecting the binomial distribution to the Bernoulli distribution means: In the special case where n = 1 , the binomial distribution is equivalent to a Bernoulli distribution.

Special cases

The distribution diagram shows the main interrelationships between the distributions in the statistics section. The relationships shown are of four types:

EXPLANATION OF THE DISTRIBUTION DIAGRAM

3

STATISTICAL METHODS

3.1

SAMPLE MEAN AND VARIANCE The random sample ( x1, x2 ,K, xn ) has the following sample moments: Sample mean:

1 n x = å xi n i =1

ü 1 ïì n 2 2ï Sample variance: s = íå xi - nx ý n - 1 ïîi =1 ïþ 2

3.2

PARAMETRIC INFERENCE (NORMAL MODEL) One sample For a single sample of size n under the normal model X ~ N (m, s 2 ) : (n - 1) S 2 X -m ~ c n2 -1 ~ tn -1 and 2 S n s

Two samples For two independent samples of sizes m and n under the normal models X ~ N (m X , s 2X ) and Y ~ N (mY , sY2 ) : S X2 s 2X SY2 sY2

22

~ Fm -1, n -1

Under the additional assumption that s 2X = sY2 : ( X - Y ) - (m X - mY ) ~ tm + n - 2 1 1 Sp + m n

where S 2p = variance. 3.3

{

}

1 (m - 1) S X2 + (n - 1) SY2 is the pooled sample m+n-2

MAXIMUM LIKELIHOOD ESTIMATORS Asymptotic distribution If qˆ is the maximum likelihood estimator of a parameter q based on a sample X , then qˆ is asymptotically normally distributed with mean q and variance equal to the Cramér-Rao lower bound CRLB (q) = -1

é ¶2 ù E ê 2 log L(q, X ) ú êë ¶q úû

Likelihood ratio test æ max L ö H0 ÷ ~ c 2 approximately (under H ) -2(l p - l p + q ) = -2log ç q 0 ç max L ÷ è H 0 È H1 ø

where l p = max log L H0

is the maximum log-likelihood for the model under H 0 (in which there are p free parameters)

and

l p + q = max log L H 0 È H1

is the maximum log-likelihood for the model under H 0 È H1 (in which there are p + q free parameters).

23

3.4

LINEAR REGRESSION MODEL WITH NORMAL ERRORS Model Yi ~ N (a + bxi , s 2 ) , i = 1, 2,K, n

Intermediate calculations n

n

s xx = å ( xi - x ) = å xi2 - nx 2 2

i =1

i =1

n

n

s yy = å ( yi - y ) = å yi2 - ny 2 2

i =1

i =1

n

n

i =1

i =1

s xy = å ( xi - x )( yi - y ) = å xi yi - n x y

Parameter estimates s xy aˆ = y - bˆ x , bˆ = s xx 2 ö s xy 1 n 1 æ 2 s = å ( yi - yˆi ) = n - 2 ç s yy - s ÷ n - 2 i =1 xx ø è

ˆ2

Distribution of bˆ bˆ - b ˆ2

s s xx

24

~ tn - 2

Variance of predicted mean response ìï 1 ( x0 - x )2 üï 2 ˆ var(aˆ + bx0 ) = í + ýs s xx ïþ ïî n

An additional s 2 must be added to obtain the variance of the predicted individual response. Testing the correlation coefficient r=

s xy s xx s yy

If r = 0 , then

r n-2 1- r

2

~ tn - 2 .

Fisher Z transformation 1 ö æ zr ~ N ç zr , approximately è n - 3 ÷ø æ1 + rö æ1 + r ö and zr = tanh -1 r = 12 log ç . where zr = tanh -1 r = 12 log ç ÷ è1 - r ø è 1 - r ÷ø

Sum of squares relationship n

n

n

i =1

i =1

i =1

å ( yi - y )2 = å ( yi - yˆi )2 + å ( yˆi - y )2

25

3.5

ANALYSIS OF VARIANCE Single factor normal model Yij ~ N (m + ti , s 2 ) , i = 1, 2,K, k , j = 1,2,K, ni k

where n = å ni , with i =1

k

å ni ti = 0 i =1

Intermediate calculations (sums of squares) k ni

k ni

SST = å å ( yij - ygg ) = å å

Total:

2

i =1 j =1

i =1 j =1 k 2 yi g

yij2

ygg2 n

ygg2 Between treatments: SS B = å ni ( yi g - ygg ) = å n n i i =1 i =1 k

SS R = SST - SS B

Residual: Variance estimate sˆ 2 =

SS R n-k

Statistical test Under the appropriate null hypothesis: SS B SS R ~ Fk -1,n - k k -1 n - k

26

2

3.6

GENERALISED LINEAR MODELS Exponential family For a random variable Y from the exponential family, with natural parameter q and scale parameter f : Probability (density) function: Mean:

E (Y ) = b ¢(q)

Variance:

var(Y ) = a(f)b ¢¢(q)

é yq - b(q) ù fY ( y; q, f) = exp ê + c ( y , f) ú ë a ( f) û

Canonical link functions m 1- m

Binomial:

g (m ) = log

Poisson:

g (m ) = log m

Normal:

g (m ) = m

Gamma:

g (m ) =

1 m

27

3.7

BAYESIAN METHODS Relationship between posterior and prior distributions Posterior µ Prior ´ Likelihood

The posterior distribution f (q | x ) for the parameter q is related to the prior distribution f (q) via the likelihood function f ( x | q) : f (q | x ) µ f (q) ´ f ( x | q)

Normal / normal model If x is a random sample of size n from a N (m, s 2 ) distribution, where s 2 is known, and the prior distribution for the parameter m is N (m 0 , s 02 ) , then the posterior distribution for m is:

m | x ~ N (m* , s*2 ) æ nx m ö where m* = ç 2 + 02 ÷ s0 ø ès

28

æ n æ n 1 ö 1 ö 2 + s = + and 1 * ç 2 ÷ ç 2 ÷ s 02 ø s 02 ø ès ès

3.8

EMPIRICAL BAYES CREDIBILITY – MODEL 1 Data requirements { X ij , i = 1, 2,K, N , j = 1, 2,K, n} X ij represents the aggregate claims in the j th year from the i th risk.

Intermediate calculations 1 n 1 N X i = å X ij , X = å X i n j =1 N i =1

Parameter estimation Quantity

Estimator

E[m(q)]

X

E[ s (q)]

ü 1 N ìï 1 n 2ï ( ) X X å í å ij i ý N i =1 ï n - 1 j =1 ïþ î

var[m(q)]

ü 1 N 1 N ìï 1 n 2 2ï ( ) ( ) X X X X å i å í å ij i ý N - 1 i =1 Nn i =1ï n - 1 j =1 ïþ î

2

Credibility factor n

Z= n+

E[ s 2 (q)] var[m(q)]

29

3.9

EMPIRICAL BAYES CREDIBILITY – MODEL 2 Data requirements {Yij , i = 1,2,K, N , j = 1,2,K, n} , {Pij , i = 1, 2,K, N , j = 1,2,K, n} Yij represents the aggregate claims in the j th year from the i th risk; Pij is the corresponding risk volume.

Intermediate calculations n

1 N æ Pi ö Pi = å Pij , P = å Pi , P* = å Pi çè1 - P ÷ø 1 Nn j =1 i =1 i =1 X ij =

Yij Pij

N

, Xi =

n

Pij X ij

j =1

Pi

å

N n

Pij X ij

, X =åå

P

i =1 j =1

Parameter estimation Quantity

Estimator

E[m(q)]

X

ü 1 N ìï 1 n 2ï ( ) P X X å í å ij ij i ý N i =1ï n - 1 j =1 ïþ î

2

E[ s (q)]

æ 1 N n å å Pij ( X ij P * çè Nn - 1 1

var[m(q)]

i =1 j =1

Credibility factor n

å Pij Zi =

j =1

n

å Pij + j =1

30

E[ s 2 (q)] var[m(q)]

- X)

2

ì 1 n - åí å Pij ( X ij N i =1 î n - 1 j =1 1

N

- Xi )

2

üö ý÷ þø

4

COMPOUND INTEREST Increasing/decreasing annuity functions ( Ia )n =

a&&n - nv n i

, ( Da ) n =

n - an i

Accumulation factor for variable interest rates æ t2 ö A(t1, t2 ) = exp ç ò d(t )dt ÷ èç t1 ø÷

31

5

SURVIVAL MODELS

5.1

MORTALITY “LAWS” Survival probabilities æ t ö t p x = exp ç - ò m x + s ds ÷ è 0 ø

Gompertz’ Law m x = Bc x ,

c t px = g

x

(ct -1)

where g = e - B log c

Makeham’s Law m x = A + Bc x ,

t c t px = s g

x

(ct -1)

where s = e - A

Gompertz-Makeham formula The Gompertz-Makeham graduation formula, denoted by GM(r , s) , states that m x = poly1(t ) + exp[ poly2 (t )]

where t is a linear function of x and poly1(t ) and poly2 (t ) are polynomials of degree r and s respectively.

32

5.2

EMPIRICAL ESTIMATION Greenwood’s formula for the variance of the Kaplan-Meier estimator 2 var[ F% (t )] = éë1 - Fˆ (t ) ùû

å

t j £t

dj n j (n j - d j )

Variance of the Nelson-Aalen estimate of the integrated hazard % ]= var[ L t

5.3

å

d j (n j - d j )

t j £t

n3j

MORTALITY ASSUMPTIONS Balducci assumption 1- t q x + t

5.4

= (1 - t )q x (x is an integer, 0 £ t £ 1)

GENERAL MARKOV MODEL Kolmogorov forward differential equation ¶ gh t px = ¶t

å ( t pxgj m xjh+t - t pxghm hjx +t ) j¹h

33

5.5

GRADUATION TESTS Grouping of signs test If there are n1 positive signs and n2 negative signs and G denotes the observed number of positive runs, then: æ n1 - 1ö æ n2 + 1ö çè t - 1 ÷ø çè t ÷ø and, approximately, P (G = t ) = æ n1 + n2 ö çè n ÷ø 1

æ n1(n2 + 1) (n1n2 )2 ö G~ Nç , ÷ è n1 + n2 (n1 + n2 )3 ø

Critical values for the grouping of signs test are tabulated in the statistical tables section for small values of n1 and n2 . For larger values of n1 and n2 the normal approximation can be used. Serial correlation test

rj »

1 m- j

m- j

å ( zi - z )( zi + j - z ) i =1

1 m ( zi - z ) 2 å m i =1

1 m where z = å zi m i =1

r j ´ m ~ N (0,1) approximately.

Variance adjustment factor

rx =

å i 2pi i

å ip i i

where pi is the proportion of lives at age x who have exactly i policies. 34

5.6

MULTIPLE DECREMENT TABLES For a multiple decrement table with three decrements a , b and g , each uniform over the year of age ( x, x + 1) in its single decrement table, then (aq )ax = q xa é1 - 12 (qbx + q xg ) + 13 qbx q xg ù ë û

5.7

POPULATION PROJECTION MODELS Logistic model 1 dP(t ) r = r - kP(t ) has general solution P(t ) = P (t ) dt Cr e -rt + k

where C is a constant.

35

6

ANNUITIES AND ASSURANCES

6.1

APPROXIMATIONS FOR NON ANNUAL ANNUITIES a&&x(m ) » a&&x -

m -1 2m

a&&( m) » a&&x:n x:n

6.2

m - 1 æ Dx + n ö 12m èç Dx ø÷

MOMENTS OF ANNUITIES AND ASSURANCES Let K x and Tx denote the curtate and complete future lifetimes (respectively) of a life aged exactly x . Whole life assurances E[v K x +1] = Ax , var[v K x +1] = 2 Ax - ( Ax )2 E[vTx ] = Ax , var[vTx ] = 2 Ax - ( Ax )2

Similar relationships hold for endowment assurances (with status L x:n ), pure endowments (with status x:1n ), term assurances (with status

1 x:n

) and deferred whole life assurances (with status

m| L x ).

Whole life annuities E[a&&K

x +1

] = a&&x , var[a&&K

x +1

E[aT ] = a x , var[aT ] = x

x

2

]=

2

Ax - ( Ax )2 d2

Ax - ( Ax )2 d2

Similar relationships hold for temporary annuities (with status L x:n ).

36

6.3

PREMIUMS AND RESERVES Premium conversion relationship between annuities and assurances Ax = 1 - da&&x , Ax = 1 - d a x

Similar relationships hold for endowment assurance policies (with status L x:n ). Net premium reserve tVx = 1 -

a&&x + t a , tVx = 1 - x + t a&&x ax

Similar formulae hold for endowment assurance policies (with statuses L x:n and L x + t:n - t ). 6.4

THIELE’S DIFFERENTIAL EQUATION Whole life assurance ¶ V = d tVx + Px - (1 - tVx )m x + t ¶t t x

Similar formulae hold for other types of policies. Multiple state model

(

¶ j j j jk jk k j tVx = dtVx + bx + t - å m x + t bx + t + tVx - tVx ¶t k¹ j

)

37

7

STOCHASTIC PROCESSES

7.1

MARKOV “JUMP” PROCESSES Kolmogorov differential equations Forward equation:

¶ p ( s, t ) = å pik ( s, t )s kj (t ) ¶t ij k ÎS

Backward equation:

¶ p ( s, t ) = - å s ik ( s ) pkj ( s, t ) ¶s ij k ÎS

where s ij (t ) is the transition rate from state i to state j ( j ¹ i ) at time t , and s ii = - å s ij . j ¹i

Expected time to reach a subsequent state k mi =

7.2

1 + li

å

j ¹ i, j ¹ k

s ij li

m j , where l i = å s ij j ¹i

BROWNIAN MOTION AND RELATED PROCESSES Martingales for standard Brownian motion If {Bt , t ³ 0} is a standard Brownian motion, then the following processes are martingales: Bt , Bt2 - t and exp(lBt - 12 l 2t )

Distribution of the maximum value P éê max ( Bs + ms ) > ë 0£ s £ t

38

æ - y + mt ö æ - y - mt ö y ùú = F ç + e2my F ç ÷ ÷, y>0 è è û t ø t ø

Hitting times If t y = min{s : Bs + ms = y} where m > 0 and y < 0 , then s³0

E[e

-lt y

] = e y (m+

m2 + 2l )

, l>0

Ornstein-Uhlenbeck process dX t = -gX t dt + sdBt , g > 0

7.3

MONTE CARLO METHODS Box-Muller formulae If U1 and U 2 are independent random variables from the U (0,1) distribution then Z1 = -2log U1 cos(2pU 2 ) and Z 2 = -2log U1 sin(2pU 2 )

are independent standard normal variables. Polar method If V1 and V2 are independent random variables from the U ( -1,1) distribution and S = V12 + V22 then, conditional on 0 < S £ 1 , Z1 = V1

-2log S -2log S and Z 2 = V2 S S

are independent standard normal variables. Pseudorandom values from the U (0,1) distribution and the N (0,1) distribution are included in the statistical tables section.

39

8

TIME SERIES

8.1

TIME SERIES – TIME DOMAIN Sample autocovariance and autocorrelation function Autocovariance:

1 n 1 n gˆ k = å ( xt - mˆ )( xt - k - mˆ ) , where mˆ = å xt n t = k +1 n t =1

Autocorrelation:

rˆ k =

gˆ k gˆ 0

Autocorrelation function for ARMA(1,1) For the process X t = aX t -1 + et + bet -1 : rk =

(1 + ab)(a + b) 2

(1 + b + 2ab)

a k -1 , k = 1, 2,3,K

Partial autocorrelation function f1 = r1 , f2 =

fk =

r2 - r12

1 - r12

det Pk* , k = 2,3,K , det Pk

r1 r2 æ 1 ç r 1 r1 ç 1 1 r1 where Pk = ç r2 ç M M M ç çè r k -1 rk - 2 rk - 3

L rk -1 ö L rk - 2 ÷ ÷ L rk - 3 ÷ O M ÷ ÷ L 1 ÷ø

and Pk* equals Pk , but with the last column replaced with (r1, r2 , r3 , ..., rk )T .

40

Partial autocorrelation function for MA(1) For the process X t = m + et + bet -1 : 2 k k +1 (1 - b )b fk = ( -1) 2( k +1)

1- b

8.2

, k = 1, 2,3,K

TIME SERIES – FREQUENCY DOMAIN Spectral density function f (w ) =

1 ¥ - ik w å e g k , -p < w < p 2p k =-¥

Inversion formula gk = ò

p

-p

eik w f (w )d w

Spectral density function for ARMA(p,q) The spectral density function of the process f( B )( X t - m ) = q( B )et , where var(et ) = s 2 , is s 2 q(eiw )q(e - iw ) f (w ) = 2 p f ( ei w ) f ( e - i w )

Linear filters For the linear filter Yt =

¥

å

k =-¥

ak X t - k :

2

fY (w ) = A(w ) f X (w ) ,

where A(w ) =

¥

å

k =-¥

e - ik w ak is the transfer function for the filter.

41

8.3

TIME SERIES – BOX-JENKINS METHODOLOGY Ljung and Box “portmanteau” test of the residuals for an ARMA(p,q) model rk2 2 ~ cm n(n + 2) å -( p + q) k =1 n - k m

where rk ( k = 1,2,K, m ) is the estimated value of the k th autocorrelation coefficient of the residuals and n is the number of data values used in the ARMA( p, q ) series. Turning point test In a sequence of n independent random variables the number of turning points T is such that: E (T ) = 23 (n - 2) and var(T ) =

42

16n - 29 90

9

ECONOMIC MODELS

9.1

UTILITY THEORY Utility functions Exponential: U ( w) = - e - aw ,

a>0

Logarithmic: U ( w) = log w Power:

U ( w) = g -1 ( wg - 1) , g £ 1 , g ¹ 0

Quadratic:

U ( w) = w + dw2 , d < 0

Measures of risk aversion

9.2

U ¢¢( w) U ¢( w)

Absolute risk aversion:

A( w) = -

Relative risk aversion:

R( w) = w A( w)

CAPITAL ASSET PRICING MODEL (CAPM) Security market line Ei - r = bi ( EM - r ) where bi =

cov( Ri , RM ) var( RM )

Capital market line (for efficient portfolios) E P - r = ( EM - r )

sP sM

43

9.3

INTEREST RATE MODELS Spot rates and forward rates for zero-coupon bonds Let P(t ) be the price at time 0 of a zero-coupon bond that pays 1 unit at time t . Let s (t ) be the spot rate for the period (0, t ) . Let f (t ) be the instantaneous forward rate at time 0 for time t . Spot rate 1 P(t ) = e -ts ( t ) or s (t ) = - log P (t) t

Instantaneous forward rate t d P(t ) = exp æ - ò f ( s )ds ö or f (t ) = - log P (t ) è 0 ø dt

Vasicek model Instantaneous forward rate f (t ) = e -at R + (1 - e -at ) L +

b -at e (1 - e -at ) a

Price of a zero-coupon bond b é ù P(t ) = exp ê - D(t ) R - (t - D(t )) L - D(t ) 2 ú 2 ë û

1 - e -at where D(t ) = a

44

10

FINANCIAL DERIVATIVES Note. In this section, q denotes the (continuously-payable) dividend rate.

10.1

PRICE OF A FORWARD OR FUTURES CONTRACT For an asset with fixed income of present value I: F = ( S0 - I )erT

For an asset with dividends: F = S0e( r - q )T

10.2

BINOMIAL PRICING (“TREE”) MODEL Risk-neutral probabilities er Dt - d , Up-step probability = u-d

where u » es and

d » e -s

Dt + q Dt Dt + q Dt

.

45

10.3

STOCHASTIC DIFFERENTIAL EQUATIONS Generalised Wiener process dx = adt + bdz

where a and b are constant and dz is the increment for a Wiener process (standard Brownian motion). Ito process dx = a ( x, t )dt + b( x, t )dz

Ito’s lemma for a function G(x, t) æ ¶G 1 2 ¶ 2G ¶G ö ¶G dG = ç a + b + dt + b dz ÷ 2 ¶ x 2 ¶ t ¶ x ¶x è ø

Models for the short rate rt

10.4

Ho-Lee:

dr = q(t )dt + sdz

Hull-White:

dr = [q(t ) - ar ]dt + sdz

Vasicek:

dr = a (b - r )dt + sdz

Cox-Ingersoll-Ross:

dr = a (b - r )dt + s rdz

BLACK-SCHOLES FORMULAE FOR EUROPEAN OPTIONS Geometric Brownian motion model for a stock price S t dSt = St (mdt + sdz )

Black-Scholes partial differential equation ¶f ¶f 1 2 2 ¶ 2 f + (r - q ) St + s St = rf ¶t ¶St 2 ¶St2

46

Garman-Kohlhagen formulae for the price of call and put options Call: ct = St e - q (T - t ) F(d1) - Ke - r (T -t )F(d 2 ) Put:

pt = Ke - r (T - t )F( - d 2 ) - St e - q (T - t ) F( - d1)

log( St K ) + (r - q + ½s 2 )(T - t ) where d1 = s T -t

and 10.5

log( St K ) + (r - q - ½s 2 )(T - t ) d2 = = d1 - s T - t s T -t

PUT-CALL PARITY RELATIONSHIP ct + Ke - r (T - t ) = pt + St e - q (T -t )

47

48

COMPOUND INTEREST TABLES

49

Compound Interest ½% i

0.005 000 (2)

0.004 994

i (4)

0.004 991

i i

(12)

0.004 989

d

0.004 988

(1 + i )1 2

1.002 497

(1 + i )1 4

1.001 248

1/12

(1 + i )

1.000 416

v

0.995 025

12

v

14

v

v1 12

d d

(2)

d (4) d (12) i / i (2)

0.997 509 0.998 754 0.999 584 0.004 975 0.004 981 0.004 984 0.004 987

i / i (12)

1.001 248 1.001 873 1.002 290

i/d

1.002 498

i / i (4)

i/d

(2)

1.003 748

i/d

(4)

1.003 123

i / d (12)

50

1.002 706

n

(1 + i )n

1 2 3 4 5

vn

sn

an

( Ia )n

( Da) n

n

1.005 00 1.010 03 1.015 08 1.020 15 1.025 25

0.995 02 0.990 07 0.985 15 0.980 25 0.975 37

1.000 0 2.005 0 3.015 0 4.030 1 5.050 3

0.995 0 1.985 1 2.970 2 3.950 5 4.925 9

0.995 0 2.975 2 5.930 6 9.851 6 14.728 5

0.995 0 2.980 1 5.950 4 9.900 9 14.826 7

1 2 3 4 5

6 7 8 9 10

1.030 38 1.035 53 1.040 71 1.045 91 1.051 14

0.970 52 0.965 69 0.960 89 0.956 10 0.951 35

6.075 5 7.105 9 8.141 4 9.182 1 10.228 0

5.896 4 6.862 1 7.823 0 8.779 1 9.730 4

20.551 6 27.311 4 34.998 5 43.603 4 53.116 9

20.723 1 27.585 2 35.408 2 44.187 2 53.917 6

6 7 8 9 10

11 12 13 14 15

1.056 40 1.061 68 1.066 99 1.072 32 1.077 68

0.946 61 0.941 91 0.937 22 0.932 56 0.927 92

11.279 2 12.335 6 13.397 2 14.464 2 15.536 5

10.677 0 11.618 9 12.556 2 13.488 7 14.416 6

63.529 7 74.832 5 87.016 4 100.072 2 113.990 9

64.594 7 76.213 6 88.769 7 102.258 4 116.675 1

11 12 13 14 15

16 17 18 19 20

1.083 07 1.088 49 1.093 93 1.099 40 1.104 90

0.923 30 0.918 71 0.914 14 0.909 59 0.905 06

16.614 2 17.697 3 18.785 8 19.879 7 20.979 1

15.339 9 16.258 6 17.172 8 18.082 4 18.987 4

128.763 7 144.381 7 160.836 2 178.118 4 196.219 6

132.015 0 148.273 6 165.446 4 183.528 8 202.516 2

16 17 18 19 20

21 22 23 24 25

1.110 42 1.115 97 1.121 55 1.127 16 1.132 80

0.900 56 0.896 08 0.891 62 0.887 19 0.882 77

22.084 0 23.194 4 24.310 4 25.432 0 26.559 1

19.888 0 20.784 1 21.675 7 22.562 9 23.445 6

215.131 4 234.845 1 255.352 4 276.644 9 298.714 2

222.404 1 243.188 2 264.863 9 287.426 8 310.872 4

21 22 23 24 25

26 27 28 29 30

1.138 46 1.144 15 1.149 87 1.155 62 1.161 40

0.878 38 0.874 01 0.869 66 0.865 33 0.861 03

27.691 9 28.830 4 29.974 5 31.124 4 32.280 0

24.324 0 25.198 0 26.067 7 26.933 0 27.794 1

321.552 1 345.150 3 369.500 9 394.595 6 420.426 5

335.196 4 360.394 4 386.462 1 413.395 2 441.189 2

26 27 28 29 30

31 32 33 34 35

1.167 21 1.173 04 1.178 91 1.184 80 1.190 73

0.856 75 0.852 48 0.848 24 0.844 02 0.839 82

33.441 4 34.608 6 35.781 7 36.960 6 38.145 4

28.650 8 29.503 3 30.351 5 31.195 5 32.035 4

446.985 6 474.265 1 502.257 1 530.953 8 560.347 6

469.840 0 499.343 3 529.694 8 560.890 4 592.925 7

31 32 33 34 35

36 37 38 39 40

1.196 68 1.202 66 1.208 68 1.214 72 1.220 79

0.835 64 0.831 49 0.827 35 0.823 23 0.819 14

39.336 1 40.532 8 41.735 4 42.944 1 44.158 8

32.871 0 33.702 5 34.529 9 35.353 1 36.172 2

590.430 8 621.195 9 652.635 2 684.741 4 717.506 9

625.796 8 659.499 3 694.029 1 729.382 2 765.554 4

36 37 38 39 40

41 42 43 44 45

1.226 90 1.233 03 1.239 20 1.245 39 1.251 62

0.815 06 0.811 01 0.806 97 0.802 96 0.798 96

45.379 6 46.606 5 47.839 6 49.078 8 50.324 2

36.987 3 37.798 3 38.605 3 39.408 2 40.207 2

750.924 5 784.986 9 819.686 7 855.016 9 890.970 3

802.541 7 840.340 0 878.945 3 918.353 5 958.560 7

41 42 43 44 45

46 47 48 49 50

1.257 88 1.264 17 1.270 49 1.276 84 1.283 23

0.794 99 0.791 03 0.787 10 0.783 18 0.779 29

51.575 8 52.833 7 54.097 8 55.368 3 56.645 2

41.002 2 927.539 8 41.793 2 964.718 4 42.580 3 1 002.499 1 43.363 5 1 040.875 1 44.142 8 1 079.839 4

999.562 9 1 041.356 1 1 083.936 4 1 127.299 9 1 171.442 7

46 47 48 49 50

60 70 80 90 100

1.348 85 1.417 83 1.490 34 1.566 55 1.646 67

0.741 37 69.770 0 0.705 30 83.566 1 0.670 99 98.067 7 0.638 34 113.310 9 0.607 29 129.333 7

51.725 6 58.939 4 65.802 3 72.331 3 78.542 6

1 654.887 8 60 2 212.116 5 70 2 839.538 9 80 3 533.740 1 90 4 291.471 0 100

1 500.371 4 1 972.582 2 2 490.447 8 3 048.408 2 3 641.336 1

Compound Interest n

(1 + i )n

vn

sn

an

( Ia )n

( Da) n

n

1 2 3 4 5

1.010 00 1.020 10 1.030 30 1.040 60 1.051 01

0.990 10 0.980 30 0.970 59 0.960 98 0.951 47

1.000 0 2.010 0 3.030 1 4.060 4 5.101 0

0.990 1 1.970 4 2.941 0 3.902 0 4.853 4

0.990 1 2.950 7 5.862 5 9.706 4 14.463 7

0.990 1 2.960 5 5.901 5 9.803 4 14.656 9

1 2 3 4 5

6 7 8 9 10

1.061 52 1.072 14 1.082 86 1.093 69 1.104 62

0.942 05 0.932 72 0.923 48 0.914 34 0.905 29

6.152 0 7.213 5 8.285 7 9.368 5 10.462 2

5.795 5 6.728 2 7.651 7 8.566 0 9.471 3

20.116 0 26.645 0 34.032 9 42.261 9 51.314 8

20.452 4 27.180 5 34.832 2 43.398 2 52.869 5

6 7 8 9 10

11 12 13 14 15

1.115 67 1.126 83 1.138 09 1.149 47 1.160 97

0.896 32 0.887 45 0.878 66 0.869 96 0.861 35

11.566 8 12.682 5 13.809 3 14.947 4 16.096 9

10.367 6 11.255 1 12.133 7 13.003 7 13.865 1

61.174 4 71.823 8 83.246 4 95.425 8 108.346 1

63.237 2 74.492 3 86.626 0 99.629 7 113.494 7

11 12 13 14 15

16 17 18 19 20

1.172 58 1.184 30 1.196 15 1.208 11 1.220 19

0.852 82 0.844 38 0.836 02 0.827 74 0.819 54

17.257 9 18.430 4 19.614 7 20.810 9 22.019 0

14.717 9 15.562 3 16.398 3 17.226 0 18.045 6

121.991 2 136.345 6 151.394 0 167.121 0 183.511 9

128.212 6 143.774 9 160.173 1 177.399 2 195.444 7

16 17 18 19 20

21 22 23 24 25

1.232 39 1.244 72 1.257 16 1.269 73 1.282 43

0.811 43 0.803 40 0.795 44 0.787 57 0.779 77

23.239 2 24.471 6 25.716 3 26.973 5 28.243 2

18.857 0 19.660 4 20.455 8 21.243 4 22.023 2

200.551 9 218.226 7 236.521 8 255.423 4 274.917 6

214.301 7 233.962 1 254.417 9 275.661 3 297.684 4

21 22 23 24 25

26 27 28 29 30

1.295 26 1.308 21 1.321 29 1.334 50 1.347 85

0.772 05 0.764 40 0.756 84 0.749 34 0.741 92

29.525 6 30.820 9 32.129 1 33.450 4 34.784 9

22.795 2 23.559 6 24.316 4 25.065 8 25.807 7

294.990 9 315.629 8 336.821 2 358.552 1 380.809 8

320.479 6 344.039 2 368.355 7 393.421 5 419.229 2

26 27 28 29 30

31 32 33 34 35

1.361 33 1.374 94 1.388 69 1.402 58 1.416 60

0.734 58 0.727 30 0.720 10 0.712 97 0.705 91

36.132 7 37.494 1 38.869 0 40.257 7 41.660 3

26.542 3 27.269 6 27.989 7 28.702 7 29.408 6

403.581 7 426.855 4 450.618 8 474.859 9 499.566 9

445.771 5 473.041 1 501.030 7 529.733 4 559.142 0

31 32 33 34 35

36 37 38 39 40

1.430 77 1.445 08 1.459 53 1.474 12 1.488 86

0.698 92 0.692 00 0.685 15 0.678 37 0.671 65

43.076 9 44.507 6 45.952 7 47.412 3 48.886 4

30.107 5 30.799 5 31.484 7 32.163 0 32.834 7

524.728 2 550.332 4 576.368 2 602.824 6 629.690 7

589.249 5 620.049 0 651.533 7 683.696 7 716.531 4

36 37 38 39 40

41 42 43 44 45

1.503 75 1.518 79 1.533 98 1.549 32 1.564 81

0.665 00 0.658 42 0.651 90 0.645 45 0.639 05

50.375 2 51.879 0 53.397 8 54.931 8 56.481 1

33.499 7 34.158 1 34.810 0 35.455 5 36.094 5

656.955 9 684.609 5 712.641 2 741.040 8 769.798 2

750.031 1 784.189 2 818.999 2 854.454 6 890.549 2

41 42 43 44 45

46 47 48 49 50

1.580 46 1.596 26 1.612 23 1.628 35 1.644 63

0.632 73 0.626 46 0.620 26 0.614 12 0.608 04

58.045 9 59.626 3 61.222 6 62.834 8 64.463 2

36.727 2 37.353 7 37.974 0 38.588 1 39.196 1

798.903 7 927.276 4 828.347 5 964.630 1 858.120 0 1 002.604 1 888.211 8 1 041.192 1 918.613 7 1 080.388 2

46 47 48 49 50

60 70 80 90 100

1.816 70 2.006 76 2.216 72 2.448 63 2.704 81

0.550 45 0.498 31 0.451 12 0.408 39 0.369 71

81.669 7 100.676 3 121.671 5 144.863 3 170.481 4

44.955 0 50.168 5 54.888 2 59.160 9 63.028 9

1 237.761 2 1 578.816 0 1 934.765 3 2 299.728 4 2 668.804 6

1% i

0.010 000

i

(2)

0.009 975

i

(4)

0.009 963

i

(12)

0.009 954

d

0.009 950

(1 + i )1 2

1.004 988

(1 + i )1 4

1.002 491

1/12

(1 + i )

1.000 830

v

0.990 099

12

0.995 037

14

0.997 516

1 12

0.999 171

v v v

d

0.009 901 (2)

0.009 926

d (4)

0.009 938

d

(12)

0.009 946

i / i (2)

1.002 494

i / i (4)

1.003 742

d

i/i

(12)

i/d

1.004 575 1.004 992

i / d (2)

1.007 494

(4)

1.006 242

i/d

i / d (12)

1.005 408

1 504.496 2 60 1 983.148 6 70 2 511.179 4 80 3 083.911 9 90 3 697.112 1 100

51

Compound Interest 1½% 0.015 000

i (2)

0.014 944

i (4)

0.014 916

i i

(12)

0.014 898

d

0.014 889

(1 + i )1 2

1.007 472

(1 + i )1 4

1.003 729

1/12

(1 + i )

1.001 241

v

0.985 222

12

0.992 583

14

v

0.996 285

v1 12

0.998 760

d

0.014 778

v

(2)

0.014 833

d (4)

0.014 861

d (12)

0.014 879

i / i (2)

1.003 736

i / i (4)

1.005 608

i / i (12)

1.006 857

i/d

1.007 481

d

i/d

(2)

1.011 236

i/d

(4)

1.009 358

i / d (12)

52

1.008 107

n

(1 + i )n

vn

sn

an

( Ia )n

( Da) n

n

1 2 3 4 5

1.015 00 1.030 23 1.045 68 1.061 36 1.077 28

0.985 22 0.970 66 0.956 32 0.942 18 0.928 26

1.000 0 2.015 0 3.045 2 4.090 9 5.152 3

0.985 2 1.955 9 2.912 2 3.854 4 4.782 6

0.985 2 2.926 5 5.795 5 9.564 2 14.205 5

0.985 2 2.941 1 5.853 3 9.707 7 14.490 3

1 2 3 4 5

6 7 8 9 10

1.093 44 1.109 84 1.126 49 1.143 39 1.160 54

0.914 54 0.901 03 0.887 71 0.874 59 0.861 67

6.229 6 7.323 0 8.432 8 9.559 3 10.702 7

5.697 2 6.598 2 7.485 9 8.360 5 9.222 2

19.692 8 26.000 0 33.101 7 40.973 0 49.589 7

20.187 5 26.785 7 34.271 7 42.632 2 51.854 4

6 7 8 9 10

11 12 13 14 15

1.177 95 1.195 62 1.213 55 1.231 76 1.250 23

0.848 93 0.836 39 0.824 03 0.811 85 0.799 85

11.863 3 13.041 2 14.236 8 15.450 4 16.682 1

10.071 1 10.907 5 11.731 5 12.543 4 13.343 2

58.927 9 68.964 6 79.676 9 91.042 8 103.040 6

61.925 5 72.833 0 84.564 5 97.107 9 110.451 1

11 12 13 14 15

16 17 18 19 20

1.268 99 1.288 02 1.307 34 1.326 95 1.346 86

0.788 03 0.776 39 0.764 91 0.753 61 0.742 47

17.932 4 19.201 4 20.489 4 21.796 7 23.123 7

14.131 3 14.907 6 15.672 6 16.426 2 17.168 6

115.649 1 128.847 6 142.616 0 156.934 6 171.784 0

124.582 4 139.490 0 155.162 6 171.588 8 188.757 4

16 17 18 19 20

21 22 23 24 25

1.367 06 1.387 56 1.408 38 1.429 50 1.450 95

0.731 50 0.720 69 0.710 04 0.699 54 0.689 21

24.470 5 25.837 6 27.225 1 28.633 5 30.063 0

17.900 1 18.620 8 19.330 9 20.030 4 20.719 6

187.145 5 203.000 6 219.331 4 236.120 5 253.350 6

206.657 6 225.278 4 244.609 2 264.639 6 285.359 3

21 22 23 24 25

26 27 28 29 30

1.472 71 1.494 80 1.517 22 1.539 98 1.563 08

0.679 02 0.668 99 0.659 10 0.649 36 0.639 76

31.514 0 32.986 7 34.481 5 35.998 7 37.538 7

21.398 6 22.067 6 22.726 7 23.376 1 24.015 8

271.005 2 289.067 8 307.522 6 326.354 0 345.546 8

306.757 9 328.825 5 351.552 2 374.928 3 398.944 1

26 27 28 29 30

31 32 33 34 35

1.586 53 1.610 32 1.634 48 1.659 00 1.683 88

0.630 31 0.620 99 0.611 82 0.602 77 0.593 87

39.101 8 40.688 3 42.298 6 43.933 1 45.592 1

24.646 1 25.267 1 25.879 0 26.481 7 27.075 6

365.086 4 384.958 2 405.148 1 425.642 4 446.427 7

423.590 3 448.857 4 474.736 4 501.218 1 528.293 7

31 32 33 34 35

36 37 38 39 40

1.709 14 1.734 78 1.760 80 1.787 21 1.814 02

0.585 09 0.576 44 0.567 92 0.559 53 0.551 26

47.276 0 48.985 1 50.719 9 52.480 7 54.267 9

27.660 7 28.237 1 28.805 1 29.364 6 29.915 8

467.490 9 488.819 3 510.400 5 532.222 2 554.272 7

555.954 4 584.191 5 612.996 6 642.361 1 672.277 0

36 37 38 39 40

41 42 43 44 45

1.841 23 1.868 85 1.896 88 1.925 33 1.954 21

0.543 12 0.535 09 0.527 18 0.519 39 0.511 71

56.081 9 57.923 1 59.792 0 61.688 9 63.614 2

30.459 0 30.994 1 31.521 2 32.040 6 32.552 3

576.540 4 599.014 2 621.683 0 644.536 1 667.563 3

702.735 9 733.730 0 765.251 2 797.291 9 829.844 2

41 42 43 44 45

46 47 48 49 50

1.983 53 2.013 28 2.043 48 2.074 13 2.105 24

0.504 15 0.496 70 0.489 36 0.482 13 0.475 00

65.568 4 67.551 9 69.565 2 71.608 7 73.682 8

33.056 5 33.553 2 34.042 6 34.524 7 34.999 7

690.754 3 862.900 7 714.099 3 896.453 9 737.588 7 930.496 4 761.213 1 965.021 1 784.963 3 1 000.020 8

46 47 48 49 50

60 70 80 90 100

2.443 22 2.835 46 3.290 66 3.818 95 4.432 05

0.409 30 0.352 68 0.303 89 0.261 85 0.225 63

96.214 7 122.363 8 152.710 9 187.929 9 228.803 0

39.380 3 43.154 9 46.407 3 49.209 9 51.624 7

1 027.547 7 1 274.320 7 1 519.481 4 1 758.753 7 1 989.075 3

1 374.648 7 60 1 789.675 2 70 2 239.511 8 80 2 719.343 0 90 3 225.019 8 100

Compound Interest n

(1 + i )n

1 2 3 4 5

1.020 00 1.040 40 1.061 21 1.082 43 1.104 08

6 7 8 9 10

vn

sn

an

( Ia )n

( Da) n

n

0.980 39 0.961 17 0.942 32 0.923 85 0.905 73

1.000 0 2.020 0 3.060 4 4.121 6 5.204 0

0.980 4 1.941 6 2.883 9 3.807 7 4.713 5

0.980 4 2.902 7 5.729 7 9.425 1 13.953 7

0.980 4 2.922 0 5.805 8 9.613 6 14.327 0

1 2 3 4 5

1.126 16 1.148 69 1.171 66 1.195 09 1.218 99

0.887 97 0.870 56 0.853 49 0.836 76 0.820 35

6.308 1 7.434 3 8.583 0 9.754 6 10.949 7

5.601 4 6.472 0 7.325 5 8.162 2 8.982 6

19.281 6 25.375 5 32.203 4 39.734 2 47.937 7

19.928 5 26.400 4 33.725 9 41.888 2 50.870 7

6 7 8 9 10

11 12 13 14 15

1.243 37 1.268 24 1.293 61 1.319 48 1.345 87

0.804 26 0.788 49 0.773 03 0.757 88 0.743 01

12.168 7 13.412 1 14.680 3 15.973 9 17.293 4

9.786 8 10.575 3 11.348 4 12.106 2 12.849 3

56.784 6 66.246 5 76.295 9 86.906 2 98.051 4

60.657 6 71.232 9 82.581 3 94.687 6 107.536 8

11 12 13 14 15

16 17 18 19 20

1.372 79 1.400 24 1.428 25 1.456 81 1.485 95

0.728 45 0.714 16 0.700 16 0.686 43 0.672 97

18.639 3 20.012 1 21.412 3 22.840 6 24.297 4

13.577 7 14.291 9 14.992 0 15.678 5 16.351 4

109.706 5 121.847 3 134.450 2 147.492 3 160.951 8

121.114 5 135.406 4 150.398 4 166.076 9 182.428 3

16 17 18 19 20

21 22 23 24 25

1.515 67 1.545 98 1.576 90 1.608 44 1.640 61

0.659 78 0.646 84 0.634 16 0.621 72 0.609 53

25.783 3 27.299 0 28.845 0 30.421 9 32.030 3

17.011 2 17.658 0 18.292 2 18.913 9 19.523 5

174.807 1 189.037 5 203.623 1 218.544 4 233.782 7

199.439 5 217.097 6 235.389 8 254.303 7 273.827 2

21 22 23 24 25

26 27 28 29 30

1.673 42 1.706 89 1.741 02 1.775 84 1.811 36

0.597 58 0.585 86 0.574 37 0.563 11 0.552 07

33.670 9 35.344 3 37.051 2 38.792 2 40.568 1

20.121 0 20.706 9 21.281 3 21.844 4 22.396 5

249.319 8 265.138 0 281.220 5 297.550 8 314.112 9

293.948 2 314.655 1 335.936 4 357.780 8 380.177 2

26 27 28 29 30

31 32 33 34 35

1.847 59 1.884 54 1.922 23 1.960 68 1.999 89

0.541 25 0.530 63 0.520 23 0.510 03 0.500 03

42.379 4 44.227 0 46.111 6 48.033 8 49.994 5

22.937 7 23.468 3 23.988 6 24.498 6 24.998 6

330.891 5 347.871 8 365.039 3 382.380 3 399.881 3

403.114 9 426.583 3 450.571 8 475.070 4 500.069 0

31 32 33 34 35

36 37 38 39 40

2.039 89 2.080 69 2.122 30 2.164 74 2.208 04

0.490 22 0.480 61 0.471 19 0.461 95 0.452 89

51.994 4 54.034 3 56.114 9 58.237 2 60.402 0

25.488 8 25.969 5 26.440 6 26.902 6 27.355 5

417.529 3 435.311 9 453.217 0 471.233 0 489.348 6

525.557 9 551.527 3 577.968 0 604.870 6 632.226 0

36 37 38 39 40

41 42 43 44 45

2.252 20 2.297 24 2.343 19 2.390 05 2.437 85

0.444 01 0.435 30 0.426 77 0.418 40 0.410 20

62.610 0 64.862 2 67.159 5 69.502 7 71.892 7

27.799 5 28.234 8 28.661 6 29.080 0 29.490 2

507.553 0 525.835 8 544.186 9 562.596 5 581.055 3

660.025 5 688.260 3 716.921 9 746.001 8 775.492 0

41 42 43 44 45

46 47 48 49 50

2.486 61 2.536 34 2.587 07 2.638 81 2.691 59

0.402 15 0.394 27 0.386 54 0.378 96 0.371 53

74.330 6 76.817 2 79.353 5 81.940 6 84.579 4

29.892 3 30.286 6 30.673 1 31.052 1 31.423 6

599.554 4 618.085 0 636.638 8 655.207 8 673.784 2

805.384 3 835.670 9 866.344 0 897.396 1 928.819 7

46 47 48 49 50

60 70 80 90 100

3.281 03 3.999 56 4.875 44 5.943 13 7.244 65

0.304 78 0.250 03 0.205 11 0.168 26 0.138 03

114.051 5 149.977 9 193.772 0 247.156 7 312.232 3

34.760 9 37.498 6 39.744 5 41.586 9 43.098 4

858.458 4 1 037.332 9 1 206.531 3 1 363.757 0 1 507.851 1

2% i

0.020 000

i

(2)

0.019 901

i

(4)

0.019 852

i

(12)

0.019 819

d

0.019 803

(1 + i )1 2

1.009 950

(1 + i )1 4

1.004 963

1/12

(1 + i )

1.001 652

v

0.980 392

12

0.990 148

14

0.995 062

1 12

0.998 351

v v v

d

0.019 608 (2)

0.019 705

d (4)

0.019 754

d

(12)

0.019 786

i / i (2)

1.004 975

i / i (4)

1.007 469

d

i/i

(12)

i/d

1.009 134 1.009 967

i / d (2)

1.014 975

(4)

1.012 469

i/d

i / d (12)

1.010 801

1 261.955 7 60 1 625.069 0 70 2 012.774 3 80 2 420.653 5 90 2 845.082 4 100

53

Compound Interest 2½% i

0.025 000 (2)

0.024 846

i (4)

0.024 769

i i

(12)

0.024 718

d

0.024 693

(1 + i )1 2

1.012 423

(1 + i )1 4

1.006 192

1/12

(1 + i )

1.002 060

v

0.975 610

12

0.987 730

14

v

0.993 846

v1 12

0.997 944

d

0.024 390

v

(2)

0.024 541

d (4)

0.024 617

d (12)

0.024 667

i / i (2)

1.006 211

i / i (4)

1.009 327

i / i (12)

1.011 407

i/d

1.012 449

d

i/d

(2)

1.018 711

i/d

(4)

1.015 577

i / d (12)

54

1.013 491

n

(1 + i )n

vn

1 2 3 4 5

1.025 00 1.050 63 1.076 89 1.103 81 1.131 41

0.975 61 0.951 81 0.928 60 0.905 95 0.883 85

6 7 8 9 10

1.159 69 1.188 69 1.218 40 1.248 86 1.280 08

11 12 13 14 15

an

( Ia )n

( Da) n

n

1.000 0 2.025 0 3.075 6 4.152 5 5.256 3

0.975 6 1.927 4 2.856 0 3.762 0 4.645 8

0.975 6 2.879 2 5.665 0 9.288 8 13.708 1

0.975 6 2.903 0 5.759 1 9.521 0 14.166 9

1 2 3 4 5

0.862 30 0.841 27 0.820 75 0.800 73 0.781 20

6.387 7 7.547 4 8.736 1 9.954 5 11.203 4

5.508 1 6.349 4 7.170 1 7.970 9 8.752 1

18.881 9 24.770 7 31.336 7 38.543 3 46.355 3

19.675 0 26.024 4 33.194 5 41.165 4 49.917 4

6 7 8 9 10

1.312 09 1.344 89 1.378 51 1.412 97 1.448 30

0.762 14 0.743 56 0.725 42 0.707 73 0.690 47

12.483 5 13.795 6 15.140 4 16.519 0 17.931 9

9.514 2 10.257 8 10.983 2 11.690 9 12.381 4

54.738 9 63.661 5 73.092 0 83.000 2 93.357 2

59.431 7 69.689 4 80.672 6 92.363 5 104.744 9

11 12 13 14 15

16 17 18 19 20

1.484 51 1.521 62 1.559 66 1.598 65 1.638 62

0.673 62 0.657 20 0.641 17 0.625 53 0.610 27

19.380 2 20.864 7 22.386 3 23.946 0 25.544 7

13.055 0 13.712 2 14.353 4 14.978 9 15.589 2

104.135 2 115.307 5 126.848 5 138.733 5 150.938 9

117.799 9 131.512 1 145.865 5 160.844 3 176.433 5

16 17 18 19 20

21 22 23 24 25

1.679 58 1.721 57 1.764 61 1.808 73 1.853 94

0.595 39 0.580 86 0.566 70 0.552 88 0.539 39

27.183 3 28.862 9 30.584 4 32.349 0 34.157 8

16.184 5 16.765 4 17.332 1 17.885 0 18.424 4

163.442 0 176.221 0 189.255 1 202.524 1 216.008 8

192.618 1 209.383 5 226.715 6 244.600 6 263.024 9

21 22 23 24 25

26 27 28 29 30

1.900 29 1.947 80 1.996 50 2.046 41 2.097 57

0.526 23 0.513 40 0.500 88 0.488 66 0.476 74

36.011 7 37.912 0 39.859 8 41.856 3 43.902 7

18.950 6 19.464 0 19.964 9 20.453 5 20.930 3

229.690 9 243.552 7 257.577 3 271.748 5 286.050 8

281.975 6 301.439 6 321.404 5 341.858 0 362.788 3

26 27 28 29 30

31 32 33 34 35

2.150 01 2.203 76 2.258 85 2.315 32 2.373 21

0.465 11 0.453 77 0.442 70 0.431 91 0.421 37

46.000 3 48.150 3 50.354 0 52.612 9 54.928 2

21.395 4 21.849 2 22.291 9 22.723 8 23.145 2

300.469 3 314.990 0 329.599 2 344.284 0 359.032 0

384.183 7 406.032 9 428.324 8 451.048 5 474.193 7

31 32 33 34 35

36 37 38 39 40

2.432 54 2.493 35 2.555 68 2.619 57 2.685 06

0.411 09 0.401 07 0.391 28 0.381 74 0.372 43

57.301 4 59.733 9 62.227 3 64.783 0 67.402 6

23.556 3 23.957 3 24.348 6 24.730 3 25.102 8

373.831 3 388.670 8 403.539 6 418.427 6 433.324 8

497.750 0 521.707 3 546.055 9 570.786 2 595.889 0

36 37 38 39 40

41 42 43 44 45

2.752 19 2.821 00 2.891 52 2.963 81 3.037 90

0.363 35 0.354 48 0.345 84 0.337 40 0.329 17

70.087 6 72.839 8 75.660 8 78.552 3 81.516 1

25.466 1 25.820 6 26.166 4 26.503 8 26.833 0

448.222 0 463.110 4 477.981 4 492.827 2 507.640 1

621.355 1 647.175 7 673.342 2 699.846 0 726.679 0

41 42 43 44 45

46 47 48 49 50

3.113 85 3.191 70 3.271 49 3.353 28 3.437 11

0.321 15 0.313 31 0.305 67 0.298 22 0.290 94

84.554 0 87.667 9 90.859 6 94.131 1 97.484 3

27.154 2 27.467 5 27.773 2 28.071 4 28.362 3

522.412 8 537.138 5 551.810 7 566.423 3 580.970 4

753.833 2 781.300 7 809.073 9 837.145 2 865.507 5

46 47 48 49 50

60 4.399 79 70 5.632 10 80 7.209 57 90 9.228 86 100 11.813 72

0.227 28 0.177 55 0.138 70 0.108 36 0.084 65

135.991 6 185.284 1 248.382 7 329.154 3 432.548 7

sn

30.908 7 721.774 3 32.897 9 851.662 1 34.451 8 968.669 9 35.665 8 1 072.215 7 36.614 1 1 162.588 8

1 163.653 7 60 1 484.085 7 70 1 821.927 3 80 2 173.369 3 90 2 535.435 8 100

Compound Interest n

(1 + i )n

vn

sn

an

( Ia )n

( Da) n

n

1 2 3 4 5

1.030 00 1.060 90 1.092 73 1.125 51 1.159 27

0.970 87 0.942 60 0.915 14 0.888 49 0.862 61

1.000 0 2.030 0 3.090 9 4.183 6 5.309 1

0.970 9 1.913 5 2.828 6 3.717 1 4.579 7

0.970 9 2.856 1 5.601 5 9.155 4 13.468 5

0.970 9 2.884 3 5.713 0 9.430 1 14.009 8

1 2 3 4 5

6 7 8 9 10

1.194 05 1.229 87 1.266 77 1.304 77 1.343 92

0.837 48 0.813 09 0.789 41 0.766 42 0.744 09

6.468 4 7.662 5 8.892 3 10.159 1 11.463 9

5.417 2 6.230 3 7.019 7 7.786 1 8.530 2

18.493 4 24.185 0 30.500 3 37.398 1 44.839 0

19.427 0 25.657 2 32.676 9 40.463 0 48.993 2

6 7 8 9 10

11 12 13 14 15

1.384 23 1.425 76 1.468 53 1.512 59 1.557 97

0.722 42 0.701 38 0.680 95 0.661 12 0.641 86

12.807 8 9.252 6 14.192 0 9.954 0 15.617 8 10.635 0 17.086 3 11.296 1 18.598 9 11.937 9

52.785 6 61.202 2 70.054 6 79.310 2 88.938 1

58.245 9 68.199 9 78.834 8 90.130 9 102.068 8

11 12 13 14 15

16 17 18 19 20

1.604 71 1.652 85 1.702 43 1.753 51 1.806 11

0.623 17 0.605 02 0.587 39 0.570 29 0.553 68

20.156 9 21.761 6 23.414 4 25.116 9 26.870 4

98.908 8 109.194 1 119.767 2 130.602 6 141.676 1

114.629 9 127.796 1 141.549 6 155.873 4 170.750 8

16 17 18 19 20

12.561 1 13.166 1 13.753 5 14.323 8 14.877 5

21 22 23 24 25

1.860 29 1.916 10 1.973 59 2.032 79 2.093 78

0.537 55 0.521 89 0.506 69 0.491 93 0.477 61

28.676 5 30.536 8 32.452 9 34.426 5 36.459 3

15.415 0 15.936 9 16.443 6 16.935 5 17.413 1

152.964 7 164.446 3 176.100 2 187.906 6 199.846 8

186.165 9 202.102 8 218.546 4 235.481 9 252.895 1

21 22 23 24 25

26 27 28 29 30

2.156 59 2.221 29 2.287 93 2.356 57 2.427 26

0.463 69 0.450 19 0.437 08 0.424 35 0.411 99

38.553 0 40.709 6 42.930 9 45.218 9 47.575 4

17.876 8 18.327 0 18.764 1 19.188 5 19.600 4

211.902 8 224.057 9 236.296 1 248.602 1 260.961 7

270.771 9 289.099 0 307.863 1 327.051 5 346.652 0

26 27 28 29 30

31 32 33 34 35

2.500 08 2.575 08 2.652 34 2.731 91 2.813 86

0.399 99 0.388 34 0.377 03 0.366 04 0.355 38

50.002 7 52.502 8 55.077 8 57.730 2 60.462 1

20.000 4 20.388 8 20.765 8 21.131 8 21.487 2

273.361 3 285.788 1 298.230 0 310.675 5 323.113 9

366.652 4 387.041 1 407.806 9 428.938 8 450.426 0

31 32 33 34 35

36 37 38 39 40

2.898 28 2.985 23 3.074 78 3.167 03 3.262 04

0.345 03 0.334 98 0.325 23 0.315 75 0.306 56

63.275 9 66.174 2 69.159 4 72.234 2 75.401 3

21.832 3 22.167 2 22.492 5 22.808 2 23.114 8

335.535 1 347.929 5 360.288 1 372.602 4 384.864 7

472.258 3 494.425 5 516.917 9 539.726 2 562.840 9

36 37 38 39 40

41 42 43 44 45

3.359 90 3.460 70 3.564 52 3.671 45 3.781 60

0.297 63 0.288 96 0.280 54 0.272 37 0.264 44

78.663 3 82.023 2 85.483 9 89.048 4 92.719 9

23.412 4 23.701 4 23.981 9 24.254 3 24.518 7

397.067 5 409.203 8 421.267 1 433.251 5 445.151 2

586.253 3 609.954 7 633.936 6 658.190 9 682.709 6

41 42 43 44 45

46 47 48 49 50

3.895 04 4.011 90 4.132 25 4.256 22 4.383 91

0.256 74 0.249 26 0.242 00 0.234 95 0.228 11

96.501 5 100.396 5 104.408 4 108.540 6 112.796 9

24.775 4 25.024 7 25.266 7 25.501 7 25.729 8

456.961 1 468.676 2 480.292 2 491.804 7 503.210 1

707.485 0 732.509 7 757.776 4 783.278 1 809.007 9

46 47 48 49 50

60 5.891 60 0.169 73 70 7.917 82 0.126 30 80 10.640 89 0.093 98 90 14.300 47 0.069 93 100 19.218 63 0.052 03

163.053 4 230.594 1 321.363 0 443.348 9 607.287 7

27.675 6 29.123 4 30.200 8 31.002 4 31.598 9

610.728 2 705.210 3 786.287 3 854.632 6 911.453 0

3% i

0.030 000

i

(2)

0.029 778

i

(4)

0.029 668

i

(12)

0.029 595

d

0.029 559

(1 + i )1 2

1.014 889

(1 + i )1 4

1.007 417

1/12

(1 + i )

1.002 466

v

0.970 874

12

0.985 329

14

0.992 638

1 12

0.997 540

v v v

d

0.029 126 (2)

0.029 341

d (4)

0.029 450

d

(12)

0.029 522

i / i (2)

1.007 445

i / i (4)

1.011 181

d

i/i

(12)

i/d

1.013 677 1.014 926

i / d (2)

1.022 445

(4)

1.018 681

i/d

i / d (12)

1.016 177

1 077.481 2 60 1 362.552 6 70 1 659.974 6 80 1 966.586 4 90 2 280.036 5 100

55

Compound Interest 4% i

0.040 000 (2)

0.039 608

i (4)

0.039 414

i i

(12)

0.039 285

d

0.039 221

(1 + i )1 2

1.019 804

(1 + i )1 4

1.009 853

1/12

(1 + i )

1.003 274

v

0.961 538

12

0.980 581

14

v

0.990 243

v1 12

0.996 737

d

0.038 462

v

(2)

0.038 839

d (4)

0.039 029

d (12)

0.039 157

i / i (2)

1.009 902

i / i (4)

1.014 877

i / i (12)

1.018 204

i/d

1.019 869

d

i/d

(2)

1.029 902

i/d

(4)

1.024 877

i / d (12)

56

1.021 537

n

(1 + i )n

vn

sn

an

( Ia )n

( Da) n

n

1 2 3 4 5

1.040 00 1.081 60 1.124 86 1.169 86 1.216 65

0.961 54 0.924 56 0.889 00 0.854 80 0.821 93

1.000 0 2.040 0 3.121 6 4.246 5 5.416 3

0.961 5 1.886 1 2.775 1 3.629 9 4.451 8

0.961 5 2.810 7 5.477 6 8.896 9 13.006 5

0.961 5 2.847 6 5.622 7 9.252 6 13.704 4

1 2 3 4 5

6 7 8 9 10

1.265 32 1.315 93 1.368 57 1.423 31 1.480 24

0.790 31 0.759 92 0.730 69 0.702 59 0.675 56

6.633 0 7.898 3 9.214 2 10.582 8 12.006 1

5.242 1 6.002 1 6.732 7 7.435 3 8.110 9

17.748 4 23.067 8 28.913 3 35.236 6 41.992 2

18.946 6 24.948 6 31.681 4 39.116 7 47.227 6

6 7 8 9 10

11 12 13 14 15

1.539 45 1.601 03 1.665 07 1.731 68 1.800 94

0.649 58 0.624 60 0.600 57 0.577 48 0.555 26

13.486 4 8.760 5 15.025 8 9.385 1 16.626 8 9.985 6 18.291 9 10.563 1 20.023 6 11.118 4

49.137 6 56.632 8 64.440 3 72.524 9 80.853 9

55.988 1 65.373 2 75.358 8 85.921 9 97.040 3

11 12 13 14 15

16 17 18 19 20

1.872 98 1.947 90 2.025 82 2.106 85 2.191 12

0.533 91 0.513 37 0.493 63 0.474 64 0.456 39

21.824 5 23.697 5 25.645 4 27.671 2 29.778 1

11.652 3 89.396 4 12.165 7 98.123 8 12.659 3 107.009 1 13.133 9 116.027 3 13.590 3 125.155 0

108.692 6 120.858 3 133.517 6 146.651 5 160.241 8

16 17 18 19 20

21 22 23 24 25

2.278 77 2.369 92 2.464 72 2.563 30 2.665 84

0.438 83 0.421 96 0.405 73 0.390 12 0.375 12

31.969 2 34.248 0 36.617 9 39.082 6 41.645 9

14.029 2 14.451 1 14.856 8 15.247 0 15.622 1

134.370 5 143.653 5 152.985 2 162.348 2 171.726 1

174.271 0 188.722 1 203.579 0 218.825 9 234.448 0

21 22 23 24 25

26 27 28 29 30

2.772 47 2.883 37 2.998 70 3.118 65 3.243 40

0.360 69 0.346 82 0.333 48 0.320 65 0.308 32

44.311 7 47.084 2 49.967 6 52.966 3 56.084 9

15.982 8 16.329 6 16.663 1 16.983 7 17.292 0

181.104 0 190.468 0 199.805 4 209.104 3 218.353 9

250.430 8 266.760 4 283.423 4 300.407 1 317.699 2

26 27 28 29 30

31 32 33 34 35

3.373 13 3.508 06 3.648 38 3.794 32 3.946 09

0.296 46 0.285 06 0.274 09 0.263 55 0.253 42

59.328 3 62.701 5 66.209 5 69.857 9 73.652 2

17.588 5 17.873 6 18.147 6 18.411 2 18.664 6

227.544 1 236.666 0 245.711 1 254.671 9 263.541 4

335.287 7 353.161 2 371.308 9 389.720 1 408.384 7

31 32 33 34 35

36 37 38 39 40

4.103 93 4.268 09 4.438 81 4.616 37 4.801 02

0.243 67 0.234 30 0.225 29 0.216 62 0.208 29

77.598 3 81.702 2 85.970 3 90.409 1 95.025 5

18.908 3 19.142 6 19.367 9 19.584 5 19.792 8

272.313 5 280.982 5 289.543 3 297.991 5 306.323 1

427.293 0 446.435 5 465.803 4 485.387 9 505.180 7

36 37 38 39 40

41 42 43 44 45

4.993 06 5.192 78 5.400 50 5.616 52 5.841 18

0.200 28 0.192 57 0.185 17 0.178 05 0.171 20

99.826 5 104.819 6 110.012 4 115.412 9 121.029 4

19.993 1 20.185 6 20.370 8 20.548 8 20.720 0

314.534 5 322.622 6 330.584 9 338.418 9 346.122 8

525.173 7 545.359 3 565.730 1 586.279 0 606.999 0

41 42 43 44 45

46 47 48 49 50

6.074 82 6.317 82 6.570 53 6.833 35 7.106 68

0.164 61 0.158 28 0.152 19 0.146 34 0.140 71

126.870 6 132.945 4 139.263 2 145.833 7 152.667 1

20.884 7 21.042 9 21.195 1 21.341 5 21.482 2

353.695 1 361.134 3 368.439 7 375.610 4 382.646 0

627.883 7 648.926 6 670.121 7 691.463 2 712.945 4

46 47 48 49 50

60 70 80 90 100

10.519 63 15.571 62 23.049 80 34.119 33 50.504 95

0.095 06 237.990 7 0.064 22 364.290 5 0.043 38 551.245 0 0.029 31 827.983 3 0.019 80 1 237.623 7

22.623 5 23.394 5 23.915 4 24.267 3 24.505 0

445.620 1 495.873 4 535.031 5 565.004 2 587.629 9

934.412 8 60 1 165.137 1 70 1 402.115 2 80 1 643.318 1 90 1 887.375 0 100

Compound Interest n

(1 + i )n

vn

sn

an

( Ia )n

( Da) n

n

1 2 3 4 5

1.050 00 1.102 50 1.157 63 1.215 51 1.276 28

0.952 38 0.907 03 0.863 84 0.822 70 0.783 53

1.000 0 2.050 0 3.152 5 4.310 1 5.525 6

0.952 4 1.859 4 2.723 2 3.546 0 4.329 5

0.952 4 2.766 4 5.358 0 8.648 8 12.566 4

0.952 4 2.811 8 5.535 0 9.081 0 13.410 5

1 2 3 4 5

6 7 8 9 10

1.340 10 1.407 10 1.477 46 1.551 33 1.628 89

0.746 22 0.710 68 0.676 84 0.644 61 0.613 91

6.801 9 8.142 0 9.549 1 11.026 6 12.577 9

5.075 7 5.786 4 6.463 2 7.107 8 7.721 7

17.043 7 22.018 5 27.433 2 33.234 7 39.373 8

18.486 2 24.272 5 30.735 7 37.843 6 45.565 3

6 7 8 9 10

11 12 13 14 15

1.710 34 1.795 86 1.885 65 1.979 93 2.078 93

0.584 68 0.556 84 0.530 32 0.505 07 0.481 02

14.206 8 8.306 4 15.917 1 8.863 3 17.713 0 9.393 6 19.598 6 9.898 6 21.578 6 10.379 7

45.805 3 52.487 3 59.381 5 66.452 4 73.667 7

53.871 7 62.735 0 72.128 5 82.027 2 92.406 8

11 12 13 14 15

16 17 18 19 20

2.182 87 2.292 02 2.406 62 2.526 95 2.653 30

0.458 11 0.436 30 0.415 52 0.395 73 0.376 89

23.657 5 25.840 4 28.132 4 30.539 0 33.066 0

10.837 8 80.997 5 11.274 1 88.414 5 11.689 6 95.893 9 12.085 3 103.412 8 12.462 2 110.950 6

103.244 6 114.518 7 126.208 3 138.293 6 150.755 8

16 17 18 19 20

21 22 23 24 25

2.785 96 2.925 26 3.071 52 3.225 10 3.386 35

0.358 94 0.341 85 0.325 57 0.310 07 0.295 30

35.719 3 38.505 2 41.430 5 44.502 0 47.727 1

12.821 2 13.163 0 13.488 6 13.798 6 14.093 9

118.488 4 126.009 1 133.497 3 140.938 9 148.321 5

163.576 9 176.739 9 190.228 5 204.027 2 218.121 1

21 22 23 24 25

26 27 28 29 30

3.555 67 3.733 46 3.920 13 4.116 14 4.321 94

0.281 24 0.267 85 0.255 09 0.242 95 0.231 38

51.113 5 54.669 1 58.402 6 62.322 7 66.438 8

14.375 2 14.643 0 14.898 1 15.141 1 15.372 5

155.633 7 162.865 6 170.008 2 177.053 7 183.995 0

232.496 3 247.139 3 262.037 5 277.178 5 292.551 0

26 27 28 29 30

31 32 33 34 35

4.538 04 4.764 94 5.003 19 5.253 35 5.516 02

0.220 36 0.209 87 0.199 87 0.190 35 0.181 29

70.760 8 75.298 8 80.063 8 85.067 0 90.320 3

15.592 8 15.802 7 16.002 5 16.192 9 16.374 2

190.826 1 197.541 9 204.137 7 210.609 7 216.954 9

308.143 8 323.946 5 339.949 0 356.141 9 372.516 1

31 32 33 34 35

36 37 38 39 40

5.791 82 6.081 41 6.385 48 6.704 75 7.039 99

0.172 66 0.164 44 0.156 61 0.149 15 0.142 05

95.836 3 101.628 1 107.709 5 114.095 0 120.799 8

16.546 9 16.711 3 16.867 9 17.017 0 17.159 1

223.170 5 229.254 7 235.205 7 241.022 4 246.704 3

389.063 0 405.774 3 422.642 1 439.659 2 456.818 3

36 37 38 39 40

41 42 43 44 45

7.391 99 7.761 59 8.149 67 8.557 15 8.985 01

0.135 28 0.128 84 0.122 70 0.116 86 0.111 30

127.839 8 135.231 8 142.993 3 151.143 0 159.700 2

17.294 4 17.423 2 17.545 9 17.662 8 17.774 1

252.250 8 257.662 1 262.938 4 268.080 3 273.088 6

474.112 6 491.535 8 509.081 8 526.744 5 544.518 6

41 42 43 44 45

46 47 48 49 50

9.434 26 9.905 97 10.401 27 10.921 33 11.467 40

0.106 00 0.100 95 0.096 14 0.091 56 0.087 20

168.685 2 178.119 4 188.025 4 198.426 7 209.348 0

17.880 1 17.981 0 18.077 2 18.168 7 18.255 9

277.964 5 282.709 1 287.323 9 291.810 5 296.170 7

562.398 7 580.379 7 598.456 8 616.625 6 634.881 5

46 47 48 49 50

0.053 54 353.583 7 0.032 87 588.528 5 0.020 18 971.228 8 0.012 39 1 594.607 3 0.007 60 2 610.025 2

18.929 3 19.342 7 19.596 5 19.752 3 19.847 9

333.272 5 360.183 6 379.242 5 392.501 1 401.597 1

60 18.679 19 70 30.426 43 80 49.561 44 90 80.730 37 100 131.501 26

5% i

0.050 000

i

(2)

0.049 390

i

(4)

0.049 089

i

(12)

0.048 889

d

0.048 790

(1 + i )1 2

1.024 695

(1 + i )1 4

1.012 272

1/12

(1 + i )

1.004 074

v

0.952 381

12

0.975 900

14

0.987 877

1 12

0.995 942

v v v

d

0.047 619 (2)

0.048 200

d (4)

0.048 494

d

(12)

0.048 691

i / i (2)

1.012 348

i / i (4)

1.018 559

d

i/i

(12)

i/d

1.022 715 1.024 797

i / d (2)

1.037 348

(4)

1.031 059

i/d

i / d (12)

1.026 881

821.414 2 60 1 013.146 5 70 1 208.070 8 80 1 404.954 8 90 1 603.041 8 100

57

Compound Interest 6% i

0.060 000 (2)

0.059 126

i (4)

0.058 695

i i

(12)

0.058 411

d

0.058 269

(1 + i )1 2 1.029 563 (1 + i )1 4 1.014 674 (1 + i )1/12 1.004 868 v

0.943 396

12

0.971 286

14

v

0.985 538

v1 12

0.995 156

d

0.056 604

v

(2)

0.057 428

d (4)

0.057 847

d (12)

0.058 128

i / i (2)

1.014 782

i / i (4)

1.022 227

i / i (12)

1.027 211

i/d

1.029 709

d

i/d

(2)

1.044 782

(4)

1.037 227 1.032 211

i/d i / d (12)

58

n

(1 + i )n

vn

1 2 3 4 5

1.060 00 1.123 60 1.191 02 1.262 48 1.338 23

0.943 40 0.890 00 0.839 62 0.792 09 0.747 26

1.000 0 2.060 0 3.183 6 4.374 6 5.637 1

6 7 8 9 10

1.418 52 1.503 63 1.593 85 1.689 48 1.790 85

0.704 96 0.665 06 0.627 41 0.591 90 0.558 39

11 12 13 14 15

1.898 30 2.012 20 2.132 93 2.260 90 2.396 56

16 17 18 19 20

( Ia )n

( Da) n

n

0.943 4 1.833 4 2.673 0 3.465 1 4.212 4

0.943 4 2.723 4 5.242 2 8.410 6 12.146 9

0.943 4 2.776 8 5.449 8 8.914 9 13.127 3

1 2 3 4 5

6.975 3 8.393 8 9.897 5 11.491 3 13.180 8

4.917 3 5.582 4 6.209 8 6.801 7 7.360 1

16.376 7 21.032 1 26.051 4 31.378 5 36.962 4

18.044 6 23.627 0 29.836 8 36.638 5 43.998 5

6 7 8 9 10

0.526 79 0.496 97 0.468 84 0.442 30 0.417 27

14.971 6 16.869 9 18.882 1 21.015 1 23.276 0

7.886 9 8.383 8 8.852 7 9.295 0 9.712 2

42.757 1 48.720 7 54.815 6 61.007 8 67.266 8

51.885 4 60.269 3 69.122 0 78.416 9 88.129 2

11 12 13 14 15

2.540 35 2.692 77 2.854 34 3.025 60 3.207 14

0.393 65 0.371 36 0.350 34 0.330 51 0.311 80

25.672 5 28.212 9 30.905 7 33.760 0 36.785 6

10.105 9 10.477 3 10.827 6 11.158 1 11.469 9

73.565 1 79.878 3 86.184 5 92.464 3 98.700 4

98.235 1 108.712 3 119.539 9 130.698 1 142.168 0

16 17 18 19 20

21 22 23 24 25

3.399 56 3.603 54 3.819 75 4.048 93 4.291 87

0.294 16 0.277 51 0.261 80 0.246 98 0.233 00

39.992 7 43.392 3 46.995 8 50.815 6 54.864 5

11.764 1 12.041 6 12.303 4 12.550 4 12.783 4

104.877 6 110.982 7 117.004 1 122.931 6 128.756 5

153.932 1 165.973 6 178.277 0 190.827 4 203.610 7

21 22 23 24 25

26 27 28 29 30

4.549 38 4.822 35 5.111 69 5.418 39 5.743 49

0.219 81 0.207 37 0.195 63 0.184 56 0.174 11

59.156 4 63.705 8 68.528 1 73.639 8 79.058 2

13.003 2 13.210 5 13.406 2 13.590 7 13.764 8

134.471 6 140.070 5 145.548 2 150.900 3 156.123 6

216.613 9 229.824 4 243.230 6 256.821 3 270.586 1

26 27 28 29 30

31 32 33 34 35

6.088 10 6.453 39 6.840 59 7.251 03 7.686 09

0.164 25 0.154 96 0.146 19 0.137 91 0.130 11

84.801 7 90.889 8 97.343 2 104.183 8 111.434 8

13.929 1 14.084 0 14.230 2 14.368 1 14.498 2

161.215 5 166.174 2 170.998 3 175.687 3 180.241 0

284.515 2 298.599 3 312.829 5 327.197 6 341.695 9

31 32 33 34 35

36 37 38 39 40

8.147 25 8.636 09 9.154 25 9.703 51 10.285 72

0.122 74 0.115 79 0.109 24 0.103 06 0.097 22

119.120 9 127.268 1 135.904 2 145.058 5 154.762 0

14.621 0 14.736 8 14.846 0 14.949 1 15.046 3

184.659 6 188.944 0 193.095 1 197.114 2 201.003 1

356.316 9 371.053 7 385.899 7 400.848 8 415.895 1

36 37 38 39 40

41 42 43 44 45

10.902 86 11.557 03 12.250 45 12.985 48 13.764 61

0.091 72 0.086 53 0.081 63 0.077 01 0.072 65

165.047 7 175.950 5 187.507 6 199.758 0 212.743 5

15.138 0 15.224 5 15.306 2 15.383 2 15.455 8

204.763 6 208.397 8 211.907 8 215.296 2 218.565 5

431.033 1 446.257 6 461.563 8 476.947 0 492.402 8

41 42 43 44 45

46 47 48 49 50

14.590 49 15.465 92 16.393 87 17.377 50 18.420 15

0.068 54 0.064 66 0.061 00 0.057 55 0.054 29

226.508 1 241.098 6 256.564 5 272.958 4 290.335 9

15.524 4 15.589 0 15.650 0 15.707 6 15.761 9

221.718 2 224.757 2 227.685 1 230.504 8 233.219 2

507.927 2 523.516 2 539.166 2 554.873 8 570.635 7

46 47 48 49 50

60 32.987 69 0.030 31 533.128 2 70 59.075 93 0.016 93 967.932 2 80 105.795 99 0.009 45 1 746.599 9 90 189.464 51 0.005 28 3 141.075 2 100 339.302 08 0.002 95 5 638.368 1

16.161 4 16.384 5 16.509 1 16.578 7 16.617 5

255.204 2 730.642 9 60 269.711 7 893.590 9 70 279.058 4 1 058.181 2 80 284.973 3 1 223.688 3 90 288.664 6 1 389.707 6 100

sn

an

Compound Interest n

(1 + i )n

vn

sn

an

( Ia )n

( Da) n

n

1 2 3 4 5

1.070 00 1.144 90 1.225 04 1.310 80 1.402 55

0.934 58 0.873 44 0.816 30 0.762 90 0.712 99

1.000 0 2.070 0 3.214 9 4.439 9 5.750 7

0.934 6 1.808 0 2.624 3 3.387 2 4.100 2

0.934 6 2.681 5 5.130 4 8.181 9 11.746 9

0.934 6 2.742 6 5.366 9 8.754 1 12.854 3

1 2 3 4 5

6 7 8 9 10

1.500 73 1.605 78 1.718 19 1.838 46 1.967 15

0.666 34 0.622 75 0.582 01 0.543 93 0.508 35

7.153 3 8.654 0 10.259 8 11.978 0 13.816 4

4.766 5 5.389 3 5.971 3 6.515 2 7.023 6

15.744 9 20.104 2 24.760 2 29.655 6 34.739 1

17.620 9 23.010 2 28.981 4 35.496 7 42.520 3

6 7 8 9 10

11 12 13 14 15

2.104 85 2.252 19 2.409 85 2.578 53 2.759 03

0.475 09 0.444 01 0.414 96 0.387 82 0.362 45

15.783 6 17.888 5 20.140 6 22.550 5 25.129 0

7.498 7 7.942 7 8.357 7 8.745 5 9.107 9

39.965 2 45.293 3 50.687 8 56.117 3 61.554 0

50.018 9 57.961 6 66.319 3 75.064 7 84.172 7

11 12 13 14 15

16 17 18 19 20

2.952 16 3.158 82 3.379 93 3.616 53 3.869 68

0.338 73 0.316 57 0.295 86 0.276 51 0.258 42

27.888 1 9.446 6 30.840 2 9.763 2 33.999 0 10.059 1 37.379 0 10.335 6 40.995 5 10.594 0

66.973 7 72.355 5 77.681 0 82.934 7 88.103 1

93.619 3 103.382 5 113.441 6 123.777 2 134.371 2

16 17 18 19 20

21 22 23 24 25

4.140 56 4.430 40 4.740 53 5.072 37 5.427 43

0.241 51 0.225 71 0.210 95 0.197 15 0.184 25

44.865 2 49.005 7 53.436 1 58.176 7 63.249 0

10.835 5 93.174 8 11.061 2 98.140 5 11.272 2 102.992 3 11.469 3 107.723 8 11.653 6 112.330 1

145.206 8 156.268 0 167.540 2 179.009 5 190.663 1

21 22 23 24 25

26 27 28 29 30

5.807 35 6.213 87 6.648 84 7.114 26 7.612 26

0.172 20 0.160 93 0.150 40 0.140 56 0.131 37

68.676 5 74.483 8 80.697 7 87.346 5 94.460 8

11.825 8 11.986 7 12.137 1 12.277 7 12.409 0

116.807 1 121.152 3 125.363 5 129.439 9 133.380 9

202.488 9 214.475 6 226.612 7 238.890 4 251.299 4

26 27 28 29 30

31 32 33 34 35

8.145 11 8.715 27 9.325 34 9.978 11 10.676 58

0.122 77 0.114 74 0.107 23 0.100 22 0.093 66

102.073 0 110.218 2 118.933 4 128.258 8 138.236 9

12.531 8 12.646 6 12.753 8 12.854 0 12.947 7

137.186 8 140.858 5 144.397 3 147.804 7 151.082 9

263.831 2 276.477 8 289.231 6 302.085 6 315.033 3

31 32 33 34 35

36 37 38 39 40

11.423 94 12.223 62 13.079 27 13.994 82 14.974 46

0.087 54 0.081 81 0.076 46 0.071 46 0.066 78

148.913 5 160.337 4 172.561 0 185.640 3 199.635 1

13.035 2 13.117 0 13.193 5 13.264 9 13.331 7

154.234 2 157.261 2 160.166 5 162.953 3 165.624 5

328.068 5 341.185 5 354.379 0 367.643 9 380.975 6

36 37 38 39 40

41 42 43 44 45

16.022 67 17.144 26 18.344 35 19.628 46 21.002 45

0.062 41 0.058 33 0.054 51 0.050 95 0.047 61

214.609 6 230.632 2 247.776 5 266.120 9 285.749 3

13.394 1 13.452 4 13.507 0 13.557 9 13.605 5

168.183 3 170.633 1 172.977 2 175.218 8 177.361 4

394.369 7 407.822 2 421.329 1 434.887 0 448.492 5

41 42 43 44 45

46 47 48 49 50

22.472 62 24.045 71 25.728 91 27.529 93 29.457 03

0.044 50 0.041 59 0.038 87 0.036 32 0.033 95

306.751 8 329.224 4 353.270 1 378.999 0 406.528 9

13.650 0 13.691 6 13.730 5 13.766 8 13.800 7

179.408 4 181.363 0 183.228 6 185.008 5 186.705 9

462.142 6 475.834 2 489.564 7 503.331 4 517.132 2

46 47 48 49 50

60 70 80 90 100

57.946 43 113.989 39 224.234 39 441.102 98 867.716 33

0.017 26 813.520 4 0.008 77 1 614.134 2 0.004 46 3 189.062 7 0.002 27 6 287.185 4 0.001 15 12 381.661 8

14.039 2 14.160 4 14.222 0 14.253 3 14.269 3

199.806 9 656.583 1 60 207.678 9 797.708 7 70 212.296 8 939.685 6 80 214.957 5 1 082.095 3 90 216.469 3 1 224.725 0 100

7% i

0.070 000

i

(2)

0.068 816

i

(4)

0.068 234

i

(12)

0.067 850

d

0.067 659

(1 + i )1 2 1.034 408 (1 + i )1 4 1.017 059 (1 + i )1/12 1.005 654 v

0.934 579

12

0.966 736

14

0.983 228

1 12

0.994 378

v v v

d

0.065 421 (2)

0.066 527

d (4)

0.067 090

d

(12)

0.067 468

i / i (2)

1.017 204

i / i (4)

1.025 880

d

i/i

(12)

i/d

1.031 691 1.034 605

i / d (2)

1.052 204

(4)

1.043 380

i/d

i / d (12)

1.037 525

59

Compound Interest 8% i

0.080 000 (2)

0.078 461

i (4)

0.077 706

i i

(12)

0.077 208

d

0.076 961

(1 + i )1 2 1.039 230 (1 + i )1 4 1.019 427 (1 + i )1/12 1.006 434 v

0.925 926

12

0.962 250

14

v

0.980 944

v1 12

0.993 607

d

0.074 074

v

(2)

0.075 499

d (4)

0.076 225

d (12)

0.076 715

i / i (2)

1.019 615

i / i (4)

1.029 519

i / i (12)

1.036 157

i/d

1.039 487

d

i/d

(2)

1.059 615

i/d

(4)

1.049 519

i / d (12)

1.042 824

n

(1 + i )n

vn

sn

an

( Ia )n

( Da) n

n

1 2 3 4 5

1.080 00 1.166 40 1.259 71 1.360 49 1.469 33

0.925 93 0.857 34 0.793 83 0.735 03 0.680 58

1.000 0 2.080 0 3.246 4 4.506 1 5.866 6

0.925 9 1.783 3 2.577 1 3.312 1 3.992 7

0.925 9 2.640 6 5.022 1 7.962 2 11.365 1

0.925 9 2.709 2 5.286 3 8.598 4 12.591 1

1 2 3 4 5

6 7 8 9 10

1.586 87 1.713 82 1.850 93 1.999 00 2.158 92

0.630 17 0.583 49 0.540 27 0.500 25 0.463 19

7.335 9 8.922 8 10.636 6 12.487 6 14.486 6

4.622 9 5.206 4 5.746 6 6.246 9 6.710 1

15.146 2 19.230 6 23.552 7 28.055 0 32.686 9

17.214 0 22.420 4 28.167 0 34.413 9 41.124 0

6 7 8 9 10

11 12 13 14 15

2.331 64 2.518 17 2.719 62 2.937 19 3.172 17

0.428 88 0.397 11 0.367 70 0.340 46 0.315 24

16.645 5 18.977 1 21.495 3 24.214 9 27.152 1

7.139 0 7.536 1 7.903 8 8.244 2 8.559 5

37.404 6 42.170 0 46.950 1 51.716 5 56.445 1

48.262 9 55.799 0 63.702 8 71.947 0 80.506 5

11 12 13 14 15

16 17 18 19 20

3.425 94 3.700 02 3.996 02 4.315 70 4.660 96

0.291 89 0.270 27 0.250 25 0.231 71 0.214 55

30.324 3 33.750 2 37.450 2 41.446 3 45.762 0

8.851 4 9.121 6 9.371 9 9.603 6 9.818 1

61.115 4 65.710 0 70.214 4 74.617 0 78.907 9

89.357 9 98.479 5 107.851 4 117.455 0 127.273 2

16 17 18 19 20

21 22 23 24 25

5.033 83 5.436 54 5.871 46 6.341 18 6.848 48

0.198 66 0.183 94 0.170 32 0.157 70 0.146 02

50.422 9 55.456 8 60.893 3 66.764 8 73.105 9

10.016 8 10.200 7 10.371 1 10.528 8 10.674 8

83.079 7 87.126 4 91.043 7 94.828 4 98.478 9

137.290 0 147.490 7 157.861 8 168.390 5 179.065 3

21 22 23 24 25

26 27 28 29 30

7.396 35 7.988 06 8.627 11 9.317 27 10.062 66

0.135 20 0.125 19 0.115 91 0.107 33 0.099 38

79.954 4 87.350 8 95.338 8 103.965 9 113.283 2

10.810 0 10.935 2 11.051 1 11.158 4 11.257 8

101.994 1 105.374 2 108.619 8 111.732 3 114.713 6

189.875 3 200.810 4 211.861 5 223.019 9 234.277 7

26 27 28 29 30

31 32 33 34 35

10.867 67 11.737 08 12.676 05 13.690 13 14.785 34

0.092 02 0.085 20 0.078 89 0.073 05 0.067 63

123.345 9 134.213 5 145.950 6 158.626 7 172.316 8

11.349 8 11.435 0 11.513 9 11.586 9 11.654 6

117.566 1 120.292 5 122.895 8 125.379 3 127.746 6

245.627 5 257.062 5 268.576 4 280.163 3 291.817 9

31 32 33 34 35

36 37 38 39 40

15.968 17 17.245 63 18.625 28 20.115 30 21.724 52

0.062 62 0.057 99 0.053 69 0.049 71 0.046 03

187.102 1 203.070 3 220.315 9 238.941 2 259.056 5

11.717 2 11.775 2 11.828 9 11.878 6 11.924 6

130.001 0 132.146 5 134.186 8 136.125 6 137.966 8

303.535 1 315.310 3 327.139 1 339.017 7 350.942 3

36 37 38 39 40

41 42 43 44 45

23.462 48 25.339 48 27.366 64 29.555 97 31.920 45

0.042 62 0.039 46 0.036 54 0.033 83 0.031 33

280.781 0 304.243 5 329.583 0 356.949 6 386.505 6

11.967 2 12.006 7 12.043 2 12.077 1 12.108 4

139.714 3 141.371 8 142.943 0 144.431 7 145.841 5

362.909 6 374.916 3 386.959 5 399.036 6 411.145 0

41 42 43 44 45

46 47 48 49 50

34.474 09 37.232 01 40.210 57 43.427 42 46.901 61

0.029 01 0.026 86 0.024 87 0.023 03 0.021 32

418.426 1 452.900 2 490.132 2 530.342 7 573.770 2

12.137 4 12.164 3 12.189 1 12.212 2 12.233 5

147.175 8 148.438 2 149.631 9 150.760 2 151.826 3

423.282 4 435.446 7 447.635 8 459.848 0 472.081 4

46 47 48 49 50

0.009 88 1 253.213 3 0.004 57 2 720.080 1 0.002 12 5 886.935 4 0.000 98 12 723.938 6 0.000 45 27 484.515 7

12.376 6 12.442 8 12.473 5 12.487 7 12.494 3

159.676 6 595.293 1 60 163.975 4 719.464 8 70 166.273 6 844.081 1 80 167.480 3 968.903 3 90 168.105 0 1 093.821 0 100

60 101.257 06 70 218.606 41 80 471.954 83 90 1 018.915 09 100 2 199.761 26

60

Compound Interest n

(1 + i )n

vn

sn

an

( Ia )n

( Da) n

n

1 2 3 4 5

1.090 00 1.188 10 1.295 03 1.411 58 1.538 62

0.917 43 0.841 68 0.772 18 0.708 43 0.649 93

1.000 0 2.090 0 3.278 1 4.573 1 5.984 7

0.917 4 1.759 1 2.531 3 3.239 7 3.889 7

0.917 4 2.600 8 4.917 3 7.751 0 11.000 7

0.917 4 2.676 5 5.207 8 8.447 6 12.337 2

1 2 3 4 5

6 7 8 9 10

1.677 10 1.828 04 1.992 56 2.171 89 2.367 36

0.596 27 0.547 03 0.501 87 0.460 43 0.422 41

7.523 3 9.200 4 11.028 5 13.021 0 15.192 9

4.485 9 5.033 0 5.534 8 5.995 2 6.417 7

14.578 3 18.407 5 22.422 5 26.566 3 30.790 4

16.823 1 21.856 1 27.390 9 33.386 1 39.803 8

6 7 8 9 10

11 12 13 14 15

2.580 43 2.812 66 3.065 80 3.341 73 3.642 48

0.387 53 0.355 53 0.326 18 0.299 25 0.274 54

17.560 3 20.140 7 22.953 4 26.019 2 29.360 9

6.805 2 7.160 7 7.486 9 7.786 2 8.060 7

35.053 3 39.319 7 43.560 0 47.749 5 51.867 6

46.609 0 53.769 7 61.256 6 69.042 8 77.103 5

11 12 13 14 15

16 17 18 19 20

3.970 31 4.327 63 4.717 12 5.141 66 5.604 41

0.251 87 0.231 07 0.211 99 0.194 49 0.178 43

33.003 4 36.973 7 41.301 3 46.018 5 51.160 1

8.312 6 8.543 6 8.755 6 8.950 1 9.128 5

55.897 5 85.416 0 59.825 7 93.959 7 63.641 6 102.715 3 67.336 9 111.665 4 70.905 5 120.793 9

16 17 18 19 20

21 22 23 24 25

6.108 81 6.658 60 7.257 87 7.911 08 8.623 08

0.163 70 0.150 18 0.137 78 0.126 40 0.115 97

56.764 5 62.873 3 69.531 9 76.789 8 84.700 9

9.292 2 9.442 4 9.580 2 9.706 6 9.822 6

74.343 2 77.647 2 80.816 2 83.849 9 86.749 1

130.086 2 139.528 6 149.108 8 158.815 4 168.638 0

21 22 23 24 25

26 27 28 29 30

9.399 16 10.245 08 11.167 14 12.172 18 13.267 68

0.106 39 0.097 61 0.089 55 0.082 15 0.075 37

93.324 0 102.723 1 112.968 2 124.135 4 136.307 5

9.929 0 10.026 6 10.116 1 10.198 3 10.273 7

89.515 3 92.150 7 94.658 0 97.040 5 99.301 7

178.567 0 188.593 6 198.709 7 208.908 0 219.181 6

26 27 28 29 30

31 32 33 34 35

14.461 77 15.763 33 17.182 03 18.728 41 20.413 97

0.069 15 0.063 44 0.058 20 0.053 39 0.048 99

149.575 2 164.037 0 179.800 3 196.982 3 215.710 8

10.342 8 10.406 2 10.464 4 10.517 8 10.566 8

101.445 2 103.475 3 105.395 9 107.211 3 108.925 8

229.524 4 239.930 7 250.395 1 260.912 9 271.479 8

31 32 33 34 35

36 37 38 39 40

22.251 23 24.253 84 26.436 68 28.815 98 31.409 42

0.044 94 0.041 23 0.037 83 0.034 70 0.031 84

236.124 7 258.375 9 282.629 8 309.066 5 337.882 4

10.611 8 10.653 0 10.690 8 10.725 5 10.757 4

110.543 7 112.069 2 113.506 6 114.860 0 116.133 5

282.091 5 292.744 5 303.435 3 314.160 9 324.918 2

36 37 38 39 40

41 42 43 44 45

34.236 27 37.317 53 40.676 11 44.336 96 48.327 29

0.029 21 0.026 80 0.024 58 0.022 55 0.020 69

369.291 9 403.528 1 440.845 7 481.521 8 525.858 7

10.786 6 10.813 4 10.838 0 10.860 5 10.881 2

117.331 1 118.456 6 119.513 7 120.506 1 121.437 3

335.704 8 346.518 2 357.356 1 368.216 6 379.097 8

41 42 43 44 45

46 47 48 49 50

52.676 74 57.417 65 62.585 24 68.217 91 74.357 52

0.018 98 0.017 42 0.015 98 0.014 66 0.013 45

574.186 0 626.862 8 684.280 4 746.865 6 815.083 6

10.900 2 10.917 6 10.933 6 10.948 2 10.961 7

122.310 5 123.129 1 123.896 0 124.614 3 125.286 7

389.998 0 400.915 6 411.849 2 422.797 4 433.759 1

46 47 48 49 50

0.005 68 1 944.792 1 0.002 40 4 619.223 2 0.001 01 10 950.574 1 0.000 43 25 939.184 2 0.000 18 61 422.675 5

11.048 0 11.084 4 11.099 8 11.106 4 11.109 1

130.016 2 132.378 6 133.530 5 134.082 1 134.342 6

543.911 2 60 654.617 2 70 765.557 2 80 876.596 1 90 987.676 6 100

60 176.031 29 70 416.730 09 80 986.551 67 90 2 335.526 58 100 5 529.040 79

9% i

0.090 000

i

(2)

0.088 061

i

(4)

0.087 113

i

(12)

0.086 488

d

0.086 178

(1 + i )1 2 1.044 031 (1 + i )1 4 1.021 778 (1 + i )1/12 1.007 207 v

0.917 431

12

0.957 826

14

0.978 686

1 12

0.992 844

v v v

d

0.082 569 (2)

0.084 347

d (4)

0.085 256

d

(12)

0.085 869

i / i (2)

1.022 015

i / i (4)

1.033 144

d

i/i

(12)

i/d

1.040 608 1.044 354

i / d (2)

1.067 015

(4)

1.055 644

i/d

i / d (12)

1.048 108

61

Compound Interest 10% i

0.100 000 (2)

0.097 618

i (4)

0.096 455

i i

(12)

0.095 690

d

0.095 310

(1 + i )1 2 1.048 809 (1 + i )1 4 1.024 114 (1 + i )1/12 1.007 974 v

0.909 091

12

0.953 463

14

v

0.976 454

v1 12

0.992 089

d

0.090 909

v

(2)

0.093 075

d (4)

0.094 184

d (12)

0.094 933

i / i (2)

1.024 404

i / i (4)

1.036 756

i / i (12)

1.045 045

i/d

1.049 206

d

i/d

(2)

1.074 404

i/d

(4)

1.061 756

i / d (12)

1.053 378

n

(1 + i )n

vn

1 2 3 4 5

1.100 00 1.210 00 1.331 00 1.464 10 1.610 51

0.909 09 0.826 45 0.751 31 0.683 01 0.620 92

6 7 8 9 10

1.771 56 1.948 72 2.143 59 2.357 95 2.593 74

11 12 13 14 15

an

( Ia )n

( Da) n

n

1.000 0 2.100 0 3.310 0 4.641 0 6.105 1

0.909 1 1.735 5 2.486 9 3.169 9 3.790 8

0.909 1 2.562 0 4.815 9 7.548 0 10.652 6

0.909 1 2.644 6 5.131 5 8.301 3 12.092 1

1 2 3 4 5

0.564 47 0.513 16 0.466 51 0.424 10 0.385 54

7.715 6 9.487 2 11.435 9 13.579 5 15.937 4

4.355 3 4.868 4 5.334 9 5.759 0 6.144 6

14.039 4 17.631 5 21.363 6 25.180 5 29.035 9

16.447 4 21.315 8 26.650 7 32.409 8 38.554 3

6 7 8 9 10

2.853 12 3.138 43 3.452 27 3.797 50 4.177 25

0.350 49 0.318 63 0.289 66 0.263 33 0.239 39

18.531 2 21.384 3 24.522 7 27.975 0 31.772 5

6.495 1 6.813 7 7.103 4 7.366 7 7.606 1

32.891 3 36.714 9 40.480 5 44.167 2 47.758 1

45.049 4 51.863 1 58.966 4 66.333 1 73.939 2

11 12 13 14 15

16 17 18 19 20

4.594 97 5.054 47 5.559 92 6.115 91 6.727 50

0.217 63 0.197 84 0.179 86 0.163 51 0.148 64

35.949 7 40.544 7 45.599 2 51.159 1 57.275 0

7.823 7 8.021 6 8.201 4 8.364 9 8.513 6

51.240 1 81.762 9 54.603 5 89.784 5 57.841 0 97.985 9 60.947 6 106.350 8 63.920 5 114.864 4

16 17 18 19 20

21 22 23 24 25

7.400 25 8.140 27 8.954 30 9.849 73 10.834 71

0.135 13 0.122 85 0.111 68 0.101 53 0.092 30

64.002 5 71.402 7 79.543 0 88.497 3 98.347 1

8.648 7 8.771 5 8.883 2 8.984 7 9.077 0

66.758 2 69.460 8 72.029 4 74.466 0 76.773 4

123.513 1 132.284 6 141.167 8 150.152 6 159.229 6

21 22 23 24 25

26 27 28 29 30

11.918 18 13.109 99 14.420 99 15.863 09 17.449 40

0.083 91 0.076 28 0.069 34 0.063 04 0.057 31

109.181 8 121.099 9 134.209 9 148.630 9 164.494 0

9.160 9 9.237 2 9.306 6 9.369 6 9.426 9

78.955 0 81.014 5 82.956 1 84.784 2 86.503 5

168.390 5 177.627 8 186.934 3 196.303 9 205.730 9

26 27 28 29 30

31 32 33 34 35

19.194 34 21.113 78 23.225 15 25.547 67 28.102 44

0.052 10 0.047 36 0.043 06 0.039 14 0.035 58

181.943 4 201.137 8 222.251 5 245.476 7 271.024 4

9.479 0 9.526 4 9.569 4 9.608 6 9.644 2

88.118 6 89.634 2 91.055 0 92.385 9 93.631 3

215.209 9 224.736 2 234.305 7 243.914 3 253.558 4

31 32 33 34 35

36 37 38 39 40

30.912 68 34.003 95 37.404 34 41.144 78 45.259 26

0.032 35 0.029 41 0.026 73 0.024 30 0.022 09

299.126 8 330.039 5 364.043 4 401.447 8 442.592 6

9.676 5 9.705 9 9.732 7 9.757 0 9.779 1

94.795 9 95.884 0 96.899 9 97.847 8 98.731 6

263.234 9 272.940 8 282.673 5 292.430 4 302.209 5

36 37 38 39 40

41 42 43 44 45

49.785 18 54.763 70 60.240 07 66.264 08 72.890 48

0.020 09 0.018 26 0.016 60 0.015 09 0.013 72

487.851 8 537.637 0 592.400 7 652.640 8 718.904 8

9.799 1 9.817 4 9.834 0 9.849 1 9.862 8

99.555 1 100.322 1 101.035 9 101.699 9 102.317 2

312.008 6 321.826 0 331.660 0 341.509 1 351.371 9

41 42 43 44 45

46 47 48 49 50

80.179 53 88.197 49 97.017 23 106.718 96 117.390 85

0.012 47 0.011 34 0.010 31 0.009 37 0.008 52

791.795 3 871.974 9 960.172 3 1 057.189 6 1 163.908 5

9.875 3 9.886 6 9.896 9 9.906 3 9.914 8

102.891 0 103.423 8 103.918 6 104.377 8 104.803 7

361.247 2 371.133 8 381.030 7 390.937 0 400.851 9

46 47 48 49 50

0.003 28 3 034.816 4 0.001 27 7 887.469 6 0.000 49 20 474.002 1 0.000 19 53 120.226 1 0.000 07 137 796.123 4

9.967 2 9.987 3 9.995 1 9.998 1 9.999 3

107.668 2 108.974 4 109.555 8 109.809 9 109.919 5

500.328 4 60 600.126 6 70 700.048 8 80 800.018 8 90 900.007 3 100

60 304.481 64 70 789.746 96 80 2 048.400 21 90 5 313.022 61 100 13 780.612 34

62

sn

Compound Interest n

(1 + i )n

vn

1 2 3 4 5

1.120 00 1.254 40 1.404 93 1.573 52 1.762 34

0.892 86 0.797 19 0.711 78 0.635 52 0.567 43

1.000 0 2.120 0 3.374 4 4.779 3 6.352 8

6 7 8 9 10

1.973 82 2.210 68 2.475 96 2.773 08 3.105 85

0.506 63 0.452 35 0.403 88 0.360 61 0.321 97

11 12 13 14 15

3.478 55 3.895 98 4.363 49 4.887 11 5.473 57

16 17 18 19 20

6.130 39 6.866 04 7.689 97 8.612 76 9.646 29

( Da) n

n

0.892 9 0.892 9 1.690 1 2.487 2 2.401 8 4.622 6 3.037 3 7.164 7 3.604 8 10.001 8

0.892 9 2.582 9 4.984 7 8.022 1 11.626 9

1 2 3 4 5

8.115 2 10.089 0 12.299 7 14.775 7 17.548 7

4.111 4 4.563 8 4.967 6 5.328 2 5.650 2

13.041 6 16.208 0 19.439 1 22.684 6 25.904 3

15.738 3 20.302 0 25.269 7 30.597 9 36.248 1

6 7 8 9 10

0.287 48 0.256 68 0.229 17 0.204 62 0.182 70

20.654 6 24.133 1 28.029 1 32.392 6 37.279 7

5.937 7 6.194 4 6.423 5 6.628 2 6.810 9

29.066 5 32.146 7 35.125 9 37.990 6 40.731 0

42.185 8 48.380 2 54.803 8 61.431 9 68.242 8

11 12 13 14 15

0.163 12 0.145 64 0.130 04 0.116 11 0.103 67

42.753 3 48.883 7 55.749 7 63.439 7 72.052 4

6.974 0 7.119 6 7.249 7 7.365 8 7.469 4

43.341 0 75.216 8 45.816 9 82.336 4 48.157 6 89.586 1 50.363 7 96.951 9 52.437 0 104.421 3

16 17 18 19 20

sn

an

( Ia )n

21 22 23 24 25

10.803 85 12.100 31 13.552 35 15.178 63 17.000 06

0.092 56 0.082 64 0.073 79 0.065 88 0.058 82

81.698 7 92.502 6 104.602 9 118.155 2 133.333 9

7.562 0 7.644 6 7.718 4 7.784 3 7.843 1

54.380 8 56.198 9 57.896 0 59.477 2 60.947 8

111.983 3 119.628 0 127.346 4 135.130 7 142.973 8

21 22 23 24 25

26 27 28 29 30

19.040 07 21.324 88 23.883 87 26.749 93 29.959 92

0.052 52 0.046 89 0.041 87 0.037 38 0.033 38

150.333 9 169.374 0 190.698 9 214.582 8 241.332 7

7.895 7 7.942 6 7.984 4 8.021 8 8.055 2

62.313 3 63.579 4 64.751 8 65.835 9 66.837 2

150.869 5 158.812 1 166.796 5 174.818 3 182.873 5

26 27 28 29 30

31 32 33 34 35

33.555 11 37.581 73 42.091 53 47.142 52 52.799 62

0.029 80 0.026 61 0.023 76 0.021 21 0.018 94

271.292 6 304.847 7 342.429 4 384.521 0 431.663 5

8.085 0 8.111 6 8.135 4 8.156 6 8.175 5

67.761 1 68.612 6 69.396 6 70.117 8 70.780 7

190.958 5 199.070 0 207.205 4 215.362 0 223.537 5

31 32 33 34 35

36 37 38 39 40

59.135 57 66.231 84 74.179 66 83.081 22 93.050 97

0.016 91 0.015 10 0.013 48 0.012 04 0.010 75

484.463 1 543.598 7 609.830 5 684.010 2 767.091 4

8.192 4 8.207 5 8.221 0 8.233 0 8.243 8

71.389 4 71.948 1 72.460 4 72.929 8 73.359 6

231.729 9 239.937 4 248.158 4 256.391 4 264.635 2

36 37 38 39 40

41 42 43 44 45

104.217 09 116.723 14 130.729 91 146.417 50 163.987 60

0.009 60 0.008 57 0.007 65 0.006 83 0.006 10

860.142 4 964.359 5 1 081.082 6 1 211.812 5 1 358.230 0

8.253 4 8.261 9 8.269 6 8.276 4 8.282 5

73.753 1 74.112 9 74.441 8 74.742 3 75.016 7

272.888 6 281.150 5 289.420 1 297.696 5 305.979 0

41 42 43 44 45

46 47 48 49 50

183.666 12 205.706 05 230.390 78 258.037 67 289.002 19

0.005 44 0.004 86 0.004 34 0.003 88 0.003 46

1 522.217 6 1 705.883 8 1 911.589 8 2 141.980 6 2 400.018 2

8.288 0 8.292 8 8.297 2 8.301 0 8.304 5

75.267 2 75.495 7 75.704 0 75.893 9 76.066 9

314.267 0 322.559 8 330.857 0 339.158 0 347.462 5

46 47 48 49 50

0.001 11 7 471.641 1 0.000 36 23 223.331 9 0.000 12 72 145.692 5 0.000 04 224 091.118 5 0.000 01 696 010.547 7

8.324 0 8.330 3 8.332 4 8.333 0 8.333 2

77.134 1 77.540 6 77.691 8 77.747 0 77.766 9

430.632 9 60 513.913 8 70 597.230 2 80 680.558 1 90 763.889 7 100

60 897.596 93 70 2 787.799 83 80 8 658.483 10 90 26 891.934 22 100 83 522.265 73

12% i

0.120 000

i

(2)

0.116 601

i

(4)

0.114 949

i

(12)

0.113 866

d

0.113 329

(1 + i )1 2 1.058 301 (1 + i )1 4 1.028 737 (1 + i )1/12 1.009 489 v

0.892 857

12

0.944 911

14

0.972 065

1 12

0.990 600

v v v

d

0.107 143 (2)

0.110 178

d (4)

0.111 738

d

(12)

0.112 795

i / i (2)

1.029 150

i / i (4)

1.043 938

d

i/i

(12)

i/d

1.053 875 1.058 867

i / d (2)

1.089 150

(4)

1.073 938

i/d

i / d (12)

1.063 875

63

Compound Interest 15% i

0.150 000 (2)

0.144 761

i (4)

0.142 232

i i

(12)

0.140 579

d

0.139 762

(1 + i )1 2

1.072 381

(1 + i )1 4

1.035 558

1/12

(1 + i )

1.011 715

v

0.869 565

12

0.932 505

14

v

0.965 663

v1 12

0.988 421

d

0.130 435

v

(2)

0.134 990

d (4)

0.137 348

d (12)

0.138 951

i / i (2)

1.036 190

i / i (4)

1.054 613

i / i (12)

1.067 016

i/d

1.073 254

d

i/d

(2)

1.111 190

i/d

(4)

1.092 113

i / d (12)

64

1.079 516

n

(1 + i )n

vn

1 2 3 4 5

1.150 00 1.322 50 1.520 88 1.749 01 2.011 36

0.869 57 0.756 14 0.657 52 0.571 75 0.497 18

1.000 0 2.150 0 3.472 5 4.993 4 6.742 4

6 7 8 9 10

2.313 06 2.660 02 3.059 02 3.517 88 4.045 56

0.432 33 0.375 94 0.326 90 0.284 26 0.247 18

11 12 13 14 15

4.652 39 5.350 25 6.152 79 7.075 71 8.137 06

16 17 18 19 20

( Ia )n

( Da) n

n

0.869 6 1.625 7 2.283 2 2.855 0 3.352 2

0.869 6 2.381 9 4.354 4 6.641 4 9.127 3

0.869 6 2.495 3 4.778 5 7.633 5 10.985 6

1 2 3 4 5

8.753 7 11.066 8 13.726 8 16.785 8 20.303 7

3.784 5 4.160 4 4.487 3 4.771 6 5.018 8

11.721 3 14.352 8 16.968 0 19.526 4 21.998 2

14.770 1 18.930 5 23.417 9 28.189 4 33.208 2

6 7 8 9 10

0.214 94 0.186 91 0.162 53 0.141 33 0.122 89

24.349 3 29.001 7 34.351 9 40.504 7 47.580 4

5.233 7 5.420 6 5.583 1 5.724 5 5.847 4

24.362 6 26.605 5 28.718 4 30.697 0 32.540 4

38.441 9 43.862 5 49.445 7 55.170 2 61.017 5

11 12 13 14 15

9.357 62 10.761 26 12.375 45 14.231 77 16.366 54

0.106 86 0.092 93 0.080 81 0.070 27 0.061 10

55.717 5 65.075 1 75.836 4 88.211 8 102.443 6

5.954 2 6.047 2 6.128 0 6.198 2 6.259 3

34.250 2 35.830 0 37.284 5 38.619 5 39.841 5

66.971 8 73.018 9 79.146 9 85.345 1 91.604 5

16 17 18 19 20

21 22 23 24 25

18.821 52 21.644 75 24.891 46 28.625 18 32.918 95

0.053 13 0.046 20 0.040 17 0.034 93 0.030 38

118.810 1 137.631 6 159.276 4 184.167 8 212.793 0

6.312 5 6.358 7 6.398 8 6.433 8 6.464 1

40.957 2 41.973 7 42.897 7 43.736 1 44.495 5

97.916 9 104.275 6 110.674 4 117.108 2 123.572 3

21 22 23 24 25

26 27 28 29 30

37.856 80 43.535 31 50.065 61 57.575 45 66.211 77

0.026 42 0.022 97 0.019 97 0.017 37 0.015 10

245.712 0 283.568 8 327.104 1 377.169 7 434.745 1

6.490 6 6.513 5 6.533 5 6.550 9 6.566 0

45.182 3 45.802 5 46.361 8 46.865 5 47.318 6

130.062 9 136.576 4 143.109 9 149.660 8 156.226 8

26 27 28 29 30

31 32 33 34 35

76.143 54 87.565 07 100.699 83 115.804 80 133.175 52

0.013 13 0.011 42 0.009 93 0.008 64 0.007 51

500.956 9 577.100 5 664.665 5 765.365 4 881.170 2

6.579 1 6.590 5 6.600 5 6.609 1 6.616 6

47.725 7 48.091 1 48.418 8 48.712 4 48.975 2

162.805 9 169.396 4 175.996 9 182.606 0 189.222 6

31 32 33 34 35

36 37 38 39 40

153.151 85 176.124 63 202.543 32 232.924 82 267.863 55

0.006 53 0.005 68 0.004 94 0.004 29 0.003 73

1 014.345 7 1 167.497 5 1 343.622 2 1 546.165 5 1 779.090 3

6.623 1 6.628 8 6.633 8 6.638 0 6.641 8

49.210 3 49.420 4 49.608 0 49.775 4 49.924 8

195.845 8 202.474 6 209.108 3 215.746 4 222.388 1

36 37 38 39 40

41 42 43 44 45

308.043 08 354.249 54 407.386 97 468.495 02 538.769 27

0.003 25 0.002 82 0.002 45 0.002 13 0.001 86

2 046.953 9 2 354.996 9 2 709.246 5 3 116.633 4 3 585.128 5

6.645 0 6.647 8 6.650 3 6.652 4 6.654 3

50.057 9 50.176 4 50.282 0 50.375 9 50.459 4

229.033 2 235.681 0 242.331 3 248.983 8 255.638 0

41 42 43 44 45

46 47 48 49 50

619.584 66 712.522 36 819.400 71 942.310 82 1 083.657 44

0.001 61 0.001 40 0.001 22 0.001 06 0.000 92

4 123.897 7 4 743.482 4 5 456.004 7 6 275.405 5 7 217.716 3

6.655 9 6.657 3 6.658 5 6.659 6 6.660 5

50.533 7 50.599 6 50.658 2 50.710 2 50.756 3

262.294 0 268.951 3 275.609 8 282.269 4 288.929 9

46 47 48 49 50

sn

an

Compound Interest n

(1 + i )n

vn

sn

an

( Ia )n

( Da) n

n

1 2 3 4 5

1.200 00 1.440 00 1.728 00 2.073 60 2.488 32

0.833 33 0.694 44 0.578 70 0.482 25 0.401 88

1.000 0 2.200 0 3.640 0 5.368 0 7.441 6

0.833 3 1.527 8 2.106 5 2.588 7 2.990 6

0.833 3 2.222 2 3.958 3 5.887 3 7.896 7

0.833 3 2.361 1 4.467 6 7.056 3 10.046 9

1 2 3 4 5

6 7 8 9 10

2.985 98 3.583 18 4.299 82 5.159 78 6.191 74

0.334 90 0.279 08 0.232 57 0.193 81 0.161 51

9.929 9 12.915 9 16.499 1 20.798 9 25.958 7

3.325 5 3.604 6 3.837 2 4.031 0 4.192 5

9.906 1 11.859 7 13.720 2 15.464 5 17.079 6

13.372 4 6 16.977 0 7 20.814 2 8 24.845 2 9 29.037 6 10

11 12 13 14 15

7.430 08 8.916 10 10.699 32 12.839 18 15.407 02

0.134 59 0.112 16 0.093 46 0.077 89 0.064 91

32.150 4 39.580 5 48.496 6 59.195 9 72.035 1

4.327 1 4.439 2 4.532 7 4.610 6 4.675 5

18.560 0 19.905 9 21.120 9 22.211 3 23.184 9

33.364 7 37.803 9 42.336 6 46.947 2 51.622 6

11 12 13 14 15

16 17 18 19 20

18.488 43 22.186 11 26.623 33 31.948 00 38.337 60

0.054 09 0.045 07 0.037 56 0.031 30 0.026 08

87.442 1 105.930 6 128.116 7 154.740 0 186.688 0

4.729 6 4.774 6 4.812 2 4.843 5 4.869 6

24.050 3 24.816 6 25.492 7 26.087 4 26.609 1

56.352 2 61.126 8 65.939 0 70.782 5 75.652 1

16 17 18 19 20

21 22 23 24 25

46.005 12 55.206 14 66.247 37 79.496 85 95.396 22

0.021 74 0.018 11 0.015 09 0.012 58 0.010 48

225.025 6 271.030 7 326.236 9 392.484 2 471.981 1

4.891 3 4.909 4 4.924 5 4.937 1 4.947 6

27.065 5 80.543 4 27.464 1 85.452 8 27.811 2 90.377 4 28.113 1 95.314 5 28.375 2 100.262 1

21 22 23 24 25

26 27 28 29 30

114.475 46 137.370 55 164.844 66 197.813 59 237.376 31

0.008 74 0.007 28 0.006 07 0.005 06 0.004 21

567.377 3 681.852 8 819.223 3 984.068 0 1 181.881 6

4.956 3 4.963 6 4.969 7 4.974 7 4.978 9

28.602 3 28.798 9 28.968 7 29.115 3 29.241 7

105.218 4 110.182 0 115.151 7 120.126 4 125.105 3

26 27 28 29 30

31 32 33 34 35

284.851 58 341.821 89 410.186 27 492.223 52 590.668 23

0.003 51 0.002 93 0.002 44 0.002 03 0.001 69

1 419.257 9 1 704.109 5 2 045.931 4 2 456.117 6 2 948.341 1

4.982 4 4.985 4 4.987 8 4.989 8 4.991 5

29.350 5 29.444 2 29.524 6 29.593 7 29.652 9

130.087 8 135.073 1 140.060 9 145.050 8 150.042 3

31 32 33 34 35

36 708.801 87 37 850.562 25 38 1 020.674 70 39 1 224.809 64 40 1 469.771 57

0.001 41 0.001 18 0.000 98 0.000 82 0.000 68

3 539.009 4 4 247.811 2 5 098.373 5 6 119.048 2 7 343.857 8

4.992 9 4.994 1 4.995 1 4.995 9 4.996 6

29.703 7 29.747 2 29.784 5 29.816 3 29.843 5

155.035 3 160.029 4 165.024 5 170.020 4 175.017 0

36 37 38 39 40

41 42 43 44 45

1 763.725 88 2 116.471 06 2 539.765 27 3 047.718 32 3 657.261 99

0.000 57 0.000 47 0.000 39 0.000 33 0.000 27

8 813.629 4 10 577.355 3 12 693.826 3 15 233.591 6 18 281.309 9

4.997 2 4.997 6 4.998 0 4.998 4 4.998 6

29.866 8 29.886 6 29.903 5 29.918 0 29.930 3

180.014 2 185.011 8 190.009 8 195.008 2 200.006 8

41 42 43 44 45

46 47 48 49 50

4 388.714 39 5 266.457 26 6 319.748 72 7 583.698 46 9 100.438 15

0.000 23 0.000 19 0.000 16 0.000 13 0.000 11

21 938.571 9 26 327.286 3 31 593.743 6 37 913.492 3 45 497.190 8

4.998 9 4.999 1 4.999 2 4.999 3 4.999 5

29.940 8 29.949 7 29.957 3 29.963 7 29.969 2

205.005 7 210.004 7 215.004 0 220.003 3 225.002 7

46 47 48 49 50

20% i

0.200 000

i

(2)

0.190 890

i

(4)

0.186 541

i

(12)

0.183 714

d

0.182 322

(1 + i )1 2

1.095 445

(1 + i )1 4

1.046 635

1/12

(1 + i )

1.015 309

v

0.833 333

12

0.912 871

14

0.955 443

1 12

0.984 921

v v v

d

0.166 667 (2)

0.174 258

d (4)

0.178 229

d

(12)

0.180 943

i / i (2)

1.047 723

i / i (4)

1.072 153

d

i/i

(12)

i/d

1.088 651 1.096 963

i / d (2)

1.147 723

(4)

1.122 153

i/d

i / d (12)

1.105 317

65

Compound Interest 25% i

0.250 000 (2)

0.236 068

i (4)

0.229 485

i i

(12)

0.225 231

d

0.223 144

(1 + i )1 2

1.118 034

(1 + i )1 4

1.057 371

1/12

(1 + i )

1.018 769

v

0.800 000

12

0.894 427

14

v

0.945 742

v1 12

0.981 577

d

0.200 000

v

(2)

0.211 146

d (4)

0.217 034

d (12)

0.221 082

i / i (2)

1.059 017

i / i (4)

1.089 396

i / i (12)

1.109 971

i/d

1.120 355

d

i/d

(2)

1.184 017

i/d

(4)

1.151 896

i / d (12)

66

1.130 804

n

(1 + i )n

1 2 3 4 5

1.250 00 1.562 50 1.953 13 2.441 41 3.051 76

6 7 8 9 10

vn

sn

an

( Ia )n

0.800 00 0.640 00 0.512 00 0.409 60 0.327 68

1.000 0 2.250 0 3.812 5 5.765 6 8.207 0

0.800 0 1.440 0 1.952 0 2.361 6 2.689 3

0.800 0 2.080 0 3.616 0 5.254 4 6.892 8

3.814 70 4.768 37 5.960 46 7.450 58 9.313 23

0.262 14 0.209 72 0.167 77 0.134 22 0.107 37

11.258 8 15.073 5 19.841 9 25.802 3 33.252 9

11 12 13 14 15

11.641 53 14.551 92 18.189 89 22.737 37 28.421 71

0.085 90 0.068 72 0.054 98 0.043 98 0.035 18

16 17 18 19 20

35.527 14 44.408 92 55.511 15 69.388 94 86.736 17

21 22 23 24 25

( Da) n

n

0.800 0 2.240 0 4.192 0 6.553 6 9.242 9

1 2 3 4 5

2.951 4 8.465 7 3.161 1 9.933 7 3.328 9 11.275 8 3.463 1 12.483 8 3.570 5 13.557 5

12.194 3 6 15.355 4 7 18.684 4 8 22.147 5 9 25.718 0 10

42.566 1 54.207 7 68.759 6 86.949 5 109.686 8

3.656 4 3.725 1 3.780 1 3.824 1 3.859 3

14.502 4 15.327 1 16.041 8 16.657 5 17.185 3

29.374 4 33.099 5 36.879 6 40.703 7 44.562 9

11 12 13 14 15

0.028 15 0.022 52 0.018 01 0.014 41 0.011 53

138.108 5 173.635 7 218.044 6 273.555 8 342.944 7

3.887 4 3.909 9 3.927 9 3.942 4 3.953 9

17.635 6 18.018 4 18.342 7 18.616 5 18.847 1

48.450 4 52.360 3 56.288 2 60.230 6 64.184 5

16 17 18 19 20

108.420 22 135.525 27 169.406 59 211.758 24 264.697 80

0.009 22 0.007 38 0.005 90 0.004 72 0.003 78

429.680 9 538.101 1 673.626 4 843.032 9 1 054.791 2

3.963 1 3.970 5 3.976 4 3.981 1 3.984 9

19.040 8 19.203 1 19.338 9 19.452 2 19.546 7

68.147 6 72.118 1 76.094 4 80.075 6 84.060 4

21 22 23 24 25

26 27 28 29 30

330.872 25 413.590 31 516.987 88 646.234 85 807.793 57

0.003 02 0.002 42 0.001 93 0.001 55 0.001 24

1 319.489 0 1 650.361 2 2 063.951 5 2 580.939 4 3 227.174 3

3.987 9 3.990 3 3.992 3 3.993 8 3.995 0

19.625 2 88.048 4 19.690 5 92.038 7 19.744 7 96.030 9 19.789 6 100.024 8 19.826 7 104.019 8

26 27 28 29 30

31 32 33 34 35

1 009.741 96 1 262.177 45 1 577.721 81 1 972.152 26 2 465.190 33

0.000 99 0.000 79 0.000 63 0.000 51 0.000 41

4 034.967 8 5 044.709 8 6 306.887 2 7 884.609 1 9 856.761 3

3.996 0 3.996 8 3.997 5 3.998 0 3.998 4

19.857 4 19.882 7 19.903 7 19.920 9 19.935 1

108.015 8 112.012 7 116.010 1 120.008 1 124.006 5

31 32 33 34 35

36 37 38 39 40

3 081.487 91 3 851.859 89 4 814.824 86 6 018.531 08 7 523.163 85

0.000 32 0.000 26 0.000 21 0.000 17 0.000 13

12 321.951 6 15 403.439 6 19 255.299 4 24 070.124 3 30 088.655 4

3.998 7 3.999 0 3.999 2 3.999 3 3.999 5

19.946 8 19.956 4 19.964 3 19.970 8 19.976 1

128.005 2 132.004 2 136.003 3 140.002 7 144.002 1

36 37 38 39 40

41 42 43 44 45

9 403.954 81 11 754.943 51 14 693.679 39 18 367.099 23 22 958.874 04

0.000 11 0.000 09 0.000 07 0.000 05 0.000 04

37 611.819 2 47 015.774 0 58 770.717 5 73 464.396 9 91 831.496 2

3.999 6 3.999 7 3.999 7 3.999 8 3.999 8

19.980 4 19.984 0 19.986 9 19.989 3 19.991 3

148.001 7 152.001 4 156.001 1 160.000 9 164.000 7

41 42 43 44 45

46 47 48 49 50

28 698.592 55 35 873.240 69 44 841.550 86 56 051.938 57 70 064.923 22

0.000 03 0.000 03 0.000 02 0.000 02 0.000 01

114 790.370 2 143 488.962 7 179 362.203 4 224 203.754 3 280 255.692 9

3.999 9 3.999 9 3.999 9 3.999 9 3.999 9

19.992 9 19.994 2 19.995 3 19.996 1 19.996 9

168.000 6 172.000 4 176.000 4 180.000 3 184.000 2

46 47 48 49 50

POPULATION MORTALITY TABLE ELT15 (Males) and ELT15 (Females) This table is based on the mortality of the population of England and Wales during the years 1990, 1991, and 1992. Full details are given in English Life Tables No. 15 published by The Stationery Office. Note that no m0 values have been included because of the difficulty of calculating reasonable estimates from observed data.

67

ELT15 (Males)

68

o

x

lx

dx

qx

mx

0 1 2 3 4

100 000 99 186 99 124 99 086 99 056

814 62 38 30 24

0.008 14 0.000 62 0.000 38 0.000 30 0.000 24

0.000 80 0.000 43 0.000 33 0.000 27

73.413 73.019 72.064 71.091 70.113

0 1 2 3 4

5 6 7 8 9

99 032 99 010 98 990 98 972 98 953

22 20 18 19 18

0.000 22 0.000 20 0.000 19 0.000 18 0.000 18

0.000 23 0.000 21 0.000 19 0.000 18 0.000 18

69.130 68.145 67.158 66.171 65.183

5 6 7 8 9

10 11 12 13 14

98 935 98 917 98 899 98 880 98 857

18 18 19 23 29

0.000 18 0.000 18 0.000 19 0.000 23 0.000 29

0.000 18 0.000 18 0.000 19 0.000 21 0.000 26

64.195 63.206 62.218 61.230 60.244

10 11 12 13 14

15 16 17 18 19

98 828 98 789 98 737 98 663 98 577

39 52 74 86 81

0.000 40 0.000 52 0.000 75 0.000 87 0.000 83

0.000 34 0.000 45 0.000 64 0.000 83 0.000 85

59.261 58.285 57.315 56.358 55.406

15 16 17 18 19

20 21 22 23 24

98 496 98 413 98 328 98 241 98 154

83 85 87 87 87

0.000 84 0.000 86 0.000 89 0.000 89 0.000 88

0.000 83 0.000 85 0.000 88 0.000 89 0.000 89

54.452 53.497 52.543 51.589 50.635

20 21 22 23 24

25 26 27 28 29

98 067 97 983 97 900 97 817 97 732

84 83 83 85 87

0.000 86 0.000 85 0.000 85 0.000 87 0.000 90

0.000 87 0.000 85 0.000 84 0.000 86 0.000 88

49.679 48.721 47.762 46.802 45.842

25 26 27 28 29

30 31 32 33 34

97 645 97 556 97 465 97 370 97 273

89 91 95 97 103

0.000 91 0.000 94 0.000 97 0.000 99 0.001 06

0.000 90 0.000 92 0.000 96 0.000 98 0.001 02

44.883 43.923 42.964 42.005 41.046

30 31 32 33 34

35 36 37 38 39

97 170 97 057 96 933 96 800 96 655

113 124 133 145 155

0.001 16 0.001 27 0.001 38 0.001 49 0.001 60

0.001 11 0.001 22 0.001 33 0.001 44 0.001 55

40.090 39.136 38.185 37.237 36.292

35 36 37 38 39

40 41 42 43 44

96 500 96 334 96 155 95 961 95 751

166 179 194 210 230

0.001 72 0.001 86 0.002 01 0.002 19 0.002 40

0.001 66 0.001 79 0.001 93 0.002 10 0.002 29

35.349 34.409 33.473 32.539 31.609

40 41 42 43 44

45 46 47 48 49

95 521 95 266 94 983 94 668 94 316

255 283 315 352 391

0.002 66 0.002 97 0.003 32 0.003 71 0.004 15

0.002 53 0.002 81 0.003 14 0.003 52 0.003 93

30.684 29.765 28.852 27.947 27.049

45 46 47 48 49

50 51 52 53 54

93 925 93 489 93 004 92 467 91 873

436 485 537 594 656

0.004 64 0.005 19 0.005 77 0.006 42 0.007 14

0.004 40 0.004 92 0.005 49 0.006 10 0.006 79

26.159 25.279 24.408 23.547 22.696

50 51 52 53 54

ex

x

ELT15 (Males) o

qx

mx

727 806 892 987 1 091

0.007 97 0.008 90 0.009 95 0.011 12 0.012 43

0.007 57 0.008 45 0.009 45 0.010 57 0.011 82

21.856 21.027 20.211 19.409 18.622

55 56 57 58 59

86 714 85 507 84 173 82 701 81 076

1 207 1 334 1 472 1 625 1 783

0.013 92 0.015 60 0.017 49 0.019 65 0.021 99

0.013 23 0.014 83 0.016 64 0.018 70 0.021 01

17.850 17.095 16.357 15.640 14.943

60 61 62 63 64

65 66 67 68 69

79 293 77 353 75 256 73 001 70 598

1 940 2 097 2 255 2 403 2 543

0.024 47 0.027 11 0.029 97 0.032 92 0.036 02

0.023 48 0.026 10 0.028 93 0.031 92 0.035 05

14.267 13.612 12.978 12.363 11.767

65 66 67 68 69

70 71 72 73 74

68 055 65 381 62 562 59 593 56 484

2 674 2 819 2 969 3 109 3 218

0.039 30 0.043 11 0.047 45 0.052 17 0.056 97

0.038 33 0.041 98 0.046 26 0.051 05 0.056 09

11.187 10.624 10.080 9.557 9.056

70 71 72 73 74

75 76 77 78 79

53 266 49 965 46 579 43 124 39 630

3 301 3 386 3 455 3 494 3 502

0.061 97 0.067 77 0.074 18 0.081 01 0.088 38

0.061 23 0.066 94 0.073 52 0.080 68 0.088 40

8.572 8.106 7.658 7.232 6.825

75 76 77 78 79

80 81 82 83 84

36 128 32 654 29 254 25 954 22 779

3 474 3 400 3 300 3 175 3 023

0.096 16 0.104 11 0.112 79 0.122 35 0.132 70

0.096 75 0.105 44 0.114 64 0.124 91 0.136 27

6.438 6.070 5.718 5.382 5.063

80 81 82 83 84

85 86 87 88 89

19 756 16 917 14 280 11 874 9 730

2 839 2 637 2 406 2 144 1 873

0.143 72 0.155 85 0.168 48 0.180 61 0.192 46

0.148 57 0.162 08 0.176 89 0.191 90 0.206 47

4.762 4.478 4.213 3.968 3.734

85 86 87 88 89

90 91 92 93 94

7 857 6 249 4 880 3 726 2 773

1 608 1 369 1 154 953 762

0.204 65 0.219 11 0.236 55 0.255 75 0.274 83

0.221 14 0.237 54 0.257 93 0.282 26 0.308 37

3.508 3.285 3.071 2.872 2.693

90 91 92 93 94

95 96 97 98 99

2 011 1 421 979 657 428

590 442 322 229 157

0.293 11 0.311 04 0.329 19 0.347 83 0.367 12

0.334 24 0.359 74 0.385 79 0.413 13 0.442 16

2.531 2.383 2.244 2.114 1.991

95 96 97 98 99

100 101 102 103 104

271 166 98 56 31

105 68 42 25 15

0.387 05 0.407 60 0.428 70 0.450 30 0.474 28

0.473 12 0.506 09 0.541 17 0.578 32 0.619 01

1.874 1.764 1.660 1.562 1.468

100 101 102 103 104

105 106 107 108 109

16 8 4 2 1

8 4 2 1 1

0.496 34 0.518 41 0.540 41 0.562 25 0.583 85

0.664 18 0.706 30 0.751 11 0.797 41 0.844 99

1.384 1.306 1.234 1.166 1.104

105 106 107 108 109

x

lx

55 56 57 58 59

91 217 90 490 89 684 88 792 87 805

60 61 62 63 64

dx

ex

x

69

ELT15 (Females)

70

o

x

lx

dx

qx

mx

ex

x

0 1 2 3 4

100 000 99 368 99 313 99 283 99 261

632 55 30 22 18

0.006 32 0.000 55 0.000 30 0.000 22 0.000 18

0.000 73 0.000 35 0.000 25 0.000 20

78.956 78.462 77.505 76.528 75.545

0 1 2 3 4

5 6 7 8 9

99 243 99 228 99 213 99 199 99 185

15 15 14 14 13

0.000 16 0.000 15 0.000 14 0.000 14 0.000 13

0.000 17 0.000 15 0.000 14 0.000 14 0.000 14

74.559 73.570 72.581 71.591 70.601

5 6 7 8 9

10 11 12 13 14

99 172 99 159 99 145 99 131 99 116

13 14 14 15 18

0.000 13 0.000 14 0.000 14 0.000 15 0.000 18

0.000 13 0.000 14 0.000 14 0.000 14 0.000 17

69.610 68.620 67.629 66.638 65.649

10 11 12 13 14

15 16 17 18 19

99 098 99 077 99 051 99 020 98 989

21 26 31 31 32

0.000 22 0.000 26 0.000 31 0.000 31 0.000 32

0.000 20 0.000 24 0.000 29 0.000 31 0.000 32

64.660 63.674 62.691 61.710 60.729

15 16 17 18 19

20 21 22 23 24

98 957 98 926 98 894 98 862 98 829

31 32 32 33 32

0.000 31 0.000 32 0.000 33 0.000 33 0.000 33

0.000 32 0.000 32 0.000 32 0.000 33 0.000 33

59.748 58.767 57.786 56.805 55.823

20 21 22 23 24

25 26 27 28 29

98 797 98 763 98 729 98 694 98 656

34 34 35 38 39

0.000 34 0.000 35 0.000 36 0.000 38 0.000 40

0.000 33 0.000 34 0.000 35 0.000 37 0.000 39

54.842 53.860 52.878 51.897 50.917

25 26 27 28 29

30 31 32 33 34

98 617 98 574 98 528 98 477 98 420

43 46 51 57 61

0.000 43 0.000 47 0.000 52 0.000 57 0.000 63

0.000 42 0.000 45 0.000 50 0.000 54 0.000 60

49.937 48.958 47.981 47.006 46.032

30 31 32 33 34

35 36 37 38 39

98 359 98 291 98 217 98 136 98 048

68 74 81 88 96

0.000 69 0.000 75 0.000 82 0.000 90 0.000 98

0.000 66 0.000 72 0.000 79 0.000 86 0.000 94

45.061 44.092 43.124 42.160 41.197

35 36 37 38 39

40 41 42 43 44

97 952 97 847 97 733 97 607 97 469

105 114 126 138 154

0.001 07 0.001 17 0.001 29 0.001 42 0.001 58

0.001 02 0.001 12 0.001 23 0.001 35 0.001 49

40.237 39.279 38.325 37.374 36.426

40 41 42 43 44

45 46 47 48 49

97 315 97 142 96 950 96 738 96 504

173 192 212 234 257

0.001 77 0.001 98 0.002 19 0.002 41 0.002 66

0.001 67 0.001 87 0.002 08 0.002 30 0.002 53

35.483 34.545 33.612 32.685 31.763

45 46 47 48 49

50 51 52 53 54

96 247 95 964 95 652 95 310 94 938

283 312 342 372 406

0.002 94 0.003 26 0.003 57 0.003 90 0.004 28

0.002 80 0.003 10 0.003 42 0.003 74 0.004 08

30.846 29.936 29.032 28.134 27.242

50 51 52 53 54

ELT15 (Females) o

x

lx

dx

qx

mx

ex

x

55 56 57 58 59

94 532 94 082 93 583 93 029 92 415

450 499 554 614 683

0.004 75 0.005 31 0.005 92 0.006 60 0.007 39

0.004 51 0.005 03 0.005 62 0.006 26 0.007 00

26.357 25.481 24.614 23.757 22.912

55 56 57 58 59

60 61 62 63 64

91 732 90 971 90 132 89 217 88 210

761 839 915 1007 1117

0.008 30 0.009 22 0.010 15 0.011 29 0.012 66

0.007 86 0.008 80 0.009 72 0.010 74 0.012 03

22.079 21.259 20.452 19.657 18.875

60 61 62 63 64

65 66 67 68 69

87 093 85 875 84 567 83 150 81 617

1218 1308 1417 1533 1647

0.013 99 0.015 23 0.016 76 0.018 44 0.020 17

0.013 42 0.014 70 0.016 09 0.017 74 0.019 49

18.111 17.361 16.621 15.896 15.185

65 66 67 68 69

70 71 72 73 74

79 970 78 219 76 343 74 287 72 048

1751 1876 2056 2239 2366

0.021 90 0.023 99 0.026 93 0.030 14 0.032 84

0.021 23 0.023 11 0.025 69 0.028 97 0.032 03

14.487 13.800 13.127 12.476 11.848

70 71 72 73 74

75 76 77 78 79

69 682 67 195 64 561 61 749 58 765

2487 2634 2812 2984 3158

0.035 69 0.039 19 0.043 56 0.048 33 0.053 73

0.034 80 0.038 03 0.042 14 0.046 94 0.052 28

11.234 10.631 10.044 9.478 8.934

75 76 77 78 79

80 81 82 83 84

55 607 52 293 48 858 45 332 41 736

3314 3435 3526 3596 3655

0.059 61 0.065 68 0.072 16 0.079 33 0.087 57

0.058 27 0.064 64 0.071 31 0.078 61 0.086 91

8.413 7.914 7.435 6.974 6.532

80 81 82 83 84

85 86 87 88 89

38 081 34 375 30 651 27 017 23 542

3706 3724 3634 3475 3330

0.097 31 0.108 33 0.118 59 0.128 60 0.141 46

0.096 74 0.108 41 0.120 52 0.131 74 0.144 62

6.111 5.715 5.349 5.002 4.667

85 86 87 88 89

90 91 92 93 94

20 212 17 069 14 166 11 535 9 214

3143 2903 2631 2321 2008

0.155 50 0.170 06 0.185 73 0.201 26 0.217 90

0.160 53 0.177 51 0.195 73 0.214 98 0.234 90

4.354 4.065 3.797 3.551 3.322

90 91 92 93 94

95 96 97 98 99

7 206 5 504 4 109 3 007 2 154

1702 1395 1102 853 653

0.236 19 0.253 44 0.268 20 0.283 52 0.303 31

0.257 32 0.281 14 0.302 67 0.322 41 0.346 28

3.112 2.925 2.754 2.588 2.422

95 96 97 98 99

100 101 102 103 104

1 501 1 013 663 423 262

488 350 240 161 105

0.324 89 0.345 62 0.361 86 0.379 92 0.400 45

0.376 71 0.408 87 0.437 69 0.462 73 0.493 00

2.269 2.133 2.011 1.887 1.758

100 101 102 103 104

105 106 107 108 109

157 89 48 25 12

68 41 23 13 6

0.436 18 0.459 94 0.483 89 0.507 91 0.531 90

0.537 29 0.599 08 0.637 85 0.683 88 0.731 91

1.621 1.518 1.425 1.338 1.257

105 106 107 108 109

110 111 112

6 3 1

3 2 1

0.555 74 0.579 32 0.602 55

0.781 81 0.833 37 0.886 29

1.183 1.114 1.050

110 111 112

71

72

ASSURED LIVES MORTALITY TABLE AM92 This table is based on the mortality of assured male lives in the UK during the years 1991, 1992, 1993, and 1994. Full details are given in C.M.I.R. 17. Due to potential rounding errors at high ages, the commutation functions ( Dx , N x , S x , C x , M x and Rx ) are tabulated here to age 110 only.

73

AM92

74

x

l[ x ]

l[ x -1]+1

17 18 19

9 997.809 1 9 991.890 4 9 986.035 1

9 993.540 0 9 987.633 8

10 000.000 0 9 994.000 0 9 988.063 6

17 18 19

20 21 22 23 24

9 980.243 2 9 974.504 6 9 968.839 1 9 963.196 7 9 957.577 5

9 981.791 1 9 976.001 6 9 970.265 4 9 964.582 4 9 958.922 5

9 982.200 6 9 976.390 9 9 970.634 6 9 964.931 3 9 959.261 3

20 21 22 23 24

25 26 27 28 29

9 951.991 3 9 946.398 2 9 940.798 4 9 935.181 8 9 929.508 8

9 953.285 8 9 947.662 2 9 942.021 8 9 936.354 9 9 930.661 3

9 953.614 4 9 947.980 7 9 942.340 2 9 936.673 0 9 930.969 4

25 26 27 28 29

30 31 32 33 34

9 923.749 7 9 917.914 5 9 911.953 8 9 905.828 2 9 899.498 4

9 924.891 6 9 919.026 0 9 913.054 7 9 906.928 5 9 900.607 8

9 925.209 4 9 919.353 5 9 913.382 1 9 907.265 5 9 900.964 5

30 31 32 33 34

35 36 37 38 39

9 892.915 1 9 886.039 5 9 878.812 8 9 871.166 5 9 863.022 7

9 894.053 6 9 887.206 9 9 880.028 8 9 872.450 8 9 864.404 7

9 894.429 9 9 887.612 6 9 880.454 0 9 872.895 4 9 864.868 8

35 36 37 38 39

40 41 42 43 44

9 854.303 6 9 844.902 5 9 834.703 0 9 823.599 4 9 811.447 3

9 855.793 1 9 846.538 4 9 836.524 5 9 825.635 4 9 813.746 3

9 856.286 3 9 847.051 0 9 837.066 1 9 826.206 0 9 814.335 9

40 41 42 43 44

45 46 47 48 49

9 798.083 7 9 783.337 1 9 766.998 3 9 748.860 3 9 728.649 9

9 800.693 9 9 786.316 2 9 770.423 1 9 752.787 4 9 733.193 8

9 801.312 3 9 786.953 4 9 771.078 9 9 753.471 4 9 733.886 5

45 46 47 48 49

50 51 52 53 54

9 706.097 7 9 680.899 0 9 652.696 5 9 621.100 6 9 585.691 6

9 711.352 4 9 686.966 9 9 659.707 5 9 629.211 5 9 595.056 3

9 712.072 8 9 687.714 9 9 660.502 1 9 630.052 2 9 595.971 5

50 51 52 53 54

55 56 57 58 59

9 545.992 9 9 501.483 9 9 451.593 8 9 395.697 1 9 333.128 4

9 556.800 3 9 513.937 5 9 465.929 3 9 412.171 2 9 352.016 5

9 557.817 9 9 515.104 0 9 467.290 6 9 413.800 4 9 354.004 0

55 56 57 58 59

60 61 62 63 64

9 263.142 2 9 184.968 7 9 097.740 5 9 000.588 4 8 892.574 1

9 284.764 1 9 209.656 8 9 125.881 8 9 032.564 2 8 928.817 7

9 287.216 4 9 212.714 3 9 129.717 0 9 037.397 3 8 934.877 1

60 61 62 63 64

lx

x

AM92 x

l[ x ]

l[ x -1]+1

lx

x

65 66 67 68 69

8 772.735 9 8 640.048 1 8 493.518 7 8 332.139 6 8 154.931 8

8 813.688 1 8 686.201 6 8 545.353 2 8 390.161 1 8 219.639 0

8 821.261 2 8 695.619 9 8 557.011 8 8 404.491 6 8 237.132 9

65 66 67 68 69

70 71 72 73 74

7 960.977 6 7 749.465 9 7 519.702 7 7 271.146 1 7 003.521 6

8 032.860 6 7 828.968 6 7 607.240 0 7 367.082 8 7 108.105 2

8 054.054 4 7 854.450 8 7 637.620 8 7 403.008 4 7 150.240 1

70 71 72 73 74

75 76 77 78 79

6 716.823 1 6 411.345 9 6 087.808 4 5 747.362 4 5 391.640 0

6 830.184 4 6 533.500 8 6 218.575 9 5 886.362 8 5 538.279 1

6 879.167 3 6 589.925 8 6 282.980 3 5 959.168 0 5 619.757 7

75 76 77 78 79

80 81 82 83 84

5 022.793 1 4 643.512 9 4 257.005 6 3 866.988 4 3 477.592 9

5 176.222 4 4 802.629 0 4 420.452 5 4 033.146 7 3 644.632 7

5 266.460 4 4 901.478 9 4 527.496 0 4 147.670 8 3 765.599 8

80 81 82 83 84

85 86 87 88 89

3 093.286 3 2 718.712 8 2 358.529 9 2 017.229 8 1 698.908 9

3 259.186 2 2 881.346 7 2 515.731 0 2 166.880 5 1 839.045 8

3 385.247 9 3 010.839 5 2 646.741 6 2 297.297 6 1 966.649 9

85 86 87 88 89

90 91 92 93 94

1 407.055 0

1 535.980 1 1 260.735 4

1 658.554 5 1 376.190 6 1 121.988 9 897.502 5 703.324 2

90 91 92 93 94

95 96 97 98 99

539.064 3 403.402 3 294.206 1 208.706 0 143.712 0

95 96 97 98 99

100 101 102 103 104

95.847 6 61.773 3 38.379 6 22.928 4 13.135 9

100 101 102 103 104

105 106 107 108 109

7.196 8 3.759 6 1.866 9 0.878 4 0.390 3

105 106 107 108 109

110 111 112 113 114

0.163 2 0.064 0 0.023 4 0.008 0 0.002 5

110 111 112 113 114

115 116 117 118 119

0.000 7 0.000 2 0.000 0 0.000 0 0.000 0

115 116 117 118 119

120

0.000 0

120

75

AM92

76

x

d[ x ]

d[ x -1]+1

dx

x

17 18 19

4.269 1 4.256 5 4.244 1

5.476 5 5.433 3

6.000 0 5.936 4 5.863 0

17 18 19

20 21 22 23 24

4.241 6 4.239 2 4.256 7 4.274 2 4.291 7

5.400 1 5.367 1 5.334 1 5.321 1 5.308 1

5.809 6 5.756 4 5.703 2 5.670 0 5.646 9

20 21 22 23 24

25 26 27 28 29

4.329 1 4.376 4 4.443 5 4.520 5 4.617 2

5.305 1 5.322 0 5.348 8 5.385 5 5.451 9

5.633 7 5.640 5 5.667 1 5.703 7 5.760 0

25 26 27 28 29

30 31 32 33 34

4.723 7 4.859 8 5.025 4 5.220 4 5.444 7

5.538 1 5.643 9 5.789 2 5.964 0 6.178 0

5.855 9 5.971 5 6.116 6 6.301 0 6.534 6

30 31 32 33 34

35 36 37 38 39

5.708 2 6.010 7 6.362 0 6.761 7 7.229 6

6.441 0 6.753 0 7.133 4 7.582 0 8.118 4

6.817 3 7.158 6 7.558 5 8.026 7 8.582 4

35 36 37 38 39

40 41 42 43 44

7.765 2 8.378 0 9.067 6 9.853 1 10.753 3

8.742 1 9.472 4 10.318 5 11.299 5 12.434 0

9.235 3 9.984 9 10.860 1 11.870 1 13.023 6

40 41 42 43 44

45 46 47 48 49

11.767 5 12.914 0 14.211 0 15.666 4 17.297 5

13.740 6 15.237 3 16.951 7 18.900 9 21.121 0

14.358 9 15.874 4 17.607 5 19.585 0 21.813 6

45 46 47 48 49

50 51 52 53 54

19.130 7 21.191 5 23.485 0 26.044 3 28.891 3

23.637 4 26.464 8 29.655 3 33.240 0 37.238 4

24.357 9 27.212 8 30.449 9 34.080 8 38.153 6

50 51 52 53 54

55 56 57 58 59

32.055 4 35.554 6 39.422 6 43.680 6 48.364 3

41.696 3 46.646 8 52.128 9 58.167 2 64.800 1

42.713 9 47.813 4 53.490 2 59.796 5 66.787 6

55 56 57 58 59

60 61 62 63 64

53.485 4 59.086 9 65.176 2 71.770 7 78.886 0

72.049 8 79.939 8 88.484 6 97.687 2 107.556 5

74.502 0 82.997 3 92.319 7 102.520 2 113.615 9

60 61 62 63 64

AM92 x

d[ x ]

d[ x -1]+1

dx

x

65 66 67 68 69

86.534 3 94.694 9 103.357 6 112.500 5 122.071 2

118.068 2 129.189 9 140.861 6 153.028 1 165.584 6

125.641 2 138.608 2 152.520 2 167.358 6 183.078 5

65 66 67 68 69

70 71 72 73 74

132.008 9 142.225 9 152.619 9 163.040 9 173.337 2

178.409 8 191.347 8 204.231 6 216.842 7 228.937 9

199.603 6 216.830 0 234.612 4 252.768 3 271.072 8

70 71 72 73 74

75 76 77 78 79

183.322 3 192.769 9 201.445 6 209.083 3 215.417 6

240.258 6 250.520 6 259.407 9 266.605 1 271.818 7

289.241 5 306.945 6 323.812 2 339.410 4 353.297 3

75 76 77 78 79

80 81 82 83 84

220.164 1 223.060 4 223.858 9 222.355 7 218.406 7

274.743 5 275.133 0 272.781 7 267.546 8 259.384 9

364.981 5 373.982 8 379.825 2 382.071 0 380.351 9

80 81 82 83 84

85 86 87 88 89

211.939 6 202.981 8 191.649 4 178.183 9 162.928 8

248.346 7 234.605 0 218.433 4 200.230 6 180.491 3

374.408 4 364.097 8 349.444 0 330.647 8 308.095 4

85 86 87 88 89

90 91 92 93 94

146.319 7

159.789 5 138.746 4

282.363 9 254.201 7 224.486 4 194.178 3 164.260 0

90 91 92 93 94

95 96 97 98 99

135.662 0 109.196 2 85.500 1 64.994 0 47.864 4

95 96 97 98 99

100 101 102 103 104

34.074 3 23.393 7 15.451 2 9.792 5 5.939 1

100 101 102 103 104

105 106 107 108 109

3.437 3 1.892 7 .988 5 .488 1 .227 1

105 106 107 108 109

110 111 112 113 114

.099 2 .040 5 .015 4 .005 5 .001 8

110 111 112 113 114

115 116 117 118 119

.000 5 .000 1 .000 0 .000 0 .000 0

115 116 117 118 119

120

.000 0

120

77

AM92

78

x

q[ x ]

q[ x -1]+1

qx

x

17 18 19

.000 427 .000 426 .000 425

.000 548 .000 544

.000 600 .000 594 .000 587

17 18 19

20 21 22 23 24

.000 425 .000 425 .000 427 .000 429 .000 431

.000 541 .000 538 .000 535 .000 534 .000 533

.000 582 .000 577 .000 572 .000 569 .000 567

20 21 22 23 24

25 26 27 28 29

.000 435 .000 440 .000 447 .000 455 .000 465

.000 533 .000 535 .000 538 .000 542 .000 549

.000 566 .000 567 .000 570 .000 574 .000 580

25 26 27 28 29

30 31 32 33 34

.000 476 .000 490 .000 507 .000 527 .000 550

.000 558 .000 569 .000 584 .000 602 .000 624

.000 590 .000 602 .000 617 .000 636 .000 660

30 31 32 33 34

35 36 37 38 39

.000 577 .000 608 .000 644 .000 685 .000 733

.000 651 .000 683 .000 722 .000 768 .000 823

.000 689 .000 724 .000 765 .000 813 .000 870

35 36 37 38 39

40 41 42 43 44

.000 788 .000 851 .000 922 .001 003 .001 096

.000 887 .000 962 .001 049 .001 150 .001 267

.000 937 .001 014 .001 104 .001 208 .001 327

40 41 42 43 44

45 46 47 48 49

.001 201 .001 320 .001 455 .001 607 .001 778

.001 402 .001 557 .001 735 .001 938 .002 170

.001 465 .001 622 .001 802 .002 008 .002 241

45 46 47 48 49

50 51 52 53 54

.001 971 .002 189 .002 433 .002 707 .003 014

.002 434 .002 732 .003 070 .003 452 .003 881

.002 508 .002 809 .003 152 .003 539 .003 976

50 51 52 53 54

55 56 57 58 59

.003 358 .003 742 .004 171 .004 649 .005 182

.004 363 .004 903 .005 507 .006 180 .006 929

.004 469 .005 025 .005 650 .006 352 .007 140

55 56 57 58 59

60 61 62 63 64

.005 774 .006 433 .007 164 .007 974 .008 871

.007 760 .008 680 .009 696 .010 815 .012 046

.008 022 .009 009 .010 112 .011 344 .012 716

60 61 62 63 64

AM92 x

q[ x ]

q[ x -1]+1

qx

x

65 66 67 68 69

.009 864 .010 960 .012 169 .013 502 .014 969

.013 396 .014 873 .016 484 .018 239 .020 145

.014 243 .015 940 .017 824 .019 913 .022 226

65 66 67 68 69

70 71 72 73 74

.016 582 .018 353 .020 296 .022 423 .024 750

.022 210 .024 441 .026 847 .029 434 .032 208

.024 783 .027 606 .030 718 .034 144 .037 911

70 71 72 73 74

75 76 77 78 79

.027 293 .030 067 .033 090 .036 379 .039 954

.035 176 .038 344 .041 715 .045 292 .049 080

.042 046 .046 578 .051 538 .056 956 .062 867

75 76 77 78 79

80 81 82 83 84

.043 833 .048 037 .052 586 .057 501 .062 804

.053 078 .057 288 .061 709 .066 337 .071 169

.069 303 .076 300 .083 893 .092 117 .101 007

80 81 82 83 84

85 86 87 88 89

.068 516 .074 661 .081 258 .088 331 .095 902

.076 199 .081 422 .086 827 .092 405 .098 144

.110 600 .120 929 .132 028 .143 929 .156 660

85 86 87 88 89

90 91 92 93 94

.103 990

.104 031 .110 052

.170 247 .184 714 .200 079 .216 354 .233 548

90 91 92 93 94

95 96 97 98 99

.251 662 .270 688 .290 613 .311 414 .333 058

95 96 97 98 99

100 101 102 103 104

.355 505 .378 702 .402 588 .427 090 .452 127

100 101 102 103 104

105 106 107 108 109

.477 608 .503 432 .529 493 .555 674 .581 857

105 106 107 108 109

110 111 112 113 114

.607 918 .633 731 .659 171 .684 114 .708 442

110 111 112 113 114

115 116 117 118 119

.732 042 .754 809 .776 648 .797 477 .817 225

115 116 117 118 119

120

1.000 000

120

79

AM92

80

x

m[ x ]

m[ x-1]+1

mx

x

17 18 19

0.000 367 0.000 367 0.000 367

0.000 488 0.000 485

0.000 603 0.000 597 0.000 591

17 18 19

20 21 22 23 24

0.000 369 0.000 370 0.000 374 0.000 377 0.000 380

0.000 483 0.000 482 0.000 480 0.000 481 0.000 481

0.000 585 0.000 580 0.000 574 0.000 570 0.000 568

20 21 22 23 24

25 26 27 28 29

0.000 385 0.000 391 0.000 400 0.000 408 0.000 419

0.000 482 0.000 485 0.000 489 0.000 495 0.000 502

0.000 566 0.000 566 0.000 568 0.000 572 0.000 577

25 26 27 28 29

30 31 32 33 34

0.000 430 0.000 443 0.000 460 0.000 479 0.000 500

0.000 512 0.000 523 0.000 537 0.000 555 0.000 576

0.000 585 0.000 596 0.000 609 0.000 626 0.000 647

30 31 32 33 34

35 36 37 38 39

0.000 524 0.000 551 0.000 582 0.000 616 0.000 656

0.000 601 0.000 630 0.000 665 0.000 706 0.000 754

0.000 674 0.000 706 0.000 744 0.000 788 0.000 840

35 36 37 38 39

40 41 42 43 44

0.000 701 0.000 752 0.000 808 0.000 871 0.000 943

0.000 810 0.000 875 0.000 950 0.001 037 0.001 136

0.000 902 0.000 974 0.001 057 0.001 154 0.001 265

40 41 42 43 44

45 46 47 48 49

0.001 023 0.001 113 0.001 214 0.001 326 0.001 451

0.001 250 0.001 380 0.001 529 0.001 698 0.001 890

0.001 394 0.001 541 0.001 709 0.001 902 0.002 122

45 46 47 48 49

50 51 52 53 54

0.001 592 0.001 750 0.001 925 0.002 122 0.002 342

0.002 108 0.002 354 0.002 633 0.002 947 0.003 300

0.002 372 0.002 656 0.002 978 0.003 343 0.003 756

50 51 52 53 54

55 56 57 58 59

0.002 588 0.002 862 0.003 170 0.003 513 0.003 898

0.003 696 0.004 139 0.004 636 0.005 189 0.005 806

0.004 221 0.004 747 0.005 340 0.006 005 0.006 754

55 56 57 58 59

60 61 62 63 64

0.004 327 0.004 809 0.005 348 0.005 949 0.006 623

0.006 493 0.007 254 0.008 099 0.009 032 0.010 063

0.007 593 0.008 533 0.009 586 0.010 763 0.012 078

60 61 62 63 64

AM92 x

m[ x ]

m[ x-1]+1

mx

x

65 66 67 68 69

0.007 377 0.008 220 0.009 162 0.010 216 0.011 393

0.011 199 0.012 449 0.013 821 0.015 326 0.016 972

0.013 544 0.015 176 0.016 993 0.019 012 0.021 255

65 66 67 68 69

70 71 72 73 74

0.012 709 0.014 178 0.015 819 0.017 648 0.019 687

0.018 771 0.020 733 0.022 869 0.025 190 0.027 708

0.023 741 0.026 496 0.029 543 0.032 912 0.036 631

70 71 72 73 74

75 76 77 78 79

0.021 959 0.024 487 0.027 300 0.030 423 0.033 892

0.030 436 0.033 385 0.036 569 0.040 000 0.043 691

0.040 732 0.045 251 0.050 223 0.055 689 0.061 689

75 76 77 78 79

80 81 82 83 84

0.037 737 0.041 996 0.046 709 0.051 916 0.057 665

0.047 656 0.051 909 0.056 462 0.061 329 0.066 524

0.068 271 0.075 481 0.083 372 0.091 999 0.101 417

80 81 82 83 84

85 86 87 88 89

0.064 000 0.070 978 0.078 646 0.087 067 0.096 302

0.072 061 0.077 952 0.084 213 0.090 853 0.097 889

0.111 691 0.122 884 0.135 066 0.148 309 0.162 691

85 86 87 88 89

90 91 92 93 94

0.106 409

0.105 333 0.113 198

0.178 289 0.195 190 0.213 482 0.233 257 0.254 610

90 91 92 93 94

95 96 97 98 99

0.277 645 0.302 462 0.329 170 0.357 882 0.388 711

95 96 97 98 99

100 101 102 103 104

0.421 777 0.457 202 0.495 111 0.535 631 0.578 890

100 101 102 103 104

105 106 107 108 109

0.625 023 0.674 162 0.726 443 0.782 002 0.840 973

105 106 107 108 109

110 111 112 113 114

0.903 494 0.969 700 1.039 723 1.113 695 1.191 744

110 111 112 113 114

115 116 117 118 119

1.274 000 1.360 581 1.451 603 1.547 178 1.647 417

115 116 117 118 119

120

2.000 000

120

81

AM92

82

x

e[ x ]

e[ x -1]+1

ex

x

17 18 19

61.353 60.389 59.424

60.379 59.414

61.339 60.376 59.412

17 18 19

20 21 22 23 24

58.458 57.492 56.524 55.556 54.587

58.449 57.483 56.516 55.548 54.580

58.447 57.481 56.514 55.546 54.578

20 21 22 23 24

25 26 27 28 29

53.618 52.648 51.677 50.706 49.735

53.611 52.641 51.671 50.700 49.729

53.609 52.639 51.669 50.699 49.728

25 26 27 28 29

30 31 32 33 34

48.764 47.792 46.821 45.850 44.879

48.758 47.787 46.816 45.845 44.874

48.757 47.785 46.814 45.843 44.872

30 31 32 33 34

35 36 37 38 39

43.909 42.939 41.970 41.003 40.036

43.904 42.934 41.965 40.997 40.031

43.902 42.932 41.963 40.995 40.029

35 36 37 38 39

40 41 42 43 44

39.071 38.108 37.148 36.189 35.234

39.066 38.102 37.141 36.182 35.226

39.064 38.100 37.139 36.180 35.224

40 41 42 43 44

45 46 47 48 49

34.282 33.333 32.388 31.448 30.513

34.273 33.323 32.377 31.436 30.499

34.271 33.321 32.375 31.433 30.497

45 46 47 48 49

50 51 52 53 54

29.583 28.660 27.742 26.833 25.931

29.567 28.642 27.722 26.810 25.905

29.565 28.639 27.720 26.808 25.903

50 51 52 53 54

55 56 57 58 59

25.037 24.153 23.279 22.415 21.563

25.009 24.122 23.244 22.376 21.520

25.006 24.119 23.240 22.373 21.516

55 56 57 58 59

60 61 62 63 64

20.724 19.897 19.084 18.286 17.503

20.676 19.844 19.026 18.222 17.433

20.670 19.837 19.018 18.212 17.421

60 61 62 63 64

AM92 x

e[ x ]

e[ x -1]+1

ex

x

65 66 67 68 69

16.736 15.987 15.255 14.541 13.847

16.660 15.903 15.164 14.443 13.740

16.645 15.886 15.143 14.418 13.711

65 66 67 68 69

70 71 72 73 74

13.172 12.517 11.883 11.270 10.679

13.057 12.394 11.751 11.129 10.529

13.023 12.354 11.704 11.075 10.467

70 71 72 73 74

75 76 77 78 79

10.110 9.562 9.037 8.534 8.053

9.950 9.393 8.859 8.346 7.856

9.879 9.313 8.768 8.244 7.742

75 76 77 78 79

80 81 82 83 84

7.594 7.157 6.741 6.347 5.974

7.388 6.942 6.518 6.116 5.734

7.261 6.802 6.364 5.947 5.550

80 81 82 83 84

85 86 87 88 89

5.620 5.287 4.972 4.676 4.397

5.374 5.034 4.713 4.412 4.129

5.174 4.817 4.480 4.161 3.861

85 86 87 88 89

90 91 92 93 94

4.136

3.864 3.616

3.578 3.312 3.063 2.829 2.610

90 91 92 93 94

95 96 97 98 99

2.405 2.214 2.035 1.869 1.715

95 96 97 98 99

100 101 102 103 104

1.571 1.437 1.314 1.199 1.093

100 101 102 103 104

105 106 107 108 109

0.994 0.904 0.820 0.743 0.672

105 106 107 108 109

110 111 112 113 114

0.606 0.546 0.491 0.440 0.394

110 111 112 113 114

115 116 117 118 119

0.352 0.313 0.277 0.240 0.183

115 116 117 118 119

120

0.000

120

83

AM92 4%

84

x

D[ x ]

D[ x -1]+1

Dx

x

17 18 19

5 132.61 4 932.28 4 739.80

4 933.09 4 740.55

5 133.73 4 933.32 4 740.76

17 18 19

20 21 22 23 24

4 554.85 4 377.15 4 206.41 4 042.33 3 884.66

4 555.56 4 377.80 4 207.01 4 042.89 3 885.19

4 555.75 4 377.98 4 207.16 4 043.04 3 885.32

20 21 22 23 24

25 26 27 28 29

3 733.16 3 587.56 3 447.63 3 313.16 3 183.91

3 733.64 3 588.01 3 448.06 3 313.55 3 184.28

3 733.77 3 588.13 3 448.17 3 313.66 3 184.38

25 26 27 28 29

30 31 32 33 34

3 059.68 2 940.27 2 825.48 2 715.13 2 609.03

3 060.03 2 940.60 2 825.79 2 715.43 2 609.33

3 060.13 2 940.69 2 825.89 2 715.52 2 609.42

30 31 32 33 34

35 36 37 38 39

2 507.02 2 408.92 2 314.57 2 223.83 2 136.53

2 507.31 2 409.20 2 314.86 2 224.12 2 136.83

2 507.40 2 409.30 2 314.96 2 224.22 2 136.93

35 36 37 38 39

40 41 42 43 44

2 052.54 1 971.72 1 893.92 1 819.02 1 746.89

2 052.85 1 972.04 1 894.27 1 819.40 1 747.30

2 052.96 1 972.15 1 894.37 1 819.50 1 747.41

40 41 42 43 44

45 46 47 48 49

1 677.42 1 610.47 1 545.95 1 483.73 1 423.70

1 677.86 1 610.96 1 546.49 1 484.32 1 424.37

1 677.97 1 611.07 1 546.59 1 484.43 1 424.47

45 46 47 48 49

50 51 52 53 54

1 365.77 1 309.83 1 255.78 1 203.53 1 152.98

1 366.51 1 310.65 1 256.70 1 204.55 1 154.11

1 366.61 1 310.75 1 256.80 1 204.65 1 154.22

50 51 52 53 54

55 56 57 58 59

1 104.05 1 056.63 1 010.66 966.04 922.70

1 105.30 1 058.02 1 012.19 967.73 924.57

1 105.41 1 058.15 1 012.34 967.90 924.76

55 56 57 58 59

60 61 62 63 64

880.56 839.55 799.59 760.62 722.59

882.61 841.80 802.06 763.33 725.54

882.85 842.08 802.40 763.74 726.03

60 61 62 63 64

AM92 x

D[ x ]

D[ x -1]+1

Dx

x

65 66 67 68 69

685.44 649.11 613.56 578.75 544.65

688.64 652.57 617.30 582.78 548.97

689.23 653.28 618.14 583.77 550.14

65 66 67 68 69

70 71 72 73 74

511.25 478.53 446.48 415.12 384.46

515.87 483.43 451.68 420.59 390.20

517.23 485.01 453.48 422.64 392.51

70 71 72 73 74

75 76 77 78 79

354.54 325.40 297.09 269.69 243.27

360.52 331.60 303.48 276.21 249.89

363.11 334.46 306.62 279.63 253.56

75 76 77 78 79

80 81 82 83 84

217.91 193.71 170.75 149.14 128.97

224.57 200.35 177.31 155.55 135.16

228.48 204.47 181.60 159.97 139.65

80 81 82 83 84

85 86 87 88 89

110.30 93.22 77.76 63.95 51.78

116.22 98.79 82.94 68.69 56.06

120.71 103.23 87.26 72.83 59.95

85 86 87 88 89

90 91 92 93 94

41.24

45.02 35.53

48.61 38.78 30.40 23.38 17.62

90 91 92 93 94

95 96 97 98 99

12.99 9.34 6.55 4.47 2.96

95 96 97 98 99

100 101 102 103 104

1.90 1.18 .70 .40 .22

100 101 102 103 104

105 106 107 108 109

.12 .06 .03 .01 .01

105 106 107 108 109

110

.00

110

4%

85

AM92 4%

86

x

N[ x ]

N[ x -1]+1

Nx

x

17 18 19

119 958.58 114 824.96 109 891.73

114 825.98 109 892.68

119 959.94 114 826.20 109 892.88

17 18 19

20 21 22 23 24

105 151.06 100 595.40 96 217.50 92 010.40 87 967.43

105 151.94 100 596.21 96 218.25 92 011.10 87 968.07

105 152.13 100 596.38 96 218.40 92 011.24 87 968.21

20 21 22 23 24

25 26 27 28 29

84 082.16 80 348.43 76 760.35 73 312.22 69 998.60

84 082.76 80 349.00 76 760.88 73 312.71 69 999.06

84 082.88 80 349.12 76 760.99 73 312.82 69 999.16

25 26 27 28 29

30 31 32 33 34

66 814.23 63 754.13 60 813.46 57 987.58 55 272.07

66 814.68 63 754.56 60 813.87 57 987.98 55 272.45

66 814.78 63 754.65 60 813.96 57 988.07 55 272.55

30 31 32 33 34

35 36 37 38 39

52 662.65 50 155.24 47 745.94 45 430.98 43 206.74

52 663.03 50 155.63 47 746.33 45 431.37 43 207.15

52 663.13 50 155.73 47 746.43 45 431.47 43 207.25

35 36 37 38 39

40 41 42 43 44

41 069.80 39 016.82 37 044.65 35 150.25 33 330.72

41 070.21 39 017.25 37 045.10 35 150.73 33 331.23

41 070.31 39 017.36 37 045.21 35 150.84 33 331.34

40 41 42 43 44

45 46 47 48 49

31 583.27 29 905.26 28 294.14 26 747.50 25 263.01

31 583.82 29 905.86 28 294.79 26 748.20 25 263.77

31 583.93 29 905.96 28 294.89 26 748.30 25 263.87

45 46 47 48 49

50 51 52 53 54

23 838.46 22 471.77 21 160.92 19 904.01 18 699.23

23 839.30 22 472.69 21 161.94 19 905.14 18 700.48

23 839.41 22 472.79 21 162.04 19 905.24 18 700.59

50 51 52 53 54

55 56 57 58 59

17 544.87 16 439.29 15 380.96 14 368.41 13 400.27

17 546.25 16 440.82 15 382.66 14 370.30 13 402.37

17 546.37 16 440.95 15 382.81 14 370.47 13 402.57

55 56 57 58 59

60 61 62 63 64

12 475.24 11 592.08 10 749.66 9 946.87 9 182.71

12 477.57 11 594.68 10 752.54 9 950.07 9 186.25

12 477.80 11 594.96 10 752.88 9 950.48 9 186.74

60 61 62 63 64

AM92 x

N[ x ]

N[ x -1]+1

Nx

x

65 66 67 68 69

8 456.21 7 766.46 7 112.62 6 493.86 5 909.43

8 460.12 7 770.77 7 117.36 6 499.06 5 915.12

8 460.71 7 771.48 7 118.20 6 500.06 5 916.29

65 66 67 68 69

70 71 72 73 74

5 358.59 4 840.63 4 354.86 3 900.59 3 477.14

5 364.78 4 847.34 4 362.10 3 908.38 3 485.47

5 366.14 4 848.92 4 363.91 3 910.43 3 487.78

70 71 72 73 74

75 76 77 78 79

3 083.84 2 719.96 2 384.76 2 077.47 1 797.25

3 092.69 2 729.30 2 394.56 2 087.67 1 807.78

3 095.27 2 732.16 2 397.70 2 091.08 1 811.45

75 76 77 78 79

80 81 82 83 84

1 543.20 1 314.35 1 109.67 928.03 768.19

1 553.98 1 325.29 1 120.65 938.92 778.88

1 557.89 1 329.41 1 124.94 943.34 783.37

80 81 82 83 84

85 86 87 88 89

628.87 508.67 406.14 319.75 247.93

639.22 518.57 415.45 328.38 255.80

643.72 523.01 419.77 332.51 259.69

85 86 87 88 89

90 91 92 93 94

189.12

196.15 147.88

199.74 151.13 112.35 81.95 58.56

90 91 92 93 94

95 96 97 98 99

40.94 27.95 18.61 12.06 7.59

95 96 97 98 99

100 101 102 103 104

4.63 2.73 1.55 .85 .45

100 101 102 103 104

105 106 107 108 109

.23 .11 .05 .02 .01

105 106 107 108 109

110

.00

110

4%

87

AM92 4%

88

x

S[ x ]

17 18 19

2 398 085.62 2 278 125.81 2 163 299.72

2 278 127.03 2 163 300.85

2 398 087.20 2 278 127.26 2 163 301.06

17 18 19

20 21 22 23 24

2 053 406.94 1 948 254.91 1 847 658.63 1 751 440.30 1 659 429.12

2 053 407.99 1 948 255.88 1 847 659.51 1 751 441.12 1 659 429.89

2 053 408.17 1 948 256.05 1 847 659.67 1 751 441.27 1 659 430.03

20 21 22 23 24

25 26 27 28 29

1 571 460.98 1 487 378.14 1 407 029.07 1 330 268.14 1 256 955.35

1 571 461.70 1 487 378.82 1 407 029.71 1 330 268.73 1 256 955.92

1 571 461.82 1 487 378.94 1 407 029.82 1 330 268.83 1 256 956.02

25 26 27 28 29

30 31 32 33 34

1 186 956.21 1 120 141.46 1 056 386.83 995 572.87 937 584.81

1 186 956.76 1 120 141.98 1 056 387.32 995 573.36 937 585.29

1 186 956.85 1 120 142.07 1 056 387.42 995 573.46 937 585.38

30 31 32 33 34

35 36 37 38 39

882 312.25 829 649.12 779 493.40 731 746.96 686 315.48

882 312.74 829 649.61 779 493.88 731 747.45 686 315.99

882 312.84 829 649.71 779 493.98 731 747.56 686 316.09

35 36 37 38 39

40 41 42 43 44

643 108.22 602 037.89 563 020.51 525 975.27 490 824.40

643 108.74 602 038.43 563 021.07 525 975.86 490 825.02

643 108.84 602 038.53 563 021.17 525 975.96 490 825.13

40 41 42 43 44

45 46 47 48 49

457 493.03 425 909.06 396 003.05 367 708.11 340 959.74

457 493.69 425 909.76 396 003.80 367 708.91 340 960.61

457 493.79 425 909.86 396 003.90 367 709.01 340 960.71

45 46 47 48 49

50 51 52 53 54

315 695.79 291 856.30 269 383.41 248 221.26 228 315.88

315 696.73 291 857.33 269 384.53 248 222.49 228 317.24

315 696.84 291 857.43 269 384.64 248 222.60 228 317.35

50 51 52 53 54

55 56 57 58 59

209 615.14 192 068.59 175 627.43 160 244.38 145 873.64

209 616.65 192 070.27 175 629.30 160 246.47 145 875.97

209 616.77 192 070.40 175 629.44 160 246.64 145 876.17

55 56 57 58 59

60 61 62 63 64

132 470.75 119 992.59 108 397.21 97 643.87 87 692.86

132 473.37 119 995.52 108 400.50 97 647.55 87 696.99

132 473.60 119 995.80 108 400.84 97 647.96 87 697.49

60 61 62 63 64

S[ x -1]+1

Sx

x

AM92 x

S[ x ]

S[ x -1]+1

Sx

x

65 66 67 68 69

78 505.54 70 044.17 62 271.97 55 152.99 48 652.08

78 510.15 70 049.32 62 277.71 55 159.35 48 659.12

78 510.74 70 050.03 62 278.55 55 160.35 48 660.29

65 66 67 68 69

70 71 72 73 74

42 734.88 37 367.77 32 517.84 28 152.89 24 241.39

42 742.64 37 376.29 32 527.14 28 162.99 24 252.30

42 744.01 37 377.86 32 528.95 28 165.04 24 254.61

70 71 72 73 74

75 76 77 78 79

20 752.53 17 656.21 14 923.03 12 524.40 10 432.48

20 764.24 17 668.69 14 936.25 12 538.27 10 446.93

20 766.83 17 671.56 14 939.39 12 541.69 10 450.60

75 76 77 78 79

80 81 82 83 84

8 620.33 7 061.91 5 732.17 4 607.11 3 663.90

8 635.24 7 077.14 5 747.56 4 622.49 3 679.09

8 639.15 7 081.26 5 751.85 4 626.91 3 683.57

80 81 82 83 84

85 86 87 88 89

2 880.92 2 237.83 1 715.71 1 297.05 965.85

2 895.71 2 252.05 1 729.16 1 309.57 977.30

2 900.21 2 256.49 1 733.48 1 313.71 981.19

85 86 87 88 89

90 91 92 93 94

707.63

717.91 518.51

721.51 521.76 370.63 258.28 176.34

90 91 92 93 94

95 96 97 98 99

117.78 76.84 48.88 30.28 18.22

95 96 97 98 99

100 101 102 103 104

10.63 6.00 3.27 1.72 .87

100 101 102 103 104

105 106 107 108 109

.42 .19 .09 .04 .01

105 106 107 108 109

110

.01

110

4%

89

AM92 4%

90

x

C[ x ]

C[ x -1]+1

Cx

x

17 18 19

2.11 2.02 1.94

2.60 2.48

2.96 2.82 2.68

17 18 19

20 21 22 23 24

1.86 1.79 1.73 1.67 1.61

2.37 2.26 2.16 2.08 1.99

2.55 2.43 2.31 2.21 2.12

20 21 22 23 24

25 26 27 28 29

1.56 1.52 1.48 1.45 1.42

1.91 1.85 1.78 1.73 1.68

2.03 1.96 1.89 1.83 1.78

25 26 27 28 29

30 31 32 33 34

1.40 1.39 1.38 1.38 1.38

1.64 1.61 1.59 1.57 1.57

1.74 1.70 1.68 1.66 1.66

30 31 32 33 34

35 36 37 38 39

1.39 1.41 1.43 1.46 1.51

1.57 1.58 1.61 1.64 1.69

1.66 1.68 1.70 1.74 1.79

35 36 37 38 39

40 41 42 43 44

1.56 1.61 1.68 1.75 1.84

1.75 1.82 1.91 2.01 2.13

1.85 1.92 2.01 2.11 2.23

40 41 42 43 44

45 46 47 48 49

1.94 2.04 2.16 2.29 2.43

2.26 2.41 2.58 2.77 2.97

2.36 2.51 2.68 2.87 3.07

45 46 47 48 49

50 51 52 53 54

2.59 2.76 2.94 3.13 3.34

3.20 3.44 3.71 4.00 4.31

3.30 3.54 3.81 4.10 4.41

50 51 52 53 54

55 56 57 58 59

3.56 3.80 4.05 4.32 4.60

4.64 4.99 5.36 5.75 6.16

4.75 5.11 5.50 5.91 6.35

55 56 57 58 59

60 61 62 63 64

4.89 5.19 5.51 5.83 6.16

6.59 7.03 7.48 7.94 8.40

6.81 7.29 7.80 8.33 8.88

60 61 62 63 64

AM92 x

C[ x ]

C[ x -1]+1

Cx

x

65 66 67 68 69

6.50 6.84 7.18 7.51 7.84

8.87 9.33 9.78 10.22 10.63

9.44 10.01 10.59 11.18 11.76

65 66 67 68 69

70 71 72 73 74

8.15 8.44 8.71 8.95 9.15

11.02 11.36 11.66 11.90 12.08

12.33 12.87 13.39 13.88 14.31

70 71 72 73 74

75 76 77 78 79

9.30 9.41 9.45 9.43 9.35

12.19 12.23 12.17 12.03 11.79

14.68 14.98 15.19 15.31 15.33

75 76 77 78 79

80 81 82 83 84

9.18 8.95 8.63 8.25 7.79

11.46 11.04 10.52 9.92 9.25

15.23 15.00 14.65 14.17 13.56

80 81 82 83 84

85 86 87 88 89

7.27 6.69 6.08 5.43 4.78

8.52 7.73 6.92 6.10 5.29

12.84 12.00 11.08 10.08 9.03

85 86 87 88 89

90 91 92 93 94

4.12

4.50 3.76

7.96 6.89 5.85 4.86 3.96

90 91 92 93 94

95 96 97 98 99

3.14 2.43 1.83 1.34 .95

95 96 97 98 99

100 101 102 103 104

.65 .43 .27 .17 .10

100 101 102 103 104

105 106 107 108 109

.05 .03 .01 .01 .00

105 106 107 108 109

110

.00

110

4%

91

AM92 4%

92

x

M[ x]

17 18 19

518.82 515.93 513.19

516.71 513.91

519.89 516.93 514.11

17 18 19

20 21 22 23 24

510.58 508.09 505.73 503.47 501.30

511.25 508.72 506.31 504.01 501.80

511.43 508.88 506.46 504.14 501.93

20 21 22 23 24

25 26 27 28 29

499.23 497.23 495.31 493.46 491.66

499.69 497.67 495.72 493.83 492.01

499.81 497.78 495.82 493.93 492.10

25 26 27 28 29

30 31 32 33 34

489.90 488.19 486.50 484.84 483.18

490.23 488.50 486.80 485.12 483.46

490.33 488.59 486.89 485.21 483.55

30 31 32 33 34

35 36 37 38 39

481.53 479.87 478.19 476.48 474.74

481.80 480.14 478.46 476.76 475.02

481.90 480.24 478.56 476.86 475.12

35 36 37 38 39

40 41 42 43 44

472.94 471.07 469.12 467.09 464.94

473.23 471.38 469.46 467.44 465.33

473.33 471.48 469.56 467.55 465.43

40 41 42 43 44

45 46 47 48 49

462.68 460.27 457.71 454.98 452.05

463.10 460.74 458.23 455.55 452.68

463.20 460.84 458.33 455.65 452.78

45 46 47 48 49

50 51 52 53 54

448.91 445.53 441.90 437.99 433.78

449.61 446.32 442.78 438.96 434.86

449.71 446.42 442.88 439.07 434.97

50 51 52 53 54

55 56 57 58 59

429.24 424.35 419.08 413.41 407.30

430.44 425.68 420.55 415.03 409.09

430.55 425.80 420.69 415.19 409.28

55 56 57 58 59

60 61 62 63 64

400.74 393.70 386.14 378.05 369.41

402.71 395.85 388.50 380.63 372.22

402.93 396.12 388.83 381.02 372.69

60 61 62 63 64

M [ x -1]+1

Mx

x

AM92 x

M[ x]

65 66 67 68 69

360.20 350.40 339.99 328.98 317.37

363.25 353.70 343.56 332.81 321.47

363.82 354.38 344.37 333.77 322.59

65 66 67 68 69

70 71 72 73 74

305.15 292.35 278.98 265.09 250.72

309.53 297.00 283.90 270.27 256.14

310.84 298.51 285.64 272.24 258.37

70 71 72 73 74

75 76 77 78 79

235.93 220.78 205.37 189.79 174.14

241.57 226.63 211.38 195.92 180.36

244.06 229.38 214.40 199.20 183.89

75 76 77 78 79

80 81 82 83 84

158.56 143.16 128.07 113.45 99.42

164.80 149.37 134.21 119.44 105.20

168.56 153.34 138.34 123.69 109.52

80 81 82 83 84

85 86 87 88 89

86.12 73.65 62.14 51.65 42.25

91.63 78.85 66.96 56.06 46.22

95.96 83.12 71.11 60.04 49.96

85 86 87 88 89

90 91 92 93 94

33.97

37.47 29.84

40.93 32.97 26.08 20.23 15.37

90 91 92 93 94

95 96 97 98 99

11.41 8.27 5.84 4.01 2.67

95 96 97 98 99

100 101 102 103 104

1.72 1.07 .64 .37 .21

100 101 102 103 104

105 106 107 108 109

.11 .05 .03 .01 .01

105 106 107 108 109

110

.00

110

M [ x -1]+1

Mx

x

4%

93

AM92 4%

94

x

R[ x ]

17 18 19

27 724.52 27 204.73 26 687.90

27 205.71 26 688.80

27 725.81 27 205.92 26 689.00

17 18 19

20 21 22 23 24

26 173.87 25 662.51 25 153.71 24 647.32 24 143.23

26 174.71 25 663.29 25 154.42 24 647.98 24 143.85

26 174.89 25 663.45 25 154.57 24 648.11 24 143.97

20 21 22 23 24

25 26 27 28 29

23 641.35 23 141.58 22 643.84 22 148.06 21 654.16

23 641.93 23 142.12 22 644.35 22 148.53 21 654.60

23 642.04 23 142.23 22 644.45 22 148.63 21 654.70

25 26 27 28 29

30 31 32 33 34

21 162.07 20 671.77 20 183.20 19 696.32 19 211.11

21 162.50 20 672.17 20 183.59 19 696.70 19 211.48

21 162.60 20 672.27 20 183.68 19 696.79 19 211.57

30 31 32 33 34

35 36 37 38 39

18 727.56 18 245.66 17 765.43 17 286.86 16 809.99

18 727.93 18 246.03 17 765.79 17 287.23 16 810.38

18 728.02 18 246.12 17 765.89 17 287.33 16 810.47

35 36 37 38 39

40 41 42 43 44

16 334.87 15 861.52 15 390.01 14 920.43 14 452.85

16 335.26 15 861.93 15 390.45 14 920.89 14 453.35

16 335.36 15 862.03 15 390.55 14 920.99 14 453.45

40 41 42 43 44

45 46 47 48 49

13 987.39 13 524.14 13 063.26 12 604.88 12 149.17

13 987.91 13 524.71 13 063.87 12 605.55 12 149.90

13 988.01 13 524.81 13 063.97 12 605.65 12 150.00

45 46 47 48 49

50 51 52 53 54

11 696.32 11 246.53 10 800.02 10 357.04 9 917.85

11 697.12 11 247.41 10 800.99 10 358.12 9 919.05

11 697.22 11 247.51 10 801.09 10 358.22 9 919.15

50 51 52 53 54

55 56 57 58 59

9 482.75 9 052.04 8 626.06 8 205.17 7 789.75

9 484.07 9 053.51 8 627.69 8 206.98 7 791.76

9 484.19 9 053.63 8 627.83 8 207.14 7 791.95

55 56 57 58 59

60 61 62 63 64

7 380.21 6 976.98 6 580.53 6 191.34 5 809.91

7 382.44 6 979.47 6 583.29 6 194.39 5 813.29

7 382.67 6 979.73 6 583.61 6 194.79 5 813.76

60 61 62 63 64

R[ x -1]+1

Rx

x

AM92 x

R[ x ]

R[ x -1]+1

Rx

x

65 66 67 68 69

5 436.77 5 072.46 4 717.54 4 372.60 4 038.20

5 440.50 5 076.57 4 722.06 4 377.55 4 043.61

5 441.07 5 077.25 4 722.87 4 378.51 4 044.74

65 66 67 68 69

70 71 72 73 74

3 714.94 3 403.41 3 104.17 2 817.78 2 544.78

3 720.83 3 409.79 3 111.06 2 825.19 2 552.69

3 722.14 3 411.31 3 112.79 2 827.16 2 554.91

70 71 72 73 74

75 76 77 78 79

2 285.66 2 040.87 1 810.80 1 595.76 1 396.00

2 294.06 2 049.74 1 820.09 1 605.43 1 405.97

2 296.55 2 052.49 1 823.11 1 608.71 1 409.51

75 76 77 78 79

80 81 82 83 84

1 211.64 1 042.74 889.21 750.83 627.27

1 221.85 1 053.09 899.59 761.13 637.38

1 225.62 1 057.05 903.72 765.38 641.69

80 81 82 83 84

85 86 87 88 89

518.06 422.60 340.15 269.86 210.79

527.85 431.95 348.95 278.01 218.21

532.17 436.22 353.10 281.99 221.95

85 86 87 88 89

90 91 92 93 94

161.90

168.54 127.94

171.99 131.06 98.09 72.01 51.78

90 91 92 93 94

95 96 97 98 99

36.41 25.00 16.73 10.89 6.89

95 96 97 98 99

100 101 102 103 104

4.22 2.50 1.43 .79 .41

100 101 102 103 104

105 106 107 108 109

.21 .10 .05 .02 .01

105 106 107 108 109

110

.00

110

4%

95

AM92 4% 2

Ax

0.016 96 0.017 78 0.018 67

23.367 23.276 23.180

0.101 27 0.104 78 0.108 44

0.017 16 0.017 97 0.018 85

17 18 19

0.112 10 0.116 08 0.120 23 0.124 55 0.129 05

0.019 64 0.020 70 0.021 84 0.023 08 0.024 43

23.081 22.978 22.870 22.758 22.641

0.112 26 0.116 24 0.120 38 0.124 69 0.129 19

0.019 82 0.020 86 0.022 00 0.023 24 0.024 58

20 21 22 23 24

22.523 22.396 22.265 22.128 21.985

0.133 73 0.138 60 0.143 67 0.148 94 0.154 42

0.025 89 0.027 47 0.029 17 0.031 02 0.033 01

22.520 22.393 22.261 22.124 21.982

0.133 86 0.138 73 0.143 79 0.149 06 0.154 54

0.026 03 0.027 61 0.029 31 0.031 15 0.033 14

25 26 27 28 29

30 31 32 33 34

21.837 21.683 21.523 21.357 21.185

0.160 11 0.166 03 0.172 18 0.178 57 0.185 20

0.035 15 0.037 47 0.039 96 0.042 64 0.045 52

21.834 21.680 21.520 21.354 21.182

0.160 23 0.166 15 0.172 30 0.178 68 0.185 31

0.035 28 0.037 59 0.040 08 0.042 76 0.045 65

30 31 32 33 34

35 36 37 38 39

21.006 20.821 20.628 20.429 20.223

0.192 07 0.199 21 0.206 60 0.214 26 0.222 20

0.048 61 0.051 93 0.055 49 0.059 30 0.063 38

21.003 20.818 20.625 20.426 20.219

0.192 19 0.199 33 0.206 72 0.214 39 0.222 34

0.048 74 0.052 07 0.055 63 0.059 45 0.063 54

35 36 37 38 39

40 41 42 43 44

20.009 19.788 19.560 19.324 19.080

0.230 41 0.238 91 0.247 70 0.256 78 0.266 15

0.067 75 0.072 41 0.077 38 0.082 67 0.088 32

20.005 19.784 19.555 19.319 19.075

0.230 56 0.239 07 0.247 87 0.256 96 0.266 36

0.067 92 0.072 59 0.077 58 0.082 89 0.088 56

40 41 42 43 44

45 46 47 48 49

18.829 18.569 18.302 18.027 17.745

0.275 83 0.285 80 0.296 07 0.306 64 0.317 52

0.094 31 0.100 68 0.107 44 0.114 60 0.122 17

18.823 18.563 18.295 18.019 17.736

0.276 05 0.286 05 0.296 35 0.306 95 0.317 86

0.094 58 0.100 98 0.107 78 0.114 98 0.122 60

45 46 47 48 49

50 51 52 53 54

17.454 17.156 16.851 16.538 16.218

0.328 68 0.340 14 0.351 89 0.363 92 0.376 23

0.130 17 0.138 61 0.147 49 0.156 84 0.166 65

17.444 17.145 16.838 16.524 16.202

0.329 07 0.340 58 0.352 38 0.364 48 0.376 85

0.130 65 0.139 15 0.148 11 0.157 55 0.167 45

50 51 52 53 54

55 56 57 58 59

15.891 15.558 15.219 14.874 14.523

0.388 79 0.401 61 0.414 66 0.427 94 0.441 43

0.176 93 0.187 69 0.198 93 0.210 64 0.222 82

15.873 15.537 15.195 14.847 14.493

0.389 50 0.402 40 0.415 56 0.428 96 0.442 58

0.177 85 0.188 74 0.200 12 0.212 00 0.224 37

55 56 57 58 59

60 61 62 63 64

14.167 13.808 13.444 13.077 12.708

0.455 10 0.468 94 0.482 92 0.497 03 0.511 23

0.235 47 0.248 57 0.262 11 0.276 08 0.290 46

14.134 13.769 13.401 13.029 12.653

0.456 40 0.470 41 0.484 58 0.498 90 0.513 33

0.237 23 0.250 58 0.264 40 0.278 68 0.293 40

60 61 62 63 64

a&&[ x ]

A[ x ]

17 18 19

23.372 23.280 23.185

0.101 08 0.104 60 0.108 27

20 21 22 23 24

23.086 22.982 22.874 22.762 22.645

25 26 27 28 29

A[ x ]

Note. 2 A[ x ] = A[ x ] at 8.16% and 2 Ax = Ax at 8.16%.

96

2

a&&x

x

Ax

x

AM92 4% 2

2

a&&x

Ax

0.305 22 0.320 33 0.335 78 0.351 51 0.367 51

12.276 11.896 11.515 11.135 10.754

0.527 86 0.542 46 0.557 10 0.571 75 0.586 38

0.308 55 0.324 10 0.340 03 0.356 30 0.372 89

65 66 67 68 69

0.596 87 0.610 93 0.624 85 0.638 60 0.652 14

0.383 72 0.400 12 0.416 65 0.433 27 0.449 93

10.375 9.998 9.623 9.252 8.886

0.600 97 0.615 48 0.629 88 0.644 14 0.658 24

0.389 75 0.406 86 0.424 16 0.441 62 0.459 19

70 71 72 73 74

8.698 8.359 8.027 7.703 7.388

0.665 45 0.678 51 0.691 27 0.703 73 0.715 85

0.466 59 0.483 20 0.499 71 0.516 09 0.532 27

8.524 8.169 7.820 7.478 7.144

0.672 14 0.685 81 0.699 24 0.712 38 0.725 23

0.476 83 0.494 48 0.512 10 0.529 65 0.547 07

75 76 77 78 79

80 81 82 83 84

7.082 6.785 6.499 6.222 5.957

0.727 62 0.739 03 0.750 05 0.760 68 0.770 90

0.548 22 0.563 90 0.579 27 0.594 30 0.608 95

6.818 6.502 6.194 5.897 5.610

0.737 75 0.749 93 0.761 75 0.773 19 0.784 25

0.564 32 0.581 36 0.598 14 0.614 61 0.630 75

80 81 82 83 84

85 86 87 88 89

5.701 5.457 5.223 5.000 4.788

0.780 72 0.790 12 0.799 11 0.807 69 0.815 85

0.623 20 0.637 01 0.650 38 0.663 29 0.675 73

5.333 5.066 4.811 4.566 4.332

0.794 90 0.805 14 0.814 98 0.824 39 0.833 38

0.646 52 0.661 88 0.676 80 0.691 27 0.705 25

85 86 87 88 89

90 91 92 93 94

4.586

0.823 62

0.687 68

4.109 3.897 3.695 3.504 3.323

0.841 96 0.850 12 0.857 87 0.865 22 0.872 18

0.718 74 0.731 72 0.744 17 0.756 09 0.767 48

90 91 92 93 94

95 96 97 98 99

3.153 2.992 2.840 2.698 2.564

0.878 75 0.884 94 0.890 77 0.896 25 0.901 39

0.778 34 0.788 67 0.798 47 0.807 76 0.816 54

95 96 97 98 99

100 101 102 103 104

2.439 2.321 2.212 2.110 2.015

0.906 21 0.910 71 0.914 92 0.918 85 0.922 51

0.824 83 0.832 63 0.839 97 0.846 86 0.853 31

100 101 102 103 104

105 106 107 108 109

1.926 1.844 1.768 1.697 1.632

0.925 91 0.929 07 0.932 01 0.934 72 0.937 24

0.859 34 0.864 98 0.870 23 0.875 12 0.879 66

105 106 107 108 109

110 111 112 113 114

1.571 1.516 1.464 1.417 1.374

0.939 56 0.941 70 0.943 67 0.945 49 0.947 15

0.883 87 0.887 77 0.891 37 0.894 69 0.897 75

110 111 112 113 114

115 116 117 118 119

1.334 1.298 1.264 1.229 1.176

0.948 68 0.950 08 0.951 39 0.952 73 0.954 78

0.900 56 0.903 15 0.905 57 0.908 04 0.911 81

115 116 117 118 119

120

1.000

0.961 54

0.924 56

120

x

a&&[ x ]

A[ x ]

65 66 67 68 69

12.337 11.965 11.592 11.221 10.850

0.525 50 0.539 81 0.554 14 0.568 44 0.582 70

70 71 72 73 74

10.481 10.116 9.754 9.396 9.044

75 76 77 78 79

A[ x ]

Ax

x

Note. 2 A[ x ] = A[ x ] at 8.16% and 2 Ax = Ax at 8.16%. 97

AM92 4%

98

x

( Ia&&)[ x ]

( IA)[ x ]

( Ia&&) x

( IA) x

x

17 18 19

467.226 461.881 456.412

5.401 64 5.515 65 5.630 60

467.124 461.784 456.320

5.400 71 5.514 73 5.629 69

17 18 19

20 21 22 23 24

450.817 445.097 439.249 433.275 427.174

5.746 37 5.862 84 5.979 86 6.097 30 6.215 01

450.729 445.013 439.170 433.200 427.102

5.745 47 5.861 95 5.978 99 6.096 44 6.214 15

20 21 22 23 24

25 26 27 28 29

420.947 414.593 408.114 401.510 394.783

6.332 80 6.450 51 6.567 94 6.684 88 6.801 12

420.878 414.528 408.051 401.450 394.726

6.331 95 6.449 67 6.567 10 6.684 05 6.800 29

25 26 27 28 29

30 31 32 33 34

387.935 380.966 373.879 366.676 359.361

6.916 44 7.030 57 7.143 28 7.254 28 7.363 31

387.878 380.911 373.825 366.623 359.308

6.915 59 7.029 72 7.142 42 7.253 40 7.362 39

30 31 32 33 34

35 36 37 38 39

351.937 344.407 336.776 329.048 321.228

7.470 05 7.574 21 7.675 46 7.773 46 7.867 88

351.883 344.353 336.720 328.991 321.169

7.469 09 7.573 20 7.674 38 7.772 31 7.866 63

35 36 37 38 39

40 41 42 43 44

313.323 305.337 297.278 289.153 280.970

7.958 35 8.044 52 8.126 02 8.202 46 8.273 47

313.260 305.271 297.207 289.077 280.888

7.956 99 8.043 03 8.124 35 8.200 60 8.271 37

40 41 42 43 44

45 46 47 48 49

272.737 264.462 256.156 247.828 239.488

8.338 65 8.397 62 8.450 01 8.495 42 8.533 51

272.647 264.365 256.049 247.711 239.360

8.336 28 8.394 93 8.446 95 8.491 93 8.529 50

45 46 47 48 49

50 51 52 53 54

231.149 222.820 214.514 206.244 198.022

8.563 90 8.586 24 8.600 22 8.605 54 8.601 90

231.007 222.664 214.342 206.053 197.811

8.559 29 8.580 95 8.594 12 8.598 51 8.593 81

50 51 52 53 54

55 56 57 58 59

189.861 181.774 173.775 165.878 158.094

8.589 08 8.566 87 8.535 08 8.493 60 8.442 34

189.627 181.516 173.489 165.561 157.744

8.579 76 8.556 11 8.522 68 8.479 31 8.425 88

55 56 57 58 59

60 61 62 63 64

150.440 142.926 135.566 128.373 121.359

8.381 28 8.310 44 8.229 90 8.139 81 8.040 36

150.053 142.499 135.096 127.856 120.790

8.362 34 8.288 67 8.204 91 8.111 17 8.007 60

60 61 62 63 64

AM92 x

( Ia&&)[ x ]

( IA)[ x ]

( Ia&&) x

( IA) x

65 66 67 68 69

114.533 107.909 101.494 95.297 89.327

7.931 82 7.814 53 7.688 86 7.555 27 7.414 26

113.911 107.228 100.751 94.489 88.450

7.894 42 7.771 92 7.640 43 7.500 35 7.352 15

65 66 67 68 69

70 71 72 73 74

83.589 78.089 72.832 67.819 63.053

7.266 40 7.112 29 6.952 57 6.787 95 6.619 14

82.641 77.067 71.732 66.640 61.793

7.196 35 7.033 51 6.864 24 6.689 22 6.509 13

70 71 72 73 74

75 76 77 78 79

58.534 54.260 50.230 46.440 42.885

6.446 87 6.271 92 6.095 04 5.916 97 5.738 48

57.192 52.836 48.723 44.851 41.215

6.324 70 6.136 69 5.945 86 5.752 98 5.558 83

75 76 77 78 79

80 81 82 83 84

39.559 36.457 33.570 30.890 28.410

5.560 29 5.383 08 5.207 53 5.034 26 4.863 82

37.811 34.633 31.673 28.924 26.378

5.364 17 5.169 76 4.976 31 4.784 53 4.595 08

80 81 82 83 84

85 86 87 88 89

26.118 24.007 22.065 20.283 18.651

4.696 75 4.533 50 4.374 48 4.220 03 4.070 43

24.025 21.858 19.866 18.039 16.368

4.408 56 4.225 55 4.046 57 3.872 08 3.702 50

85 86 87 88 89

90 91 92 93 94

17.159

3.925 89

14.843 13.453 12.191 11.045 10.007

3.538 17 3.379 39 3.226 40 3.079 39 2.938 48

90 91 92 93 94

95 96 97 98 99

9.070 8.223 7.460 6.774 6.156

2.803 78 2.675 30 2.553 06 2.437 01 2.327 08

95 96 97 98 99

100 101 102 103 104

5.602 5.104 4.659 4.259 3.902

2.223 16 2.125 12 2.032 81 1.946 07 1.864 71

100 101 102 103 104

105 106 107 108 109

3.582 3.295 3.039 2.811 2.606

1.788 53 1.717 34 1.650 92 1.589 07 1.531 58

105 106 107 108 109

110 111 112 113 114

2.424 2.261 2.115 1.985 1.869

1.478 23 1.428 82 1.383 15 1.341 02 1.302 22

110 111 112 113 114

115 116 117 118 119

1.765 1.672 1.584 1.492 1.351

1.266 54 1.233 70 1.202 99 1.171 57 1.123 76

115 116 117 118 119

120

1.000

0.961 54

120

x

4%

99

AM92 4%

100

x

a&&[ x ]:n

A[ x ]:n

17 18 19

20.941 20.750 20.552

0.194 59 0.201 90 0.209 53

20 21 22 23 24

20.346 20.131 19.908 19.675 19.433

25 26 27 28 29

a&&x:n

Ax:n

x

43 42 41

20.936 20.746 20.548

0.194 75 0.202 06 0.209 68

17 18 19

0.217 46 0.225 72 0.234 32 0.243 27 0.252 59

40 39 38 37 36

20.342 20.128 19.904 19.672 19.430

0.217 60 0.225 86 0.234 45 0.243 40 0.252 71

20 21 22 23 24

19.181 18.918 18.645 18.361 18.066

0.262 28 0.272 37 0.282 87 0.293 79 0.305 15

35 34 33 32 31

19.178 18.916 18.643 18.359 18.064

0.262 40 0.272 48 0.282 97 0.293 89 0.305 25

25 26 27 28 29

30 31 32 33 34

17.759 17.439 17.107 16.762 16.402

0.316 97 0.329 26 0.342 04 0.355 33 0.369 14

30 29 28 27 26

17.756 17.437 17.105 16.759 16.400

0.317 06 0.329 35 0.342 12 0.355 41 0.369 23

30 31 32 33 34

35 36 37 38 39

16.029 15.641 15.237 14.818 14.383

0.383 50 0.398 43 0.413 95 0.430 07 0.446 82

25 24 23 22 21

16.027 15.639 15.235 14.816 14.380

0.383 59 0.398 52 0.414 03 0.430 16 0.446 92

35 36 37 38 39

40 41 42 43 44

13.930 13.460 12.971 12.464 11.937

0.464 23 0.482 31 0.501 10 0.520 61 0.540 88

20 19 18 17 16

13.927 13.457 12.969 12.461 11.934

0.464 33 0.482 42 0.501 21 0.520 73 0.541 00

40 41 42 43 44

45 46 47 48 49

11.390 10.821 10.231 9.617 8.980

0.561 93 0.583 80 0.606 51 0.630 10 0.654 61

15 14 13 12 11

11.386 10.818 10.227 9.613 8.976

0.562 06 0.583 93 0.606 65 0.630 25 0.654 77

45 46 47 48 49

50 51 52 53 54

8.318 7.630 6.914 6.170 5.396

0.680 07 0.706 54 0.734 06 0.762 68 0.792 46

10 9 8 7 6

8.314 7.625 6.910 6.166 5.391

0.680 24 0.706 72 0.734 24 0.762 86 0.792 64

50 51 52 53 54

55 56 57 58 59

4.590 3.749 2.873 1.957 1.000

0.823 48 0.855 80 0.889 52 0.924 73 0.961 54

5 4 3 2 1

4.585 3.745 2.870 1.955 1.000

0.823 65 0.855 95 0.889 63 0.924 79 0.961 54

55 56 57 58 59

n = 60 - x

AM92 x

a&&[ x ]:n

A[ x ]:n

17 18 19

21.723 21.565 21.400

0.164 48 0.170 58 0.176 93

20 21 22 23 24

21.228 21.049 20.863 20.669 20.467

25 26 27 28 29

a&&x:n

Ax:n

x

48 47 46

21.719 21.561 21.396

0.164 66 0.170 74 0.177 09

17 18 19

0.183 54 0.190 42 0.197 59 0.205 05 0.212 81

45 44 43 42 41

21.224 21.045 20.859 20.665 20.464

0.183 69 0.190 57 0.197 73 0.205 18 0.212 94

20 21 22 23 24

20.257 20.038 19.811 19.574 19.328

0.220 90 0.229 31 0.238 05 0.247 16 0.256 62

40 39 38 37 36

20.254 20.035 19.808 19.571 19.325

0.221 02 0.229 42 0.238 17 0.247 26 0.256 73

25 26 27 28 29

30 31 32 33 34

19.072 18.806 18.529 18.241 17.942

0.266 47 0.276 71 0.287 35 0.298 42 0.309 92

35 34 33 32 31

19.069 18.803 18.526 18.239 17.940

0.266 57 0.276 81 0.287 45 0.298 52 0.310 02

30 31 32 33 34

35 36 37 38 39

17.631 17.308 16.973 16.625 16.263

0.321 87 0.334 29 0.347 19 0.360 59 0.374 51

30 29 28 27 26

17.629 17.306 16.970 16.622 16.260

0.321 97 0.334 39 0.347 29 0.360 70 0.374 62

35 36 37 38 39

40 41 42 43 44

15.887 15.497 15.092 14.672 14.237

0.388 96 0.403 95 0.419 52 0.435 67 0.452 43

25 24 23 22 21

15.884 15.494 15.089 14.669 14.233

0.389 07 0.404 07 0.419 65 0.435 81 0.452 58

40 41 42 43 44

45 46 47 48 49

13.785 13.316 12.829 12.325 11.802

0.469 82 0.487 86 0.506 56 0.525 96 0.546 08

20 19 18 17 16

13.780 13.311 12.824 12.320 11.796

0.469 98 0.488 03 0.506 75 0.526 17 0.546 30

45 46 47 48 49

50 51 52 53 54

11.259 10.697 10.113 9.508 8.880

0.566 95 0.588 58 0.611 02 0.634 30 0.658 46

15 14 13 12 11

11.253 10.690 10.106 9.500 8.872

0.567 19 0.588 84 0.611 30 0.634 60 0.658 78

50 51 52 53 54

55 56 57 58 59

8.228 7.551 6.847 6.115 5.353

0.683 54 0.709 58 0.736 64 0.764 79 0.794 10

10 9 8 7 6

8.219 7.542 6.838 6.106 5.344

0.683 88 0.709 93 0.737 01 0.765 16 0.794 46

55 56 57 58 59

60 61 62 63 64

4.559 3.730 2.863 1.954 1.000

0.824 65 0.856 54 0.889 90 0.924 85 0.961 54

5 4 3 2 1

4.550 3.722 2.857 1.951 1.000

0.824 99 0.856 85 0.890 13 0.924 98 0.961 54

60 61 62 63 64

n = 65 - x

4%

101

AM92 6% 2

A[ x ]

a&&x

Ax

0.039 02 0.040 80 0.042 70

0.006 11 0.006 30 0.006 52

16.974 16.943 16.909

16.877 16.839 16.798 16.756 16.710

0.044 72 0.046 86 0.049 14 0.051 57 0.054 14

0.006 77 0.007 05 0.007 38 0.007 75 0.008 16

25 26 27 28 29

16.662 16.611 16.557 16.499 16.439

0.056 86 0.059 76 0.062 82 0.066 07 0.069 51

30 31 32 33 34

16.374 16.306 16.234 16.158 16.078

35 36 37 38 39

x

a&&[ x ]

A[ x ]

17 18 19

16.977 16.946 16.912

20 21 22 23 24

Ax

x

0.039 21 0.040 99 0.042 88

0.006 30 0.006 48 0.006 69

17 18 19

16.874 16.836 16.796 16.753 16.708

0.044 89 0.047 03 0.049 30 0.051 72 0.054 28

0.006 93 0.007 21 0.007 53 0.007 90 0.008 31

20 21 22 23 24

0.008 63 0.009 16 0.009 75 0.010 41 0.011 15

16.660 16.609 16.554 16.497 16.436

0.057 01 0.059 90 0.062 96 0.066 20 0.069 64

0.008 77 0.009 30 0.009 88 0.010 54 0.011 28

25 26 27 28 29

0.073 16 0.077 01 0.081 09 0.085 40 0.089 95

0.011 97 0.012 89 0.013 90 0.015 03 0.016 27

16.372 16.304 16.232 16.156 16.075

0.073 28 0.077 14 0.081 21 0.085 52 0.090 07

0.012 10 0.013 01 0.014 03 0.015 15 0.016 40

30 31 32 33 34

15.993 15.903 15.809 15.709 15.605

0.094 75 0.099 82 0.105 16 0.110 79 0.116 72

0.017 65 0.019 16 0.020 84 0.022 67 0.024 69

15.990 15.901 15.806 15.707 15.602

0.094 88 0.099 95 0.105 30 0.110 94 0.116 88

0.017 78 0.019 30 0.020 98 0.022 82 0.024 85

35 36 37 38 39

40 41 42 43 44

15.494 15.378 15.257 15.129 14.995

0.122 96 0.129 52 0.136 41 0.143 65 0.151 23

0.026 90 0.029 33 0.031 98 0.034 87 0.038 02

15.491 15.375 15.253 15.125 14.991

0.123 13 0.129 70 0.136 60 0.143 85 0.151 46

0.027 07 0.029 51 0.032 18 0.035 09 0.038 26

40 41 42 43 44

45 46 47 48 49

14.855 14.708 14.554 14.393 14.226

0.159 18 0.167 50 0.176 19 0.185 28 0.194 76

0.041 45 0.045 17 0.049 21 0.053 59 0.058 32

14.850 14.703 14.548 14.387 14.219

0.159 43 0.167 78 0.176 51 0.185 63 0.195 16

0.041 72 0.045 48 0.049 56 0.053 98 0.058 76

45 46 47 48 49

50 51 52 53 54

14.051 13.870 13.681 13.485 13.282

0.204 63 0.214 91 0.225 60 0.236 69 0.248 18

0.063 42 0.068 92 0.074 83 0.081 18 0.087 97

14.044 13.861 13.671 13.474 13.269

0.205 08 0.215 42 0.226 17 0.237 34 0.248 92

0.063 92 0.069 49 0.075 48 0.081 92 0.088 82

50 51 52 53 54

55 56 57 58 59

13.072 12.855 12.631 12.400 12.163

0.260 08 0.272 37 0.285 06 0.298 12 0.311 55

0.095 24 0.102 98 0.111 23 0.119 98 0.129 26

13.057 12.838 12.612 12.378 12.138

0.260 92 0.273 33 0.286 14 0.299 35 0.312 94

0.096 21 0.104 09 0.112 50 0.121 44 0.130 93

55 56 57 58 59

60 61 62 63 64

11.919 11.670 11.415 11.155 10.890

0.325 33 0.339 45 0.353 88 0.368 61 0.383 60

0.139 07 0.149 41 0.160 29 0.171 71 0.183 66

11.891 11.638 11.379 11.114 10.844

0.326 92 0.341 25 0.355 92 0.370 91 0.386 20

0.140 98 0.151 60 0.162 80 0.174 57 0.186 92

60 61 62 63 64

Note. 2 A[ x ] = A[ x ] at 12.36% and 2 Ax = Ax at 12.36%.

102

2

AM92 6% 2

A[ x ]

a&&x

Ax

0.398 83 0.414 27 0.429 88 0.445 64 0.461 50

0.196 14 0.209 13 0.222 62 0.236 58 0.251 00

10.569 10.289 10.006 9.720 9.431

9.232 8.950 8.669 8.388 8.109

0.477 43 0.493 38 0.509 33 0.525 21 0.541 01

0.265 83 0.281 06 0.296 64 0.312 54 0.328 70

75 76 77 78 79

7.832 7.559 7.289 7.024 6.763

0.556 67 0.572 15 0.587 42 0.602 44 0.617 17

80 81 82 83 84

6.509 6.260 6.018 5.783 5.556

85 86 87 88 89 90 91 92 93 94

x

a&&[ x ]

A[ x ]

65 66 67 68 69

10.621 10.348 10.072 9.794 9.513

70 71 72 73 74

2

Ax

x

0.401 77 0.417 58 0.433 61 0.449 82 0.466 17

0.199 85 0.213 35 0.227 40 0.242 00 0.257 12

65 66 67 68 69

9.140 8.848 8.555 8.262 7.969

0.482 65 0.499 19 0.515 78 0.532 36 0.548 90

0.272 74 0.288 82 0.305 34 0.322 26 0.339 55

70 71 72 73 74

0.345 09 0.361 64 0.378 33 0.395 08 0.411 86

7.679 7.390 7.105 6.822 6.544

0.565 35 0.581 69 0.597 86 0.613 83 0.629 56

0.357 14 0.375 01 0.393 09 0.411 33 0.429 69

75 76 77 78 79

0.631 59 0.645 66 0.659 35 0.672 65 0.685 53

0.428 60 0.445 25 0.461 77 0.478 11 0.494 22

6.271 6.004 5.742 5.487 5.239

0.645 01 0.660 16 0.674 97 0.689 42 0.703 46

0.448 11 0.466 52 0.484 88 0.503 13 0.521 21

80 81 82 83 84

5.336 5.124 4.920 4.724 4.537

0.697 97 0.709 97 0.721 50 0.732 58 0.743 18

0.510 05 0.525 57 0.540 75 0.555 55 0.569 94

4.998 4.765 4.540 4.323 4.114

0.717 10 0.730 29 0.743 04 0.755 31 0.767 11

0.539 07 0.556 67 0.573 96 0.590 88 0.607 41

85 86 87 88 89

4.358

0.753 32

0.583 90

3.914 3.723 3.541 3.367 3.201

0.778 43 0.789 25 0.799 59 0.809 44 0.818 80

0.623 50 0.639 13 0.654 26 0.668 88 0.682 96

90 91 92 93 94

95 96 97 98 99

3.044 2.896 2.755 2.622 2.498

0.827 69 0.836 10 0.844 06 0.851 56 0.858 63

0.696 49 0.709 46 0.721 87 0.733 70 0.744 96

95 96 97 98 99

100 101 102 103 104

2.380 2.270 2.167 2.070 1.980

0.865 27 0.871 51 0.877 36 0.882 83 0.887 94

0.755 65 0.765 79 0.775 37 0.784 42 0.792 93

100 101 102 103 104

105 106 107 108 109

1.895 1.817 1.744 1.676 1.614

0.892 71 0.897 15 0.901 28 0.905 11 0.908 66

0.800 94 0.808 45 0.815 48 0.822 05 0.828 17

105 106 107 108 109

110 111 112 113 114

1.556 1.502 1.452 1.407 1.365

0.911 95 0.914 99 0.917 79 0.920 37 0.922 75

0.833 87 0.839 17 0.844 08 0.848 61 0.852 80

110 111 112 113 114

115 116 117 118 119

1.326 1.291 1.258 1.224 1.172

0.924 92 0.926 93 0.928 80 0.930 72 0.933 64

0.856 66 0.860 22 0.863 55 0.866 94 0.872 10

115 116 117 118 119

120

1.000

0.943 40

0.890 00

120

Note. 2 A[ x ] = A[ x ] at 12.36% and 2 Ax = Ax at 12.36%. 103

AM92 6%

104

x

( Ia&&)[ x ]

( IA)[ x ]

( Ia&&) x

( IA) x

x

17 18 19

268.142 266.392 264.567

1.799 55 1.867 08 1.936 81

268.083 266.336 264.514

1.799 40 1.866 92 1.936 64

17 18 19

20 21 22 23 24

262.666 260.687 258.626 256.482 254.253

2.008 74 2.082 89 2.159 25 2.237 82 2.318 58

262.615 260.638 258.579 256.437 254.210

2.008 56 2.082 70 2.159 06 2.237 62 2.318 37

20 21 22 23 24

25 26 27 28 29

251.936 249.531 247.034 244.444 241.759

2.401 51 2.486 57 2.573 73 2.662 93 2.754 10

251.896 249.491 246.996 244.407 241.724

2.401 29 2.486 35 2.573 50 2.662 70 2.753 86

25 26 27 28 29

30 31 32 33 34

238.978 236.099 233.120 230.041 226.861

2.847 18 2.942 06 3.038 64 3.136 81 3.236 43

238.943 236.065 233.087 230.008 226.827

2.846 92 2.941 80 3.038 37 3.136 53 3.236 13

30 31 32 33 34

35 36 37 38 39

223.579 220.194 216.706 213.116 209.424

3.337 35 3.439 40 3.542 39 3.646 13 3.750 37

223.545 220.159 216.671 213.079 209.385

3.337 02 3.439 04 3.542 00 3.645 69 3.749 89

35 36 37 38 39

40 41 42 43 44

205.630 201.736 197.744 193.654 189.471

3.854 89 3.959 42 4.063 68 4.167 36 4.270 14

205.589 201.692 197.696 193.603 189.416

3.854 35 3.958 80 4.062 97 4.166 55 4.269 22

40 41 42 43 44

45 46 47 48 49

185.197 180.834 176.388 171.863 167.264

4.371 70 4.471 66 4.569 65 4.665 29 4.758 18

185.136 180.768 176.315 171.783 167.175

4.370 62 4.470 41 4.568 20 4.663 59 4.756 18

45 46 47 48 49

50 51 52 53 54

162.597 157.867 153.082 148.249 143.376

4.847 89 4.934 00 5.016 09 5.093 72 5.166 47

162.497 157.757 152.959 148.113 143.224

4.845 55 4.931 26 5.012 87 5.089 94 5.162 03

50 51 52 53 54

55 56 57 58 59

138.472 133.545 128.605 123.662 118.726

5.233 89 5.295 58 5.351 13 5.400 16 5.442 29

138.302 133.356 128.394 123.427 118.464

5.228 68 5.289 47 5.343 97 5.391 76 5.432 47

55 56 57 58 59

60 61 62 63 64

113.808 108.918 104.067 99.267 94.528

5.477 20 5.504 57 5.524 16 5.535 74 5.539 13

113.516 108.594 103.707 98.868 94.087

5.465 72 5.491 18 5.508 56 5.517 59 5.518 08

60 61 62 63 64

AM92 x

( Ia&&)[ x ]

( IA)[ x ]

( Ia&&) x

( IA) x

x

65 66 67 68 69

89.861 85.277 80.785 76.397 72.121

5.534 21 5.520 93 5.499 28 5.469 31 5.431 14

89.374 84.740 80.196 75.752 71.416

5.509 85 5.492 80 5.466 88 5.432 09 5.388 51

65 66 67 68 69

70 71 72 73 74

67.965 63.939 60.048 56.300 52.700

5.384 97 5.331 01 5.269 59 5.201 07 5.125 86

67.198 63.105 59.146 55.326 51.652

5.336 28 5.275 60 5.206 73 5.129 99 5.045 77

70 71 72 73 74

75 76 77 78 79

49.251 45.958 42.822 39.846 37.028

5.044 44 4.957 31 4.865 04 4.768 19 4.667 37

48.128 44.758 41.545 38.491 35.596

4.954 52 4.856 72 4.752 91 4.643 69 4.529 64

75 76 77 78 79

80 81 82 83 84

34.369 31.866 29.517 27.320 25.268

4.563 20 4.456 30 4.347 29 4.236 78 4.125 36

32.860 30.283 27.861 25.594 23.475

4.411 42 4.289 68 4.165 09 4.038 31 3.910 00

80 81 82 83 84

85 86 87 88 89

23.359 21.586 19.944 18.426 17.026

4.013 61 3.902 05 3.791 19 3.681 49 3.573 36

21.503 19.671 17.974 16.406 14.962

3.780 82 3.651 39 3.522 31 3.394 16 3.267 46

85 86 87 88 89

90 91 92 93 94

15.738

3.467 16

13.634 12.417 11.303 10.287 9.361

3.142 70 3.020 33 2.900 75 2.784 31 2.671 32

90 91 92 93 94

95 96 97 98 99

8.518 7.754 7.061 6.435 5.869

2.562 02 2.456 63 2.355 32 2.258 21 2.165 37

95 96 97 98 99

100 101 102 103 104

5.358 4.898 4.483 4.111 3.776

2.076 86 1.992 70 1.912 86 1.837 31 1.765 98

100 101 102 103 104

105 106 107 108 109

3.475 3.205 2.963 2.746 2.551

1.698 78 1.635 63 1.576 39 1.520 96 1.469 20

105 106 107 108 109

110 111 112 113 114

2.377 2.221 2.081 1.956 1.845

1.420 96 1.376 11 1.334 50 1.295 98 1.260 40

110 111 112 113 114

115 116 117 118 119

1.744 1.654 1.570 1.481 1.345

1.227 60 1.197 34 1.169 04 1.140 18 1.096 31

115 116 117 118 119

120

1.000

0.943 40

120

6%

105

AM92 6%

106

x

a&&[ x ]:n

A[ x ]:n

17 18 19

16.076 15.990 15.898

20 21 22 23 24

n = 60 - x

a&&x:n

Ax:n

x

0.090 05 0.094 93 0.100 11

43 42 41

16.072 15.986 15.895

0.090 24 0.095 11 0.100 28

17 18 19

15.801 15.698 15.588 15.472 15.349

0.105 61 0.111 45 0.117 64 0.124 22 0.131 19

40 39 38 37 36

15.798 15.695 15.586 15.470 15.347

0.105 77 0.111 60 0.117 79 0.124 36 0.131 33

20 21 22 23 24

25 26 27 28 29

15.218 15.080 14.933 14.777 14.612

0.138 59 0.146 43 0.154 75 0.163 57 0.172 92

35 34 33 32 31

15.216 15.078 14.931 14.775 14.610

0.138 72 0.146 56 0.154 87 0.163 69 0.173 03

25 26 27 28 29

30 31 32 33 34

14.437 14.251 14.054 13.846 13.625

0.182 83 0.193 33 0.204 46 0.216 26 0.228 75

30 29 28 27 26

14.435 14.249 14.053 13.844 13.624

0.182 94 0.193 44 0.204 57 0.216 36 0.228 85

30 31 32 33 34

35 36 37 38 39

13.392 13.144 12.882 12.605 12.311

0.241 98 0.255 99 0.270 82 0.286 53 0.303 16

25 24 23 22 21

13.390 13.142 12.880 12.603 12.309

0.242 08 0.256 09 0.270 93 0.286 64 0.303 27

35 36 37 38 39

40 41 42 43 44

12.000 11.671 11.323 10.954 10.564

0.320 76 0.339 38 0.359 10 0.379 96 0.402 03

20 19 18 17 16

11.998 11.669 11.320 10.952 10.561

0.320 88 0.339 51 0.359 23 0.380 10 0.402 19

40 41 42 43 44

45 46 47 48 49

10.151 9.715 9.253 8.764 8.246

0.425 39 0.450 11 0.476 26 0.503 94 0.533 24

15 14 13 12 11

10.149 9.712 9.249 8.760 8.242

0.425 56 0.450 28 0.476 45 0.504 15 0.533 46

45 46 47 48 49

50 51 52 53 54

7.698 7.118 6.503 5.851 5.160

0.564 26 0.597 11 0.631 91 0.668 79 0.707 91

10 9 8 7 6

7.694 7.114 6.499 5.847 5.156

0.564 49 0.597 35 0.632 16 0.669 04 0.708 15

50 51 52 53 54

55 56 57 58 59

4.427 3.648 2.820 1.939 1.000

0.749 41 0.793 50 0.840 36 0.890 24 0.943 40

5 4 3 2 1

4.423 3.645 2.817 1.937 1.000

0.749 65 0.793 70 0.840 52 0.890 34 0.943 40

55 56 57 58 59

AM92 6% x

a&&[ x ]:n

A[ x ]:n

17 18 19

16.409 16.343 16.272

20 21 22 23 24

n = 65 - x

a&&x:n

Ax:n

x

0.071 21 0.074 95 0.078 92

48 47 46

16.405 16.339 16.269

0.071 40 0.075 13 0.079 09

17 18 19

16.198 16.119 16.035 15.946 15.852

0.083 13 0.087 61 0.092 36 0.097 40 0.102 74

45 44 43 42 41

16.195 16.116 16.032 15.943 15.849

0.083 30 0.087 77 0.092 51 0.097 54 0.102 88

20 21 22 23 24

25 26 27 28 29

15.751 15.645 15.532 15.413 15.286

0.108 42 0.114 43 0.120 81 0.127 58 0.134 75

40 39 38 37 36

15.749 15.643 15.530 15.411 15.284

0.108 55 0.114 56 0.120 94 0.127 70 0.134 86

25 26 27 28 29

30 31 32 33 34

15.152 15.010 14.859 14.700 14.531

0.142 34 0.150 39 0.158 92 0.167 95 0.177 51

35 34 33 32 31

15.150 15.008 14.857 14.698 14.529

0.142 46 0.150 50 0.159 03 0.168 06 0.177 62

30 31 32 33 34

35 36 37 38 39

14.352 14.163 13.963 13.751 13.527

0.187 63 0.198 33 0.209 67 0.221 65 0.234 33

30 29 28 27 26

14.350 14.161 13.960 13.749 13.525

0.187 74 0.198 45 0.209 79 0.221 78 0.234 46

35 36 37 38 39

40 41 42 43 44

13.290 13.040 12.775 12.495 12.200

0.247 74 0.261 91 0.276 89 0.292 72 0.309 44

25 24 23 22 21

13.288 13.037 12.772 12.492 12.197

0.247 87 0.262 06 0.277 05 0.292 89 0.309 63

40 41 42 43 44

45 46 47 48 49

11.888 11.558 11.210 10.842 10.454

0.327 11 0.345 78 0.365 49 0.386 30 0.408 28

20 19 18 17 16

11.884 11.554 11.206 10.837 10.449

0.327 31 0.345 99 0.365 72 0.386 56 0.408 57

45 46 47 48 49

50 51 52 53 54

10.044 9.610 9.153 8.669 8.159

0.431 50 0.456 02 0.481 91 0.509 27 0.538 19

15 14 13 12 11

10.038 9.604 9.146 8.662 8.151

0.431 81 0.456 35 0.482 28 0.509 67 0.538 62

50 51 52 53 54

55 56 57 58 59

7.618 7.047 6.442 5.801 5.121

0.568 77 0.601 12 0.635 36 0.671 65 0.710 15

10 9 8 7 6

7.610 7.038 6.433 5.792 5.112

0.569 22 0.601 60 0.635 86 0.672 16 0.710 66

55 56 57 58 59

60 61 62 63 64

4.398 3.630 2.811 1.936 1.000

0.751 04 0.794 54 0.840 90 0.890 42 0.943 40

5 4 3 2 1

4.390 3.622 2.805 1.933 1.000

0.751 52 0.794 97 0.841 23 0.890 60 0.943 40

60 61 62 63 64

107

108

PENSIONER MORTALITY TABLES PMA92 and PFA92 (Base tables) and PMA92C20 and PFA92C20 (Projected tables) The Base tables are based on the mortality of pensioners insured by UK life offices during the years 1991, 1992, 1993, and 1994. Mortality is measured by amounts of annuities held. The projected tables are projected to the calendar year 2020. Full details are given in C.M.I.R. 16 and 17.

PROJECTION FORMULAE The projected mortality rate applicable in a particular calendar year is calculated using the formula: qYear ( projected ) = q xBase ´ RF ( x, t ) where t = Year - 1992 x

The reduction factor is calculated as: RF ( x, t ) = a + (1 - a)(1 - f )t

20

The parameters used are: Age range

a

f

x < 60

0.13

0.55

60 £ x £ 110

æ 110 - x ö 1 - 0.87 ç è 50 ÷ø

æ 110 - x ö æ x - 60 ö 0.55 ç + 0.29 ç ÷ è 50 ø è 50 ÷ø

x > 110

1

0.29

109

PMA92Base

110

x

qx

50 51 52 53 54

0.001 315 0.001 519 0.001 761 0.002 045 0.002 379

55 56 57 58 59

0.002 771 0.003 228 0.003 759 0.004 376 0.005 090

60 61 62 63 64

0.005 914 0.006 861 0.007 947 0.009 189 0.010 604

65 66 67 68 69

0.012 211 0.014 032 0.016 088 0.018 402 0.020 998

70 71 72 73 74

0.023 901 0.027 137 0.030 732 0.034 713 0.039 105

75 76 77 78 79

0.043 935 0.049 227 0.055 006 0.061 292 0.068 106

80 81 82 83 84

0.075 464 0.083 379 0.091 862 0.100 917 0.110 544

85 86 87 88 89

0.120 739 0.131 492 0.142 786 0.154 599 0.166 903

90 91 92 93 94

0.179 664 0.192 841 0.206 389 0.220 257 0.234 389

95 96 97 98 99

0.248 727 0.263 206 0.277 762 0.292 327 0.306 832

100 101 102 103 104

0.321 209 0.335 389 0.349 305 0.362 893 0.376 091

105

0.388 838

PFA92base x

qx

50 51 52 53 54

0.001 271 0.001 456 0.001 670 0.001 917 0.002 200

55 56 57 58 59

0.002 524 0.002 894 0.003 317 0.003 799 0.004 345

60 61 62 63 64

0.004 965 0.005 667 0.006 458 0.007 350 0.008 352

65 66 67 68 69

0.009 476 0.010 734 0.012 138 0.013 703 0.015 442

70 71 72 73 74

0.017 371 0.019 505 0.021 861 0.024 455 0.027 306

75 76 77 78 79

0.030 432 0.033 849 0.037 577 0.041 632 0.046 035

80 81 82 83 84

0.050 800 0.055 946 0.061 488 0.067 441 0.073 817

85 86 87 88 89

0.080 629 0.087 885 0.095 594 0.103 761 0.112 386

90 91 92 93 94

0.121 470 0.131 009 0.140 996 0.151 420 0.162 267

95 96 97 98 99

0.173 519 0.185 155 0.197 150 0.209 477 0.222 103

100 101 102 103 104

0.234 995 0.248 115 0.261 424 0.274 879 0.288 437

105

0.302 054

111

PMA92C20 x

112

lx

dx

qx

mx

o

ex

x

50 51 52 53 54

9 941.923 9 936.504 9 930.244 9 922.995 9 914.580

5.418 6.260 7.249 8.415 9.776

0.000 545 0.000 630 0.000 730 0.000 848 0.000 986

0.000 507 0.000 585 0.000 677 0.000 786 0.000 914

34.10 33.12 32.14 31.17 30.19

50 51 52 53 54

55 56 57 58 59

9 904.805 9 893.434 9 880.196 9 864.803 9 846.908

11.371 13.237 15.393 17.895 20.777

0.001 148 0.001 338 0.001 558 0.001 814 0.002 110

0.001 063 0.001 239 0.001 444 0.001 681 0.001 957

29.22 28.25 27.29 26.33 25.38

55 56 57 58 59

60 61 62 63 64

9 826.131 9 802.048 9 773.083 9 738.388 9 696.990

24.084 28.965 34.694 41.398 49.193

0.002 451 0.002 955 0.003 550 0.004 251 0.005 073

0.002 266 0.002 685 0.003 241 0.003 889 0.004 651

24.43 23.49 22.56 21.64 20.73

60 61 62 63 64

65 66 67 68 69

9 647.797 9 589.602 9 521.065 9 440.717 9 346.970

58.195 68.537 80.348 93.746 108.836

0.006 032 0.007 147 0.008 439 0.009 930 0.011 644

0.005 543 0.006 583 0.007 792 0.009 191 0.010 806

19.83 18.95 18.08 17.23 16.40

65 66 67 68 69

70 71 72 73 74

9 238.134 9 112.449 8 968.099 8 803.265 8 616.170

125.685 144.350 164.834 187.096 211.010

0.013 605 0.015 841 0.018 380 0.021 253 0.024 490

0.012 661 0.014 783 0.017 204 0.019 956 0.023 072

15.59 14.79 14.02 13.28 12.55

70 71 72 73 74

75 76 77 78 79

8 405.160 8 168.798 7 905.934 7 615.818 7 298.223

236.362 262.864 290.116 317.595 344.688

0.028 121 0.032 179 0.036 696 0.041 702 0.047 229

0.026 587 0.030 537 0.034 962 0.039 899 0.045 390

11.86 11.18 10.54 9.92 9.33

75 76 77 78 79

80 81 82 83 84

6 953.536 6 582.891 6 188.234 5 772.378 5 339.057

370.644 394.658 415.856 433.321 446.180

0.053 303 0.059 952 0.067 201 0.075 068 0.083 569

0.051 473 0.058 188 0.065 576 0.073 676 0.082 522

8.77 8.23 7.73 7.25 6.80

80 81 82 83 84

85 86 87 88 89

4 892.878 4 439.230 3 984.138 3 534.054 3 095.591

453.648 455.092 450.084 438.463 420.387

0.092 716 0.102 516 0.112 969 0.124 068 0.135 802

0.092 149 0.102 590 0.113 873 0.126 023 0.139 060

6.37 5.97 5.59 5.24 4.91

85 86 87 88 89

90 91 92 93 94

2 675.203 2 278.869 1 911.771 1 578.012 1 280.416

396.334 367.099 333.759 297.596 260.008

0.148 151 0.161 088 0.174 581 0.188 589 0.203 065

0.152 998 0.167 846 0.183 606 0.200 273 0.217 836

4.61 4.32 4.06 3.81 3.59

90 91 92 93 94

95 96 97 98 99

1 020.409 798.003 611.905 459.696 338.101

222.405 186.098 152.209 121.595 94.813

0.217 957 0.233 205 0.248 746 0.264 511 0.280 429

0.236 273 0.255 556 0.275 647 0.296 499 0.318 054

3.38 3.18 3.00 2.84 2.68

95 96 97 98 99

100 101 102 103 104

243.288 171.171 117.693 79.050 51.848

72.117 53.478 38.644 27.202 18.647

0.296 425 0.312 423 0.328 344 0.344 113 0.359 653

0.340 247 0.363 002 0.386 232 0.409 842 0.433 729

2.54 2.41 2.29 2.18 2.08

100 101 102 103 104

105

33.200

12.446

0.374 887

0.457 778

1.99

105

PFA92C20 x

lx

dx

qx

mx

o

ex

x

50 51 52 53 54

9 952.697 9 947.452 9 941.454 9 934.574 9 926.676

5.245 5.998 6.879 7.898 9.053

0.000 527 0.000 603 0.000 692 0.000 795 0.000 912

0.000 492 0.000 563 0.000 645 0.000 741 0.000 851

37.08 36.10 35.12 34.15 33.17

50 51 52 53 54

55 56 57 58 59

9 917.623 9 907.249 9 895.370 9 881.764 9 866.200

10.374 11.879 13.606 15.564 17.769

0.001 046 0.001 199 0.001 375 0.001 575 0.001 801

0.000 976 0.001 120 0.001 284 0.001 472 0.001 685

32.20 31.24 30.27 29.31 28.36

55 56 57 58 59

60 61 62 63 64

9 848.431 9 828.163 9 804.173 9 775.888 9 742.640

20.268 23.991 28.285 33.248 38.932

0.002 058 0.002 441 0.002 885 0.003 401 0.003 996

0.001 918 0.002 236 0.002 655 0.003 135 0.003 691

27.41 26.46 25.53 24.60 23.68

60 61 62 63 64

65 66 67 68 69

9 703.708 9 658.285 9 605.483 9 544.325 9 473.745

45.423 52.802 61.158 70.580 81.124

0.004 681 0.005 467 0.006 367 0.007 395 0.008 563

0.004 332 0.005 069 0.005 914 0.006 882 0.007 986

22.78 21.88 21.00 20.13 19.28

65 66 67 68 69

70 71 72 73 74

9 392.621 9 299.747 9 193.860 9 073.650 8 937.791

92.874 105.887 120.210 135.860 152.836

0.009 888 0.011 386 0.013 075 0.014 973 0.017 100

0.009 240 0.010 663 0.012 272 0.014 086 0.016 126

18.44 17.62 16.81 16.03 15.27

70 71 72 73 74

75 76 77 78 79

8 784.955 8 613.841 8 423.243 8 212.080 7 979.465

171.113 190.598 211.162 232.615 254.729

0.019 478 0.022 127 0.025 069 0.028 326 0.031 923

0.018 414 0.020 974 0.023 829 0.027 004 0.030 527

14.52 13.80 13.10 12.42 11.77

75 76 77 78 79

80 81 82 83 84

7 724.737 7 447.558 7 147.965 6 826.442 6 483.987

277.179 299.593 321.523 342.455 361.832

0.035 882 0.040 227 0.044 981 0.050 166 0.055 804

0.034 425 0.038 728 0.043 464 0.048 664 0.054 357

11.14 10.54 9.96 9.41 8.88

80 81 82 83 84

85 86 87 88 89

6 122.154 5 743.101 5 349.595 4 945.000 4 533.230

379.053 393.506 404.595 411.770 414.537

0.061 915 0.068 518 0.075 631 0.083 270 0.091 444

0.060 576 0.067 349 0.074 708 0.082 686 0.091 308

8.37 7.89 7.43 7.00 6.59

85 86 87 88 89

90 91 92 93 94

4 118.693 3 706.149 3 300.559 2 906.914 2 530.033

412.545 405.590 393.644 376.882 355.677

0.100 164 0.109 437 0.119 266 0.129 650 0.140 582

0.100 604 0.110 601 0.121 325 0.132 801 0.145 048

6.20 5.84 5.49 5.17 4.87

90 91 92 93 94

95 96 97 98 99

2 174.356 1 843.738 1 541.271 1 269.152 1 028.591

330.617 302.467 272.119 240.562 208.795

0.152 053 0.164 051 0.176 555 0.189 545 0.202 991

0.158 084 0.171 926 0.186 586 0.202 071 0.218 386

4.58 4.32 4.07 3.84 3.62

95 96 97 98 99

100 101 102 103 104

819.796 642.013 493.627 372.325 275.277

177.783 148.385 121.303 97.048 75.930

0.216 863 0.231 125 0.245 737 0.260 654 0.275 830

0.235 531 0.253 502 0.272 288 0.291 872 0.312 234

3.41 3.22 3.05 2.89 2.73

100 101 102 103 104

105

199.347

58.053

0.291 217

0.333 348

2.59

105

113

PMA92C20 4%

x

a&&x

50 51 52 53 54

2

Ax

x

a&&x

18.843 18.567 18.281 17.985 17.680

0.088 02 0.094 71 0.101 87 0.109 54 0.117 73

50 51 52 53 54

19.539 19.291 19.034 18.768 18.494

0.074 21 0.079 78 0.085 74 0.092 11 0.098 91

55 56 57 58 59

17.364 17.038 16.702 16.356 15.999

0.126 47 0.135 80 0.145 74 0.156 32 0.167 56

55 56 57 58 59

18.210 17.917 17.615 17.303 16.982

0.106 16 0.113 90 0.122 14 0.130 91 0.140 24

60 61 62 63 64

15.632 15.254 14.868 14.475 14.073

0.179 50 0.192 17 0.205 50 0.219 50 0.234 16

60 61 62 63 64

16.652 16.311 15.963 15.606 15.242

0.150 15 0.160 68 0.171 77 0.183 43 0.195 66

65 66 67 68 69

13.666 13.252 12.834 12.412 11.988

0.249 46 0.265 38 0.281 90 0.298 99 0.316 60

65 66 67 68 69

14.871 14.494 14.111 13.723 13.330

0.208 47 0.221 83 0.235 76 0.250 22 0.265 21

70 71 72 73 74

11.562 11.136 10.711 10.288 9.870

0.334 69 0.353 20 0.372 08 0.391 25 0.410 65

70 71 72 73 74

12.934 12.535 12.135 11.734 11.333

0.280 69 0.296 64 0.313 02 0.329 80 0.346 93

75 76 77 78 79

9.456 9.049 8.649 8.258 7.877

0.430 21 0.449 84 0.469 47 0.489 03 0.508 44

75 76 77 78 79

10.933 10.536 10.142 9.752 9.367

0.364 37 0.382 07 0.399 97 0.418 02 0.436 16

80 81 82 83 84

7.506 7.148 6.801 6.468 6.148

0.527 62 0.546 50 0.565 01 0.583 10 0.600 71

80 81 82 83 84

8.989 8.618 8.254 7.900 7.555

0.454 33 0.472 47 0.490 53 0.508 45 0.526 16

85 86 87 88 89

5.842 5.551 5.273 5.010 4.762

0.617 79 0.634 29 0.650 19 0.665 45 0.680 06

85 86 87 88 89

7.220 6.896 6.582 6.281 5.991

0.543 63 0.560 80 0.577 62 0.594 05 0.610 06

90 91 92 93 94

4.527 4.306 4.098 3.903 3.721

0.693 99 0.707 25 0.719 83 0.731 74 0.742 97

90 91 92 93 94

5.713 5.447 5.193 4.951 4.722

0.625 60 0.640 66 0.655 20 0.669 21 0.682 68

95 96 97 98 99

3.551 3.393 3.245 3.109 2.982

0.753 56 0.763 50 0.772 82 0.781 55 0.789 69

95 96 97 98 99

4.504 4.297 4.102 3.918 3.744

0.695 59 0.707 94 0.719 73 0.730 97 0.741 64

100 101 102 103 104

2.864 2.755 2.655 2.562 2.477

0.797 28 0.804 34 0.810 89 0.816 96 0.822 57

100 101 102 103 104

3.581 3.428 3.284 3.149 3.023

0.751 77 0.761 36 0.770 43 0.778 99 0.787 05

105

2.399

0.827 74

105

2.905

0.794 63

Note. 2 Ax = Ax at 8.16%.

114

2

PFA92C20 Ax

115

10.978 8.833 6.876 5.235 3.963 3.039 2.400

10.494 8.357 6.441 4.864 3.664 2.808 2.223

10.368 8.238 6.336 4.777 3.597 2.757 2.186

12.560 12.125 11.687 11.247 10.807 10.233 8.110 6.224 4.687 3.528 2.706 2.149

12.427 11.991 11.552 11.112 10.672 10.088 7.975 6.107 4.593 3.456 2.654 2.112

12.282 11.845 11.406 10.966 10.526

14.393 13.985 13.569 13.145 12.716

9.932 7.831 5.985 4.496 3.384 2.602 2.075

12.126 11.688 11.248 10.808 10.369

14.248 13.837 13.418 12.992 12.561

9.766 7.679 5.857 4.396 3.310 2.549 2.038

11.958 11.520 11.080 10.640 10.201

14.090 13.675 13.254 12.826 12.394

16.016 15.651 15.276 14.891 14.495

9.590 7.520 5.725 4.294 3.235 2.496 2.001

11.778 11.339 10.900 10.460 10.023

13.918 13.501 13.078 12.648 12.215

15.864 15.495 15.116 14.727 14.327

9.404 7.355 5.588 4.189 3.160 2.444 1.965

11.586 11.147 10.708 10.270 9.835

13.734 13.314 12.888 12.458 12.023

15.699 15.326 14.942 14.549 14.145

9.209 7.182 5.449 4.084 3.084 2.391 1.930

11.382 10.944 10.506 10.070 9.637

13.536 13.114 12.686 12.255 11.819

15.521 15.143 14.755 14.357 13.950

17.258 16.931 16.594 16.246 15.888

9.005 7.005 5.306 3.977 3.008 2.339 1.895

11.167 10.729 10.293 9.859 9.429

13.325 12.901 12.472 12.039 11.604

15.328 14.945 14.553 14.151 13.742

17.090 16.758 16.416 16.064 15.701

8.792 6.822 5.161 3.870 2.933 2.288 1.861

10.940 10.504 10.070 9.639 9.213

13.101 12.675 12.245 11.812 11.376

15.121 14.733 14.337 13.932 13.520

16.909 16.572 16.225 15.867 15.499

7.636 5.860 4.422 3.340 2.571 2.049 1.708

9.662 9.243 8.830 8.423 8.025

11.796 11.368 10.939 10.511 10.085

13.880 13.473 13.061 12.644 12.222

15.801 15.433 15.057 14.672 14.279

+10

5.250 4.027 3.108 2.449 1.998 1.708 1.000

6.790 6.457 6.137 5.829 5.533

8.605 8.224 7.851 7.487 7.133

10.595 10.189 9.786 9.387 8.993

12.638 12.232 11.823 11.413 11.004

+20 d

70 75 80 85 90 95 100

65 66 67 68 69

60 61 62 63 64

55 56 57 58 59

11.412 9.295 7.335 5.660 4.339 3.361 2.670

12.682 12.248 11.811 11.372 10.933

14.526 14.121 13.708 13.287 12.859

16.156 15.795 15.423 15.041 14.650

17.413 17.091 16.758 16.415 16.062

+5

70 75 80 85 90 95 100

13.136 12.711 12.282 11.849 11.414

14.646 14.244 13.834 13.416 12.991

16.284 15.926 15.558 15.180 14.791

17.556 17.238 16.910 16.572 16.223

+4

13.529 13.112 12.692 12.267 11.840

14.756 14.356 13.949 13.533 13.111

16.400 16.046 15.681 15.306 14.921

17.688 17.374 17.050 16.716 16.371

+3

65 66 67 68 69

15.161 14.772 14.374 13.968 13.555

16.506 16.155 15.793 15.421 15.039

17.808 17.498 17.178 16.848 16.507

+2

15.509 15.129 14.740 14.343 13.939

16.602 16.253 15.894 15.525 15.146

17.918 17.612 17.295 16.968 16.631

+1

60 61 62 63 64

16.955 16.617 16.269 15.910 15.541

18.019 17.715 17.402 17.078 16.744

0

17.255 16.926 16.587 16.238 15.879

18.110 17.809 17.499 17.178 16.847

–1

55 56 57 58 59

18.192 17.894 17.586 17.269 16.941

–2

50 51 52 53 54

18.493 18.206 17.908 17.601 17.283

–3

18.746 18.467 18.179 17.881 17.573

–4

50 51 52 53 54

–5

x

–10

x

d –20

Age difference d ( = y - x )

a&&xy for male (x) and female (y)

PMA92C20 and PFA92C20 4%

116

INTERNATIONAL ACTUARIAL NOTATION Reproduced from Bulletin of the Permanent Committee of the International Congress of Actuaries, 46, 207 (1949), Journal of the Institute of Actuaries, 75, 121 (1949) and Transactions of the Faculty of Actuaries, 19, 89 (1949–50).

117

The existing international actuarial notation was founded on the “Key to the Notation” given in the Institute of Actuaries Text Book, Part II, Life Contingencies by George King (1887), and was adopted by the Second International Actuarial Congress, London, 1898 (Transactions, pp. 618–640) with minor revisions approved by the Third International Congress, Paris, 1900 (Transactions, pp. 622– 651). Further revisions were discussed during 1937–1939, and were introduced by the Institute and the Faculty in 1949 (J.I.A., 75, 121 and T.F.A., 19, 89). These revisions were finally adopted internationally at the Fourteenth International Actuarial Congress, Madrid, 1954 (Bulletin of the Permanent Committee of the International Congress of Actuaries (1949), 46, pp. 207–217). The general principles on which the system is based are as follows: To each fundamental symbolic letter are attached signs and letters each having its own signification. The lower space to the left is reserved for signs indicating the conditions relative to the duration of the operations and to their position with regard to time. The lower space to the right is reserved for signs indicating the conditions relative to ages and the order of succession of the events. The upper space to the right is reserved for signs indicating the periodicity of events. The upper space to the left is free, and in it can be placed signs corresponding to other notions. In what follows these two conventions are used: A letter enclosed in brackets, thus (x), denotes “a person aged x”. A letter or number enclosed in a right angle, thus n or 15 , denotes a term-certain of years.

118

1

FUNDAMENTAL SYMBOLIC LETTERS

1.1

INTEREST i = the effective rate of interest, namely, the total interest earned on 1 in a year on the assumption that the actual interest (if receivable otherwise than yearly) is invested forthwith as it becomes due on the same terms as the original principal. v = (1 + i) -1 = the present value of 1 due one year hence. d = 1 - v = the discount on 1 due one year hence. d = log e (1 + i ) = - log e (1 - d ) = the force of interest or the force of discount.

1.2

MORTALITY TABLES l = number living. d = number dying. p = probability of living. q = probability of dying. m = force of mortality. m = central death rate. a = present value of an annuity. s = amount of an annuity. e = expectation of life. A = present value of an assurance. E = present value of an endowment. P = premium per annum. P generally refers to net premiums, p to special premiums. p = premium per annum. V = policy value. W = paid-up policy.

119

The methods of using the foregoing principal letters and their precise meaning when added to by suffixes, etc., follow. 1.3

INTEREST i ( m ) = m{(1 + i )1/ m - 1} = the nominal rate of interest, convertible m times a year. an = v + v 2 + ... + v n = the value of an annuity-certain of 1 per annum

for n years, the payments being made at the end of each year. a&&n = 1 + v + v 2 + ... + v n -1 = the value of a similar annuity, the

payments being made at the beginning of each year. sn = 1 + (1 + i ) + (1 + i ) 2 + ... + (1 + i )n -1 = the amount of an annuity-

certain of 1 per annum for n years, the payments being made at the end of each year. && sn = (1 + i ) + (1 + i ) 2 + ... + (1 + i )n = the amount of a similar annuity,

the payments being made at the beginning of each year. The diaeresis or trema ( ¨ ) above the letters a and s is used as a symbol of acceleration of payments. 1.4

MORTALITY TABLES The ages of the lives involved are denoted by letters placed as suffixes in the lower space to the right. Thus: l x = the number of persons who attain age x according to the mortality table. d x = l x - lx +1 = the number of persons who die between ages x and x + 1 according to the mortality table. p x = the probability that (x) will live 1 year.

120

q x = the probability that (x) will die within 1 year. mx = -

1 dl x = the force of mortality at age x. l x dx

mx = the central death-rate for the year of age x to x + 1 = dx

1

ò 0 lx +t dt.

ex = the curtate “expectation of life” (or average after-lifetime) of (x).

In the following it is always to be understood (unless otherwise expressed) that the annual payment of an annuity is 1, that the sum assured in any case is 1, and that the symbols indicate the present values: a x = an annuity, first payment at the end of a year, to continue during the life of (x). a&&x = 1 + a x = an “annuity-due” to continue during the life of (x), the first payment to be made at once. Ax = an assurance payable at the end of the year of death of (x).

Note. ex = a x at rate of interest i = 0. A letter or number at the lower left corner of the principal symbol denotes the number of years involved in the probability or benefit in question. Thus: = the probability that (x) will live n years. n q x = the probability that (x) will die within n years. n px

Note. When n = 1 it is customary to omit it (as shown above) provided no ambiguity is introduced. = v n n p x = the value of an endowment on (x) payable at the end of n years if (x) be then alive. n Ex

121

If the letter or number comes before a perpendicular bar it shows that a period of deferment is meant. Thus: q n½ x

= the probability that (x) will die in a year, deferred n years; that

is, that he will die in the (n + 1)th year. a n½ x

= an annuity on (x) deferred n years; that is, that the first

payment is to be made at the end of (n + 1) years. a n½t x

= an intercepted, or deferred, temporary annuity on (x) deferred

n years and, after that, to run for t years. A letter or number in brackets at the upper right corner of the principal symbol shows the number of intervals into which the year is to be divided. Thus: a x( m ) = an annuity of (x) payable by m instalments of 1/m each throughout the year, the first payment being one of 1/m at the end of the first 1/mth of a year. a&&x( m ) = a similar annuity but the first payment of 1/m is to be made at

once, so that a&&x( m ) = 1/ m + a x( m ) . Ax( m) = an assurance payable at the end of that fraction 1/m of a year in which (x) dies.

If m ® ¥ then instead of writing (¥) a bar is placed over the principal symbol. Thus: a = a continuous or momently annuity. A = an assurance payable at the moment of death.

122

A small circle placed over the principal symbol shows that the benefit is to be complete. Thus: ao = a complete annuity. eo = the complete expectation of life.

Note. Some consider that e would be as appropriate as eo . As ex = a x at rate of interest i = 0, so also the complete expectation of life = a x at rate of interest i = 0. When more than one life is involved the following rules are observed: If there are two or more letters or numbers in a suffix without any distinguishing mark, joint lives are intended. Thus: l xy = l x ´ l y ,

d xy = l xy - l x +1: y +1.

Note. When, for the sake of distinctness, it is desired to separate letters or numbers in a suffix, a colon is placed between them. A colon is used instead of a point or comma to avoid confusion with decimals when numbers are involved. a xyz = an annuity, first payment at the end of a year, to continue

during the joint lives of (x), (y) and (z). Axyz = an assurance payable at the end of the year of the failure of the

joint lives (x), (y) and (z). In place of a life a term-certain may be involved. Thus: a x:n = an annuity to continue during the joint duration of the life of

(x) and a term of n years certain; that is, a temporary annuity for n years on the life of (x). Ax:n = an assurance payable at the end of the year of death of (x) if he

dies within n years, or at the end of n years if (x) be then alive; that is, an endowment assurance for n years. 123

If a perpendicular bar separates the letters in the suffix, then the status after the bar is to follow the status before the bar. Thus: a y½x = a reversionary annuity, that is, an annuity on the life of (x)

after the death of (y). Az½xy = an assurance payable on the failure of the joint lives (x) and

(y) provided both these lives survive (z). If a horizontal bar appears above the suffix then survivors of the lives, and not joint lives, are intended. The number of survivors can be denoted by a letter or number over the right end of the bar. If that letter, say r, is not distinguished by any mark, then the meaning is at least r survivors; but if it is enclosed in square brackets, [r], then the meaning is exactly r survivors. If no letter or number appears over the bar, then unity is supposed and the meaning is at least one survivor. Thus: a xyz = an annuity payable so long as at least one of the three lives (x),

(y) and (z) is alive. a xyz2 = an annuity payable so long as at least two of the three lives

(x), (y) and (z) are alive. p[2] = probability that exactly two of the three lives (x), (y) and (z) xyz

will survive a year. n q xy

= probability that the survivor of the two lives (x) and (y) will

die within n years = n q x ´ n q y . n Axy

= an assurance payable at the end of the year of death of the

survivor of the lives (x) and (y) provided the death occurs within n years. When numerals are placed above or below the letters of the suffix, they designate the order in which the lives are to fail. The numeral 124

placed over the suffix points out the life whose failure will finally determine the event; and the numerals placed under the suffix indicate the order in which the other lives involved are to fail. Thus: A1xy = an assurance payable at the end of the year of death of (x) if he

dies first of the two lives (x) and (y). 2 Axyz = an assurance payable at the end of the year of death of (x) if

he dies second of the three lives (x), (y) and (z). 2 Axyz = an assurance payable at the end of the year of death of (x) if 1

he dies second of the three lives, (y) having died first. Axy:z = an assurance payable at the end of the year of death of the 3

survivor of (x) and (y) if he dies before (z). A1x:n = an assurance payable at the end of the year of death of (x) if

he dies within a term of n years. a yz½x ü 1 ï ï an annuity to ( x) after the failure of the survivor of ( y ) and ( z ), or ý = ï provided ( z ) fails before ( y ). a 2yz½x ï þ

Note. Sometimes to make quite clear that a joint-life status is G H involved a symbol is placed above the lives included. Thus 1 A GH = a joint-life temporary assurance on (x) and (y). xy:n

125

In the case of reversionary annuities, distinction has sometimes to be made between those where the times of year at which payments are to take place are determined at the outset and those where the times depend on the failure of the preceding status. Thus: a y½x = annuity to (x), first payment at the end of the year of death of

(y) or, on the average, about 6 months after his death. aˆ y½x = annuity to (x), first payment 1 year after the death of (y). aoˆ y½x = complete annuity to (x), first payment 1 year after the death of

(y).

126

2

ANNUAL PREMIUMS The symbol P with the appropriate suffix or suffixes is used in simple cases, where no misunderstanding can occur, to denote the annual premium for a benefit. Thus: Px = the annual premium for an assurance payable at the end of the year of death of (x). Px:n = the annual premium for an endowment assurance on (x)

payable after n years or at the end of the year of death of (x) if he dies within n years. 1 Pxy = the annual premium for a contingent assurance payable at the

end of the year of death of (x) if he dies before (y). In all cases it is optional to use the symbol P in conjunction with the principal symbol denoting the benefit. Thus instead of Px:n we may write P( Ax:n ) . In the more complicated cases it is necessary to use the two symbols in this way. Suffixes, etc., showing the conditions of the benefit are to be attached to the principal letter, and those showing the condition of payment of the premium are to be attached to the subsidiary symbol P. Thus: n P ( Ax ) =

the annual premium payable for n years only for an assurance payable at the moment of death of (x). Pxy ( Ax ) = the annual premium payable during the joint lives of (x)

and (y) for an assurance payable at the end of the year of death of (x). n P ( n½a x ) =

the annual premium payable for n years only for an annuity on (x) deferred n years. tP

( m)

( Ax:n ) = the annual premium payable for t years only, by m

instalments throughout the year, for an endowment assurance for n years on (x) (see below as to P ( m) ). 127

Notes. (1) As a general rule the symbol P could be used without the principal symbol in the case of assurances where the sum assured is payable at the end of the year of death, but if it is payable at other times, or if the benefit is an annuity, then the principal symbol should be used. (2) Px( m) . A point which was not brought out when the international system was adopted is that there are two kinds of premiums payable m times a year, namely those which cease on payment of the instalment immediately preceding death and those which continue to be payable to the end of the year of death. To distinguish the latter, the m is sometimes enclosed in square brackets, thus P[ m] .

128

3

POLICY VALUES AND PAID-UP POLICIES = the value of an ordinary whole-life assurance on (x) which has been t years in force, the premium then just due being unpaid. tVx

tWx

= the paid-up policy the present value of which is tVx .

The symbols V and W may, in simple cases, be used alone, but in the more complicated cases it is necessary to insert the full symbol for the benefit thus: tV

(m)

( Ax:n ) (corresponding to P ( m) ( Ax:n )),

tV ( n½ a x ).

Note. As a general rule V or W can be used as the main symbol if the sum assured is payable at the end of the year of death and the premium is payable periodically throughout the duration of the assurance. If the premium is payable for a limited number of years, say n, the policy value after t years could be written tV [ n P( A)], or, if desired, tnV ( A). In investigations where modified premiums and policy values are in question such modification may be denoted by adding accents to the symbols. Thus, when a premium other than the net premium (a valuation premium) is used in a valuation it may be denoted by P ¢ and the corresponding policy value by V ¢. Similarly, the office (or commercial) premium may be denoted by P ¢¢ and the corresponding paid-up policy by W ¢¢ .

129

4

COMPOUND SYMBOLS ( Ia ) = an annuity

ü ý commencing at 1 and increasing 1 per annum. ( IA) = an assurance þ

If the whole benefit is to be temporary the symbol of limitation is placed outside the brackets. Thus: ( Ia ) x:n = a temporary increasing annuity. ( IA)1x:n = a temporary increasing assurance.

If only the increase is to be temporary but the benefit is to continue thereafter, then the symbol of limitation is placed immediately after the symbol I. Thus: ( I n a ) x = a whole-life annuity

üï increasing for n years and thereafter ý ( I n A) x = a whole-life assurance ïþ stationary.

If the benefit is a decreasing one, the corresponding symbol is D. From the nature of the case this decrease must have a limit, as otherwise negative values might be implied. Thus: ( Dn A)1x:n = a temporary assurance commencing at n and decreasing

by 1 in each successive year. If the benefit is a varying one the corresponding symbol is v. Thus: (va ) = a varying annuity.

130

5

COMMUTATION COLUMNS

5.1

SINGLE LIVES Dx = v xlx , N x = Dx + Dx +1 + Dx + 2 + etc., S x = N x + N x +1 + N x + 2 + etc.,

C x = v x +1d x , M x = C x + C x +1 + C x + 2 + etc., Rx = M x + M x +1 + M x + 2 + etc.

When it is desired to construct the assurance columns so as to give directly assurances payable at the moment of death, the symbols are distinguished by a bar placed over them. Thus: C x = v x +½ d x which is an approximation to

ò

1 x +t v m x + t l x + t dt. 0

M x = C x + C x +1 + C x + 2 + etc. Rx = M x + M x +1 + M x + 2 + etc.

5.2

JOINT LIVES Dxy = v½( x + y )l xy , N xy = Dxy + Dx +1: y +1 + Dx + 2: y + 2 + etc. C xy = v½( x + y ) +1d xy , M xy = C xy + C x +1: y +1 + C x + 2: y + 2 + etc. C1xy = v½( x + y ) +1d xl y +½ , M 1xy = C1xy + C x 1+1: y +1 + C x 1+ 2: y + 2 + etc.

131

6

SELECTION If the suffix to a symbol which denotes the age is enclosed in a square bracket it indicates the age at which the life was selected. To this may be added, outside the bracket, the number of years which have elapsed since selection, so that the total suffix denotes the present age. Thus: l[ x ]+ t = the number in the select life table who were selected at age x

and have attained age x + t. d[ x]+ t = l[ x ]+ t - l[ x ]+ t +1. a[ x] = value of an annuity on a life now aged x and now select. a[ x - n ]+ n = value of an annuity on a life now aged x and select n years

ago at age x - n. N[ x ] = D[ x ] + D[ x]+1 + D[ x ]+ 2 + ... a&&[ x ] = N[ x ] ¸ D[ x ] = 1 + a[ x ],

and similarly for other functions. When Dr Sprague presented his statement [in 1900] he mentioned that an objection had been raised that the notation in some cases offers the choice of two symbols for the same benefit. For instance, a temporary annuity may be denoted either by n a x or by a x:n . This is, he says, a necessary consequence of the principles underlying the system, and neither of the alternative forms could have been suppressed without injury to the symmetry of the system.

132

SICKNESS TABLE (MANCHESTER UNITY METHODOLOGY) S(MU) This table was produced using the methodology underlying that of the Manchester Unity Sickness Experience 1893–97. The underlying rates of sickness have, however, been updated to reflect more modern experience, and have been combined with the mortality of English Life Tables No. 15 (Males).

133

S(MU) Central rates of sickness (weeks per annum) Duration of sickness in weeks 0–13

13–26

26–52

52–104

Age

134

³104

All Age

16 17 18 19

0.315 0 0.332 3 0.348 2 0.357 6

0.004 8 0.008 0 0.008 8 0.009 7

0.001 2 0.004 4 0.005 0 0.005 6

0.000 0 0.002 0 0.003 9 0.004 4

0.000 0 0.000 0 0.001 1 0.003 0

0.321 0 0.346 7 0.367 0 0.380 3

16 17 18 19

20 21 22 23 24

0.366 5 0.374 9 0.383 0 0.390 5 0.397 7

0.010 6 0.011 6 0.012 7 0.013 9 0.015 1

0.006 3 0.007 0 0.007 8 0.008 7 0.009 7

0.005 1 0.005 8 0.006 6 0.007 4 0.008 4

0.004 8 0.006 8 0.008 9 0.011 3 0.014 0

0.393 3 0.406 1 0.419 0 0.431 8 0.444 9

20 21 22 23 24

25 26 27 28 29

0.402 6 0.410 9 0.417 1 0.423 0 0.428 7

0.016 4 0.017 8 0.019 3 0.020 9 0.022 5

0.010 8 0.011 9 0.013 2 0.014 6 0.016 1

0.009 5 0.010 7 0.012 0 0.013 5 0.015 1

0.017 0 0.020 3 0.024 1 0.028 4 0.033 2

0.456 3 0.471 6 0.485 7 0.500 4 0.515 6

25 26 27 28 29

30 31 32 33 34

0.434 4 0.439 8 0.445 4 0.451 0 0.456 7

0.024 3 0.026 2 0.028 3 0.030 4 0.032 8

0.017 7 0.019 5 0.021 5 0.023 6 0.025 9

0.016 9 0.018 9 0.021 1 0.023 6 0.026 3

0.038 6 0.044 8 0.051 8 0.059 6 0.068 6

0.531 9 0.549 2 0.568 1 0.588 2 0.610 3

30 31 32 33 34

35 36 37 38 39

0.462 6 0.468 8 0.475 2 0.482 2 0.489 8

0.035 3 0.037 9 0.040 8 0.043 9 0.047 3

0.028 4 0.031 2 0.034 2 0.037 6 0.041 2

0.029 3 0.032 7 0.036 4 0.040 5 0.045 2

0.078 7 0.090 1 0.103 1 0.117 9 0.134 6

0.634 3 0.660 7 0.689 7 0.722 1 0.758 1

35 36 37 38 39

40 41 42 43 44

0.497 9 0.506 7 0.516 3 0.526 9 0.538 6

0.050 9 0.054 8 0.059 1 0.063 8 0.068 9

0.045 3 0.049 7 0.054 6 0.060 1 0.066 1

0.050 3 0.056 1 0.062 5 0.069 7 0.077 8

0.153 6 0.175 2 0.199 7 0.227 7 0.259 5

0.798 0 0.842 5 0.892 2 0.948 2 1.010 9

40 41 42 43 44

45 46 47 48 49

0.551 4 0.565 6 0.581 2 0.598 6 0.617 8

0.074 5 0.080 6 0.087 4 0.094 8 0.103 1

0.072 9 0.080 4 0.088 8 0.098 2 0.108 8

0.086 9 0.097 2 0.108 8 0.122 0 0.137 0

0.295 9 0.337 4 0.385 0 0.439 5 0.502 0

1.081 6 1.161 2 1.251 2 1.353 1 1.468 7

45 46 47 48 49

50 51 52 53 54

0.639 0 0.662 6 0.688 8 0.717 8 0.749 9

0.112 3 0.122 5 0.133 9 0.146 6 0.160 9

0.120 7 0.134 1 0.149 3 0.166 6 0.186 2

0.154 0 0.173 4 0.195 6 0.221 0 0.250 3

0.574 0 0.656 9 0.752 7 0.863 6 0.992 1

1.600 0 1.749 5 1.920 3 2.115 6 2.339 4

50 51 52 53 54

55 56 57 58 59

0.785 6 0.825 1 0.869 1 0.917 7 0.971 7

0.176 9 0.194 9 0.215 3 0.238 2 0.264 2

0.208 5 0.234 0 0.263 2 0.296 7 0.335 1

0.283 9 0.322 8 0.367 7 0.419 9 0.480 4

1.141 6 1.315 8 1.519 3 1.757 8 2.037 8

2.596 5 2.892 6 3.234 6 3.630 3 4.089 2

55 56 57 58 59

60 61 62 63 64

1.031 1 1.096 8 1.169 0 1.247 8 1.333 5

0.293 5 0.326 8 0.364 3 0.406 7 0.454 3

0.379 3 0.430 0 0.488 4 0.555 5 0.632 5

0.550 8 0.632 8 0.728 5 0.840 0 0.970 0

2.367 7 2.757 4 3.218 9 3.767 0 4.419 8

4.622 4 5.243 8 5.969 1 6.817 0 7.810 1

60 61 62 63 64

S(MU) Present value of a sickness benefit payable at the rate of 1 per week during sickness of the following durations. All benefits cease at the earlier of death or attainment of age 65.

4%

Duration of sickness in weeks 0–13

13–26

26–52

52–104

Age

³104

All Age

16 17 18 19

10.236 10.329 10.412 10.482

1.113 1.153 1.192 1.232

1.171 1.217 1.262 1.309

1.515 1.576 1.639 1.702

5.786 6.021 6.266 6.522

19.821 20.297 20.771 21.246

16 17 18 19

20 21 22 23 24

10.546 10.603 10.654 10.699 10.739

1.272 1.313 1.355 1.398 1.441

1.357 1.406 1.456 1.508 1.560

1.767 1.834 1.903 1.974 2.047

6.785 7.057 7.339 7.630 7.931

21.726 22.213 22.707 23.209 23.718

20 21 22 23 24

25 26 27 28 29

10.772 10.802 10.825 10.842 10.853

1.484 1.528 1.573 1.617 1.662

1.614 1.669 1.725 1.783 1.841

2.122 2.199 2.278 2.359 2.442

8.241 8.561 8.890 9.229 9.578

24.235 24.760 25.291 25.830 26.376

25 26 27 28 29

30 31 32 33 34

10.860 10.862 10.858 10.849 10.834

1.707 1.752 1.797 1.842 1.887

1.899 1.959 2.020 2.080 2.142

2.527 2.613 2.701 2.790 2.880

9.936 10.303 10.680 11.065 11.458

26.929 27.489 28.055 28.626 29.201

30 31 32 33 34

35 36 37 38 39

10.813 10.787 10.754 10.715 10.668

1.931 1.974 2.017 2.059 2.100

2.203 2.265 2.327 2.388 2.449

2.972 3.064 3.158 3.251 3.345

11.859 12.267 12.682 13.103 13.527

29.778 30.358 30.939 31.517 32.089

35 36 37 38 39

40 41 42 43 44

10.613 10.548 10.473 10.387 10.288

2.139 2.176 2.212 2.245 2.274

2.509 2.568 2.625 2.680 2.732

3.438 3.531 3.622 3.710 3.796

13.953 14.380 14.804 15.223 15.634

32.653 33.203 33.735 34.245 34.725

40 41 42 43 44

45 46 47 48 49

10.176 10.048 9.904 9.740 9.556

2.301 2.323 2.341 2.353 2.360

2.780 2.825 2.864 2.898 2.925

3.878 3.955 4.026 4.090 4.145

16.034 16.418 16.781 17.117 17.419

35.169 35.569 35.916 36.199 36.405

45 46 47 48 49

50 51 52 53 54

9.348 9.114 8.851 8.554 8.219

2.359 2.350 2.331 2.302 2.259

2.944 2.952 2.949 2.932 2.899

4.189 4.219 4.233 4.228 4.200

17.678 17.884 18.025 18.085 18.046

36.517 36.520 36.390 36.101 35.624

50 51 52 53 54

55 56 57 58 59

7.842 7.417 6.938 6.397 5.786

2.202 2.127 2.033 1.915 1.769

2.846 2.770 2.667 2.532 2.358

4.143 4.053 3.922 3.743 3.506

17.888 17.584 17.104 16.409 15.455

34.921 33.951 32.663 30.995 28.875

55 56 57 58 59

60 61 62 63 64

5.096 4.316 3.433 2.431 1.293

1.592 1.378 1.120 0.810 0.441

2.140 1.867 1.531 1.118 0.613

3.199 2.808 2.316 1.702 0.941

14.184 12.528 10.401 7.698 4.286

26.211 22.897 18.800 13.759 7.574

60 61 62 63 64

135

Annuity values, allowing for mortality only, on the basis of ELT15 (Males) 4%

136

x

a x:65 - x

16 17 18 19

21.231 21.072 20.911 20.746

20 21 22 23 24

20.573 20.394 20.208 20.015 19.813

25 26 27 28 29

19.604 19.385 19.157 18.920 18.674

30 31 32 33 34

18.418 18.152 17.875 17.588 17.289

35 36 37 38 39

16.979 16.658 16.326 15.982 15.626

40 41 42 43 44

15.256 14.873 14.476 14.064 13.638

45 46 47 48 49

13.197 12.740 12.268 11.779 11.274

50 51 52 53 54

10.752 10.212 9.653 9.075 8.475

55 56 57 58 59

7.854 7.210 6.541 5.846 5.123

60 61 62 63 64

4.368 3.580 2.754 1.886 0.970

SICKNESS TABLE (INCEPTION RATE / DISABILITY ANNUITY METHODOLOGY) S(ID) This table was produced using an inception rate/disability annuity method based on results presented in C.M.I.R. 12. The following are tabulated: · claim inception rates · present values of current claim sickness annuities · present values of annuities payable during sickness for lives currently healthy The annuities cease at the earliest of: death; attainment of age 65; recovery from sickness.

137

S(ID) Claim inception rates, (ia) x,d , for the given ages x and deferred periods d years.

(These rates are central, and (when d = 0) allow for the possibility of falling sick more than once during the year of age from x to x + 1. It was assumed in the construction of this table that all lives are healthy at exact age 30.) Deferred period in years, d

138

Age, x

0

1

2

30 31 32 33 34

0.322 744 0.318 254 0.313 615 0.308 879 0.304 097

0.000 521 0.000 578 0.000 641 0.000 709

0.000 294 0.000 330 0.000 371

35 36 37 38 39

0.299 317 0.294 583 0.289 937 0.285 418 0.281 061

0.000 785 0.000 869 0.000 961 0.001 063 0.001 176

0.000 416 0.000 467 0.000 524 0.000 588 0.000 659

40 41 42 43 44

0.276 901 0.272 968 0.269 290 0.265 896 0.262 810

0.001 301 0.001 440 0.001 594 0.001 767 0.001 959

0.000 739 0.000 829 0.000 930 0.001 044 0.001 172

45 46 47 48 49

0.260 057 0.257 659 0.255 639 0.254 018 0.252 816

0.002 175 0.002 416 0.002 688 0.002 994 0.003 340

0.001 317 0.001 482 0.001 669 0.001 882 0.002 125

50 51 52 53 54

0.252 056 0.251 758 0.251 943 0.252 630 0.253 841

0.003 732 0.004 177 0.004 682 0.005 259 0.005 918

0.002 403 0.002 721 0.003 086 0.003 507 0.003 992

55 56 57 58 59

0.255 594 0.257 906 0.260 793 0.264 262 0.268 316

0.006 674 0.007 541 0.008 539 0.009 690 0.011 018

0.004 554 0.005 205 0.005 962 0.006 843 0.007 873

60 61 62 63 64

0.272 945 0.278 123 0.283 800 0.289 890 0.296 263

0.012 554 0.014 332 0.016 390 0.018 772 0.021 524

0.009 076 0.010 487 0.012 141 0.014 083 0.016 362

S(ID) Present values of sickness benefit payable continuously at the rate of 1 per annum during sickness of the specified duration. All benefits cease at the earlier of death or attainment of age 65.

6% CURRENT STATUS = HEALTHY

CURRENT STATUS = SICK The table below gives the present value,

axSS, z , of a “current claim” sickness annuity for a sick life now aged x with current duration of sickness z years. (The annuity does not allow for the possibility of future new episodes of sickness.) Current duration of sickness, z years 0 1 2 Age, x

The table below gives the present value, axHS ( d / all) , of sickness benefit payable during sickness of duration at least d years for a life aged x who is currently healthy. (The value allows for the possibility of more than one episode of sickness.)

Age, x

Deferred period, d years 0 1

2

30 31 32 33 34

0.033 3 0.035 0 0.036 8 0.038 8 0.041 0

3.570 2 3.660 4 3.751 9 3.844 3 3.937 5

5.418 0 5.505 1 5.591 5 5.676 9 5.761 0

30 31 32 33 34

0.330 580 0.339 378 0.348 311 0.357 354 0.366 480

0.142 025 0.148 808 0.155 754 0.162 837 0.170 038

0.111 543 0.116 826 0.122 226 0.127 714 0.133 274

35 36 37 38 39

0.043 5 0.046 2 0.049 2 0.052 5 0.056 2

4.031 1 4.124 6 4.217 8 4.309 9 4.400 6

5.843 2 5.923 0 5.999 7 6.072 8 6.141 3

35 36 37 38 39

0.375 647 0.384 822 0.393 952 0.402 981 0.411 815

0.177 324 0.184 665 0.192 016 0.199 327 0.206 529

0.138 875 0.144 486 0.150 067 0.155 573 0.160 944

40 41 42 43 44

0.060 3 0.064 9 0.069 9 0.075 4 0.081 5

4.488 9 4.574 3 4.655 7 4.732 1 4.802 3

6.204 4 6.261 2 6.310 6 6.351 2 6.381 9

40 41 42 43 44

0.420 352 0.428 479 0.436 077 0.443 010 0.449 125

0.213 550 0.220 304 0.226 698 0.232 611 0.237 925

0.166 111 0.171 001 0.175 528 0.179 594 0.183 090

45 46 47 48 49

0.088 3 0.095 7 0.103 8 0.112 6 0.122 1

4.865 1 4.918 9 4.961 9 4.992 3 5.008 0

6.401 1 6.407 1 6.398 1 6.372 1 6.326 9

45 46 47 48 49

0.454 221 0.458 091 0.460 523 0.461 260 0.460 010

0.242 488 0.246 146 0.248 719 0.250 010 0.249 788

0.185 885 0.187 843 0.188 814 0.188 628 0.187 096

50 51 52 53 54

0.132 4 0.143 3 0.154 9 0.167 0 0.179 3

5.006 4 4.984 9 4.940 5 4.869 7 4.768 8

6.259 9 6.168 6 6.049 8 5.900 4 5.716 9

50 51 52 53 54

0.456 447 0.450 241 0.440 992 0.428 296 0.411 745

0.247 810 0.243 825 0.237 558 0.228 736 0.217 100

0.184 025 0.179 219 0.172 462 0.163 569 0.152 372

55 56 57 58 59

0.191 7 0.203 5 0.214 4 0.223 4 0.229 5

4.633 7 4.459 6 4.241 4 3.973 3 3.649 0

5.495 2 5.231 2 4.920 2 4.557 1 4.136 3

55 56 57 58 59

0.390 935 0.365 518 0.335 193 0.299 804 0.259 410

0.202 426 0.184 575 0.163 508 0.139 390 0.112 669

0.138 768 0.122 748 0.104 447 0.084 219 0.062 755

60 61 62 63 64

0.231 2 0.226 7 0.213 4 0.187 5 0.142 9

3.261 4 2.802 9 2.264 3 1.633 6 0.892 5

3.651 8 3.097 0 2.464 8 1.746 9 0.931 5

60 61 62 63 64

0.214 401 0.165 680 0.114 894 0.064 864 0.020 334

0.084 217 0.055 536 0.029 046 0.008 533 0.000 000

0.041 213 0.021 441 0.006 275 0.000 000 0.000 000

139

Annuity values, allowing for mortality only, on the basis of ELT15 (Males) 6%

140

x

a x:65 - x

16 17 18 19

15.881 15.813 15.744 15.673

20 21 22 23 24

15.597 15.517 15.432 15.342 15.247

25 26 27 28 29

15.146 15.038 14.924 14.803 14.674

30 31 32 33 34

14.538 14.394 14.242 14.081 13.911

35 36 37 38 39

13.731 13.541 13.342 13.131 12.909

40 41 42 43 44

12.675 12.428 12.168 11.893 11.604

45 46 47 48 49

11.299 10.978 10.640 10.284 9.910

50 51 52 53 54

9.516 9.102 8.666 8.207 7.722

55 56 57 58 59

7.211 6.671 6.101 5.496 4.856

60 61 62 63 64

4.176 3.452 2.679 1.851 0.961

EXAMPLE PENSION SCHEME TABLE PEN

141

PEN Service table and relative salary scale Age x

lx

wx

dx

ix

rx

s*x

s x = (1.02) x s*x

zx

z x +½

Age x

16 17 18 19

100 000 89 950 80 910 72 778

10 000 8 995 8 091 7 278

50 45 41 36

1.000 1.177 1.349 1.513

1.373 1.648 1.927 2.204

16 17 18 19

20 21 22 23 24

65 464 58 885 52 973 47 656 42 874

6 546 5 888 5 296 4 763 4 070

33 24 21 19 17

1.672 1.823 1.970 2.108 2.241

2.485 2.763 3.045 3.324 3.605

20 21 22 23 24

25 26 27 28 29

38 787 35 284 32 279 29 692 27 462

3 487 2 994 2 577 2 221 1 916

16 11 10 9 8

2.366 2.483 2.595 2.707 2.810

3.882 4.155 4.429 4.713 4.991

25 26 27 28 29

30 31 32 33 34

25 538 23 841 22 347 21 035 19 881

1 679 1 472 1 290 1 131 989

8 10 9 8 8

10 12 13 15 18

2.914 3.004 3.095 3.181 3.259

5.278 5.551 5.832 6.115 6.389

4.711 4.994 5.273 5.554 5.833

4.852 5.133 5.413 5.693 5.972

30 31 32 33 34

35 36 37 38 39

18 866 17 973 17 190 16 506 15 911

863 751 650 558 474

9 11 12 12 13

21 21 22 25 27

3.328 3.392 3.448 3.491 3.522

6.655 6.920 7.175 7.410 7.623

6.112 6.386 6.655 6.916 7.168

6.249 6.520 6.786 7.042 7.285

35 36 37 38 39

40 41 42 43 44

15 397 14 939 14 536 14 181 13 870

413 356 303 254 207

14 13 14 16 17

31 34 38 41 44

3.539 3.543 3.539 3.522 3.504

7.814 7.980 8.129 8.252 8.375

7.403 7.616 7.806 7.974 8.120

7.509 7.711 7.890 8.047 8.186

40 41 42 43 44

45 46 47 48 49

13 602 13 375 13 185 13 029 12 889

162 120 79 52 26

18 19 22 26 28

47 51 55 62 72

3.487 3.470 3.457 3.440 3.422

8.501 8.628 8.768 8.899 9.031

8.252 8.376 8.502 8.632 8.765

8.314 8.439 8.567 8.699 8.832

45 46 47 48 49

50 51 52 53 54

12 763 12 649 12 520 12 373 12 205

32 35 39 43 47

82 94 108 125 145

3.405 3.392 3.375 3.358 3.345

9.165 9.313 9.451 9.591 9.745

8.899 9.032 9.170 9.310 9.452

8.965 9.101 9.240 9.381 9.524

50 51 52 53 54

55 56 57 58 59

12 013 11 794 11 546 11 268 10 957

51 55 58 63 67

168 193 220 248 278

3.328 3.310 3.297 3.280 3.267

9.889 10.034 10.195 10.344 10.510

9.596 9.742 9.889 10.039 10.191

9.669 9.815 9.964 10.115 10.270

55 56 57 58 59

60 61 62 63 64

10 612 6 548 5 763 5 038 4 371

73 50 49 48 47

310 219 223 224 225

3.250 3.233 3.220 3.203 3.190

10.663 10.819 10.991 11.151 11.328

10.350 10.506 10.664 10.824 10.987

10.428 10.585 10.744 10.906 11.072

60 61 62 63 64

65

3 757

3 681 516 453 395 342 3 757

1 1 z x = ( sx -3 + sx -2 + sx -1 ) and z x +½ = ( z x + z x +1 ) 3 2

142

11.157

65

PEN Contribution functions Age x

4%

Dx =

Dx =

Nx =

s

v x lx

½( Dx + Dx +1 )

S Dx

s x Dx

S s Dx

s x Dx

Dx =

s

Nx =

s

Dx =

Age x

16 17 18 19

53 391 46 178 39 939 34 544

49 784 43 059 37 241 32 210

413 287 363 503 320 444 283 203

68 343 70 948 71 761 70 993

1 513 322 1 444 979 1 374 031 1 302 270

73 294 76 087 76 959 76 136

16 17 18 19

20 21 22 23 24

29 877 25 841 22 352 19 335 16 726

27 859 24 096 20 844 18 031 15 638

250 992 223 134 199 037 178 193 160 163

69 232 66 590 63 476 59 929 56 376

1 231 277 1 162 045 1 095 455 1 031 979 972 050

74 248 71 410 68 070 64 265 60 299

20 21 22 23 24

25 26 27 28 29

14 550 12 727 11 195 9 902 8 806

13 638 11 961 10 548 9 354 8 340

144 525 130 887 118 926 108 378 99 024

52 947 49 693 46 719 44 082 41 622

915 673 862 726 813 033 766 314 722 232

56 486 52 875 49 583 46 664 43 947

25 26 27 28 29

30 31 32 33 34

7 874 7 068 6 370 5 766 5 240

7 471 6 719 6 068 5 503 5 010

90 684 83 213 76 494 70 427 64 924

39 431 37 296 35 390 33 647 32 011

680 611 641 180 603 884 568 494 534 848

41 558 39 232 37 153 35 255 33 477

30 31 32 33 34

35 36 37 38 39

4 781 4 379 4 028 3 719 3 447

4 580 4 204 3 873 3 583 3 327

59 914 55 333 51 130 47 257 43 674

30 480 29 087 27 788 26 546 25 361

502 836 472 356 443 269 415 480 388 934

31 816 30 305 28 897 27 554 26 275

35 36 37 38 39

40 41 42 43 44

3 207 2 992 2 799 2 626 2 470

3 099 2 896 2 713 2 548 2 399

40 347 37 248 34 352 31 640 29 092

24 219 23 106 22 052 21 023 20 093

363 573 339 354 316 248 294 196 273 173

25 059 23 875 22 757 21 668 20 683

40 41 42 43 44

45 46 47 48 49

2 329 2 202 2 087 1 983 1 886

2 265 2 144 2 035 1 935 1 841

26 693 24 428 22 283 20 248 18 314

19 256 18 502 17 842 17 215 16 627

253 080 233 824 215 322 197 480 180 265

19 796 18 997 18 298 17 645 17 034

45 46 47 48 49

50 51 52 53 54

1 796 1 711 1 629 1 548 1 468

1 754 1 670 1 588 1 508 1 429

16 473 14 719 13 049 11 461 9 953

16 073 15 554 15 011 14 462 13 923

163 638 147 565 132 011 117 000 102 538

16 460 15 939 15 394 14 845 14 306

50 51 52 53 54

55 56 57 58 59

1 389 1 312 1 235 1 159 1 083

1 350 1 273 1 197 1 121 1 046

8 524 7 174 5 901 4 704 3 583

13 354 12 775 12 199 11 595 10 993

88 615 75 261 62 486 50 287 38 692

13 739 13 161 12 587 11 984 11 385

55 56 57 58 59

60 61 62 63 64

1 009 599 507 426 355

804 553 466 390 324

2 537 1 733 1 181 715 324

8 570 5 978 5 123 4 354 3 674

27 699 19 129 13 152 8 028 3 674

10 757 6 475 5 567 4 748 4 023

60 61 62 63 64

65

294

65

143

PEN 4%

Ill health retirement functions C xi =

M xi =

Rxi =

C xia =

M xia =

Rxia =

v x +½ ix

S C xi

S ( M xi - ½C xi )

C xi axi +½

S C xia

S ( M xia - ½C xia )

16 17 18 19

414 414 414 414

15 416 15 002 14 588 14 173

7 023 7 023 7 023 7 023

252 924 245 901 238 878 231 855

16 17 18 19

20 21 22 23 24

414 414 414 414 414

13 759 13 345 12 930 12 516 12 102

7 023 7 023 7 023 7 023 7 023

224 831 217 808 210 785 203 762 196 739

20 21 22 23 24

25 26 27 28 29

414 414 414 414 414

11 688 11 273 10 859 10 445 10 030

7 023 7 023 7 023 7 023 7 023

189 715 182 692 175 669 168 646 161 622

25 26 27 28 29

Age x

axi +½

Age x

30 31 32 33 34

21.852 21.720 21.583 21.441 21.294

3 3 4 4 5

414 411 408 404 400

9 616 9 203 8 794 8 388 7 986

66 76 78 86 99

7 023 6 957 6 881 6 803 6 717

154 599 147 609 140 690 133 848 127 088

30 31 32 33 34

35 36 37 38 39

21.142 20.985 20.822 20.654 20.481

5 5 5 6 6

395 390 385 380 375

7 588 7 195 6 807 6 425 6 047

110 105 105 114 117

6 617 6 507 6 402 6 297 6 183

120 421 113 859 107 404 101 055 94 815

35 36 37 38 39

40 41 42 43 44

20.302 20.118 19.929 19.734 19.534

6 7 7 7 8

369 363 356 349 341

5 676 5 310 4 951 4 598 4 253

129 134 143 147 150

6 065 5 937 5 802 5 659 5 512

88 691 82 691 76 821 71 091 65 505

40 41 42 43 44

45 46 47 48 49

19.330 19.120 18.906 18.669 18.407

8 8 9 9 10

334 326 317 309 300

3 916 3 586 3 265 2 951 2 647

153 157 161 173 190

5 362 5 210 5 052 4 891 4 718

60 068 54 782 49 651 44 679 39 875

45 46 47 48 49

50 51 52 53 54

18.135 17.853 17.561 17.259 16.948

11 12 14 15 17

289 278 266 252 236

2 353 2 069 1 797 1 538 1 294

205 223 242 265 290

4 528 4 323 4 100 3 858 3 594

35 251 30 826 26 615 22 635 18 909

50 51 52 53 54

55 56 57 58 59

16.625 16.292 15.949 15.594 15.229

19 21 23 25 27

219 200 179 156 131

1 066 856 667 499 355

317 343 368 390 410

3 304 2 987 2 644 2 276 1 886

15 461 12 315 9 500 7 040 4 958

55 56 57 58 59

60 61 62 63 64

14.855 14.472 14.081 13.682 13.277

29 20 19 19 18

104 75 56 36 18

238 148 82 36 9

429 284 271 254 238

1 476 1 047 763 492 238

3 277 2 016 1 111 484 119

60 61 62 63 64

144

PEN Ill health retirement functions Age x

s

M xia =

s

Rxia =

s x ( M xia - ½C xia )

S s M xia

16 17 18 19

9 641 11 572 13 533 15 480

20 21 22 23 24

z

C xia =

z

M xia =

4% z

Rxia =

Age x

S zC xia

S ( z M xia - ½ zC xia )

1 533 946 1 524 304 1 512 732 1 499 199

64 061 64 061 64 061 64 061

2 399 660 2 335 599 2 271 539 2 207 478

16 17 18 19

17 454 19 409 21 388 23 343 25 320

1 483 720 1 466 266 1 446 858 1 425 470 1 402 126

64 061 64 061 64 061 64 061 64 061

2 143 417 2 079 357 2 015 296 1 951 235 1 887 175

20 21 22 23 24

25 26 27 28 29

27 266 29 179 31 106 33 099 35 051

1 376 807 1 349 541 1 320 361 1 289 255 1 256 156

64 061 64 061 64 061 64 061 64 061

1 823 114 1 759 054 1 694 993 1 630 932 1 566 872

25 26 27 28 29

30 31 32 33 34

36 894 38 407 39 906 41 334 42 596

1 221 105 1 184 211 1 145 804 1 105 898 1 064 565

321 389 425 492 592

64 061 63 740 63 351 62 927 62 434

1 502 811 1 438 911 1 375 365 1 312 226 1 249 546

30 31 32 33 34

35 36 37 38 39

43 671 44 664 45 554 46 234 46 683

1 021 969 978 298 933 634 888 080 841 846

689 687 714 803 856

61 843 61 153 60 467 59 753 58 949

1 187 407 1 125 909 1 065 099 1 004 989 945 638

35 36 37 38 39

40 41 42 43 44

46 889 46 836 46 587 46 092 45 540

795 163 748 274 701 438 654 850 608 759

965 1 036 1 128 1 182 1 228

58 094 57 128 56 093 54 964 53 782

887 117 829 506 772 895 717 367 662 994

40 41 42 43 44

45 46 47 48 49

44 936 44 271 43 590 42 754 41 752

563 219 518 283 474 013 430 422 387 668

1 268 1 328 1 383 1 503 1 680

52 554 51 286 49 957 48 575 47 072

609 826 557 906 507 285 458 019 410 195

45 46 47 48 49

50 51 52 53 54

40 560 39 222 37 608 35 735 33 607

345 917 305 356 266 134 228 526 192 792

1 840 2 026 2 236 2 482 2 760

45 392 43 553 41 527 39 291 36 809

363 963 319 490 276 951 236 542 198 492

50 51 52 53 54

55 56 57 58 59

31 104 28 252 25 081 21 530 17 668

159 184 128 080 99 828 74 747 53 218

3 063 3 366 3 666 3 944 4 215

34 048 30 986 27 620 23 954 20 011

163 063 130 546 101 243 75 455 53 473

55 56 57 58 59

60 61 62 63 64

13 449 9 787 6 895 4 070 1 348

35 550 22 100 12 313 5 418 1 348

4 476 3 007 2 907 2 770 2 635

15 795 11 319 8 313 5 405 2 635

35 570 22 013 12 197 5 338 1 318

60 61 62 63 64

z x +½ C xia

145

PEN 4% Age x

146

Age retirement functions axr+½

C xr =

M xr =

Rxr =

C xra =

M xra =

Rxra =

r ( a65

v x +½ rx

S C xr

S ( M xr - ½C xr )

C xr axr+½

S C xra

S ( M xra - ½C xra )

at 65)

( v 65 r65

r ( v 65 r 65 a65

at 65)

at 65)

Age x

16 17 18 19

782 782 782 782

36 449 35 667 34 885 34 103

11 915 11 915 11 915 11 915

553 630 541 715 529 800 517 885

16 17 18 19

20 21 22 23 24

782 782 782 782 782

33 321 32 539 31 757 30 975 30 193

11 915 11 915 11 915 11 915 11 915

505 970 494 055 482 140 470 225 458 310

20 21 22 23 24

25 26 27 28 29

782 782 782 782 782

29 411 28 629 27 847 27 065 26 284

11 915 11 915 11 915 11 915 11 915

446 395 434 479 422 564 410 649 398 734

25 26 27 28 29

30 31 32 33 34

782 782 782 782 782

25 502 24 720 23 938 23 156 22 374

11 915 11 915 11 915 11 915 11 915

386 819 374 904 362 989 351 074 339 159

30 31 32 33 34

35 36 37 38 39

782 782 782 782 782

21 592 20 810 20 028 19 246 18 464

11 915 11 915 11 915 11 915 11 915

327 244 315 328 303 413 291 498 279 583

35 36 37 38 39

40 41 42 43 44

782 782 782 782 782

17 682 16 900 16 118 15 336 14 554

11 915 11 915 11 915 11 915 11 915

267 668 255 753 243 838 231 923 220 008

40 41 42 43 44

45 46 47 48 49

782 782 782 782 782

13 773 12 991 12 209 11 427 10 645

11 915 11 915 11 915 11 915 11 915

208 093 196 177 184 262 172 347 160 432

45 46 47 48 49

50 51 52 53 54

782 782 782 782 782

9 863 9 081 8 299 7 517 6 735

11 915 11 915 11 915 11 915 11 915

148 517 136 602 124 687 112 772 100 857

50 51 52 53 54

55 56 57 58 59

782 782 782 782 782

5 953 5 171 4 389 3 607 2 825

11 915 11 915 11 915 11 915 11 915

88 942 77 027 65 111 53 196 41 281

55 56 57 58 59

2 043 1 433 1 017 644 307

5 590 738 609 498 405

11 915 6 325 5 587 4 979 4 480

29 366 20 246 14 290 9 007 4 278

60 61 62 63 64

4 075

4 075

60 61 62 63 64

16.292 15.949 15.594 15.229 14.855

343 46 39 33 27

782 439 393 354 321

65

13.883

294

294

65

PEN Age retirement functions Age x

s

M xra =

s x ( M xra - ½C xra )

s

Rxra =

S s M xra

z

C xra =

z x +½ C xra

z

M xra =

4% z

Rxra =

S zC xra

S ( z M xra - ½ zC xra )

Age x

ra ( z65C65 at 65)

16 17 18 19

16 357 19 632 22 959 26 262

3 801 411 3 785 055 3 765 422 3 742 463

128 026 128 026 128 026 128 026

5 956 885 5 828 859 5 700 833 5 572 807

16 17 18 19

20 21 22 23 24

29 610 32 927 36 285 39 602 42 955

3 716 201 3 686 591 3 653 664 3 617 379 3 577 776

128 026 128 026 128 026 128 026 128 026

5 444 781 5 316 755 5 188 729 5 060 703 4 932 677

20 21 22 23 24

25 26 27 28 29

46 258 49 504 52 773 56 153 59 465

3 534 821 3 488 563 3 439 059 3 386 286 3 330 133

128 026 128 026 128 026 128 026 128 026

4 804 651 4 676 625 4 548 599 4 420 573 4 292 547

25 26 27 28 29

30 31 32 33 34

62 887 66 137 69 493 72 857 76 127

3 270 668 3 207 781 3 141 643 3 072 151 2 999 294

128 026 128 026 128 026 128 026 128 026

4 164 521 4 036 495 3 908 469 3 780 443 3 652 417

30 31 32 33 34

35 36 37 38 39

79 293 82 450 85 488 88 288 90 832

2 923 167 2 843 874 2 761 424 2 675 936 2 587 648

128 026 128 026 128 026 128 026 128 026

3 524 390 3 396 364 3 268 338 3 140 312 3 012 286

35 36 37 38 39

40 41 42 43 44

93 102 95 080 96 863 98 319 99 795

2 496 816 2 403 714 2 308 634 2 211 771 2 113 452

128 026 128 026 128 026 128 026 128 026

2 884 260 2 756 234 2 628 208 2 500 182 2 372 156

40 41 42 43 44

45 46 47 48 49

101 290 102 805 104 470 106 028 107 607

2 013 657 1 912 367 1 809 562 1 705 092 1 599 064

128 026 128 026 128 026 128 026 128 026

2 244 130 2 116 104 1 988 078 1 860 052 1 732 026

45 46 47 48 49

50 51 52 53 54

109 206 110 967 112 611 114 276 116 113

1 491 457 1 382 252 1 271 285 1 158 674 1 044 398

128 026 128 026 128 026 128 026 128 026

1 604 000 1 475 974 1 347 948 1 219 921 1 091 895

50 51 52 53 54

55 56 57 58 59

117 825 119 559 121 473 123 255 125 224

928 285 810 460 690 902 569 428 446 173

128 026 128 026 128 026 128 026 128 026

963 869 835 843 707 817 579 791 451 765

55 56 57 58 59

60 61 62 63 64

97 250 64 439 58 066 52 736 48 458

320 949 223 699 159 260 101 194 48 458

58 293 7 807 6 541 5 436 4 482

128 026 69 733 61 926 55 385 49 949

323 739 224 859 159 030 100 374 47 708

60 61 62 63 64

45 467

45 467

65

65

147

PEN 4%

Age x

148

Functions for return of contributions, accumulated with interest at 2% p.a., on death j

C xd =

j

M xd =

j

Rxd =

sj

( M(1 +-j½) C ) j

d x

j

d x

Rxd =

Age x

( M(1 +-j½) C ) j

d x

j

d x

v x +½ (1 + j ) x +½ d x

S jC xd

16 17 18 19

36 32 29 25

601 565 533 504

7 617 7 196 6 808 6 449

39 369 38 791 38 152 37 459

16 17 18 19

20 21 22 23 24

22 16 14 12 11

480 457 442 428 416

6 114 5 802 5 508 5 230 4 965

36 722 35 946 35 134 34 286 33 405

20 21 22 23 24

25 26 27 28 29

10 7 6 5 5

406 396 389 383 378

4 712 4 470 4 238 4 014 3 797

32 494 31 555 30 590 29 598 28 577

25 26 27 28 29

30 31 32 33 34

4 5 5 4 4

374 369 364 359 355

3 588 3 385 3 188 2 998 2 815

27 531 26 460 25 369 24 262 23 138

30 31 32 33 34

35 36 37 38 39

5 5 6 6 6

351 346 341 335 329

2 636 2 464 2 297 2 136 1 981

22 000 20 852 19 698 18 544 17 396

35 36 37 38 39

40 41 42 43 44

6 6 6 7 7

323 317 311 305 298

1 832 1 689 1 551 1 418 1 290

16 258 15 136 14 035 12 956 11 904

40 41 42 43 44

45 46 47 48 49

7 8 9 10 11

291 283 276 267 257

1 168 1 052 940 834 734

10 882 9 891 8 930 8 001 7 109

45 46 47 48 49

50 51 52 53 54

12 13 14 15 16

246 234 221 207 192

640 551 469 394 324

6 256 5 446 4 681 3 965 3 302

50 51 52 53 54

55 56 57 58 59

17 18 19 20 21

176 158 140 121 101

262 206 157 116 81

2 693 2 142 1 653 1 227 867

55 56 57 58 59

60 61 62 63 64

23 15 15 14 13

80 57 42 27 13

53 32 18 8 2

575 355 197 86 21

60 61 62 63 64

S

x +½

Ss x

x +½

PEN Functions for return of contributions, accumulated with interest at 2% p.a., on withdrawal Age x

j

C xw =

j

M xw =

j

Rxw =

sj

( M(1 +-j½) C ) j

w x

j

w x

Rxw =

4%

Age x

( M(1 +-j½) C ) j

w x

j

w x

v x +½ (1 + j ) x +½ wx

S jC xw

16 17 18 19

7 259 6 404 5 649 4 984

55 286 48 027 41 624 35 974

230 458 193 200 161 503 134 605

622 984 571 836 519 609 467 779

16 17 18 19

20 21 22 23 24

4 396 3 878 3 421 3 018 2 529

30 991 26 594 22 716 19 294 16 277

111 848 92 662 76 556 63 103 51 935

417 622 369 943 325 433 284 465 247 347

20 21 22 23 24

25 26 27 28 29

2 125 1 790 1 511 1 277 1 080

13 747 11 622 9 832 8 322 7 045

42 694 35 038 28 691 23 425 19 056

214 031 184 310 157 939 134 617 114 025

25 26 27 28 29

30 31 32 33 34

929 798 686 590 506

5 964 5 036 4 237 3 551 2 961

15 429 12 423 9 938 7 892 6 215

95 926 80 058 66 267 54 334 44 080

30 31 32 33 34

35 36 37 38 39

433 370 314 264 220

2 454 2 021 1 652 1 338 1 074

4 848 3 740 2 849 2 137 1 575

35 344 27 971 21 802 16 699 12 531

35 36 37 38 39

40 41 42 43 44

188 159 133 109 87

853 665 506 374 264

1 134 794 536 346 212

9 171 6 510 4 454 2 913 1 800

40 41 42 43 44

45 46 47 48 49

67 49 31 20 10

177 110 62 30 10

120 62 27 10 2

1 034 538 242 85 17

45 46 47 48 49

S

x +½

Ss x

x +½

149

150

SAMPLE TIME SERIES Contents RPI NAEI FTSE 100 Death Counts Bank Base Rates National Lottery

Page 152 153 154 155 156 157

This section shows the data values and related summary statistics for various observed time series. These could be used in discussions of time series modelling focusing on the following concepts: · · · · · · · · ·

stationarity differencing seasonality autocorrelation choice of model ARIMA models parameter estimation residual analysis forecasting

The left-hand side of each table shows an extract of the data values for the time series. The right-hand side of each table shows summary statistics based on the full range of values for the series over the stated period.

151

Time Series – RPI This dataset shows the monthly Retail Prices Index for the 10-year period from January 1992 to December 2001. These figures represent the prices of a representative “basket” of goods purchased in the UK. Month

t

Jan-92 Feb-92 Mar-92 Apr-92 May-92 Jun-92 Jul-92 Aug-92 Sep-92 Oct-92 ... Mar-01 Apr-01 May-01 Jun-01 Jul-01 Aug-01 Sep-01 Oct-01 Nov-01 Dec-01

0 1 2 3 4 5 6 7 8 9 ... 110 111 112 113 114 115 116 117 118 119

Data values Qt ÑQt 135.6 136.3 136.7 138.8 139.3 139.3 138.8 138.9 139.4 139.9 ... 172.2 173.1 174.2 174.4 173.3 174.0 174.6 174.3 173.6 173.4

0.7 0.4 2.1 0.5 0.0 -0.5 0.1 0.5 0.5 ... 0.2 0.9 1.1 0.2 -1.1 0.7 0.6 –0.3 –0.7 –0.2

Summary statistics Qt ÑQt

Ñ2 Qt

-0.3 1.7 -1.6 -0.5 -0.5 0.6 0.4 0.0 ... -0.7 0.7 0.2 -0.9 -1.3 1.8 -0.1 –0.9 –0.4 0.5

Ñ2 Qt

n mean s.d. min max

120 155.4 11.9 135.6 174.6

119 0.3 0.6 -1.3 2.1

118 0.0 0.8 -1.6 2.2

r1 r2 r3 r4 r5 r6 r12

0.977 0.954 0.930 0.908 0.887 0.866 0.729

0.083 -0.101 -0.285 -0.047 -0.012 0.240 0.637

-0.404 -0.006 -0.218 0.114 -0.128 0.315 0.671

f1 f2 f3 f4 f5 f6

0.977 -0.019 -0.028 0.037 0.003 -0.011

0.083 -0.109 -0.272 -0.017 -0.066 0.179

-0.404 -0.201 -0.376 -0.235 -0.391 -0.028

180 170 160

Qt

150 140 130 0

10

20

30

40

50

60

t

152

70

80

90

100

110 120

Time Series – NAEI This dataset shows the monthly UK National Average Earnings Index for the 10-year period from January 1992 to December 2001. These figures are NOT seasonally adjusted.

Month

t

Jan-92 Feb-92 Mar-92 Apr-92 May-92 Jun-92 Jul-92 Aug-92 Sep-92 Oct-92 ... Mar-01 Apr-01 May-01 Jun-01 Jul-01 Aug-01 Sep-01 Oct-01 Nov-01 Dec-01

0 1 2 3 4 5 6 7 8 9 ... 110 111 112 113 114 115 116 117 118 119

Data values Et ÑEt 88.5 89.8 91.1 89.5 90.1 91.1 91.6 90.9 90.7 91.5 ... 134.8 128.4 127.7 129.3 128.9 127.8 127.6 128.1 128.6 134.1

1.3 1.3 -1.6 0.6 1.0 0.5 -0.7 -0.2 0.8 ... 0.9 -6.4 -0.7 1.6 -0.4 -1.1 -0.2 0.5 0.5 5.5

Summary statistics Et ÑEt

Ñ2 Et

0.0 -2.9 2.2 0.4 -0.5 -1.2 0.5 1.0 ... -4.3 -7.3 5.7 2.3 -2.0 -0.7 0.9 0.7 0.0 5.0

Ñ2 Et

n mean s.d. min max

120 108.0 12.9 88.5 134.8

119 0.4 2.2 -6.8 7.3

118 0.0 3.4 -10.8 7.8

r1 r2 r3 r4 r5 r6 r12

0.959 0.932 0.912 0.883 0.859 0.835 0.706

-0.245 -0.197 0.252 -0.170 -0.065 -0.103 0.823

-0.511 -0.163 0.358 -0.212 0.056 -0.042 0.801

f1 f2 f3 f4 f5 f6

0.959 0.160 0.103 -0.084 0.015 -0.008

-0.245 -0.274 0.141 -0.131 -0.065 -0.276

-0.511 -0.573 -0.131 -0.153 0.050 -0.140

140 130 120

Et

110 100 90 80 0

10

20

30

40

50

60

70

80

90

100

110 120

t

153

Time Series – FTSE 100 This dataset shows the monthly FTSE 100 index for the 10-year period from January 1992 to December 2001. The index is based on the average closing prices of the top 100 UK shares on the last day of each month.

Month

t

Jan-92 Feb-92 Mar-92 Apr-92 May-92 Jun-92 Jul-92 Aug-92 Sep-92 Oct-92 ... Mar-01 Apr-01 May-01 Jun-01 Jul-01 Aug-01 Sep-01 Oct-01 Nov-01 Dec-01

0 1 2 3 4 5 6 7 8 9 ... 110 111 112 113 114 115 116 117 118 119

Data values St Ñ St 2 571.2 2 562.1 2 440.1 2 654.1 2 707.6 2 521.2 2 399.6 2312.6 2 553.0 2 658.3 ... 5 633.7 5 966.9 5 796.1 5 642.5 5 529.1 5 345.0 4 903.4 5 039.7 5 203.6 5 217.4

–9.1 –122.0 214.0 53.5 –186.4 –121.6 –87.0 240.4 105.3 ... –284.2 333.2 –170.8 –153.6 –113.4 –184.1 –441.6 136.3 163.9 13.8

Summary statistics St Ñ St

2

Ñ St

–112.9 336.0 –160.5 –239.9 64.8 34.6 327.4 –135.1 ... 95.4 617.4 –504.0 17.2 40.2 –70.7 –257.5 577.9 27.6 –150.1

Ñ 2 St

n mean s.d. min max

120 4 447.4 1 394.3 2 312.6 6 930.2

119 22.2 192.3 –661.7 426.7

118 0.2 277.4 –994.7 625.8

r1 r2 r3 r4 r5 r6 r12

0.982 0.963 0.946 0.932 0.914 0.895 0.768

-0.031 -0.085 -0.049 0.094 -0.028 -0.087 0.026

-0.474 -0.043 -0.052 0.127 -0.030 -0.020 -0.010

f1 f2 f3 f4 f5 f6

0.982 -0.001 0.016 0.070 -0.090 -0.059

-0.031 -0.087 -0.055 0.084 -0.031 -0.078

-0.474 -0.345 -0.356 -0.178 -0.113 -0.083

7 000 6 000 5 000

St

4 000 3 000 2 000 0

10

20

30

40

50

60

70

80

90

100 110 120

t Copyright © FTSE International Limited 2002. All rights reserved. “FTSE™” and “Footsie®”are trade marks of the London Stock Exchange Plc and The Financial Times Limited and are used by FTSE International Limited under licence.

154

Time Series – Death Counts This dataset shows the annual number of deaths recorded in England & Wales for the 39-year period from 1961 to 1999. Data values ÑQt

Year

t

Qt

1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 ... 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999

0 1 2 3 4 5 6 7 8 9 ... 29 30 31 32 33 34 35 36 37 38

551 752 557 636 572 868 534 737 549 379 563 624 542 516 576 754 579 378 575 194 ... 564 846 570 044 558 313 578 799 553 194 569 683 560 135 555 281 555 015 556 118

5 884 15 232 –38 131 14 642 14 245 –21 108 34 238 2 624 –4 184 ... –12 026 5 198 –11 731 20 486 –25 605 16 489 –9 548 –4 854 –266 1 103

Summary statistics Qt ÑQt

Ñ 2 Qt

9 348 –53 363 52 773 –397 –35 353 55 346 –31 614 –6 808 ... –17 490 17 224 –16 929 32 217 –46 091 42 094 –26 037 4 694 4 588 1 369

Ñ 2 Qt

n mean s.d. min max

39 570 980 14 695 534 737 598 516

38 115 15 067 –38 131 34 238

37 –129 26 798 –53 363 55 346

r1 r2 r3 r4 r5 r6

0.452 0.470 0.558 0.356 0.145 0.222

-0.541 -0.033 0.204 0.059 -0.278 0.181

-0.668 0.100 0.113 0.061 -0.249 0.143

f1 f2 f3 f4 f5 f6

0.452 0.334 0.375 -0.026 -0.353 -0.089

-0.541 -0.460 -0.127 0.264 0.000 -0.064

-0.668 -0.624 -0.578 -0.163 -0.082 -0.327

650 000 600 000

Qt

550 000 500 000 450 000 0

5

10

15

20

25

30

35

40

t

155

Time Series – Bank Base Rates This dataset shows the daily Bank Base Rate for the 10-year period from 1 January 1992 to 31 December 2001. These figures act as a benchmark for interest rates in the UK.

Date

Data values Kt ÑK t

t

Summary statistics Kt ÑK t

Ñ2 K t

01-Jan-92 02-Jan-92 03-Jan-92 04-Jan-92 05-Jan-92 06-Jan-92 07-Jan-92 08-Jan-92 09-Jan-92 10-Jan-92

0 1 2 3 4 5 6 7 8 9

10.50 10.50 10.50 10.50 10.50 10.50 10.50 10.50 10.50 10.50

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

22-Dec-01 23-Dec-01 24-Dec-01 25-Dec-01 26-Dec-01 27-Dec-01 28-Dec-01 29-Dec-01 30-Dec-01 31-Dec-01

3 643 3 644 3 645 3 646 3 647 3 648 3 649 3 650 3 651 3 652

4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00 4.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

n mean s.d. min max

3 653 6.39 1.33 4.00 12.00

3 652 0.00 0.07 -2.00 2.00

3 651 0.00 0.09 -2.00 2.00

r1 r2 r3 r4 r5 r6 r365

0.997 0.994 0.992 0.989 0.987 0.984 -0.064

-0.001 -0.253 -0.001 0.125 -0.001 -0.127 -0.004

-0.374 -0.252 0.063 0.126 0.000 -0.126 0.006

f1 f2 f3 f4 f5 f6

0.997 -0.002 0.103 -0.001 -0.043 -0.001

-0.001 -0.253 -0.001 0.066 -0.001 -0.086

-0.374 -0.456 -0.359 -0.215 -0.107 -0.174

13 12 11

Kt

10 9 8 7 6 5 4 3 0

1 000

2 000

t

156

Ñ2 K t

3 000

Time Series – National Lottery This dataset shows the bonus ball number drawn in the UK National Lottery* (Saturdays only) up to 29 December 2001. Date

t

Data values Lt ÑLt

19-Nov-94 26-Nov-94 03-Dec-94 10-Dec-94 17-Dec-94 24-Dec-94 31-Dec-94 07-Jan-95 14-Jan-95 21-Jan-95

0 1 2 3 4 5 6 7 8 9

10 37 31 28 30 6 16 46 48 4

27.0 -6.0 -3.0 2.0 -24.0 10.0 30.0 2.0 -44.0

-33.0 3.0 5.0 -26.0 34.0 20.0 -28.0 -46.0

27-Oct-01 03-Nov-01 10-Nov-01 17-Nov-01 24-Nov-01 01-Dec-01 08-Dec-01 15-Dec-01 22-Dec-01 29-Dec-01

360 361 362 363 364 365 366 367 368 369

33 17 39 1 28 11 9 16 8 41

19.0 -16.0 22.0 -38.0 27.0 -17.0 -2.0 7.0 -8.0 33.0

11.0 -35.0 38.0 -60.0 65.0 -44.0 15.0 9.0 -15.0 41.0

Summary statistics Lt ÑLt

2

Ñ Lt

Ñ2 Lt

n mean s.d. min max

370 25.87 14.40 1.00 49.00

369 0.08 19.32 -46.00 46.00

368 0.02 33.10 -90.00 87.00

r1 r2 r3 r4 r5 r6 r52

0.100 0.056 0.059 0.054 -0.005 0.003 0.010

-0.471 -0.027 0.005 0.030 -0.042 -0.038 0.048

-0.648 0.139 0.003 0.036 -0.031 -0.049 0.052

f1 f2 f3 f4 f5 f6

0.100 0.047 0.049 0.041 -0.019 -0.002

-0.471 -0.320 -0.233 -0.135 -0.139 -0.192

-0.648 -0.484 -0.400 -0.266 -0.165 -0.251

50 40

Lt

30 20 10 0 0

100

200

300

t

* Note. The UK National Lottery draws seven balls (without replacement) from 49 balls numbered from 1 to 49. The bonus ball is the seventh ball drawn.

157

158

STATISTICAL TABLES Contents

Page

Standard Normal probabilities Standard Normal percentage points t percentage points c 2 probabilities

160 162 163 164

c 2 percentage points F percentage points Poisson probabilities Binomial probabilities Critical values for the grouping of signs test Pseudorandom values from U(0,1) and from N(0,1)

168 170 175 186 189 190

159

Probabilities for the Standard Normal distribution The distribution function is denoted by F(x), and the probability density function is denoted by f(x).

F ( x) = ò

x -¥

f(t )dt =

F ( x)

x 1 -½ t 2 e dt ò 2p -¥

0

160

x

x

F(x)

x

F(x)

x

F(x)

x

F(x)

x

F(x)

0.00 0.01 0.02 0.03 0.04

0.50000 0.50399 0.50798 0.51197 0.51595

0.40 0.41 0.42 0.43 0.44

0.65542 0.65910 0.66276 0.66640 0.67003

0.80 0.81 0.82 0.83 0.84

0.78814 0.79103 0.79389 0.79673 0.79955

1.20 1.21 1.22 1.23 1.24

0.88493 0.88686 0.88877 0.89065 0.89251

1.60 1.61 1.62 1.63 1.64

0.94520 0.94630 0.94738 0.94845 0.94950

0.05 0.06 0.07 0.08 0.09

0.51994 0.52392 0.52790 0.53188 0.53586

0.45 0.46 0.47 0.48 0.49

0.67364 0.67724 0.68082 0.68439 0.68793

0.85 0.86 0.87 0.88 0.89

0.80234 0.80511 0.80785 0.81057 0.81327

1.25 1.26 1.27 1.28 1.29

0.89435 0.89617 0.89796 0.89973 0.90147

1.65 1.66 1.67 1.68 1.69

0.95053 0.95154 0.95254 0.95352 0.95449

0.10 0.11 0.12 0.13 0.14

0.53983 0.54380 0.54776 0.55172 0.55567

0.50 0.51 0.52 0.53 0.54

0.69146 0.69497 0.69847 0.70194 0.70540

0.90 0.91 0.92 0.93 0.94

0.81594 0.81859 0.82121 0.82381 0.82639

1.30 1.31 1.32 1.33 1.34

0.90320 0.90490 0.90658 0.90824 0.90988

1.70 1.71 1.72 1.73 1.74

0.95543 0.95637 0.95728 0.95818 0.95907

0.15 0.16 0.17 0.18 0.19

0.55962 0.56356 0.56749 0.57142 0.57535

0.55 0.56 0.57 0.58 0.59

0.70884 0.71226 0.71566 0.71904 0.72240

0.95 0.96 0.97 0.98 0.99

0.82894 0.83147 0.83398 0.83646 0.83891

1.35 1.36 1.37 1.38 1.39

0.91149 0.91309 0.91466 0.91621 0.91774

1.75 1.76 1.77 1.78 1.79

0.95994 0.96080 0.96164 0.96246 0.96327

0.20 0.21 0.22 0.23 0.24

0.57926 0.58317 0.58706 0.59095 0.59483

0.60 0.61 0.62 0.63 0.64

0.72575 0.72907 0.73237 0.73565 0.73891

1.00 1.01 1.02 1.03 1.04

0.84134 0.84375 0.84614 0.84849 0.85083

1.40 1.41 1.42 1.43 1.44

0.91924 0.92073 0.92220 0.92364 0.92507

1.80 1.81 1.82 1.83 1.84

0.96407 0.96485 0.96562 0.96638 0.96712

0.25 0.26 0.27 0.28 0.29

0.59871 0.60257 0.60642 0.61026 0.61409

0.65 0.66 0.67 0.68 0.69

0.74215 0.74537 0.74857 0.75175 0.75490

1.05 1.06 1.07 1.08 1.09

0.85314 0.85543 0.85769 0.85993 0.86214

1.45 1.46 1.47 1.48 1.49

0.92647 0.92785 0.92922 0.93056 0.93189

1.85 1.86 1.87 1.88 1.89

0.96784 0.96856 0.96926 0.96995 0.97062

0.30 0.31 0.32 0.33 0.34

0.61791 0.62172 0.62552 0.62930 0.63307

0.70 0.71 0.72 0.73 0.74

0.75804 0.76115 0.76424 0.76730 0.77035

1.10 1.11 1.12 1.13 1.14

0.86433 0.86650 0.86864 0.87076 0.87286

1.50 1.51 1.52 1.53 1.54

0.93319 0.93448 0.93574 0.93699 0.93822

1.90 1.91 1.92 1.93 1.94

0.97128 0.97193 0.97257 0.97320 0.97381

0.35 0.36 0.37 0.38 0.39

0.63683 0.64058 0.64431 0.64803 0.65173

0.75 0.76 0.77 0.78 0.79

0.77337 0.77637 0.77935 0.78230 0.78524

1.15 1.16 1.17 1.18 1.19

0.87493 0.87698 0.87900 0.88100 0.88298

1.55 1.56 1.57 1.58 1.59

0.93943 0.94062 0.94179 0.94295 0.94408

1.95 1.96 1.97 1.98 1.99

0.97441 0.97500 0.97558 0.97615 0.97670

0.40

0.65542

0.80

0.78814

1.20

0.88493

1.60

0.94520

2.00

0.97725

Probabilities for the Standard Normal distribution x

F(x)

x

F(x)

x

F(x)

x

F(x)

x

F(x)

x

F(x)

2.00 2.01 2.02 2.03 2.04

0.97725 0.97778 0.97831 0.97882 0.97932

2.40 2.41 2.42 2.43 2.44

0.99180 0.99202 0.99224 0.99245 0.99266

2.80 2.81 2.82 2.83 2.84

0.99744 0.99752 0.99760 0.99767 0.99774

3.20 3.21 3.22 3.23 3.24

0.99931 0.99934 0.99936 0.99938 0.99940

3.60 3.61 3.62 3.63 3.64

0.99984 0.99985 0.99985 0.99986 0.99986

4.00 4.01 4.02 4.03 4.04

0.99997 0.99997 0.99997 0.99997 0.99997

2.05 2.06 2.07 2.08 2.09

0.97982 0.98030 0.98077 0.98124 0.98169

2.45 2.46 2.47 2.48 2.49

0.99286 0.99305 0.99324 0.99343 0.99361

2.85 2.86 2.87 2.88 2.89

0.99781 0.99788 0.99795 0.99801 0.99807

3.25 3.26 3.27 3.28 3.29

0.99942 0.99944 0.99946 0.99948 0.99950

3.65 3.66 3.67 3.68 3.69

0.99987 0.99987 0.99988 0.99988 0.99989

4.05 4.06 4.07 4.08 4.09

0.99997 0.99998 0.99998 0.99998 0.99998

2.10 2.11 2.12 2.13 2.14

0.98214 0.98257 0.98300 0.98341 0.98382

2.50 2.51 2.52 2.53 2.54

0.99379 0.99396 0.99413 0.99430 0.99446

2.90 2.91 2.92 2.93 2.94

0.99813 0.99819 0.99825 0.99831 0.99836

3.30 3.31 3.32 3.33 3.34

0.99952 0.99953 0.99955 0.99957 0.99958

3.70 3.71 3.72 3.73 3.74

0.99989 0.99990 0.99990 0.99990 0.99991

4.10 4.11 4.12 4.13 4.14

0.99998 0.99998 0.99998 0.99998 0.99998

2.15 2.16 2.17 2.18 2.19

0.98422 0.98461 0.98500 0.98537 0.98574

2.55 2.56 2.57 2.58 2.59

0.99461 0.99477 0.99492 0.99506 0.99520

2.95 2.96 2.97 2.98 2.99

0.99841 0.99846 0.99851 0.99856 0.99861

3.35 3.36 3.37 3.38 3.39

0.99960 0.99961 0.99962 0.99964 0.99965

3.75 3.76 3.77 3.78 3.79

0.99991 0.99992 0.99992 0.99992 0.99992

4.15 4.16 4.17 4.18 4.19

0.99998 0.99998 0.99998 0.99999 0.99999

2.20 2.21 2.22 2.23 2.24

0.98610 0.98645 0.98679 0.98713 0.98745

2.60 2.61 2.62 2.63 2.64

0.99534 0.99547 0.99560 0.99573 0.99585

3.00 3.01 3.02 3.03 3.04

0.99865 0.99869 0.99874 0.99878 0.99882

3.40 3.41 3.42 3.43 3.44

0.99966 0.99968 0.99969 0.99970 0.99971

3.80 3.81 3.82 3.83 3.84

0.99993 0.99993 0.99993 0.99994 0.99994

4.20 4.21 4.22 4.23 4.24

0.99999 0.99999 0.99999 0.99999 0.99999

2.25 2.26 2.27 2.28 2.29

0.98778 0.98809 0.98840 0.98870 0.98899

2.65 2.66 2.67 2.68 2.69

0.99598 0.99609 0.99621 0.99632 0.99643

3.05 3.06 3.07 3.08 3.09

0.99886 0.99889 0.99893 0.99896 0.99900

3.45 3.46 3.47 3.48 3.49

0.99972 0.99973 0.99974 0.99975 0.99976

3.85 3.86 3.87 3.88 3.89

0.99994 0.99994 0.99995 0.99995 0.99995

4.25 4.26 4.27 4.28 4.29

0.99999 0.99999 0.99999 0.99999 0.99999

2.30 2.31 2.32 2.33 2.34

0.98928 0.98956 0.98983 0.99010 0.99036

2.70 2.71 2.72 2.73 2.74

0.99653 0.99664 0.99674 0.99683 0.99693

3.10 3.11 3.12 3.13 3.14

0.99903 0.99906 0.99910 0.99913 0.99916

3.50 3.51 3.52 3.53 3.54

0.99977 0.99978 0.99978 0.99979 0.99980

3.90 3.91 3.92 3.93 3.94

0.99995 0.99995 0.99996 0.99996 0.99996

4.30 4.31 4.32 4.33 4.34

0.99999 0.99999 0.99999 0.99999 0.99999

2.35 2.36 2.37 2.38 2.39

0.99061 0.99086 0.99111 0.99134 0.99158

2.75 2.76 2.77 2.78 2.79

0.99702 0.99711 0.99720 0.99728 0.99736

3.15 3.16 3.17 3.18 3.19

0.99918 0.99921 0.99924 0.99926 0.99929

3.55 3.56 3.57 3.58 3.59

0.99981 0.99981 0.99982 0.99983 0.99983

3.95 3.96 3.97 3.98 3.99

0.99996 0.99996 0.99996 0.99997 0.99997

4.35 4.36 4.37 4.38 4.39

0.99999 0.99999 0.99999 0.99999 0.99999

2.40

0.99180

2.80

0.99744

3.20

0.99931

3.60

0.99984

4.00

0.99997

4.40

0.99999

161

Percentage Points for the Standard Normal distribution The table gives percentage points x defined by the equation

P=

¥ -½ t 2 1 ò e dt 2p x

P 0

x

P

x

P

x

P

x

P

x

P

x

50% 45% 40% 35% 30%

0.0000 0.1257 0.2533 0.3853 0.5244

5.0% 4.8% 4.6% 4.4% 4.2%

1.6449 1.6646 1.6849 1.7060 1.7279

3.0% 2.9% 2.8% 2.7% 2.6%

1.8808 1.8957 1.9110 1.9268 1.9431

2.0% 1.9% 1.8% 1.7% 1.6%

2.0537 2.0749 2.0969 2.1201 2.1444

1.0% 0.9% 0.8% 0.7% 0.6%

2.3263 2.3656 2.4089 2.4573 2.5121

0.10% 0.09% 0.08% 0.07% 0.06%

3.0902 3.1214 3.1559 3.1947 3.2389

25% 20% 15% 10% 5%

0.6745 0.8416 1.0364 1.2816 1.6449

4.0% 3.8% 3.6% 3.4% 3.2%

1.7507 1.7744 1.7991 1.8250 1.8522

2.5% 2.4% 2.3% 2.2% 2.1%

1.9600 1.9774 1.9954 2.0141 2.0335

1.5% 1.4% 1.3% 1.2% 1.1%

2.1701 2.1973 2.2262 2.2571 2.2904

0.5% 0.4% 0.3% 0.2% 0.1%

2.5758 2.6521 2.7478 2.8782 3.0902

0.05% 0.01% 0.005% 0.001% 0.0005%

3.2905 3.7190 3.8906 4.2649 4.4172

162

P

x

Percentage Points for the t distribution This table gives percentage points x defined by the equation

P=

1 G (½ v + ½ ) ¥ dt ò vp G (½v ) x 1 + t 2 v ½(v +1)

(

)

P

0

x

The limiting distribution of t as v tends to infinity is the standard normal distribution. When v is large, interpolation in v should be harmonic. P= v

40%

30%

25%

20%

1 2 3 4

0.3249 0.2887 0.2767 0.2707

0.7265 0.6172 0.5844 0.5686

1.000 0.8165 0.7649 0.7407

1.376 1.061 0.9785 0.9410

1.963 1.386 1.250 1.190

3.078 1.886 1.638 1.533

6.314 2.920 2.353 2.132

12.71 4.303 3.182 2.776

31.82 6.965 4.541 3.747

63.66 9.925 5.841 4.604

318.3 22.33 10.21 7.173

636.6 31.60 12.92 8.610

5 6 7 8 9

0.2672 0.2648 0.2632 0.2619 0.2610

0.5594 0.5534 0.5491 0.5459 0.5435

0.7267 0.7176 0.7111 0.7064 0.7027

0.9195 0.9057 0.8960 0.8889 0.8834

1.156 1.134 1.119 1.108 1.100

1.476 1.440 1.415 1.397 1.383

2.015 1.943 1.895 1.860 1.833

2.571 2.447 2.365 2.306 2.262

3.365 3.143 2.998 2.896 2.821

4.032 3.707 3.499 3.355 3.250

5.894 5.208 4.785 4.501 4.297

6.869 5.959 5.408 5.041 4.781

10 11 12 13 14

0.2602 0.2596 0.2590 0.2586 0.2582

0.5415 0.5399 0.5386 0.5375 0.5366

0.6998 0.6974 0.6955 0.6938 0.6924

0.8791 0.8755 0.8726 0.8702 0.8681

1.093 1.088 1.083 1.079 1.076

1.372 1.363 1.356 1.350 1.345

1.812 1.796 1.782 1.771 1.761

2.228 2.201 2.179 2.160 2.145

2.764 2.718 2.681 2.650 2.624

3.169 3.106 3.055 3.012 2.977

4.144 4.025 3.930 3.852 3.787

4.587 4.437 4.318 4.221 4.140

15 16 17 18 19

0.2579 0.2576 0.2573 0.2571 0.2569

0.5357 0.5350 0.5344 0.5338 0.5333

0.6912 0.6901 0.6892 0.6884 0.6876

0.8662 0.8647 0.8633 0.8620 0.8610

1.074 1.071 1.069 1.067 1.066

1.341 1.337 1.333 1.330 1.328

1.753 1.746 1.740 1.734 1.729

2.131 2.120 2.110 2.101 2.093

2.602 2.583 2.567 2.552 2.539

2.947 2.921 2.898 2.878 2.861

3.733 3.686 3.646 3.610 3.579

4.073 4.015 3.965 3.922 3.883

20 21 22 23 24

0.2567 0.2566 0.2564 0.2563 0.2562

0.5329 0.5325 0.5321 0.5317 0.5314

0.6870 0.6864 0.6858 0.6853 0.6848

0.8600 0.8591 0.8583 0.8575 0.8569

1.064 1.063 1.061 1.060 1.059

1.325 1.323 1.321 1.319 1.318

1.725 1.721 1.717 1.714 1.711

2.086 2.080 2.074 2.069 2.064

2.528 2.518 2.508 2.500 2.492

2.845 2.831 2.819 2.807 2.797

3.552 3.527 3.505 3.485 3.467

3.850 3.819 3.792 3.768 3.745

25 26 27 28 29

0.2561 0.2560 0.2559 0.2558 0.2557

0.5312 0.5309 0.5306 0.5304 0.5302

0.6844 0.6840 0.6837 0.6834 0.6830

0.8562 0.8557 0.8551 0.8546 0.8542

1.058 1.058 1.057 1.056 1.055

1.316 1.315 1.314 1.313 1.311

1.708 1.706 1.703 1.701 1.699

2.060 2.056 2.052 2.048 2.045

2.485 2.479 2.473 2.467 2.462

2.787 2.779 2.771 2.763 2.756

3.450 3.435 3.421 3.408 3.396

3.725 3.707 3.689 3.674 3.660

30 32 34 36 38

0.2556 0.2555 0.2553 0.2552 0.2551

0.5300 0.5297 0.5294 0.5291 0.5288

0.6828 0.6822 0.6818 0.6814 0.6810

0.8538 0.8530 0.8523 0.8517 0.8512

1.055 1.054 1.052 1.052 1.051

1.310 1.309 1.307 1.306 1.304

1.697 1.694 1.691 1.688 1.686

2.042 2.037 2.032 2.028 2.024

2.457 2.449 2.441 2.434 2.429

2.750 2.738 2.728 2.719 2.712

3.385 3.365 3.348 3.333 3.319

3.646 3.622 3.601 3.582 3.566

40 50 60 120

0.2550 0.2547 0.2545 0.2539

0.5286 0.5278 0.5272 0.5258

0.6807 0.6794 0.6786 0.6765

0.8507 0.8489 0.8477 0.8446

1.050 1.047 1.045 1.041

1.303 1.299 1.296 1.289

1.684 1.676 1.671 1.658

2.021 2.009 2.000 1.980

2.423 2.403 2.390 2.358

2.704 2.678 2.660 2.617

3.307 3.261 3.232 3.160

3.551 3.496 3.460 3.373

0.2533 0.5244

0.6745

0.8416

1.036

1.282

1.645

1.960

2.326

2.576

3.090

3.291

¥

15%

10%

5%

2.5%

1%

0.5% 0.1% 0.05%

163

Probabilities for the c 2 distribution The function tabulated is:

Fv ( x) =

Fv ( x)

1

x ½ v -1 -½ t

ò t 2½ v G (½v ) 0

e

dt

x

0

(The above shape applies for v ³ 3 only. When v < 3 the mode is at the origin.) v= x

164

1

1 x

2 x

2 x

3 x

3 x

0.0 0.1 0.2 0.3 0.4

0.0000 0.2482 0.3453 0.4161 0.4729

4.0 4.1 4.2 4.3 4.4

0.9545 0.9571 0.9596 0.9619 0.9641

0.0 0.1 0.2 0.3 0.4

0.0000 0.0488 0.0952 0.1393 0.1813

4.0 4.1 4.2 4.3 4.4

0.8647 0.8713 0.8775 0.8835 0.8892

0.0 0.1 0.2 0.3 0.4

0.0000 0.0082 0.0224 0.0400 0.0598

4.0 4.2 4.4 4.6 4.8

0.7385 0.7593 0.7786 0.7965 0.8130

0.5 0.6 0.7 0.8 0.9

0.5205 0.5614 0.5972 0.6289 0.6572

4.5 4.6 4.7 4.8 4.9

0.9661 0.9680 0.9698 0.9715 0.9731

0.5 0.6 0.7 0.8 0.9

0.2212 0.2592 0.2953 0.3297 0.3624

4.5 4.6 4.7 4.8 4.9

0.8946 0.8997 0.9046 0.9093 0.9137

0.5 0.6 0.7 0.8 0.9

0.0811 0.1036 0.1268 0.1505 0.1746

5.0 5.2 5.4 5.6 5.8

0.8282 0.8423 0.8553 0.8672 0.8782

1.0 1.1 1.2 1.3 1.4

0.6827 0.7057 0.7267 0.7458 0.7633

5.0 5.1 5.2 5.3 5.4

0.9747 0.9761 0.9774 0.9787 0.9799

1.0 1.1 1.2 1.3 1.4

0.3935 0.4231 0.4512 0.4780 0.5034

5.0 5.1 5.2 5.3 5.4

0.9179 0.9219 0.9257 0.9293 0.9328

1.0 1.1 1.2 1.3 1.4

0.1987 0.2229 0.2470 0.2709 0.2945

6.0 6.2 6.4 6.6 6.8

0.8884 0.8977 0.9063 0.9142 0.9214

1.5 1.6 1.7 1.8 1.9

0.7793 0.7941 0.8077 0.8203 0.8319

5.5 5.6 5.7 5.8 5.9

0.9810 0.9820 0.9830 0.9840 0.9849

1.5 1.6 1.7 1.8 1.9

0.5276 0.5507 0.5726 0.5934 0.6133

5.5 5.6 5.7 5.8 5.9

0.9361 0.9392 0.9422 0.9450 0.9477

1.5 1.6 1.7 1.8 1.9

0.3177 0.3406 0.3631 0.3851 0.4066

7.0 7.2 7.4 7.6 7.8

0.9281 0.9342 0.9398 0.9450 0.9497

2.0 2.1 2.2 2.3 2.4

0.8427 0.8527 0.8620 0.8706 0.8787

6.0 6.1 6.2 6.3 6.4

0.9857 0.9865 0.9872 0.9879 0.9886

2.0 2.1 2.2 2.3 2.4

0.6321 0.6501 0.6671 0.6834 0.6988

6.0 6.2 6.4 6.6 6.8

0.9502 0.9550 0.9592 0.9631 0.9666

2.0 2.1 2.2 2.3 2.4

0.4276 0.4481 0.4681 0.4875 0.5064

8.0 8.2 8.4 8.6 8.8

0.9540 0.9579 0.9616 0.9649 0.9679

2.5 2.6 2.7 2.8 2.9

0.8862 0.8931 0.8997 0.9057 0.9114

6.5 6.6 6.7 6.8 6.9

0.9892 0.9898 0.9904 0.9909 0.9914

2.5 2.6 2.7 2.8 2.9

0.7135 0.7275 0.7408 0.7534 0.7654

7.0 7.2 7.4 7.6 7.8

0.9698 0.9727 0.9753 0.9776 0.9798

2.5 2.6 2.7 2.8 2.9

0.5247 0.5425 0.5598 0.5765 0.5927

9.0 9.2 9.4 9.6 9.8

0.9707 0.9733 0.9756 0.9777 0.9797

3.0 3.1 3.2 3.3 3.4

0.9167 0.9217 0.9264 0.9307 0.9348

7.0 7.1 7.2 7.3 7.4

0.9918 0.9923 0.9927 0.9931 0.9935

3.0 3.1 3.2 3.3 3.4

0.7769 0.7878 0.7981 0.8080 0.8173

8.0 8.2 8.4 8.6 8.8

0.9817 0.9834 0.9850 0.9864 0.9877

3.0 3.1 3.2 3.3 3.4

0.6084 0.6235 0.6382 0.6524 0.6660

10.0 10.2 10.4 10.6 10.8

0.9814 0.9831 0.9845 0.9859 0.9871

3.5 3.6 3.7 3.8 3.9

0.9386 0.9422 0.9456 0.9487 0.9517

7.5 7.6 7.7 7.8 7.9

0.9938 0.9942 0.9945 0.9948 0.9951

3.5 3.6 3.7 3.8 3.9

0.8262 0.8347 0.8428 0.8504 0.8577

9.0 9.2 9.4 9.6 9.8

0.9889 0.9899 0.9909 0.9918 0.9926

3.5 3.6 3.7 3.8 3.9

0.6792 0.6920 0.7043 0.7161 0.7275

11.0 11.2 11.4 11.6 11.8

0.9883 0.9893 0.9903 0.9911 0.9919

4.0

0.9545

8.0

0.9953

4.0

0.8647

10.0

0.9933

4.0

0.7385

12.0

0.9926

Probabilities for the c 2 distribution 4

5

6

7

8

9

10

11

12

13

14

0.5 1.0 1.5 2.0

0.0265 0.0902 0.1734 0.2642

0.0079 0.0374 0.0869 0.1509

0.0022 0.0144 0.0405 0.0803

0.0006 0.0052 0.0177 0.0402

0.0001 0.0018 0.0073 0.0190

0.0000 0.0006 0.0029 0.0085

0.0000 0.0002 0.0011 0.0037

0.0000 0.0001 0.0004 0.0015

0.0000 0.0000 0.0001 0.0006

0.0000 0.0000 0.0000 0.0002

0.0000 0.0000 0.0000 0.0001

2.5 3.0 3.5 4.0 4.5

0.3554 0.4422 0.5221 0.5940 0.6575

0.2235 0.3000 0.3766 0.4506 0.5201

0.1315 0.1912 0.2560 0.3233 0.3907

0.0729 0.1150 0.1648 0.2202 0.2793

0.0383 0.0656 0.1008 0.1429 0.1906

0.0191 0.0357 0.0589 0.0886 0.1245

0.0091 0.0186 0.0329 0.0527 0.0780

0.0042 0.0093 0.0177 0.0301 0.0471

0.0018 0.0045 0.0091 0.0166 0.0274

0.0008 0.0021 0.0046 0.0088 0.0154

0.0003 0.0009 0.0022 0.0045 0.0084

5.0 5.5 6.0 6.5 7.0

0.7127 0.7603 0.8009 0.8352 0.8641

0.5841 0.6421 0.6938 0.7394 0.7794

0.4562 0.5185 0.5768 0.6304 0.6792

0.3400 0.4008 0.4603 0.5173 0.5711

0.2424 0.2970 0.3528 0.4086 0.4634

0.1657 0.2113 0.2601 0.3110 0.3629

0.1088 0.1446 0.1847 0.2283 0.2746

0.0688 0.0954 0.1266 0.1620 0.2009

0.0420 0.0608 0.0839 0.1112 0.1424

0.0248 0.0375 0.0538 0.0739 0.0978

0.0142 0.0224 0.0335 0.0477 0.0653

7.5 8.0 8.5 9.0 9.5

0.8883 0.9084 0.9251 0.9389 0.9503

0.8140 0.8438 0.8693 0.8909 0.9093

0.7229 0.7619 0.7963 0.8264 0.8527

0.6213 0.6674 0.7094 0.7473 0.7813

0.5162 0.5665 0.6138 0.6577 0.6981

0.4148 0.4659 0.5154 0.5627 0.6075

0.3225 0.3712 0.4199 0.4679 0.5146

0.2427 0.2867 0.3321 0.3781 0.4242

0.1771 0.2149 0.2551 0.2971 0.3403

0.1254 0.1564 0.1904 0.2271 0.2658

0.0863 0.1107 0.1383 0.1689 0.2022

10.0 10.5 11.0 11.5 12.0

0.9596 0.9672 0.9734 0.9785 0.9826

0.9248 0.9378 0.9486 0.9577 0.9652

0.8753 0.8949 0.9116 0.9259 0.9380

0.8114 0.8380 0.8614 0.8818 0.8994

0.7350 0.7683 0.7983 0.8251 0.8488

0.6495 0.6885 0.7243 0.7570 0.7867

0.5595 0.6022 0.6425 0.6801 0.7149

0.4696 0.5140 0.5567 0.5976 0.6364

0.3840 0.4278 0.4711 0.5134 0.5543

0.3061 0.3474 0.3892 0.4310 0.4724

0.2378 0.2752 0.3140 0.3536 0.3937

12.5 13.0 13.5 14.0 14.5

0.9860 0.9887 0.9909 0.9927 0.9941

0.9715 0.9766 0.9809 0.9844 0.9873

0.9483 0.9570 0.9643 0.9704 0.9755

0.9147 0.9279 0.9392 0.9488 0.9570

0.8697 0.8882 0.9042 0.9182 0.9304

0.8134 0.8374 0.8587 0.8777 0.8944

0.7470 0.7763 0.8030 0.8270 0.8486

0.6727 0.7067 0.7381 0.7670 0.7935

0.5936 0.6310 0.6662 0.6993 .7301

0.5129 0.5522 0.5900 0.6262 0.6604

0.4338 0.4735 0.5124 0.5503 0.5868

15.0 15.5 16.0 16.5 17.0

0.9953 0.9962 0.9970 0.9976 0.9981

0.9896 0.9916 0.9932 0.9944 0.9955

0.9797 0.9833 0.9862 0.9887 0.9907

0.9640 0.9699 0.9749 0.9791 0.9826

0.9409 0.9499 0.9576 0.9642 0.9699

0.9091 0.9219 0.9331 0.9429 0.9513

0.8679 0.8851 0.9004 0.9138 0.9256

0.8175 0.8393 0.8589 0.8764 0.8921

0.7586 0.7848 0.8088 0.8306 0.8504

0.6926 0.7228 0.7509 0.7768 0.8007

0.6218 0.6551 0.6866 0.7162 0.7438

17.5 18.0 18.5 19.0 19.5

0.9985 0.9988 0.9990 0.9992 0.9994

0.9964 0.9971 0.9976 0.9981 0.9984

0.9924 0.9938 0.9949 0.9958 0.9966

0.9856 0.9880 0.9901 0.9918 0.9932

0.9747 0.9788 0.9822 0.9851 0.9876

0.9586 0.9648 0.9702 0.9748 0.9787

0.9360 0.9450 0.9529 0.9597 0.9656

0.9061 0.9184 0.9293 0.9389 0.9473

0.8683 0.8843 0.8987 0.9115 0.9228

0.8226 0.8425 0.8606 0.8769 0.8916

0.7695 0.7932 0.8151 0.8351 0.8533

20 21 22 23 24

0.9995 0.9997 0.9998 0.9999 0.9999

0.9988 0.9992 0.9995 0.9997 0.9998

0.9972 0.9982 0.9988 0.9992 0.9995

0.9944 0.9962 0.9975 0.9983 0.9989

0.9897 0.9929 0.9951 0.9966 0.9977

0.9821 0.9873 0.9911 0.9938 0.9957

0.9707 0.9789 0.9849 0.9893 0.9924

0.9547 0.9666 0.9756 0.9823 0.9873

0.9329 0.9496 0.9625 0.9723 0.9797

0.9048 0.9271 0.9446 0.9583 0.9689

0.8699 0.8984 0.9214 0.9397 0.9542

25 26 27 28 29

0.9999 1.0000 1.0000 1.0000 1.0000

0.9999 0.9999 0.9999 1.0000 1.0000

0.9997 0.9998 0.9999 0.9999 0.9999

0.9992 0.9995 0.9997 0.9998 0.9999

0.9984 0.9989 0.9993 0.9995 0.9997

0.9970 0.9980 0.9986 0.9990 0.9994

0.9947 0.9963 0.9974 0.9982 0.9988

0.9909 0.9935 0.9954 0.9968 0.9977

0.9852 0.9893 0.9923 0.9945 0.9961

0.9769 0.9830 0.9876 0.9910 0.9935

0.9654 0.9741 0.9807 0.9858 0.9895

v= x

30 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9991 0.9984 0.9972 0.9953 0.9924

165

Probabilities for the c 2 distribution 15

v=

16

17

18

19

20

21

22

23

24

25

x 3 4

0.0004 0.0023

0.0002 0.0011

0.0001 0.0005

0.0000 0.0002

0.0000 0.0001

0.0000 0.0000

0.0000 0.0000

0.0000 0.0000

0.0000 0.0000

0.0000 0.0000

0.0000 0.0000

5 6 7 8 9

0.0079 0.0203 0.0424 0.0762 0.1225

0.0042 0.0119 0.0267 0.0511 0.0866

0.0022 0.0068 0.0165 0.0335 0.0597

0.0011 0.0038 0.0099 0.0214 0.0403

0.0006 0.0021 0.0058 0.0133 0.0265

0.0003 0.0011 0.0033 0.0081 0.0171

0.0001 0.0006 0.0019 0.0049 0.0108

0.0001 0.0003 0.0010 0.0028 0.0067

0.0000 0.0001 0.0005 0.0016 0.0040

0.0000 0.0001 0.0003 0.0009 0.0024

0.0000 0.0000 0.0001 0.0005 0.0014

10 11 12 13 14

0.1803 0.2474 0.3210 0.3977 0.4745

0.1334 0.1905 0.2560 0.3272 0.4013

0.0964 0.1434 0.1999 0.2638 0.3329

0.0681 0.1056 0.1528 0.2084 0.2709

0.0471 0.0762 0.1144 0.1614 0.2163

0.0318 0.0538 0.0839 0.1226 0.1695

0.0211 0.0372 0.0604 0.0914 0.1304

0.0137 0.0253 0.0426 0.0668 0.0985

0.0087 0.0168 0.0295 0.0480 0.0731

0.0055 0.0110 0.0201 0.0339 0.0533

0.0033 0.0071 0.0134 0.0235 0.0383

15 16 17 18 19

0.5486 0.6179 0.6811 0.7373 0.7863

0.4754 0.5470 0.6144 0.6761 0.7313

0.4045 0.4762 0.5456 0.6112 0.6715

0.3380 0.4075 0.4769 0.5443 0.6082

0.2774 0.3427 0.4101 0.4776 0.5432

0.2236 0.2834 0.3470 0.4126 0.4782

0.1770 0.2303 0.2889 0.3510 0.4149

0.1378 0.1841 0.2366 0.2940 0.3547

0.1054 0.1447 0.1907 0.2425 0.2988

0.0792 0.1119 0.1513 0.1970 0.2480

0.0586 0.0852 0.1182 0.1576 0.2029

20 21 22 23 24

0.8281 0.8632 0.8922 0.9159 0.9349

0.7798 0.8215 0.8568 0.8863 0.9105

0.7258 0.7737 0.8153 0.8507 0.8806

0.6672 0.7206 0.7680 0.8094 0.8450

0.6054 0.6632 0.7157 0.7627 0.8038

0.5421 0.6029 0.6595 0.7112 0.7576

0.4787 0.5411 0.6005 0.6560 0.7069

0.4170 0.4793 0.5401 0.5983 0.6528

0.3581 0.4189 0.4797 0.5392 0.5962

0.3032 0.3613 0.4207 0.4802 0.5384

0.2532 0.3074 0.3643 0.4224 0.4806

25 26 27 28 29

0.9501 0.9620 0.9713 0.9784 0.9839

0.9302 0.9460 0.9585 0.9684 0.9761

0.9053 0.9255 0.9419 0.9551 0.9655

0.8751 0.9002 0.9210 0.9379 0.9516

0.8395 0.8698 0.8953 0.9166 0.9340

0.7986 0.8342 0.8647 0.8906 0.9122

0.7528 0.7936 0.8291 0.8598 0.8860

0.7029 0.7483 0.7888 0.8243 0.8551

0.6497 0.6991 0.7440 0.7842 0.8197

0.5942 0.6468 0.6955 0.7400 0.7799

0.5376 0.5924 0.6441 0.6921 0.7361

30 31 32 33 34

0.9881 0.9912 0.9936 0.9953 0.9966

0.9820 0.9865 0.9900 0.9926 0.9946

0.9737 0.9800 0.9850 0.9887 0.9916

0.9626 0.9712 0.9780 0.9833 0.9874

0.9482 0.9596 0.9687 0.9760 0.9816

0.9301 0.9448 0.9567 0.9663 0.9739

0.9080 0.9263 0.9414 0.9538 0.9638

0.8815 0.9039 0.9226 0.9381 0.9509

0.8506 0.8772 0.8999 0.9189 0.9348

0.8152 0.8462 0.8730 0.8959 0.9153

0.7757 0.8110 0.8420 0.8689 0.8921

35 36 37 38 39

0.9975 0.9982 0.9987 0.9991 0.9994

0.9960 0.9971 0.9979 0.9985 0.9989

0.9938 0.9954 0.9966 0.9975 0.9982

0.9905 0.9929 0.9948 0.9961 0.9972

0.9860 0.9894 0.9921 0.9941 0.9956

0.9799 0.9846 0.9883 0.9911 0.9933

0.9718 0.9781 0.9832 0.9871 0.9902

0.9613 0.9696 0.9763 0.9817 0.9859

0.9480 0.9587 0.9675 0.9745 0.9802

0.9316 0.9451 0.9562 0.9653 0.9727

0.9118 0.9284 0.9423 0.9537 0.9632

40 41 42 43 44

0.9995 0.9997 0.9998 0.9998 0.9999

0.9992 0.9994 0.9996 0.9997 0.9998

0.9987 0.9991 0.9993 0.9995 0.9997

0.9979 0.9985 0.9989 0.9992 0.9994

0.9967 0.9976 0.9982 0.9987 0.9991

0.9950 0.9963 0.9972 0.9980 0.9985

0.9926 0.9944 0.9958 0.9969 0.9977

0.9892 0.9918 0.9937 0.9953 0.9965

0.9846 0.9882 0.9909 0.9931 0.9947

0.9786 0.9833 0.9871 0.9901 0.9924

0.9708 0.9770 0.9820 0.9860 0.9892

45 46 47 48 49

0.9999 0.9999 1.0000 1.0000 1.0000

0.9999 0.9999 0.9999 1.0000 1.0000

0.9998 0.9998 0.9999 0.9999 0.9999

0.9996 0.9997 0.9998 0.9998 0.9999

0.9993 0.9995 0.9996 0.9997 0.9998

0.9989 0.9992 0.9994 0.9996 0.9997

0.9983 0.9987 0.9991 0.9993 0.9995

0.9973 0.9980 0.9985 0.9989 0.9992

0.9960 0.9970 0.9978 0.9983 0.9988

0.9942 0.9956 0.9967 0.9975 0.9981

0.9916 0.9936 0.9951 0.9963 0.9972

50

1.0000

1.0000

1.0000

0.9999

0.9999

0.9998

0.9996

0.9994

0.9991

0.9986

0.9979

166

167

Percentage Points for the c 2 distribution This table gives percentage points x defined by the equation

P=

1 2½ v G (½v )

ò

¥ ½ v -1 -½ t t e dt x

P

0

x

(The above shape applies only for v ³ 3. When v < 3, the mode is at the origin.) P= v

99.95%

99.9%

99.5%

99%

97.5%

95%

90%

80%

70%

60%

1 3.927E–07 1.571E–06 3.927E–05 1.571E–04 9.821E–04 0.003932 0.01579 0.06418 0.1485 0.2750 2 0.001000 0.002001 0.01003 0.02010 0.05064 0.1026 0.2107 0.4463 0.7133 1.022 3 0.01528 0.02430 0.07172 0.1148 0.2158 0.3518 0.5844 1.005 1.424 1.869 4 0.06392 0.09080 0.2070 0.2971 0.4844 0.7107 1.064 1.649 2.195 2.753 5 6 7 8 9

0.1581 0.2994 0.4849 0.7104 0.9718

0.2102 0.3810 0.5985 0.8571 1.152

0.4118 0.6757 0.9893 1.344 1.735

0.5543 0.8721 1.239 1.647 2.088

0.8312 1.237 1.690 2.180 2.700

1.145 1.635 2.167 2.733 3.325

1.610 2.204 2.833 3.490 4.168

2.343 3.070 3.822 4.594 5.380

3.000 3.828 4.671 5.527 6.393

3.656 4.570 5.493 6.423 7.357

10 11 12 13 14

1.265 1.587 1.935 2.305 2.697

1.479 1.834 2.214 2.617 3.041

2.156 2.603 3.074 3.565 4.075

2.558 3.053 3.571 4.107 4.660

3.247 3.816 4.404 5.009 5.629

3.940 4.575 5.226 5.892 6.571

4.865 5.578 6.304 7.041 7.790

6.179 6.989 7.807 8.634 9.467

7.267 8.148 9.034 9.926 10.82

8.295 9.237 10.18 11.13 12.08

15 16 17 18 19

3.107 3.536 3.980 4.439 4.913

3.483 3.942 4.416 4.905 5.407

4.601 5.142 5.697 6.265 6.844

5.229 5.812 6.408 7.015 7.633

6.262 6.908 7.564 8.231 8.907

7.261 7.962 8.672 9.390 10.12

8.547 9.312 10.09 10.86 11.65

10.31 11.15 12.00 12.86 13.72

11.72 12.62 13.53 14.44 15.35

13.03 13.98 14.94 15.89 16.85

20 21 22 23 24

5.398 5.895 6.404 6.924 7.453

5.921 6.447 6.983 7.529 8.085

7.434 8.034 8.643 9.260 9.886

8.260 8.897 9.542 10.20 10.86

9.591 10.28 10.98 11.69 12.40

10.85 11.59 12.34 13.09 13.85

12.44 13.24 14.04 14.85 15.66

14.58 15.44 16.31 17.19 18.06

16.27 17.18 18.10 19.02 19.94

17.81 18.77 19.73 20.69 21.65

25 26 27 28 29

7.991 8.537 9.093 9.656 10.23

8.649 9.222 9.803 10.39 10.99

10.52 11.16 11.81 12.46 13.12

11.52 12.20 12.88 13.56 14.26

13.12 13.84 14.57 15.31 16.05

14.61 15.38 16.15 16.93 17.71

16.47 17.29 18.11 18.94 19.77

18.94 19.82 20.70 21.59 22.48

20.87 21.79 22.72 23.65 24.58

22.62 23.58 24.54 25.51 26.48

30 32 34 36 38

10.80 11.98 13.18 14.40 15.64

11.59 12.81 14.06 15.32 16.61

13.79 15.13 16.50 17.89 19.29

14.95 16.36 17.79 19.23 20.69

16.79 18.29 19.81 21.34 22.88

18.49 20.07 21.66 23.27 24.88

20.60 22.27 23.95 25.64 27.34

23.36 25.15 26.94 28.73 30.54

25.51 27.37 29.24 31.12 32.99

27.44 29.38 31.31 33.25 35.19

40 50 60 70 80

16.91 23.46 30.34 37.47 44.79

17.92 24.67 31.74 39.04 46.52

20.71 27.99 35.53 43.28 51.17

22.16 29.71 37.48 45.44 53.54

24.43 32.36 40.48 48.76 57.15

26.51 34.76 43.19 51.74 60.39

29.05 37.69 46.46 55.33 64.28

32.34 41.45 50.64 59.90 69.21

34.87 44.31 53.81 63.35 72.92

37.13 46.86 56.62 66.40 76.19

90 100

52.28 59.89

54.16 61.92

59.20 67.33

61.75 70.06

65.65 74.22

69.13 77.93

73.29 82.36

78.56 87.95

82.51 92.13

85.99 95.81

168

Percentage Points for the c 2 distribution P= v 1 2 3 4

50%

40%

0.4549 0.7083 1.386 1.833 2.366 2.946 3.357 4.045

30%

20%

10%

5%

2.5%

1%

0.5%

0.1%

0.05%

1.074 2.408 3.665 4.878

1.642 3.219 4.642 5.989

2.706 4.605 6.251 7.779

3.841 5.991 7.815 9.488

5.024 7.378 9.348 11.14

6.635 9.210 11.34 13.28

7.879 10.60 12.84 14.86

10.83 13.82 16.27 18.47

12.12 15.20 17.73 20.00

5 6 7 8 9

4.351 5.348 6.346 7.344 8.343

5.132 6.211 7.283 8.351 9.414

6.064 7.231 8.383 9.524 10.66

7.289 8.558 9.803 11.03 12.24

9.236 10.64 12.02 13.36 14.68

11.07 12.59 14.07 15.51 16.92

12.83 14.45 16.01 17.53 19.02

15.09 16.81 18.48 20.09 21.67

16.75 18.55 20.28 21.95 23.59

20.51 22.46 24.32 26.12 27.88

22.11 24.10 26.02 27.87 29.67

10 11 12 13 14

9.342 10.34 11.34 12.34 13.34

10.47 11.53 12.58 13.64 14.69

11.78 12.90 14.01 15.12 16.22

13.44 14.63 15.81 16.98 18.15

15.99 17.28 18.55 19.81 21.06

18.31 19.68 21.03 22.36 23.68

20.48 21.92 23.34 24.74 26.12

23.21 24.73 26.22 27.69 29.14

25.19 26.76 28.30 29.82 31.32

29.59 31.26 32.91 34.53 36.12

31.42 33.14 34.82 36.48 38.11

15 16 17 18 19

14.34 15.34 16.34 17.34 18.34

15.73 16.78 17.82 18.87 19.91

17.32 18.42 19.51 20.60 21.69

19.31 20.47 21.61 22.76 23.90

22.31 23.54 24.77 25.99 27.20

25.00 26.30 27.59 28.87 30.14

27.49 28.85 30.19 31.53 32.85

30.58 32.00 33.41 34.81 36.19

32.80 34.27 35.72 37.16 38.58

37.70 39.25 40.79 42.31 43.82

39.72 41.31 42.88 44.43 45.97

20 21 22 23 24

19.34 20.34 21.34 22.34 23.34

20.95 21.99 23.03 24.07 25.11

22.77 23.86 24.94 26.02 27.10

25.04 26.17 27.30 28.43 29.55

28.41 29.62 30.81 32.01 33.20

31.41 32.67 33.92 35.17 36.42

34.17 35.48 36.78 38.08 39.36

37.57 38.93 40.29 41.64 42.98

40.00 41.40 42.80 44.18 45.56

45.31 46.80 48.27 49.73 51.18

47.50 49.01 50.51 52.00 53.48

25 26 27 28 29

24.34 25.34 26.34 27.34 28.34

26.14 27.18 28.21 29.25 30.28

28.17 29.25 30.32 31.39 32.46

30.68 31.79 32.91 34.03 35.14

34.38 35.56 36.74 37.92 39.09

37.65 38.89 40.11 41.34 42.56

40.65 41.92 43.19 44.46 45.72

44.31 45.64 46.96 48.28 49.59

46.93 48.29 49.65 50.99 52.34

52.62 54.05 55.48 56.89 58.30

54.95 56.41 57.86 59.30 60.73

30 32 34 36 38

29.34 31.34 33.34 35.34 37.34

31.32 33.38 35.44 37.50 39.56

33.53 35.66 37.80 39.92 42.05

36.25 38.47 40.68 42.88 45.08

40.26 42.58 44.90 47.21 49.51

43.77 46.19 48.60 51.00 53.38

46.98 49.48 51.97 54.44 56.90

50.89 53.49 56.06 58.62 61.16

53.67 56.33 58.96 61.58 64.18

59.70 62.49 65.25 67.98 70.70

62.16 64.99 67.80 70.59 73.35

40 50 60 70 80

39.34 49.33 59.33 69.33 79.33

41.62 51.89 62.13 72.36 82.57

44.16 54.72 65.23 75.69 86.12

47.27 58.16 68.97 79.71 90.41

51.81 63.17 74.40 85.53 96.58

55.76 67.50 79.08 90.53 101.9

59.34 71.42 83.30 95.02 106.6

63.69 76.15 88.38 100.4 112.3

66.77 79.49 91.95 104.2 116.3

73.40 86.66 99.61 112.3 124.8

76.10 89.56 102.7 115.6 128.3

90 100

89.33 99.33

92.76 102.9

96.52 106.9

101.1 111.7

107.6 118.5

113.1 124.3

118.1 129.6

124.1 135.8

128.3 140.2

137.2 149.4

140.8 153.2

169

Percentage Points for the F distribution The function tabulated is x defined for the specified percentage points P by the equation

æv +v ö Gç 1 2 ÷ ¥ è 2 ø t ½ v1 -1 ½v ½v P= v1 1 v2 2 ò dt ½(v1 +v2 ) x G (½v1 ) G (½v2 ) (v2 + v1t )

P

0

x

(The above shape applies only for v1 ³ 3. When v1 < 3, the mode is at the origin. )

170

10% Points for the F distribution v1 =

8

9

10

12

24

¥

58.91 9.349 5.266 3.979

59.44 9.367 5.252 3.955

59.86 9.381 5.240 3.936

60.19 9.392 5.230 3.920

60.71 9.408 5.216 3.896

62.00 9.450 5.176 3.831

63.33 9.491 5.134 3.761

3.405 3.055 2.827 2.668 2.551

3.368 3.014 2.785 2.624 2.505

3.339 2.983 2.752 2.589 2.469

3.316 2.958 2.725 2.561 2.440

3.297 2.937 2.703 2.538 2.416

3.268 2.905 2.668 2.502 2.379

3.191 2.818 2.575 2.404 2.277

3.105 2.722 2.471 2.293 2.159

2.522 2.451 2.394 2.347 2.307

2.461 2.389 2.331 2.283 2.243

2.414 2.342 2.283 2.234 2.193

2.377 2.304 2.245 2.195 2.154

2.347 2.274 2.214 2.164 2.122

2.323 2.248 2.188 2.138 2.095

2.284 2.209 2.147 2.097 2.054

2.178 2.100 2.036 1.983 1.938

2.055 1.972 1.904 1.846 1.797

2.361 2.333 2.308 2.286 2.266

2.273 2.244 2.218 2.196 2.176

2.208 2.178 2.152 2.130 2.109

2.158 2.128 2.102 2.079 2.058

2.119 2.088 2.061 2.038 2.017

2.086 2.055 2.028 2.005 1.984

2.059 2.028 2.001 1.977 1.956

2.017 1.985 1.958 1.933 1.912

1.899 1.866 1.836 1.810 1.787

1.755 1.718 1.686 1.657 1.631

2.380 2.365 2.351 2.339 2.327

2.249 2.233 2.219 2.207 2.195

2.158 2.142 2.128 2.115 2.103

2.091 2.075 2.060 2.047 2.035

2.040 2.023 2.008 1.995 1.983

1.999 1.982 1.967 1.953 1.941

1.965 1.948 1.933 1.919 1.906

1.937 1.920 1.904 1.890 1.877

1.892 1.875 1.859 1.845 1.832

1.767 1.748 1.731 1.716 1.702

1.607 1.586 1.567 1.549 1.533

2.528 2.519 2.511 2.503 2.495

2.317 2.307 2.299 2.291 2.283

2.184 2.174 2.165 2.157 2.149

2.092 2.082 2.073 2.064 2.057

2.024 2.014 2.005 1.996 1.988

1.971 1.961 1.952 1.943 1.935

1.929 1.919 1.909 1.900 1.892

1.895 1.884 1.874 1.865 1.857

1.866 1.855 1.845 1.836 1.827

1.820 1.809 1.799 1.790 1.781

1.689 1.677 1.666 1.656 1.647

1.518 1.504 1.491 1.478 1.467

2.881 2.869 2.859 2.850 2.842

2.489 2.477 2.466 2.456 2.448

2.276 2.263 2.252 2.243 2.234

2.142 2.129 2.118 2.108 2.099

2.049 2.036 2.024 2.014 2.005

1.980 1.967 1.955 1.945 1.935

1.927 1.913 1.901 1.891 1.881

1.884 1.870 1.858 1.847 1.838

1.849 1.835 1.822 1.811 1.802

1.819 1.805 1.793 1.781 1.772

1.773 1.758 1.745 1.734 1.724

1.638 1.622 1.608 1.595 1.584

1.456 1.437 1.420 1.404 1.390

2.835 2.791 2.748 2.706

2.440 2.393 2.347 2.303

2.226 2.177 2.130 2.084

2.091 2.041 1.992 1.945

1.997 1.946 1.896 1.847

1.927 1.875 1.824 1.774

1.873 1.819 1.767 1.717

1.829 1.775 1.722 1.670

1.793 1.738 1.684 1.632

1.763 1.707 1.652 1.599

1.715 1.657 1.601 1.546

1.574 1.511 1.447 1.383

1.377 1.292 1.193 1.000

1

2

3

4

5

6

1 2 3 4

39.86 8.526 5.538 4.545

49.50 9.000 5.462 4.325

53.59 9.162 5.391 4.191

55.83 9.243 5.343 4.107

57.24 9.293 5.309 4.051

58.20 9.326 5.285 4.010

5 6 7 8 9

4.060 3.776 3.589 3.458 3.360

3.780 3.463 3.257 3.113 3.006

3.619 3.289 3.074 2.924 2.813

3.520 3.181 2.961 2.806 2.693

3.453 3.108 2.883 2.726 2.611

10 11 12 13 14

3.285 3.225 3.177 3.136 3.102

2.924 2.860 2.807 2.763 2.726

2.728 2.660 2.606 2.560 2.522

2.605 2.536 2.480 2.434 2.395

15 16 17 18 19

3.073 3.048 3.026 3.007 2.990

2.695 2.668 2.645 2.624 2.606

2.490 2.462 2.437 2.416 2.397

20 21 22 23 24

2.975 2.961 2.949 2.937 2.927

2.589 2.575 2.561 2.549 2.538

25 26 27 28 29

2.918 2.909 2.901 2.894 2.887

30 32 34 36 38 40 60 120 ¥

7

v2

171

5% Points for the F distribution v1 =

8

9

10

12

24

¥

236.8 19.35 8.887 6.094

238.9 19.37 8.845 6.041

240.5 19.38 8.812 5.999

241.9 19.40 8.785 5.964

243.9 19.41 8.745 5.912

249.1 19.45 8.638 5.774

254.3 19.50 8.527 5.628

4.950 4.284 3.866 3.581 3.374

4.876 4.207 3.787 3.500 3.293

4.818 4.147 3.726 3.438 3.230

4.772 4.099 3.677 3.388 3.179

4.735 4.060 3.637 3.347 3.137

4.678 4.000 3.575 3.284 3.073

4.527 3.841 3.410 3.115 2.900

4.365 3.669 3.230 2.928 2.707

3.326 3.204 3.106 3.025 2.958

3.217 3.095 2.996 2.915 2.848

3.135 3.012 2.913 2.832 2.764

3.072 2.948 2.849 2.767 2.699

3.020 2.896 2.796 2.714 2.646

2.978 2.854 2.753 2.671 2.602

2.913 2.788 2.687 2.604 2.534

2.737 2.609 2.505 2.420 2.349

2.538 2.405 2.296 2.206 2.131

3.056 3.007 2.965 2.928 2.895

2.901 2.852 2.810 2.773 2.740

2.790 2.741 2.699 2.661 2.628

2.707 2.657 2.614 2.577 2.544

2.641 2.591 2.548 2.510 2.477

2.588 2.538 2.494 2.456 2.423

2.544 2.494 2.450 2.412 2.378

2.475 2.425 2.381 2.342 2.308

2.288 2.235 2.190 2.150 2.114

2.066 2.010 1.960 1.917 1.878

3.098 3.072 3.049 3.028 3.009

2.866 2.840 2.817 2.796 2.776

2.711 2.685 2.661 2.640 2.621

2.599 2.573 2.549 2.528 2.508

2.514 2.488 2.464 2.442 2.423

2.447 2.420 2.397 2.375 2.355

2.393 2.366 2.342 2.320 2.300

2.348 2.321 2.297 2.275 2.255

2.278 2.250 2.226 2.204 2.183

2.082 2.054 2.028 2.005 1.984

1.843 1.812 1.783 1.757 1.733

3.385 3.369 3.354 3.340 3.328

2.991 2.975 2.960 2.947 2.934

2.759 2.743 2.728 2.714 2.701

2.603 2.587 2.572 2.558 2.545

2.490 2.474 2.459 2.445 2.432

2.405 2.388 2.373 2.359 2.346

2.337 2.321 2.305 2.291 2.278

2.282 2.265 2.250 2.236 2.223

2.236 2.220 2.204 2.190 2.177

2.165 2.148 2.132 2.118 2.104

1.964 1.946 1.930 1.915 1.901

1.711 1.691 1.672 1.654 1.638

4.171 4.149 4.130 4.113 4.098

3.316 3.295 3.276 3.259 3.245

2.922 2.901 2.883 2.866 2.852

2.690 2.668 2.650 2.634 2.619

2.534 2.512 2.494 2.477 2.463

2.421 2.399 2.380 2.364 2.349

2.334 2.313 2.294 2.277 2.262

2.266 2.244 2.225 2.209 2.194

2.211 2.189 2.170 2.153 2.138

2.165 2.142 2.123 2.106 2.091

2.092 2.070 2.050 2.033 2.017

1.887 1.864 1.843 1.824 1.808

1.622 1.594 1.569 1.547 1.527

4.085 4.001 3.920 3.841

3.232 3.150 3.072 2.996

2.839 2.758 2.680 2.605

2.606 2.525 2.447 2.372

2.449 2.368 2.290 2.214

2.336 2.254 2.175 2.099

2.249 2.167 2.087 2.010

2.180 2.097 2.016 1.938

2.124 2.040 1.959 1.880

2.077 1.993 1.910 1.831

2.003 1.917 1.834 1.752

1.793 1.700 1.608 1.517

1.509 1.389 1.254 1.000

1

2

3

4

5

6

1 2 3 4

161.4 18.51 10.13 7.709

199.5 19.00 9.552 6.944

215.7 19.16 9.277 6.591

224.6 19.25 9.117 6.388

230.2 19.30 9.013 6.256

234.0 19.33 8.941 6.163

5 6 7 8 9

6.608 5.987 5.591 5.318 5.117

5.786 5.143 4.737 4.459 4.256

5.409 4.757 4.347 4.066 3.863

5.192 4.534 4.120 3.838 3.633

5.050 4.387 3.972 3.688 3.482

10 11 12 13 14

4.965 4.844 4.747 4.667 4.600

4.103 3.982 3.885 3.806 3.739

3.708 3.587 3.490 3.411 3.344

3.478 3.357 3.259 3.179 3.112

15 16 17 18 19

4.543 4.494 4.451 4.414 4.381

3.682 3.634 3.592 3.555 3.522

3.287 3.239 3.197 3.160 3.127

20 21 22 23 24

4.351 4.325 4.301 4.279 4.260

3.493 3.467 3.443 3.422 3.403

25 26 27 28 29

4.242 4.225 4.210 4.196 4.183

30 32 34 36 38 40 60 120 ¥

7

v2

172

2½% Points for the F distribution v1 =

1

2

3

4

5

6

7

8

9

10

12

24

¥

v2 1 2 3 4

647.8 38.51 17.44 12.22

799.5 39.00 16.04 10.65

864.2 39.17 15.44 9.979

899.6 39.25 15.10 9.605

921.8 39.30 14.88 9.364

937.1 39.33 14.73 9.197

948.2 39.36 14.62 9.074

956.6 39.37 14.54 8.980

963.3 39.39 14.47 8.905

968.6 39.40 14.42 8.844

976.7 39.41 14.34 8.751

997.3 39.46 14.12 8.511

1018 39.50 13.90 8.257

5 6 7 8 9

10.01 8.813 8.073 7.571 7.209

8.434 7.260 6.542 6.059 5.715

7.764 6.599 5.890 5.416 5.078

7.388 6.227 5.523 5.053 4.718

7.146 5.988 5.285 4.817 4.484

6.978 5.820 5.119 4.652 4.320

6.853 5.695 4.995 4.529 4.197

6.757 5.600 4.899 4.433 4.102

6.681 5.523 4.823 4.357 4.026

6.619 5.461 4.761 4.295 3.964

6.525 5.366 4.666 4.200 3.868

6.278 5.117 4.415 3.947 3.614

6.015 4.849 4.142 3.670 3.333

10 11 12 13 14

6.937 6.724 6.554 6.414 6.298

5.456 5.256 5.096 4.965 4.857

4.826 4.630 4.474 4.347 4.242

4.468 4.275 4.121 3.996 3.892

4.236 4.044 3.891 3.767 3.663

4.072 3.881 3.728 3.604 3.501

3.950 3.759 3.607 3.483 3.380

3.855 3.664 3.512 3.388 3.285

3.779 3.588 3.436 3.312 3.209

3.717 3.526 3.374 3.250 3.147

3.621 3.430 3.277 3.153 3.050

3.365 3.173 3.019 2.893 2.789

3.080 2.883 2.725 2.596 2.487

15 16 17 18 19

6.200 6.115 6.042 5.978 5.922

4.765 4.687 4.619 4.560 4.508

4.153 4.077 4.011 3.954 3.903

3.804 3.729 3.665 3.608 3.559

3.576 3.502 3.438 3.382 3.333

3.415 3.341 3.277 3.221 3.172

3.293 3.219 3.156 3.100 3.051

3.199 3.125 3.061 3.005 2.956

3.123 3.049 2.985 2.929 2.880

3.060 2.986 2.922 2.866 2.817

2.963 2.889 2.825 2.769 2.720

2.701 2.625 2.560 2.503 2.452

2.395 2.316 2.248 2.187 2.133

20 21 22 23 24

5.871 5.827 5.786 5.750 5.717

4.461 4.420 4.383 4.349 4.319

3.859 3.819 3.783 3.750 3.721

3.515 3.475 3.440 3.408 3.379

3.289 3.250 3.215 3.183 3.155

3.128 3.090 3.055 3.023 2.995

3.007 2.969 2.934 2.902 2.874

2.913 2.874 2.839 2.808 2.779

2.837 2.798 2.763 2.731 2.703

2.774 2.735 2.700 2.668 2.640

2.676 2.637 2.602 2.570 2.541

2.408 2.368 2.332 2.299 2.269

2.085 2.042 2.003 1.968 1.935

25 26 27 28 29

5.686 5.659 5.633 5.610 5.588

4.291 4.265 4.242 4.221 4.201

3.694 3.670 3.647 3.626 3.607

3.353 3.329 3.307 3.286 3.267

3.129 3.105 3.083 3.063 3.044

2.969 2.945 2.923 2.903 2.884

2.848 2.824 2.802 2.782 2.763

2.753 2.729 2.707 2.687 2.669

2.677 2.653 2.631 2.611 2.592

2.613 2.590 2.568 2.547 2.529

2.515 2.491 2.469 2.448 2.430

2.242 2.217 2.195 2.174 2.154

1.906 1.878 1.853 1.829 1.807

30 32 34 36 38

5.568 5.531 5.499 5.471 5.446

4.182 4.149 4.120 4.094 4.071

3.589 3.557 3.529 3.505 3.483

3.250 3.218 3.191 3.167 3.145

3.026 2.995 2.968 2.944 2.923

2.867 2.836 2.808 2.785 2.763

2.746 2.715 2.688 2.664 2.643

2.651 2.620 2.593 2.569 2.548

2.575 2.543 2.516 2.492 2.471

2.511 2.480 2.453 2.429 2.407

2.412 2.381 2.353 2.329 2.307

2.136 2.103 2.075 2.049 2.027

1.787 1.750 1.717 1.687 1.661

40 60 120 ¥

5.424 5.286 5.152 5.024

4.051 3.925 3.805 3.689

3.463 3.343 3.227 3.116

3.126 3.008 2.894 2.786

2.904 2.786 2.674 2.567

2.744 2.627 2.515 2.408

2.624 2.507 2.395 2.288

2.529 2.412 2.299 2.192

2.452 2.334 2.222 2.114

2.388 2.270 2.157 2.048

2.288 2.169 2.055 1.945

2.007 1.882 1.760 1.640

1.637 1.482 1.311 1.000

173

1% Points for the F distribution v1 =

8

9

10

12

24

¥

5928 99.36 27.67 14.98

5981 99.38 27.49 14.80

6022 99.39 27.35 14.66

6056 99.40 27.23 14.55

6107 99.42 27.05 14.37

6234 99.46 26.60 13.93

6366 99.50 26.13 13.46

10.67 8.466 7.191 6.371 5.802

10.46 8.260 6.993 6.178 5.613

10.29 8.102 6.840 6.029 5.467

10.16 7.976 6.719 5.911 5.351

10.05 7.874 6.620 5.814 5.257

9.888 7.718 6.469 5.667 5.111

9.466 7.313 6.074 5.279 4.729

9.021 6.880 5.650 4.859 4.311

5.636 5.316 5.064 4.862 4.695

5.386 5.069 4.821 4.620 4.456

5.200 4.886 4.640 4.441 4.278

5.057 4.744 4.499 4.302 4.140

4.942 4.632 4.388 4.191 4.030

4.849 4.539 4.296 4.100 3.939

4.706 4.397 4.155 3.960 3.800

4.327 4.021 3.780 3.587 3.427

3.909 3.603 3.361 3.165 3.004

4.893 4.773 4.669 4.579 4.500

4.556 4.437 4.336 4.248 4.171

4.318 4.202 4.101 4.015 3.939

4.142 4.026 3.927 3.841 3.765

4.004 3.890 3.791 3.705 3.631

3.895 3.780 3.682 3.597 3.523

3.805 3.691 3.593 3.508 3.434

3.666 3.553 3.455 3.371 3.297

3.294 3.181 3.083 2.999 2.925

2.869 2.753 2.653 2.566 2.489

4.938 4.874 4.817 4.765 4.718

4.431 4.369 4.313 4.264 4.218

4.103 4.042 3.988 3.939 3.895

3.871 3.812 3.758 3.710 3.667

3.699 3.640 3.587 3.539 3.496

3.564 3.506 3.453 3.406 3.363

3.457 3.398 3.346 3.299 3.256

3.368 3.310 3.258 3.211 3.168

3.231 3.173 3.121 3.074 3.032

2.859 2.801 2.749 2.702 2.659

2.421 2.360 2.306 2.256 2.211

5.568 5.526 5.488 5.453 5.420

4.675 4.637 4.601 4.568 4.538

4.177 4.140 4.106 4.074 4.045

3.855 3.818 3.785 3.754 3.725

3.627 3.591 3.558 3.528 3.499

3.457 3.421 3.388 3.358 3.330

3.324 3.288 3.256 3.226 3.198

3.217 3.182 3.149 3.120 3.092

3.129 3.094 3.062 3.032 3.005

2.993 2.958 2.926 2.896 2.868

2.620 2.585 2.552 2.522 2.495

2.170 2.132 2.097 2.064 2.034

7.562 7.499 7.444 7.396 7.353

5.390 5.336 5.289 5.248 5.211

4.510 4.459 4.416 4.377 4.343

4.018 3.969 3.927 3.890 3.858

3.699 3.652 3.611 3.574 3.542

3.473 3.427 3.386 3.351 3.319

3.305 3.258 3.218 3.183 3.152

3.173 3.127 3.087 3.052 3.021

3.067 3.021 2.981 2.946 2.915

2.979 2.934 2.894 2.859 2.828

2.843 2.798 2.758 2.723 2.692

2.469 2.423 2.383 2.347 2.316

2.006 1.956 1.911 1.872 1.837

7.314 7.077 6.851 6.635

5.178 4.977 4.787 4.605

4.313 4.126 3.949 3.782

3.828 3.649 3.480 3.319

3.514 3.339 3.174 3.017

3.291 3.119 2.956 2.802

3.124 2.953 2.792 2.639

2.993 2.823 2.663 2.511

2.888 2.718 2.559 2.407

2.801 2.632 2.472 2.321

2.665 2.496 2.336 2.185

2.288 2.115 1.950 1.791

1.805 1.601 1.381 1.000

1

2

3

4

5

6

1 2 3 4

4052 98.50 34.12 21.20

4999 99.00 30.82 18.00

5403 99.17 29.46 16.69

5625 99.25 28.71 15.98

5764 99.30 28.24 15.52

5859 99.33 27.91 15.21

5 6 7 8 9

16.26 13.75 12.25 11.26 10.56

13.27 10.92 9.547 8.649 8.022

12.06 9.780 8.451 7.591 6.992

11.39 9.148 7.847 7.006 6.422

10.97 8.746 7.460 6.632 6.057

10 11 12 13 14

10.04 9.646 9.330 9.074 8.862

7.559 7.206 6.927 6.701 6.515

6.552 6.217 5.953 5.739 5.564

5.994 5.668 5.412 5.205 5.035

15 16 17 18 19

8.683 8.531 8.400 8.285 8.185

6.359 6.226 6.112 6.013 5.926

5.417 5.292 5.185 5.092 5.010

20 21 22 23 24

8.096 8.017 7.945 7.881 7.823

5.849 5.780 5.719 5.664 5.614

25 26 27 28 29

7.770 7.721 7.677 7.636 7.598

30 32 34 36 38 40 60 120 ¥

7

v2

174

0.95123 0.90484 0.86071 0.81873

0.74082 0.67032 0.60653 0.54881 0.49659

0.44933 0.40657 0.36788 0.33287 0.30119

0.27253 0.24660 0.22313 0.20190 0.18268

0.16530 0.14957 0.13534 0.12246 0.11080

0.10026 0.09072 0.08208 0.07427 0.06721

0.30 0.40 0.50 0.60 0.70

0.80 0.90 1.00 1.10 1.20

1.30 1.40 1.50 1.60 1.70

1.80 1.90 2.00 2.10 2.20

2.30 2.40 2.50 2.60 2.70

0

0.05 0.10 0.15 0.20

m

x=

0.33085 0.30844 0.28730 0.26738 0.24866

0.46284 0.43375 0.40601 0.37961 0.35457

0.62682 0.59183 0.55783 0.52493 0.49325

0.80879 0.77248 0.73576 0.69903 0.66263

0.96306 0.93845 0.90980 0.87810 0.84420

0.99879 0.99532 0.98981 0.98248

1

3

0.59604 0.56971 0.54381 0.51843 0.49362

0.73062 0.70372 0.67668 0.64963 0.62271

0.85711 0.83350 0.80885 0.78336 0.75722

0.95258 0.93714 0.91970 0.90042 0.87949

0.99640 0.99207 0.98561 0.97688 0.96586

0.79935 0.77872 0.75758 0.73600 0.71409

0.89129 0.87470 0.85712 0.83864 0.81935

0.95690 0.94627 0.93436 0.92119 0.90681

0.99092 0.98654 0.98101 0.97426 0.96623

0.99973 0.99922 0.99825 0.99664 0.99425

0.99998 0.99985 0.99950 0.99998 0.99885 0.99994

2

0.91625 0.90413 0.89118 0.87742 0.86291

0.96359 0.95592 0.94735 0.93787 0.92750

0.98934 0.98575 0.98142 0.97632 0.97039

0.99859 0.99766 0.99634 0.99456 0.99225

0.99998 0.99994 0.99983 0.99961 0.99921

4

0.97002 0.96433 0.95798 0.95096 0.94327

0.98962 0.98678 0.98344 0.97955 0.97509

0.99777 0.99680 0.99554 0.99396 0.99200

0.99982 0.99966 0.99941 0.99903 0.99850

0.99999 0.99996 0.99991

5

7

0.99064 0.98841 0.98581 0.98283 0.97943

0.99743 0.99655 0.99547 0.99414 0.99254

0.99960 0.99938 0.99907 0.99866 0.99812

0.99998 0.99996 0.99992 0.99985 0.99975

0.99741 0.99666 0.99575 0.99467 0.99338

0.99944 0.99921 0.99890 0.99851 0.99802

0.99994 0.99989 0.99983 0.99974 0.99961

0.99999 0.99998 0.99996

8

0.99936 0.99914 0.99886 0.99851 0.99809

0.99989 0.99984 0.99976 0.99966 0.99953

0.99999 0.99998 0.99997 0.99995 0.99993

All 1.00000

0.99999

6

x

0.99986 0.99980 0.99972 0.99962 0.99950

0.99998 0.99997 0.99995 0.99993 0.99990

0.99999 0.99999

9

e -m mt The function tabulated is P ( X £ x ) = å . t =0 t !

0.99997 0.99996 0.99994 0.99991 0.99988

0.99999 0.99999 0.99999 0.99998

10

Probabilities for the Poisson distribution

0.99999 0.99999 0.99999 0.99998 0.99997

11

0.99999

12

13

2.30 2.40 2.50 2.60 2.70

1.80 1.90 2.00 2.10 2.20

1.30 1.40 1.50 1.60 1.70

0.80 0.90 1.00 1.10 1.20

0.30 0.40 0.50 0.60 0.70

0.05 0.10 0.15 0.20

=x m

175

0.03688 0.03337 0.03020 0.02732 0.02472

0.02237 0.02024 0.01832 0.01657 0.01500

0.01357 0.01228 0.01111 0.01005 0.00910

0.00823 0.00745 0.00674 0.00610 0.00552

3.30 3.40 3.50 3.60 3.70

3.80 3.90 4.00 4.10 4.20

4.30 4.40 4.50 4.60 4.70

4.80 4.90 5.00 5.10 5.20

176

0.06081 0.05502 0.04979 0.04505 0.04076

0

2.80 2.90 3.00 3.10 3.20

x= m

0.04773 0.04393 0.04043 0.03719 0.03420

0.07191 0.06630 0.06110 0.05629 0.05184

0.10738 0.09919 0.09158 0.08452 0.07798

0.15860 0.14684 0.13589 0.12569 0.11620

0.23108 0.21459 0.19915 0.18470 0.17120

1

0.14254 0.13333 0.12465 0.11648 0.10879

0.19735 0.18514 0.17358 0.16264 0.15230

0.26890 0.25313 0.23810 0.22381 0.21024

0.35943 0.33974 0.32085 0.30275 0.28543

0.46945 0.44596 0.42319 0.40116 0.37990

2

0.29423 0.27934 0.26503 0.25127 0.23807

0.37715 0.35945 0.34230 0.32571 0.30968

0.47348 0.45325 0.43347 0.41418 0.39540

0.58034 0.55836 0.53663 0.51522 0.49415

0.69194 0.66962 0.64723 0.62484 0.60252

3

0.47626 0.45821 0.44049 0.42313 0.40613

0.57044 0.55118 0.53210 0.51323 0.49461

0.66784 0.64837 0.62884 0.60931 0.58983

0.76259 0.74418 0.72544 0.70644 0.68722

0.84768 0.83178 0.81526 0.79819 0.78061

4

0.65101 0.63350 0.61596 0.59842 0.58091

0.73666 0.71991 0.70293 0.68576 0.66844

0.81556 0.80056 0.78513 0.76931 0.75314

0.88288 0.87054 0.85761 0.84412 0.83009

0.93489 0.92583 0.91608 0.90567 0.89459

5

0.79080 0.77665 0.76218 0.74742 0.73239

0.85579 0.84365 0.83105 0.81803 0.80461

0.90911 0.89948 0.88933 0.87865 0.86746

0.94903 0.94215 0.93471 0.92673 0.91819

0.97559 0.97128 0.96649 0.96120 0.95538

6

0.88667 0.87686 0.86663 0.85598 0.84492

0.92897 0.92142 0.91341 0.90495 0.89603

0.95989 0.95460 0.94887 0.94269 0.93606

0.98022 0.97693 0.97326 0.96921 0.96476

0.99187 0.99012 0.98810 0.98579 0.98317

7

x

0.94418 0.93824 0.93191 0.92518 0.91806

0.96830 0.96420 0.95974 0.95493 0.94974

0.98402 0.98147 0.97864 0.97551 0.97207

0.99309 0.99171 0.99013 0.98833 0.98630

0.99757 0.99694 0.99620 0.99532 0.99429

8

e -m mt P ( X £ x) = å . t =0 t !

0.97486 0.97166 0.96817 0.96440 0.96033

0.98709 0.98511 0.98291 0.98047 0.97779

0.99420 0.99311 0.99187 0.99046 0.98887

0.99781 0.99729 0.99669 0.99598 0.99515

0.99934 0.99914 0.99890 0.99860 0.99824

9

0.98958 0.98803 0.98630 0.98440 0.98230

0.99518 0.99431 0.99333 0.99222 0.99098

0.99807 0.99765 0.99716 0.99659 0.99593

0.99936 0.99919 0.99898 0.99873 0.99843

0.99984 0.99978 0.99971 0.99962 0.99950

10

Probabilities for the Poisson distribution

0.99601 0.99532 0.99455 0.99367 0.99269

0.99833 0.99799 0.99760 0.99714 0.99661

0.99941 0.99926 0.99908 0.99887 0.99863

0.99983 0.99978 0.99971 0.99963 0.99953

0.99996 0.99995 0.99993 0.99990 0.99987

11

0.99858 0.99830 0.99798 0.99761 0.99719

0.99947 0.99934 0.99919 0.99902 0.99882

0.99983 0.99978 0.99973 0.99966 0.99957

0.99996 0.99994 0.99992 0.99990 0.99987

0.99999 0.99999 0.99998 0.99998 0.99997

12

0.99953 0.99942 0.99930 0.99916 0.99899

0.99984 0.99980 0.99975 0.99969 0.99961

0.99996 0.99994 0.99992 0.99990 0.99987

0.99999 0.99999 0.99998 0.99997 0.99997

1.00000 1.00000 1.00000 1.00000 0.99999

13

4.80 4.90 5.00 5.10 5.20

4.30 4.40 4.50 4.60 4.70

3.80 3.90 4.00 4.10 4.20

3.30 3.40 3.50 3.60 3.70

2.80 2.90 3.00 3.10 3.20

=x m

0

0.00499 0.00452 0.00409 0.00370 0.00335

0.00303 0.00274 0.00248 0.00224 0.00203

0.00184 0.00166 0.00150 0.00136 0.00123

0.00111 0.00101 0.00091 0.00071 0.00055

0.00043 0.00034 0.00026 0.00020 0.00016

x= m

5.30 5.40 5.50 5.60 5.70

5.80 5.90 6.00 6.10 6.20

6.30 6.40 6.50 6.60 6.70

6.80 6.90 7.00 7.25 7.50

7.75 8.00 8.25 8.50 8.75

0.00377 0.00302 0.00242 0.00193 0.00154

0.00869 0.00796 0.00730 0.00586 0.00470

0.01341 0.01230 0.01128 0.01034 0.00948

0.02059 0.01890 0.01735 0.01592 0.01461

0.03145 0.02891 0.02656 0.02441 0.02242

1

0.01670 0.01375 0.01131 0.00928 0.00761

0.03444 0.03195 0.02964 0.02452 0.02026

0.04985 0.04632 0.04304 0.03997 0.03711

0.07151 0.06658 0.06197 0.05765 0.05362

0.10155 0.09476 0.08838 0.08239 0.07677

2

0.05012 0.04238 0.03576 0.03011 0.02530

0.09281 0.08713 0.08177 0.06963 0.05915

0.12637 0.11892 0.11185 0.10515 0.09881

0.16996 0.16035 0.15120 0.14250 0.13423

0.22541 0.21329 0.20170 0.19062 0.18005

3

0.11487 0.09963 0.08619 0.07436 0.06401

0.19203 0.18231 0.17299 0.15138 0.13206

0.24690 0.23507 0.22367 0.21270 0.20216

0.31272 0.29866 0.28506 0.27189 0.25918

0.38952 0.37331 0.35752 0.34215 0.32721

4

0.21522 0.19124 0.16939 0.14960 0.13174

0.32698 0.31366 0.30071 0.26992 0.24144

0.39877 0.38374 0.36904 0.35467 0.34065

0.47831 0.46187 0.44568 0.42975 0.41411

0.56347 0.54613 0.52892 0.51186 0.49498

5

0.34485 0.31337 0.28380 0.25618 0.23051

0.47992 0.46472 0.44971 0.41316 0.37815

0.55823 0.54233 0.52652 0.51084 0.49530

0.63839 0.62236 0.60630 0.59024 0.57421

0.71713 0.70167 0.68604 0.67026 0.65437

6

0.48837 0.45296 0.41864 0.38560 0.35398

0.62849 0.61361 0.59871 0.56152 0.52464

0.70175 0.68732 0.67276 0.65808 0.64332

0.77103 0.75763 0.74398 0.73010 0.71602

0.83348 0.82166 0.80949 0.79698 0.78415

7

x

0.62740 0.59255 0.55770 0.52311 0.48902

0.75477 0.74203 0.72909 0.69596 0.66197

0.81477 0.80331 0.79157 0.77956 0.76728

0.86719 0.85739 0.84724 0.83674 0.82591

0.91055 0.90265 0.89436 0.88568 0.87662

8

e -m mt P ( X £ x) = å . t =0 t !

0.74712 0.71662 0.68516 0.65297 0.62031

0.85018 0.84049 0.83050 0.80427 0.77641

0.89388 0.88580 0.87738 0.86864 0.85957

0.92916 0.92279 0.91608 0.90902 0.90162

0.95594 0.95125 0.94622 0.94087 0.93518

9

0.83990 0.81589 0.79032 0.76336 0.73519

0.91507 0.90843 0.90148 0.88279 0.86224

0.94372 0.93859 0.93316 0.92743 0.92140

0.96510 0.96137 0.95738 0.95311 0.94856

0.98000 0.97749 0.97475 0.97178 0.96856

10

Probabilities for the Poisson distribution

0.90527 0.88808 0.86919 0.84866 0.82657

0.95517 0.95104 0.94665 0.93454 0.92076

0.97227 0.96930 0.96612 0.96271 0.95906

0.98405 0.98207 0.97991 0.97756 0.97502

0.99159 0.99037 0.98901 0.98751 0.98586

11

0.94749 0.93620 0.92341 0.90908 0.89320

0.97790 0.97554 0.97300 0.96581 0.95733

0.98725 0.98568 0.98397 0.98211 0.98009

0.99321 0.99224 0.99117 0.98999 0.98868

0.99671 0.99617 0.99555 0.99486 0.99408

12

0.97266 0.96582 0.95782 0.94859 0.93805

0.98979 0.98855 0.98719 0.98324 0.97844

0.99451 0.99375 0.99290 0.99196 0.99093

0.99730 0.99686 0.99637 0.99582 0.99520

0.99880 0.99857 0.99831 0.99802 0.99768

13

7.75 8.00 8.25 8.50 8.75

6.80 6.90 7.00 7.25 7.50

6.30 6.40 6.50 6.60 6.70

5.80 5.90 6.00 6.10 6.20

5.30 5.40 5.50 5.60 5.70

=x m

177

0.00012 0.00010 0.00007 0.00006 0.00005

0.00004 0.00003 0.00002 0.00002 0.00001

0.00001 0.00001 0.00001 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000

0.00000 0.00000 0.00000 0.00000 0.00000

9.00 9.25 9.50 9.75 10.00

10.25 10.50 10.75 11.00 11.25

11.50 11.75 12.00 12.25 12.50

12.75 13.00 13.25 13.50 13.75

14.00 14.25 14.50 14.75 15.00

178

0

x= m

0.00001 0.00001 0.00001 0.00001 0.00000

0.00004 0.00003 0.00003 0.00002 0.00002

0.00013 0.00010 0.00008 0.00006 0.00005

0.00040 0.00032 0.00025 0.00020 0.00016

0.00123 0.00099 0.00079 0.00063 0.00050

1

0.00009 0.00008 0.00006 0.00005 0.00004

0.00028 0.00022 0.00018 0.00014 0.00012

0.00080 0.00065 0.00052 0.00042 0.00034

0.00226 0.00183 0.00149 0.00121 0.00098

0.00623 0.00510 0.00416 0.00340 0.00277

2

0.00047 0.00039 0.00032 0.00026 0.00021

0.00128 0.00105 0.00086 0.00071 0.00058

0.00336 0.00278 0.00229 0.00189 0.00155

0.00860 0.00715 0.00593 0.00492 0.00407

0.02123 0.01777 0.01486 0.01240 0.01034

3

0.00181 0.00150 0.00125 0.00103 0.00086

0.00447 0.00374 0.00312 0.00260 0.00217

0.01075 0.00904 0.00760 0.00638 0.00535

0.02486 0.02109 0.01786 0.01510 0.01275

0.05496 0.04709 0.04026 0.03435 0.02925

4

0.00553 0.00467 0.00394 0.00332 0.00279

0.01262 0.01073 0.00911 0.00773 0.00654

0.02773 0.02377 0.02034 0.01738 0.01482

0.05820 0.05038 0.04352 0.03752 0.03228

0.11569 0.10133 0.08853 0.07716 0.06709

5

0.01423 0.01220 0.01045 0.00894 0.00763

0.02994 0.02589 0.02234 0.01925 0.01656

0.06027 0.05260 0.04582 0.03984 0.03457

0.11515 0.10163 0.08949 0.07861 0.06891

0.20678 0.18495 0.16495 0.14671 0.13014

6

0.03162 0.02753 0.02394 0.02077 0.01800

0.06148 0.05403 0.04739 0.04148 0.03625

0.11373 0.10101 0.08950 0.07914 0.06983

0.19854 0.17851 0.16008 0.14319 0.12777

0.32390 0.29544 0.26866 0.24359 0.22022

7

x

0.06206 0.05484 0.04838 0.04260 0.03745

0.11175 0.09976 0.08886 0.07900 0.07008

0.19059 0.17210 0.15503 0.13932 0.12492

0.30538 0.27941 0.25494 0.23199 0.21054

0.45565 0.42320 0.39182 0.36166 0.33282

8

e -m mt P ( X £ x) = å . t =0 t !

0.10940 0.09808 0.08776 0.07837 0.06985

0.18297 0.16581 0.14993 0.13526 0.12177

0.28879 0.26492 0.24239 0.22123 0.20143

0.42707 0.39713 0.36825 0.34051 0.31401

0.58741 0.55451 0.52183 0.48957 0.45793

9

0.17568 0.15970 0.14486 0.13113 0.11846

0.27377 0.25168 0.23083 0.21123 0.19285

0.40173 0.37397 0.34723 0.32158 0.29707

0.55179 0.52074 0.49005 0.45989 0.43041

0.70599 0.67597 0.64533 0.61428 0.58304

10

Probabilities for the Poisson distribution

0.26004 0.23952 0.22013 0.20188 0.18475

0.37901 0.35316 0.32829 0.30445 0.28169

0.51980 0.49047 0.46160 0.43332 0.40576

0.66802 0.63873 0.60908 0.57927 0.54945

0.80301 0.77810 0.75199 0.72483 0.69678

11

0.35846 0.33430 0.31108 0.28884 0.26761

0.49083 0.46310 0.43590 0.40933 0.38349

0.63295 0.60453 0.57597 0.54740 0.51898

0.76729 0.74196 0.71572 0.68870 0.66105

0.87577 0.85683 0.83643 0.81464 0.79156

12

0.46445 0.43820 0.41253 0.38751 0.36322

0.60051 0.57304 0.54558 0.51825 0.49116

0.73304 0.70763 0.68154 0.65489 0.62784

0.84556 0.82535 0.80390 0.78129 0.75763

0.92615 0.91285 0.89814 0.88200 0.86446

13

14.00 14.25 14.50 14.75 15.00

12.75 13.00 13.25 13.50 13.75

11.50 11.75 12.00 12.25 12.50

10.25 10.50 10.75 11.00 11.25

9.00 9.25 9.50 9.75 10.00

=x m

23.00 23.50 24.00 24.50 25.00

20.50 21.00 21.50 22.00 22.50

18.00 18.50 19.00 19.50 20.00

15.50 16.00 16.50 17.00 17.50

x= m

0

2

0.00003 0.00002 0.00001 0.00001 0.00000

All 0.00000

1

0.00002 0.00001 0.00001

0.00014 0.00009 0.00006 0.00004 0.00003

3

0.00001 0.00001

0.00008 0.00006 0.00004 0.00003 0.00002

0.00059 0.00040 0.00027 0.00018 0.00012

4

0.00001

0.00005 0.00003 0.00002 0.00002 0.00001

0.00032 0.00022 0.00015 0.00011 0.00007

0.00197 0.00138 0.00097 0.00067 0.00047

5

0.00003 0.00002 0.00001 0.00001 0.00001

0.00018 0.00012 0.00009 0.00006 0.00004

0.00104 0.00074 0.00052 0.00036 0.00026

0.00554 0.00401 0.00288 0.00206 0.00147

6

0.00010 0.00007 0.00005 0.00003 0.00002

0.00056 0.00039 0.00028 0.00020 0.00014

0.00289 0.00210 0.00151 0.00109 0.00078

0.01346 0.01000 0.00739 0.00543 0.00397

7

x

0.00030 0.00021 0.00015 0.00011 0.00008

0.00152 0.00111 0.00080 0.00058 0.00041

0.00706 0.00524 0.00387 0.00285 0.00209

0.02879 0.02199 0.01669 0.01260 0.00945

8

e -m mt P ( X £ x) = å . t =0 t !

0.00081 0.00059 0.00043 0.00031 0.00022

0.00373 0.00277 0.00204 0.00150 0.00110

0.01538 0.01170 0.00886 0.00667 0.00500

0.05519 0.04330 0.03374 0.02612 0.02010

9

0.00198 0.00147 0.00108 0.00080 0.00059

0.00824 0.00625 0.00472 0.00355 0.00265

0.03037 0.02366 0.01832 0.01411 0.01081

0.09612 0.07740 0.06187 0.04912 0.03875

10

Probabilities for the Poisson distribution

0.00443 0.00335 0.00252 0.00189 0.00142

0.01666 0.01290 0.00995 0.00763 0.00583

0.05489 0.04376 0.03467 0.02731 0.02139

0.15378 0.12699 0.10407 0.08467 0.06840

11

0.00912 0.00704 0.00540 0.00413 0.00314

0.03103 0.02455 0.01931 0.01512 0.01177

0.09167 0.07475 0.06056 0.04875 0.03901

0.22827 0.19312 0.16210 0.13502 0.11165

12

0.01743 0.01370 0.01072 0.00834 0.00647

0.05371 0.04336 0.03481 0.02778 0.02206

0.14260 0.11886 0.09840 0.08092 0.06613

0.31708 0.27451 0.23574 0.20087 0.16987

13

23.00 23.50 24.00 24.50 25.00

20.50 21.00 21.50 22.00 22.50

18.00 18.50 19.00 19.50 20.00

15.50 16.00 16.50 17.00 17.50

=x m

179

180

0.99999 0.99999

0.99999 0.99999 0.99998 0.99997 0.99997

0.99996 0.99994 0.99993 0.99991 0.99988

0.99985 0.99982 0.99977 0.99972 0.99966

0.99959 0.99950 0.99940 0.99928 0.99915

0.99899 0.99881 0.99860 0.99836 0.99809

3.80 3.90 4.00 4.10 4.20

4.30 4.40 4.50 4.60 4.70

4.80 4.90 5.00 5.10 5.20

5.30 5.40 5.50 5.60 5.70

5.80 5.90 6.00 6.10 6.20

14

3.30 3.40 3.50 3.60 3.70

x= m

0.99964 0.99957 0.99949 0.99939 0.99928

0.99987 0.99984 0.99980 0.99976 0.99970

0.99996 0.99995 0.99993 0.99991 0.99989

0.99999 0.99998 0.99998 0.99997 0.99997

0.99999 0.99999

15

0.99988 0.99986 0.99983 0.99979 0.99975

0.99996 0.99995 0.99994 0.99992 0.99990

0.99999 0.99998 0.99998 0.99997 0.99997

0.99999 0.99999 0.99999

16

0.99996 0.99995 0.99994 0.99993 0.99991

0.99999 0.99999 0.99998 0.99998 0.99997

0.99999 0.99999 0.99999

17

19

0.99999 0.99999 0.99998 0.99998 0.99997

0.99999 0.99999 0.99999

All 1.00000

0.99999 0.99999 0.99999

18

x

20

e -m mt P ( X £ x) = å . t =0 t ! 21

22

23

Probabilities for the Poisson distribution

24

25

5.80 5.90 6.00 6.10 6.20

5.30 5.40 5.50 5.60 5.70

4.80 4.90 5.00 5.10 5.20

4.30 4.40 4.50 4.60 4.70

3.80 3.90 4.00 4.10 4.20

3.30 3.40 3.50 3.60 3.70

=x m

14

0.99778 0.99744 0.99704 0.99661 0.99611

0.99557 0.99496 0.99428 0.99227 0.98974

0.98659 0.98274 0.97810 0.97257 0.96608

0.95853 0.94986 0.94001 0.92891 0.91654

0.90287 0.88789 0.87160 0.85404 0.83524

0.81526 0.79416 0.77202 0.74895 0.72503

x= m

6.30 6.40 6.50 6.60 6.70

6.80 6.90 7.00 7.25 7.50

7.75 8.00 8.25 8.50 8.75

9.00 9.25 9.50 9.75 10.00

10.25 10.50 10.75 11.00 11.25

11.50 11.75 12.00 12.25 12.50

0.87829 0.86194 0.84442 0.82576 0.80603

0.94203 0.93167 0.92013 0.90740 0.89345

0.97796 0.97269 0.96653 0.95941 0.95126

0.99379 0.99177 0.98925 0.98617 0.98243

0.99818 0.99791 0.99759 0.99664 0.99539

0.99916 0.99901 0.99884 0.99865 0.99843

15

0.92360 0.91172 0.89871 0.88457 0.86931

0.96712 0.96039 0.95273 0.94408 0.93438

0.98889 0.98588 0.98227 0.97799 0.97296

0.99728 0.99628 0.99500 0.99339 0.99137

0.99930 0.99918 0.99904 0.99862 0.99804

0.99970 0.99964 0.99957 0.99949 0.99940

16

0.95425 0.94612 0.93703 0.92695 0.91584

0.98224 0.97814 0.97335 0.96781 0.96146

0.99468 0.99306 0.99107 0.98864 0.98572

0.99887 0.99841 0.99779 0.99700 0.99597

0.99974 0.99969 0.99964 0.99946 0.99921

0.99990 0.99987 0.99985 0.99982 0.99978

17

0.97383 0.96858 0.96258 0.95579 0.94815

0.99085 0.98849 0.98566 0.98231 0.97839

0.99757 0.99675 0.99572 0.99442 0.99281

0.99955 0.99935 0.99907 0.99870 0.99821

0.99991 0.99989 0.99987 0.99980 0.99970

0.99997 0.99996 0.99995 0.99994 0.99993

18

0.98568 0.98247 0.97872 0.97438 0.96941

0.99550 0.99421 0.99263 0.99071 0.98841

0.99894 0.99855 0.99804 0.99738 0.99655

0.99983 0.99975 0.99963 0.99947 0.99924

0.99997 0.99996 0.99996 0.99993 0.99989

0.99999 0.99999 0.99998 0.99998 0.99998

19

20

0.99250 0.99063 0.98840 0.98577 0.98269

0.99788 0.99721 0.99637 0.99533 0.99405

0.99956 0.99938 0.99914 0.99882 0.99841

0.99994 0.99991 0.99986 0.99979 0.99969

0.99999 0.99999 0.99999 0.99998 0.99996

0.99999 0.99999

x

21

0.99623 0.99519 0.99393 0.99242 0.99060

0.99905 0.99871 0.99829 0.99775 0.99707

0.99983 0.99975 0.99964 0.99949 0.99930

0.99998 0.99997 0.99995 0.99992 0.99988

0.99999 0.99999

e -m mt P ( X £ x) = å . t =0 t ! 23

0.99818 0.99763 0.99695 0.99612 0.99509

0.99959 0.99943 0.99922 0.99896 0.99861

0.99993 0.99990 0.99985 0.99979 0.99970

0.99915 0.99888 0.99853 0.99809 0.99754

0.99983 0.99976 0.99966 0.99954 0.99937

0.99998 0.99996 0.99994 0.99992 0.99988

0.99999 0.99999 0.99998

24

0.99962 0.99949 0.99931 0.99909 0.99881

0.99993 0.99990 0.99986 0.99980 0.99972

0.99999 0.99999 0.99998 0.99997 0.99995

0.99999

All 1.00000

0.99999 0.99999 0.99998 0.99997 0.99996

22

Probabilities for the Poisson distribution

0.99984 0.99977 0.99969 0.99958 0.99944

0.99997 0.99996 0.99994 0.99992 0.99988

0.99999 0.99999 0.99998

25

11.50 11.75 12.00 12.25 12.50

10.25 10.50 10.75 11.00 11.25

9.00 9.25 9.50 9.75 10.00

7.75 8.00 8.25 8.50 8.75

6.80 6.90 7.00 7.25 7.50

6.30 6.40 6.50 6.60 6.70

=x m

181

182

14

0.70039 0.67513 0.64938 0.62327 0.59691

0.57044 0.54396 0.51760 0.49146 0.46565

0.41541 0.36753 0.32254 0.28083 0.24264

0.20808 0.17714 0.14975 0.12573 0.10486

0.08690 0.07157 0.05860 0.04769 0.03860

0.03107 0.02488 0.01983 0.01572 0.01240

x= m

12.75 13.00 13.25 13.50 13.75

14.00 14.25 14.50 14.75 15.00

15.50 16.00 16.50 17.00 17.50

18.00 18.50 19.00 19.50 20.00

20.50 21.00 21.50 22.00 22.50

23.00 23.50 24.00 24.50 25.00

0.05200 0.04241 0.03440 0.02776 0.02229

0.13227 0.11107 0.09269 0.07689 0.06341

0.28665 0.24903 0.21479 0.18398 0.15651

0.51701 0.46674 0.41802 0.37145 0.32754

0.66936 0.64443 0.61916 0.59368 0.56809

0.78529 0.76361 0.74108 0.71779 0.69385

15

0.08208 0.06814 0.05626 0.04620 0.03775

0.19040 0.16292 0.13852 0.11704 0.09830

0.37505 0.33214 0.29203 0.25497 0.22107

0.61544 0.56596 0.51648 0.46774 0.42040

0.75592 0.73391 0.71121 0.68791 0.66412

0.85294 0.83549 0.81701 0.79755 0.77716

16

0.12277 0.10372 0.08713 0.07278 0.06048

0.26050 0.22696 0.19647 0.16900 0.14447

0.46865 0.42259 0.37836 0.33639 0.29703

0.70518 0.65934 0.61205 0.56402 0.51600

0.82720 0.80891 0.78972 0.76968 0.74886

0.90368 0.89046 0.87619 0.86088 0.84454

17

0.17477 0.15017 0.12828 0.10896 0.09204

0.34034 0.30168 0.26568 0.23250 0.20219

0.56224 0.51555 0.46948 0.42461 0.38142

0.78246 0.74235 0.69965 0.65496 0.60893

0.88264 0.86829 0.85296 0.83668 0.81947

0.93962 0.93017 0.91976 0.90838 0.89601

18

0.23771 0.20761 0.18026 0.15561 0.13357

0.42648 0.38426 0.34401 0.30603 0.27054

0.65092 0.60607 0.56061 0.51514 0.47026

0.84551 0.81225 0.77572 0.73632 0.69453

0.92350 0.91282 0.90122 0.88869 0.87522

0.96374 0.95733 0.95014 0.94213 0.93326

19

20

0.31010 0.27512 0.24264 0.21276 0.18549

0.51477 0.47097 0.42821 0.38691 0.34744

0.73072 0.68979 0.64717 0.60342 0.55909

0.89437 0.86817 0.83848 0.80548 0.76943

0.95209 0.94455 0.93622 0.92705 0.91703

0.97911 0.97499 0.97027 0.96491 0.95886

x

21

0.38938 0.35065 0.31393 0.27943 0.24730

0.60095 0.55769 0.51442 0.47164 0.42983

0.79912 0.76355 0.72550 0.68538 0.64370

0.93043 0.91077 0.88780 0.86147 0.83185

0.97116 0.96608 0.96038 0.95399 0.94689

0.98845 0.98592 0.98297 0.97955 0.97563

e -m mt P ( X £ x) = å . t =0 t !

0.47227 0.43134 0.39170 0.35367 0.31753

0.68127 0.64046 0.59866 0.55638 0.51409

0.85509 0.82558 0.79314 0.75804 0.72061

0.95584 0.94176 0.92478 0.90473 0.88150

0.98329 0.98003 0.97630 0.97206 0.96726

0.99386 0.99238 0.99062 0.98854 0.98611

22

0.55515 0.51378 0.47285 0.43276 0.39388

0.75285 0.71603 0.67741 0.63742 0.59652

0.89889 0.87547 0.84902 0.81963 0.78749

0.97296 0.96331 0.95131 0.93670 0.91928

0.99067 0.98867 0.98634 0.98364 0.98054

0.99686 0.99603 0.99502 0.99382 0.99238

23

Probabilities for the Poisson distribution

0.63458 0.59451 0.55400 0.51350 0.47340

0.81399 0.78216 0.74796 0.71172 0.67379

0.93174 0.91392 0.89325 0.86968 0.84323

0.98402 0.97768 0.96955 0.95935 0.94682

0.99498 0.99380 0.99241 0.99076 0.98884

0.99845 0.99801 0.99746 0.99678 0.99597

24

0.70766 0.67039 0.63191 0.59262 0.55292

0.86413 0.83770 0.80863 0.77710 0.74334

0.95539 0.94238 0.92687 0.90872 0.88782

0.99087 0.98688 0.98159 0.97476 0.96611

0.99739 0.99673 0.99592 0.99496 0.99382

0.99926 0.99903 0.99875 0.99838 0.99794

25

23.00 23.50 24.00 24.50 25.00

20.50 21.00 21.50 22.00 22.50

18.00 18.50 19.00 19.50 20.00

15.50 16.00 16.50 17.00 17.50

14.00 14.25 14.50 14.75 15.00

12.75 13.00 13.25 13.50 13.75

=x m

0.99999

0.99999 0.99999 0.99998 0.99997 0.99995

0.99993 0.99990 0.99987 0.99982 0.99975

0.99966 0.99955 0.99940 0.99922 0.99898

0.99869 0.99833 0.99789 0.99734 0.99669

0.99496 0.99254 0.98923 0.98483 0.97908

10.25 10.50 10.75 11.00 11.25

11.50 11.75 12.00 12.25 12.50

12.75 13.00 13.25 13.50 13.75

14.00 14.25 14.50 14.75 15.00

15.50 16.00 16.50 17.00 17.50

26

9.00 9.25 9.50 9.75 10.00

x= m

0.99731 0.99589 0.99390 0.99117 0.98750

0.99936 0.99918 0.99894 0.99865 0.99828

0.99985 0.99980 0.99972 0.99963 0.99951

0.99997 0.99996 0.99994 0.99992 0.99989

0.99999 0.99999 0.99999 0.99998

27

0.99861 0.99781 0.99665 0.99502 0.99275

0.99970 0.99961 0.99948 0.99933 0.99914

0.99994 0.99991 0.99988 0.99983 0.99978

0.99999 0.99998 0.99998 0.99997 0.99995

0.99999

28

30

0.99930 0.99887 0.99822 0.99727 0.99593

0.99986 0.99982 0.99976 0.99968 0.99958

0.99997 0.99996 0.99995 0.99993 0.99990

0.99966 0.99943 0.99908 0.99855 0.99778

0.99994 0.99992 0.99989 0.99985 0.99980

0.99999 0.99998 0.99998 0.99997 0.99996

0.99999 0.99999

31

0.99984 0.99972 0.99954 0.99925 0.99882

0.99997 0.99996 0.99995 0.99993 0.99991

0.99999 0.99999 0.99999 0.99998

All 1.00000

0.99999 0.99999 0.99999 0.99998

29

32

0.99993 0.99987 0.99978 0.99963 0.99939

0.99999 0.99998 0.99998 0.99997 0.99996

0.99999 0.99999

x

33

0.99997 0.99994 0.99989 0.99982 0.99970

0.99999 0.99999 0.99999 0.99998

e -m mt P ( X £ x) = å . t =0 t !

0.99999 0.99997 0.99995 0.99991 0.99985

0.99999 0.99999

34

0.99999 0.99999 0.99998 0.99996 0.99993

35

Probabilities for the Poisson distribution

0.99999 0.99998 0.99997

36

0.99999 0.99999

37

15.50 16.00 16.50 17.00 17.50

14.00 14.25 14.50 14.75 15.00

12.75 13.00 13.25 13.50 13.75

11.50 11.75 12.00 12.25 12.50

10.25 10.50 10.75 11.00 11.25

9.00 9.25 9.50 9.75 10.00

=x m

183

184

26

0.97177 0.96263 0.95144 0.93800 0.92211

0.90366 0.88257 0.85880 0.83242 0.80353

0.77230 0.73897 0.70382 0.66717 0.62939

x= m

18.00 18.50 19.00 19.50 20.00

20.50 21.00 21.50 22.00 22.50

23.00 23.50 24.00 24.50 25.00

0.82737 0.79866 0.76774 0.73483 0.70019

0.93368 0.91746 0.89875 0.87750 0.85368

0.98268 0.97650 0.96873 0.95914 0.94752

27

0.87260 0.84876 0.82253 0.79402 0.76340

0.95565 0.94363 0.92943 0.91291 0.89399

0.98970 0.98567 0.98046 0.97387 0.96567

28

0.90848 0.88936 0.86788 0.84403 0.81790

0.97119 0.96258 0.95217 0.93978 0.92526

0.99406 0.99152 0.98815 0.98377 0.97818

29

0.93598 0.92117 0.90415 0.88487 0.86331

0.98180 0.97585 0.96847 0.95949 0.94871

0.99667 0.99512 0.99302 0.99021 0.98653

30

0.95639 0.94527 0.93224 0.91715 0.89993

0.98882 0.98483 0.97978 0.97347 0.96573

0.99819 0.99728 0.99600 0.99425 0.99191

31

32

0.97106 0.96298 0.95330 0.94187 0.92854

0.99332 0.99073 0.98737 0.98308 0.97770

0.99904 0.99852 0.99777 0.99672 0.99527

x

33

0.98128 0.97559 0.96862 0.96021 0.95022

0.99611 0.99448 0.99232 0.98949 0.98586

0.99951 0.99922 0.99879 0.99818 0.99731

e -m mt P ( X £ x) = å . t =0 t !

0.98819 0.98430 0.97943 0.97343 0.96616

0.99780 0.99680 0.99545 0.99364 0.99126

0.99975 0.99960 0.99936 0.99902 0.99851

34

0.99274 0.99015 0.98684 0.98269 0.97754

0.99878 0.99819 0.99737 0.99624 0.99473

0.99988 0.99980 0.99967 0.99948 0.99920

35

Probabilities for the Poisson distribution

0.99564 0.99397 0.99179 0.98899 0.98545

0.99934 0.99900 0.99852 0.99784 0.99690

0.99994 0.99990 0.99984 0.99973 0.99958

36

0.99745 0.99640 0.99499 0.99316 0.99079

0.99966 0.99946 0.99919 0.99879 0.99822

0.99997 0.99995 0.99992 0.99987 0.99978

37

23.00 23.50 24.00 24.50 25.00

20.50 21.00 21.50 22.00 22.50

18.00 18.50 19.00 19.50 20.00

=x m

0.99999

0.99999 0.99998 0.99996 0.99993 0.99989

0.99982 0.99972 0.99956 0.99933 0.99900

0.99854 0.99790 0.99702 0.99585 0.99430

18.00 18.50 19.00 19.50 20.00

20.50 21.00 21.50 22.00 22.50

23.00 23.50 24.00 24.50 25.00

38

15.50 16.00 16.50 17.00 17.50

x= m

0.99918 0.99880 0.99827 0.99754 0.99656

0.99991 0.99986 0.99977 0.99964 0.99945

0.99999 0.99999 0.99998 0.99997 0.99995

39

0.99955 0.99933 0.99901 0.99857 0.99796

0.99996 0.99993 0.99988 0.99981 0.99971

0.99999 0.99999 0.99997

40

0.99976 0.99963 0.99945 0.99919 0.99882

0.99998 0.99996 0.99994 0.99990 0.99985

0.99999 0.99999

41

43

0.99988 0.99981 0.99970 0.99955 0.99933

0.99999 0.99998 0.99997 0.99995 0.99992 0.99994 0.99990 0.99984 0.99976 0.99963

0.99999 0.99999 0.99998 0.99996

44

0.99997 0.99995 0.99992 0.99987 0.99980

0.99999 0.99999 0.99998

All 1.00000

0.99999

42

x

45

0.99998 0.99997 0.99996 0.99993 0.99989

0.99999 0.99999

e -m mt P ( X £ x) = å . t =0 t !

0.99999 0.99999 0.99998 0.99997 0.99994

46

0.99999 0.99999 0.99998 0.99997

47

Probabilities for the Poisson distribution

0.99999 0.99999 0.99999

48

0.99999

49

23.00 23.50 24.00 24.50 25.00

20.50 21.00 21.50 22.00 22.50

18.00 18.50 19.00 19.50 20.00

15.50 16.00 16.50 17.00 17.50

=x m

185

186

0.9801 0.9999

0 1

0 1 2

0 1 2 3

0 1 2 3 4

0 1 2 3 4 5

0 1 2 3 4 5 6

2 2

3 3 3

4 4 4 4

5 5 5 5 5

6 6 6 6 6 6

7 7 7 7 7 7 7

0.9321 0.9980 1.0000 1.0000 1.0000 1.0000 1.0000

0.9415 0.9985 1.0000 1.0000 1.0000 1.0000

0.9510 0.9990 1.0000 1.0000 1.0000

0.9606 0.9994 1.0000 1.0000

0.9703 0.9997 1.0000

0.01

p= n x

0.6983 0.9556 0.9962 0.9998 1.0000 1.0000 1.0000

0.7351 0.9672 0.9978 0.9999 1.0000 1.0000

0.7738 0.9774 0.9988 1.0000 1.0000

0.8145 0.9860 0.9995 1.0000

0.8574 0.9928 0.9999

0.9025 0.9975

0.05

0.4783 0.8503 0.9743 0.9973 0.9998 1.0000 1.0000

0.5314 0.8857 0.9842 0.9987 0.9999 1.0000

0.5905 0.9185 0.9914 0.9995 1.0000

0.6561 0.9477 0.9963 0.9999

0.7290 0.9720 0.9990

0.8100 0.9900

0.1

0.2097 0.5767 0.8520 0.9667 0.9953 0.9996 1.0000

0.2621 0.6554 0.9011 0.9830 0.9984 0.9999

0.3277 0.7373 0.9421 0.9933 0.9997

0.4096 0.8192 0.9728 0.9984

0.5120 0.8960 0.9920

0.6400 0.9600

0.2

0.1335 0.4449 0.7564 0.9294 0.9871 0.9987 0.9999

0.1780 0.5339 0.8306 0.9624 0.9954 0.9998

0.2373 0.6328 0.8965 0.9844 0.9990

0.3164 0.7383 0.9492 0.9961

0.4219 0.8438 0.9844

0.5625 0.9375

0.25

0.0824 0.3294 0.6471 0.8740 0.9712 0.9962 0.9998

0.1176 0.4202 0.7443 0.9295 0.9891 0.9993

0.1681 0.5282 0.8369 0.9692 0.9976

0.2401 0.6517 0.9163 0.9919

0.3430 0.7840 0.9730

0.4900 0.9100

0.3

0.0280 0.1586 0.4199 0.7102 0.9037 0.9812 0.9984

0.0467 0.2333 0.5443 0.8208 0.9590 0.9959

0.0778 0.3370 0.6826 0.9130 0.9898

0.1296 0.4752 0.8208 0.9744

0.2160 0.6480 0.9360

0.3600 0.8400

0.4

0.0078 0.0625 0.2266 0.5000 0.7734 0.9375 0.9922

0.0156 0.1094 0.3438 0.6563 0.8906 0.9844

0.0313 0.1875 0.5000 0.8125 0.9688

0.0625 0.3125 0.6875 0.9375

0.1250 0.5000 0.8750

0.2500 0.7500

0.5

0.0016 0.0188 0.0963 0.2898 0.5801 0.8414 0.9720

0.0041 0.0410 0.1792 0.4557 0.7667 0.9533

0.0102 0.0870 0.3174 0.6630 0.9222

0.0256 0.1792 0.5248 0.8704

0.0640 0.3520 0.7840

0.1600 0.6400

0.6

0.0002 0.0038 0.0288 0.1260 0.3529 0.6706 0.9176

0.0007 0.0109 0.0705 0.2557 0.5798 0.8824

0.0024 0.0308 0.1631 0.4718 0.8319

0.0081 0.0837 0.3483 0.7599

0.0270 0.2160 0.6570

0.0900 0.5100

0.7

æ nö The function tabulated is P( X £ x) = å ç ÷ pt q n -t . t =0 è t ø

x

0.0001 0.0013 0.0129 0.0706 0.2436 0.5551 0.8665

0.0002 0.0046 0.0376 0.1694 0.4661 0.8220

0.0010 0.0156 0.1035 0.3672 0.7627

0.0039 0.0508 0.2617 0.6836

0.0156 0.1563 0.5781

0.0625 0.4375

0.75

Probabilities for the Binomial distribution

0.0000 0.0004 0.0047 0.0333 0.1480 0.4233 0.7903

0.0001 0.0016 0.0170 0.0989 0.3446 0.7379

0.0003 0.0067 0.0579 0.2627 0.6723

0.0016 0.0272 0.1808 0.5904

0.0080 0.1040 0.4880

0.0400 0.3600

0.8

0.0000 0.0000 0.0002 0.0027 0.0257 0.1497 0.5217

0.0000 0.0001 0.0013 0.0159 0.1143 0.4686

0.0000 0.0005 0.0086 0.0815 0.4095

0.0001 0.0037 0.0523 0.3439

0.0010 0.0280 0.2710

0.0100 0.1900

0.9

0.0000 0.0000 0.0000 0.0002 0.0038 0.0444 0.3017

0.0000 0.0000 0.0001 0.0022 0.0328 0.2649

0.0000 0.0000 0.0012 0.0226 0.2262

0.0000 0.0005 0.0140 0.1855

0.0001 0.0073 0.1426

0.0025 0.0975

0.95

0.0000 0.0000 0.0000 0.0000 0.0000 0.0020 0.0679

0.0000 0.0000 0.0000 0.0000 0.0015 0.0585

0.0000 0.0000 0.0000 0.0010 0.0490

0.0000 0.0000 0.0006 0.0394

0.0000 0.0003 0.0297

0.0001 0.0199

0.99

0.9227 0.9973 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000

0 1 2 3 4 5 6 7

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8 9

8 8 8 8 8 8 8 8

9 9 9 9 9 9 9 9 9

10 10 10 10 10 10 10 10 10 10

0.9044 0.9957 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9135 0.9966 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.01

p= n x

0.5987 0.9139 0.9885 0.9990 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000

0.6302 0.9288 0.9916 0.9994 1.0000 1.0000 1.0000 1.0000 1.0000

0.6634 0.9428 0.9942 0.9996 1.0000 1.0000 1.0000 1.0000

0.05

0.3487 0.7361 0.9298 0.9872 0.9984 0.9999 1.0000 1.0000 1.0000 1.0000

0.3874 0.7748 0.9470 0.9917 0.9991 0.9999 1.0000 1.0000 1.0000

0.4305 0.8131 0.9619 0.9950 0.9996 1.0000 1.0000 1.0000

0.1

0.1074 0.3758 0.6778 0.8791 0.9672 0.9936 0.9991 0.9999 1.0000 1.0000

0.1342 0.4362 0.7382 0.9144 0.9804 0.9969 0.9997 1.0000 1.0000

0.1678 0.5033 0.7969 0.9437 0.9896 0.9988 0.9999 1.0000

0.2

0.0563 0.2440 0.5256 0.7759 0.9219 0.9803 0.9965 0.9996 1.0000 1.0000

0.0751 0.3003 0.6007 0.8343 0.9511 0.9900 0.9987 0.9999 1.0000

0.1001 0.3671 0.6785 0.8862 0.9727 0.9958 0.9996 1.0000

0.25

0.0282 0.1493 0.3828 0.6496 0.8497 0.9527 0.9894 0.9984 0.9999 1.0000

0.0404 0.1960 0.4628 0.7297 0.9012 0.9747 0.9957 0.9996 1.0000

0.0576 0.2553 0.5518 0.8059 0.9420 0.9887 0.9987 0.9999

0.3

0.0060 0.0464 0.1673 0.3823 0.6331 0.8338 0.9452 0.9877 0.9983 0.9999

0.0101 0.0705 0.2318 0.4826 0.7334 0.9006 0.9750 0.9962 0.9997

0.0168 0.1064 0.3154 0.5941 0.8263 0.9502 0.9915 0.9993

0.4

0.0010 0.0107 0.0547 0.1719 0.3770 0.6230 0.8281 0.9453 0.9893 0.9990

0.0020 0.0195 0.0898 0.2539 0.5000 0.7461 0.9102 0.9805 0.9980

0.0039 0.0352 0.1445 0.3633 0.6367 0.8555 0.9648 0.9961

0.5

0.0001 0.0017 0.0123 0.0548 0.1662 0.3669 0.6177 0.8327 0.9536 0.9940

0.0003 0.0038 0.0250 0.0994 0.2666 0.5174 0.7682 0.9295 0.9899

0.0007 0.0085 0.0498 0.1737 0.4059 0.6846 0.8936 0.9832

0.6

0.0000 0.0001 0.0016 0.0106 0.0473 0.1503 0.3504 0.6172 0.8507 0.9718

0.0000 0.0004 0.0043 0.0253 0.0988 0.2703 0.5372 0.8040 0.9596

0.0001 0.0013 0.0113 0.0580 0.1941 0.4482 0.7447 0.9424

0.7

æ nö The function tabulated is P( X £ x) = å ç ÷ pt q n -t . t =0 è t ø

x

0.0000 0.0000 0.0004 0.0035 0.0197 0.0781 0.2241 0.4744 0.7560 0.9437

0.0000 0.0001 0.0013 0.0100 0.0489 0.1657 0.3993 0.6997 0.9249

0.0000 0.0004 0.0042 0.0273 0.1138 0.3215 0.6329 0.8999

0.75

Probabilities for the Binomial distribution

0.0000 0.0000 0.0001 0.0009 0.0064 0.0328 0.1209 0.3222 0.6242 0.8926

0.0000 0.0000 0.0003 0.0031 0.0196 0.0856 0.2618 0.5638 0.8658

0.0000 0.0001 0.0012 0.0104 0.0563 0.2031 0.4967 0.8322

0.8

0.0000 0.0000 0.0000 0.0000 0.0001 0.0016 0.0128 0.0702 0.2639 0.6513

0.0000 0.0000 0.0000 0.0001 0.0009 0.0083 0.0530 0.2252 0.6126

0.0000 0.0000 0.0000 0.0004 0.0050 0.0381 0.1869 0.5695

0.9

0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0010 0.0115 0.0861 0.4013

0.0000 0.0000 0.0000 0.0000 0.0000 0.0006 0.0084 0.0712 0.3698

0.0000 0.0000 0.0000 0.0000 0.0004 0.0058 0.0572 0.3366

0.95

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0043 0.0956

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0034 0.0865

0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0027 0.0773

0.99

187

188

0.8864 0.9938 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0 1 2 3 4 5 6 7 8 9 10 11

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

12 12 12 12 12 12 12 12 12 12 12 12

20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20

0.8179 0.9831 0.9990 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.01

p= n x

0.3585 0.7358 0.9245 0.9841 0.9974 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.5404 0.8816 0.9804 0.9978 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.05

0.1216 0.3917 0.6769 0.8670 0.9568 0.9887 0.9976 0.9996 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.2824 0.6590 0.8891 0.9744 0.9957 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000

0.1

0.0115 0.0692 0.2061 0.4114 0.6296 0.8042 0.9133 0.9679 0.9900 0.9974 0.9994 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0687 0.2749 0.5583 0.7946 0.9274 0.9806 0.9961 0.9994 0.9999 1.0000 1.0000 1.0000

0.2

0.0032 0.0243 0.0913 0.2252 0.4148 0.6172 0.7858 0.8982 0.9591 0.9861 0.9961 0.9991 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0317 0.1584 0.3907 0.6488 0.8424 0.9456 0.9857 0.9972 0.9996 1.0000 1.0000 1.0000

0.25

0.0008 0.0076 0.0355 0.1071 0.2375 0.4164 0.6080 0.7723 0.8867 0.9520 0.9829 0.9949 0.9987 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.0138 0.0850 0.2528 0.4925 0.7237 0.8822 0.9614 0.9905 0.9983 0.9998 1.0000 1.0000

0.3

0.0000 0.0005 0.0036 0.0160 0.0510 0.1256 0.2500 0.4159 0.5956 0.7553 0.8725 0.9435 0.9790 0.9935 0.9984 0.9997 1.0000 1.0000 1.0000 1.0000

0.0022 0.0196 0.0834 0.2253 0.4382 0.6652 0.8418 0.9427 0.9847 0.9972 0.9997 1.0000

0.4

0.0000 0.0000 0.0002 0.0013 0.0059 0.0207 0.0577 0.1316 0.2517 0.4119 0.5881 0.7483 0.8684 0.9423 0.9793 0.9941 0.9987 0.9998 1.0000 1.0000

0.0002 0.0032 0.0193 0.0730 0.1938 0.3872 0.6128 0.8062 0.9270 0.9807 0.9968 0.9998

0.5

0.0000 0.0000 0.0000 0.0000 0.0003 0.0016 0.0065 0.0210 0.0565 0.1275 0.2447 0.4044 0.5841 0.7500 0.8744 0.9490 0.9840 0.9964 0.9995 1.0000

0.0000 0.0003 0.0028 0.0153 0.0573 0.1582 0.3348 0.5618 0.7747 0.9166 0.9804 0.9978

0.6

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0013 0.0051 0.0171 0.0480 0.1133 0.2277 0.3920 0.5836 0.7625 0.8929 0.9645 0.9924 0.9992

0.0000 0.0000 0.0002 0.0017 0.0095 0.0386 0.1178 0.2763 0.5075 0.7472 0.9150 0.9862

0.7

æ nö The function tabulated is P( X £ x) = å ç ÷ pt q n -t . t =0 è t ø

x

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0009 0.0039 0.0139 0.0409 0.1018 0.2142 0.3828 0.5852 0.7748 0.9087 0.9757 0.9968

0.0000 0.0000 0.0000 0.0004 0.0028 0.0143 0.0544 0.1576 0.3512 0.6093 0.8416 0.9683

0.75

Probabilities for the Binomial distribution

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0006 0.0026 0.0100 0.0321 0.0867 0.1958 0.3704 0.5886 0.7939 0.9308 0.9885

0.0000 0.0000 0.0000 0.0001 0.0006 0.0039 0.0194 0.0726 0.2054 0.4417 0.7251 0.9313

0.8

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 0.0024 0.0113 0.0432 0.1330 0.3231 0.6083 0.8784

0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0043 0.0256 0.1109 0.3410 0.7176

0.9

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0003 0.0026 0.0159 0.0755 0.2642 0.6415

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0022 0.0196 0.1184 0.4596

0.95

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0010 0.0169 0.1821

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0062 0.1136

0.99

1 1 1 1 1

1 1 1 1 1

1

15 16 17 18 19

20 21 22 23 24

25

2

1 1 1 1 2

1 1 1 1 1

1 1 1 1 1

1 1 1 1

3

2

2 2 2 2 2

2 2 2 2 2

1 1 1 1 2

1 1 1 1 1

4

3

2 2 2 2 3

2 2 2 2 2

2 2 2 2 2

3

3 3 3 3 3

2 2 3 3 3

2 2 2 2 2

1 1 1 1 2

1

1 1 1 1 1 1

6

5

4

3 3 3 3 3

3 3 3 3 3

2 2 2 2 3

1 1 1 2 2

1

7

The table shows the greatest integer x for which

n1

2

1 1 1 1 1

1

10 11 12 13 14

5 6 7 8 9

1 2 3 4

4

4 4 4 4 4

3 3 4 4 4

2 3 3 3 3

5

4 4 4 5 5

4 4 4 4 4

3 3 3 3 3

1 2 2 2 2

1 1

1 1 1 2 2 2 2

10

9

5

5 5 5 5 5

4 4 4 4 4

3 3 3 4 4

1 2 2 2 3

1 1

11

å çè t1- 1 ÷ø çè

æ n - 1ö æ n2 + 1ö t ÷ø t =1

x

4

4 4 4 4 4

3 3 3 3 3

2 2 3 3 3

1 1 2 2 2

1

8

6

5 5 5 6 6

4 5 5 5 5

3 3 4 4 4

2 2 2 3 3

1 1

13

6

5 6 6 6 6

5 5 5 5 5

3 4 4 4 4

2 2 2 3 3

1 1

14

7

6 6 6 6 6

5 5 5 5 6

3 4 4 4 5

2 2 2 3 3

1 1

15

æ n1 + n2 ö çè n ÷ø < 0.05 . 1

6

5 5 5 5 5

4 4 4 5 5

3 3 4 4 4

1 2 2 2 3

1 1

12

7

6 6 6 7 7

5 5 5 6 6

4 4 4 4 5

2 2 3 3 3

1 1

16

7

6 6 7 7 7

5 5 6 6 6

4 4 4 5 5

2 2 3 3 3

1 1

17

7

6 7 7 7 7

5 6 6 6 6

4 4 4 5 5

2 2 3 3 3

1 1

18

8

7 7 7 7 7

5 6 6 6 6

4 4 5 5 5

2 2 3 3 4

1 1

19

Critical values for the Grouping of Signs test n2

8

7 7 7 8 8

6 6 6 6 7

4 4 5 5 5

2 2 3 3 4

1 1

20

8

7 7 8 8 8

6 6 6 7 7

4 4 5 5 5

2 2 3 3 4

1 1

21

8

7 8 8 8 8

6 6 7 7 7

4 5 5 5 6

2 2 3 3 4

1 1

22

9

7 8 8 8 8

6 6 7 7 7

4 5 5 5 6

2 3 3 3 4

1 2

23

9

8 8 8 8 9

6 7 7 7 7

4 5 5 6 6

2 3 3 4 4

1 2

24

9

8 8 8 9 9

6 7 7 7 8

4 5 5 6 6

2 3 3 4 4

1 2

25

189

Pseudorandom values from U(0,1) 1

2

3

4

5

6

7

8

9

10

0.587 0.030 0.048 0.593 0.165

0.155 0.447 0.224 0.478 0.113

0.999 0.048 0.359 0.929 0.695

0.122 0.201 0.463 0.301 0.513

0.659 0.931 0.710 0.688 0.711

0.975 0.071 0.861 0.750 0.402

0.059 0.033 0.972 0.211 0.121

0.567 0.388 0.543 0.911 0.843

0.651 0.849 0.550 0.479 0.951

0.686 0.033 0.248 0.046 0.229

0.788 0.714 0.265 0.712 0.630

0.493 0.980 0.320 0.524 0.933

0.329 0.946 0.065 0.747 0.863

0.160 0.530 0.573 0.136 0.802

0.708 0.973 0.708 0.004 0.642

0.309 0.440 0.682 0.165 0.625

0.878 0.728 0.014 0.070 0.244

0.650 0.652 0.128 0.431 0.961

0.279 0.303 0.113 0.201 0.458

0.617 0.398 0.938 0.965 0.127

0.569 0.766 0.638 0.984 0.721

0.813 0.449 0.335 0.575 0.857

0.341 0.026 0.466 0.300 0.303

0.055 0.276 0.808 0.836 0.076

0.483 0.977 0.907 0.276 0.124

0.756 0.410 0.162 0.638 0.688

0.186 0.102 0.355 0.674 0.455

0.273 0.695 0.333 0.625 0.536

0.443 0.487 0.529 0.885 0.842

0.618 0.640 0.390 0.451 0.533

0.028 0.726 0.218 0.792 0.656

0.271 0.399 0.240 0.704 0.285

0.245 0.474 0.036 0.959 0.886

0.290 0.221 0.206 0.615 0.954

0.534 0.898 0.582 0.440 0.846

0.924 0.838 0.203 0.311 0.595

0.093 0.723 0.676 0.994 0.215

0.724 0.139 0.371 0.785 0.484

0.651 0.219 0.791 0.041 0.158

0.422 0.711 0.069 0.737 0.435

Pseudorandom values from N(0,1) 1

2

3

4

5

6

7

8

9

10

–0.603 1.469 –2.199 1.863 0.076

0.825 0.282 0.169 –1.302 0.373

1.166 –1.250 0.278 0.260 –0.448

1.880 –1.176 0.580 –1.023 0.998

1.261 –0.064 –0.875 0.114 0.149

2.542 0.860 0.373 –0.904 1.987

0.312 –1.505 –0.132 0.500 –0.405

0.611 –0.828 –0.153 –0.255 0.324

0.286 –0.965 –1.322 0.283 0.112

0.223 –0.166 2.340 0.291 –1.367

–0.667 0.495 –0.161 1.105 0.800

–0.589 –1.693 0.990 0.851 –0.867

0.080 0.647 –1.348 –1.012 0.229

1.007 0.172 2.047 0.891 –0.534

1.548 1.143 0.167 0.256 –0.602

1.204 –1.519 0.599 0.297 1.685

1.886 –2.557 –0.530 1.267 –1.210

–0.080 1.351 1.244 –0.053 –0.986

0.341 –0.466 0.278 –1.776 0.979

–0.808 0.494 0.627 1.392 0.810

–0.738 –0.489 0.170 –1.301 0.059

0.765 –0.250 –1.623 –0.901 –1.489

–2.068 –0.917 0.442 0.810 0.235

–0.660 –2.549 –0.253 –0.244 –0.230

2.704 –1.879 –0.786 0.524 1.262

0.161 0.156 –0.468 –0.622 0.751

0.790 –1.451 0.435 –0.785 –0.377

–0.284 –0.158 1.544 –0.949 0.631

–1.041 –2.252 –1.014 –0.923 0.520

–0.852 –0.309 –1.187 0.510 1.508

0.599 0.329 –0.229 –0.987 0.805

0.196 –1.008 0.446 –0.214 –0.359

–1.785 0.834 0.086 0.925 –1.179

–0.899 1.079 0.024 –0.656 0.324

–1.347 –0.101 0.555 1.991 –0.208

–0.227 –0.322 –0.360 1.030 –0.632

1.027 –0.315 0.111 –0.961 1.170

0.704 –0.254 0.589 –0.078 –0.432

1.943 –0.711 –0.325 1.023 0.716

–0.902 –0.285 –0.056 –0.070 –1.801

190