TIA MLC Formula Sheet

A. Prerequisite Review CALCULUS REVIEW d dx

ex = ex

UNIFORM DISTRIBUTION X ∼ uniform a to b f (x) =

1 b−a

D {f (g(x))} = f 0 (g(x)) · g 0 (x)

F (x) =

x−a b−a

(uv)0 = u0 · v + u · v 0 R R udv = uv − vdu

E[X] =

a+b 2

R

ex dx = ex + c

Var[X] =

(b−a)2 12

CONDITIONAL PROBABILITY Pr(A ∩ B) Pr(A|B) = Pr(B) f (x, y) fX (x|y) = fY (y) R∞ fY (y) = −∞ f (x, y) dx

v = (1 + i)−1 = e−δ v 2 = (1 + i)−2 = e−2δ v n = (1 + i)−n = e−nδ

E[X] = EY [EX [X|Y ]]     Var[X] = EY VarX [X|Y ] + VarY EX [X|Y ]

NORMAL APPROXIMATION EXPONENTIAL DISTRIBUTION

all x

E[X] =

X

Exponential distribution is memoryless

xp(x)

all x

E[g(X)] =

X

f (x) = g(x) · p(x)

all x

Var[X] = E[X 2 ] − (E[X])2 X F (x) = Pr[X ≤ x] = p(y)

1 1 −µx e µ 1x −µ

F (x) = 1 − e

Var[Y ] = (a − b)2 · p(1 − p)

CONTINUOUS RANDOM VARIABLES R x f (x) dx = 1 R E[g(X)] = x g(x) f (x) dx Rx F (x) = Pr[X ≤ x] = −∞ f (y) dy f (x) =

d F (x) dx

v → v2 i → 2i + i2

Central Limit Theorem:

d → 2d − d2

Y = X1 + X2 + · · · + Xn

i δ

Xi ’s are iid RVs(µx , σx2 )

→

2i+i2 2δ

Y ≈ Normal(E[Y ] = nµx , Var[Y ] = nσx2 ) ANNUITY CERTAINS

E[X] = µ 95th percentile of Y : Var[X] = µ2

MIXED RANDOM VARIABLES CONVERT BETWEEN i, d, and δ Example:

(1)

X=a

p(a)

d

X=b

p(b)

0

i 1 (2)

a 1, B > 0   i B cx (ct − 1) t px = exp − ln c h

MAKEHAM LAW µx = A + Bcx

c > 1, B > 0, A ≥ −B h   i B cx (ct − 1) t px = exp(−At) · exp − ln c

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2012 The Infinite Actuary

C. Insurance Benefits PURE ENDOWMENT

TERM INSURANCE ( v Tx 0 ≤ Tx ≤ n 1 ¯ Z x:n = 0 Tx > n Z n ¯1 = A v t t px µx+t dt x:n

MLC discount factor ( 0 0 ≤ Tx ≤ n Z x:n1 = v n Tx > n

0

n Ex

=

Ax:n1

=

vn

¯x ¯x = A ¯1 + n| A A x:n Z n+m ¯ v t t px µx+t dt n|m Ax =

n px

Var[Z x:n1 ] = 2n Ex − (n Ex )2 = v 2n n px n qx

n

Bernoulli shortcut

(

1 Z x:n

WHOLE LIFE INSURANCE

1 Ax:n

v Kx +1 Kx = 0, 1, . . . , n − 1 = 0 Kx = n, n + 1, . . . n−1 X = v k+1 k| qx

0

Zx = v Kx +1 , Kx = 0, 1, 2, . . . ∞ X Ax = v k+1 k| qx k=0

N-YEAR DEFERRED WHOLE LIFE ( 0 0 ≤ Tx ≤ n ¯ Z = x n| v Tx Tx > n Z ∞ ¯ v t t px µx+t dt n| Ax = n

¯ = n Ex · A ¯x+n ( 0 Kx = 0, 1, . . . , n − 1 n| Zx = v Kx +1 Kx = n, n + 1, . . .

1 Ax = Ax:n + n| Ax n+m−1 X v k+1 k| qx n|m Ax = k=n

VARYING TERM INSURANCE Z n
¯ 1 = I¯A tv t t px µx+t dt x:n Z0 n
¯A ¯ 1 = D (n − t)v t t px µx+t dt x:n

Ax = vqx + vpx Ax+1

Draw a pain curve.

Ax:n = vqx + vpx Ax+1:n−1

For constant force:

(IA)x = vqx + vpx (Ax+1 + (IA)x+1 )

= n Ex · Ax+n

INCREASING WHOLE LIFE
¯ = Tx v Tx , Tx ≥ 0 I¯Z x Z ∞
¯ = I¯A tv t t px µx+t dt x 0

Kx = 0, 1, 2, . . .

Pr(SBP is sufficient) =



µ µ+δ

µ/δ

(IA)x = Ax + vpx (IA)x+1 1 = vqx h (IA)x:n i 1 1 +vpx (IA)x+1:n−1 + Ax+1:n−1 1 1 (DA)x:n = nvqx + vpx (DA)x+1:n−1

=

¯x = A

e−n(µ+δ)

µ µ+δ

Assuming UDD: ¯x = A

CONSTANT FORCE n Ex

CONTINUOUS VS. DISCRETE

¯1 = A x:n

i 1 A δ x:n

¯x:n = A

i 1 A δ x:n

(m)

Ax Ax =

vq vq+d

¯1 = A ¯x (1 − n Ex ) A 1 = Ax (1 − n Ex ) A x:n x:n   µ −n(µ+δ) ¯ A = e n| x µ+δ
2 1 1 ¯ ¯ ¯ (I A)x = µ Ax (IA)x = vq (Ax )2

i A δ x

=

+ n Ex

i Ax i(m)


¯ = i (IA) IA x δ x
1 i 1 ¯ I A x:n = δ (IA)x:n
1 i 1 ¯ DA x:n = δ (DA)x:n
¯ = i (IA) − i I¯A x δ δ x

1 d

−

1 δ




Ax

0



¯A ¯ 1 = nA ¯1 ¯ 1 + D I¯A x:n x:n x:n 1 1 1 (IA)x:n + (DA)x:n = (n + 1)Ax:n

n| Ax

(IZ)x = (Kx + 1)v Kx +1 ,

PERCENTILES

k=0

¯x = v Tx , Tx ≥ 0 Z Z ∞ ¯x = A v t t px µx+t dt

n| Ax

RECURSION

ENDOWMENT INSURANCE ( Tx 0 ≤ Tx ≤ n ¯x:n = v Z vn Tx > n ¯x:n = A ¯1 + n Ex A (x:n v Kx +1 Kx = 0, 1, . . . , n − 1 Zx:n = vn Kx = n, n + 1, . . . 1 Ax:n = Ax:n + n Ex

Assuming death occurs in the middle of the year:

DE MOIVRE’S LAW APV is an annuity-certain for the number of years of insurance remaining divided by ω − x. All of the following formulas follow that. aω−x a ¯ ¯x = ω−x A Ax = ω−x ω−x
¯a I¯ (Ia)ω−x
ω−x ¯ = I¯A (IA)x = x ω−x ω−x a ¯n ω−x
¯a I¯
n ¯ 1 = I¯A x:n ω−x ¯ a)n
¯A ¯ 1 = (D¯ D x:n ω−x

¯1 = A x:n

1 Ax:n =

¯x = (1 + i)1/2 · Ax A (m)

Ax

= (1 + i)(m−1)/2m · Ax

an ω−x

1 (IA)x:n = 1 (DA)x:n =

(Ia)n ω−x (Da)n ω−x

(IA)x = Ax + 1| Ax + 2| Ax + · · ·

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2012 The Infinite Actuary

D. Life Annuities WHOLE LIFE ANNUITY

N-YEAR DEFERRED WLA

Twin – Whole Life Insurance

No Twin ( 0 ¯ Y = n| x a ¯Tx − a ¯n

1 − v Tx Y¯x = a ¯Tx = δ Z ∞ ¯x 1−A a ¯x = a ¯x = t Ex dt δ 0

¯x n| a

=a ¯x − a ¯x:n

¯x = 1 − δ¯ A ax

¯x n| a

= n Ex · a ¯x+n

¨x n| a

= n Ex · a ¨x+n

  Var Y¯x =

2A ¯x

¯x − A


2

δ2 1 − v Kx +1 Y¨x = a ¨Kx +1 = d ∞ X 1 − Ax a ¨x = a ¨x = k Ex d k=0

Twin – N-Year Endowment Insurance ( a ¯Tx 0 ≤ Tx ≤ n Y¯x:n = n < Tx a ¯n ¯x:n 1−Z = δ Z n ¯x:n 1−A a ¯x:n = a ¯x:n = t Ex dt δ 0 ¯x:n = 1 − δ¯ A ax:n
2A ¯x:n − A ¯x:n 2   ¯ Var Yx:n = δ2

1 − Zx:n Y¨x:n = d a ¨x:n =

1 − Ax:n d

a ¨x:n =

n−1 X k=0

a ¨x:n = a ¨x − n Ex a ¨x+n h

Var Y¨x:n

i

=

2A

x:n

k Ex

a ¨x =

¯x (1 − n Ex ) a ¯x:n = a
¯a = (¯ I¯ ax )2 x

0

1 vq + d

a ¨x:n = a ¨x (1 − n Ex )

DE MOIVRE’S LAW

DEFERRED TEMPORARY ANNUITY

a ¨x = 1 + vpx a ¨x+1

Use the most important identity.

No twin.

ax = vpx a ¨x+1 n+m

=

v t t px dt



¯a ¯a I¯ + D¯ = n¯ ax:n x:n x:n

¯x+1 a ¯x = a ¯x:1 + vpx a

ANNUITIES PAYABLE M-THLY

¯x:n =a ¯x:n+m − a

¯x:1 + vpx a ¯x+1:n−1 a ¯x:n = a

Always true:

= n| a ¯x − n+m| a ¯x

a ¨x:n = 1 + vpx a ¨x+1:n−1

Ax

= n Ex · a ¯x+n:m

(I¨ a)x = 1 + vpx (¨ ax+1 + (I¨ a)x+1 )

¨x:n Ax:n = 1 − d(m) a

(I¨ a)x = a ¨x + vpx (I¨ a)x+1   ¨x+1:n−1 + (I¨ a)x+1:n−1 (I¨ a)x:n = 1 + vpx a

Assuming UDD:

For annuity-due same 3 formulas as above but replace a ¯ with a ¨.

N-YR CERTAIN AND LIFE AFTER ( a ¯n 0 ≤ Tx ≤ n ¯ Yx:n = a ¯Tx n < Tx

(m)

(m)

= 1 − d(m) a ¨x

(m)

(m)

(m)

(D¨ a)x:n = n + vpx (D¨ a)x+1:n−1

= α(m) a ¨x − β(m) (0 Ex − ∞ Ex )

(m) a ¨x:n

= α(m) a ¨x:n − β(m) (0 Ex − n Ex )

a ¯x:n = a ¯n + n| a ¯x

ADJUSTED FORCE OF MORTALITY

¨n + n| a ¨x a ¨x:n = a

If µ∗x+t = µx+t + c and δ ∗ = δ − c, then

α(m) ≈ 1

= n Ex

(m)

a ¨x

β(m) ≈

m−1 2m

s¨x:n =

a ¨x:n n Ex

≈a ¨x −

m−1 2m

−

m2 −1 12m2

(µx + δ)

−for other annuities, write them in terms of a whole life annuity

ACCUMULATED APV a ¯x:n n Ex

= α(m) n| a ¨x − β(m) (n Ex − ∞ Ex )

Woolhouse (3 terms)

a ¯∗x = a ¯x

s¯x:n =

= α(m)· symbol w/o (m) −β(m) (“start” - “end”)

a ¨x

(m) ¨x n| a

INCREASING WHOLE LIFE

¯a I¯Y¯ x = I¯ T ≥0 Tx
x ¯ a ¯x − I¯A
x ¯a = I¯ x δ

¯ =a ¯a I¯A ¯x − δ I¯ x x

(m)

symbol with

(I¨ a)x:n = a ¨x:n + vpx (I¨ a)x+1:n−1

∗ n Ex

¯1 = 1 − δ¯ ax:n − n Ex A x:n

1 µ+δ

RECURSION

n

N-YEAR TEMPORARY ANNUITY

a ¯x =

(I¨ a)x:n + (D¨ a)x:n = (n + 1)¨ ax:n

¯x n|m a

Yx = aKx = a ¨Kx +1 − 1 ⇒ ax = a ¨x − 1

CONSTANT FORCE

(I¨ a)x = (¨ ax )2  µ/δ µ Pr(SBP is insufficient) = µ+δ

Z

Ax = 1 − d¨ ax h i 2 A − (A )2 x x Var Y¨x = d2

0 ≤ Tx ≤ n n < Tx

VARYING TEMPORARY Z n
¯a I¯ = tv t t px dt x:n Z0 n
¯a (n − t)v t t px dt = D¯ x:n

sx:n =

ax:n n Ex

(I¨ ax ) − (Ia)x = a ¨x

− (Ax:n )2 d2

ax:n = a ¨x:n − 1 + n Ex

c

2012 The Infinite Actuary

E. Premium Calculation LOSS AT ISSUE

ENDOWMENT INSURANCE

SEMICONTINUOUS INSURANCE

GROSS PREMIUMS

L ∼ loss at issue random variable

Fully Continuous:

Benefit paid at the moment of death and premiums paid at beginning of the year. ¯
¯x = Ax P A a ¨x ¯1
A x:n ¯1 P A x:n = a ¨x:n ¯
¯x:n = Ax:n P A a ¨x:n

Lg = PV of future benefits + PV of future expenses − PV of future premiums

L = PV of benefits − PV of premiums E[L] = APVFB − APVFP

¯x:n − P¯ a E[L] = S A ¯x:n  2 
 ¯ 2A ¯x:n − A ¯x:n 2 Var[L] = S + Pδ Fully Discrete: E[L] = SAx:n − P a ¨x:n  2   2 P 2A Var[L] = S + d x:n − (Ax:n )

EQUIVALENCE PRINCIPLE E[L] = 0 ⇒ APVFB = APVFP

PERCENTILE PREMIUMS N-YEAR TERM INSURANCE

Whole Life and Endowment Insurance: Var[Z] (1 − SBP)

1

¯x + δ P¯ A
¯ ¯x P A ¯x =
A ¯x + δ P¯ A





¯1 P¯ A x:n




1 P x:n =

Whole Life and Endowment Insurance Only: ¯
Ax:n ¯x = Ax P¯ A Px:n = a ¯x a ¨x:n

a ¯x =

¯1 P¯ A x:n

=

¯1 A x:n

Group of policies:

a ¯x:n

E[S] + Φ−1 (1 − x%)

2

KEY PREMIUM IDENTITIES


¯x = 1 − δ P¯ A a ¯x ¯
¯x = δ Ax P¯ A ¯x 1−A

Px:n = Px:n

Ax:n

1 −d a ¨x:n

dAx:n = 1 − Ax:n

a ¨x:n =

New Equivalence Premium: E[Lg ] = 0 ⇒ APVFB + APVFE = APVFP APV of Benefits + APV of Expenses APV Annuity  2 r Var(Z) Var[Lg ] = S + E + G−e d G=

APV benefit benefit premium = APV annuity

Var[L] =

E[Lg ] = APVFB + APVFE − APVFP



¯x:n − P¯ A ¯ 1 = P¯ A x:n

1 Ax:n

a ¨x:n

Px:n = Px:n + d

N-PAY WHOLE LIFE ¯x
A ¯ ¯ n P Ax = a ¯x:n n Px

=

N-YEAR DEFERRED WLA ¯x
n| a P¯ n| a ¯x = a ¯x:n ¨x n| a




=

Var[L] = p · 2 Ax

Fully Continuous Whole Life:

Premiums paid m-thly instead of annually. Px

(m) n Px

µ µ+δ

µ/δ

Ax

= =



(m)

a ¨x Ax:n

DE MOIVRE’S LAW

(m)

a ¨x:n Ax = (m) a ¨x:n

Use key premium identities and most important identity.

¨x n| a a ¨x:n

(P-P) / P n Px

1 − P x:n

P x:n1

Fully Continuous:

E[L] = SAx − P a ¨x  2   P 2 A − (A )2 Var[L] = S + d x x

CONSTANT FORCE
¯x = µ P¯ A Px = vq
1 ¯1 P x:n P¯ A = vq x:n = µ

Pr(benefit prem is sufficient) = (m)

Ax a ¨x:n

WHOLE LIFE

Fully Discrete:

FRACTIONAL PREMIUMS

(m) Px:n

P

¯x − P¯ a E[L] = S A ¯x  2 
 ¯ 2A ¯x − A ¯x 2 Var[L] = S + Pδ

Single policy: Find the policy you must “win” on and then solve for the premium it will take to win on that death.

¯x Var[L] = 2 A

1 Px:n + d

p Var[S] = 0

COMPARE VARIANCE Two identical policies (whole life or endowment insurance) P1 → L1

= Ax+n

Px:n − n Px = 1 − Ax+n P x:n1 1 Px:n − P x:n

P x:n1

=1

P2 → L2 Var[L2 ] = Var[L1 ]



P2 + d P1 + d

2

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2012 The Infinite Actuary

F. Reserves #1 PROSPECTIVE METHOD

#4 PAID UP

#8 (P-P) / P

#11 RESERVE CREATION FORMULA

All Policies and All Premiums. All other methods only work when premiums are determined using the equivalence principle.

Whole Life, Endowment Insurance, Term Insurance and Limited Pay Whole Life Only

Accumulated Difference in Premiums equals Difference in Reserves

Useful when death benefit is a function of the reserve.

Reserve = APV of the future benefit times the percentage of the future benefit that you are not funding with future premiums

Remember

Reserve = APVFB − APVFP Example – fully continuous whole life:

¯ ¯ ¯ ¯ ¯ ¯x+t t V Ax = Ax+t − P Ax a

Example – full continuous whole life: "
# ¯x
P¯ A ¯ ¯ ¯
1 − V A = A x t x+t ¯x+t P¯ A

1 P

1

= s¨x:n

Example Px:n − Px = n Vx:n − n Vx = 1 − n Vx P x:n1

#9 ANNUITY FORM SPINOFF

All Policies under EP.

#5 ANNUITY FORM

Whole Life and Endowment Insurance Only

Reserve = AAVPP − AAVPB

Whole Life and Endowment Insurance Only

Example – fully discrete whole life:

t Vx:n

= Px:n s¨x:t − t kx

ACCUMULATED COST OF INS t kx

=

=

t Ex

dx (1+i)t−1 +dx+1 (1+i)t−2 +···+dx+t−1 lx+t

For one year: 1 kx

=

Example – fully discrete endowment ins.: a ¨x+t:n−t t Vx:n = 1 − a ¨x:n

1 Ax:t

Thinking in terms of life tables: t kx

Reserve = 1 minus (the annuity at the dot divided by the annuity at issue)

qx px

#6 LIFE INSURANCE FORM Whole Life and Endowment Insurance Only Reserve = insurance at the dot minus the insurance at issue divided by one minus the insurance at issue Example – fully discrete whole life: Ax+t − Ax t Vx = 1 − Ax

#3 DIFFERENCE IN PREMIUM Whole Life, Endowment Insurance, Term Insurance and Limited Pay Whole Life Only Reserve = accumulated difference in premium you want to charge and premium you are actually charging Example – fully continuous whole life:


¯ ¯ ¯ ¯ ¯ ¯ a ¯x+t t V Ax = P Ax+t − P Ax

#7 BENEFIT PREMIUM FORM Whole Life and Endowment Insurance Only

1. cost of providing ensuing year’s DB 2. rest is for reserve creation tV

= P s¨t t−1 X − vqx+h (bh+1 − h+1 V ) (1 + i)t−h h=0

#2 RETROSPECTIVE METHOD

Example – fully discrete endowment ins.:

Each premium accounts for two items:

x:n

DB = Reserve: t V = P s¨t DB = 1 + Reserve: tV

= P s¨t −

t−1 X

qx+h (1 + i)t−h−1

h=0

a+b Vx = 1 − (1 − a Vx ) (1 − b Vx+a )

DB = 1 + Reserve and qx+h = q : #10 RESERVE RECURSION Start with the terminal reserve from the previous year, collect the premium and put those in the bank. If the policyholder dies pay the death benefit. If the policyholder lives setup the next reserve. Example - fully discrete 20-year endowment insurance on (40):
10 V40:20 + P40:20 (1+i) = q50 +p50 11 V40:20
NAR version: = q50 1 − 11 V40:20 + 11 V40:20

tV

= (P − vq) s¨t

VARIANCE LOSS For Whole Life and Endowment Insurance:   ¯ 2 Var(t L) = S + Pδ Var(v U ) EP ⇒

=

Var(v U ) (1 − SBP)2

Example – fully discrete endowment ins.: Var(t L) =  S+ EP ⇒

=

P d

2 

2A

x+t:n−t

2A x+t:n−t

2   − Ax+t:n−t 2

 − Ax+t:n−t

(1 − Ax:n )2

Reserve = premium at the dot minus the premium at issue divided by the premium at the dot plus delta (or d)

CONSTANT FORCE

Example – fully discrete whole life:

Under EP, level premium whole life and term insurance reserves are 0.

t Vx

=

Px+t − Px Px+t + d

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2012 The Infinite Actuary

GROSS PREMIUM RESERVES

MODIFIED RESERVES

To find the gross premium, G, solve new EP

α − lower modified prem in the first year

APVFP = APVFB + APVFE

β − higher modified prem in the renewal years

Expense Reserves:

FPT Reserves:

kV

e

= APVFE − APVFL

kV

e

= AAVPL − AAVPE

1 α = Ax:1

β = benefit premium for insurance issued to (x + 1)

Gross Premium Reserve: kV

g

= kV

n

+ kV

1V

m

=0

tV

m

= (t − 1) reserve for insurance issued to (x + 1)

e

Can also use recursion for expense and gross premium reserves.

GAIN SEMI-CONTINUOUS RESERVES Assuming UDD
i ¯ t V A x = δ t Vx
i 1 ¯1 t V Ax:n = δ t V x:n
i 1 1 ¯ t V Ax:n = δ t V x:n + t V x:n

INTERIM BENEFIT RESERVES Exact Method: (h V + πh ) (1 +

i)s

v 1−s

= s qx+h · +s px+h · h+s V

Approximation: use linear interpolation between initial reserve and the terminal reserve.

Gain = Actual Profit − Expected Profit Profit = (t V + Gt − et ) (1 + i) −qx+t (St + Et ) − px+t · t+1 V Analysis of Surplus: expenses: (et − e0t ) (1 + i) + (Et − Et0 ) qx+t interest: (i0 − i) (t V + Gt − e0t )
0 mortality: qx+t − qx+t (St + Et0 − t+1 V ) Order matters. Use actual experience if you have already accounted for that source.

POLICY ALTERATIONS Reduced Paid-Up: RP U =

THIELE’S DIFFERENTIAL EQN

t CVx Ax+t

Extended Term (solve for n): d V dt t

= δt · t V +Gt −et −(St + Et − t V ) µ[x]+t
t+h V − h Gt − et − (St + Et ) µ[x]+t
tV ≈ 1 + h µ[x]+t + δt

t CVx

1 = Ax+t:n

For endowment insurance if the cash value is large enough to cover the remaining term, then solve for reduced pure endowment benefit, P E t CVx

1 = Ax+t:n−t + P E · n−t Ex+t

c

2012 The Infinite Actuary

G. Markov Chains NOTATION ij t px

MULTIPLE DECREMENT MODEL

− probability that a subject in state i at time x will be in state j (where j may equal i) at time x + t

ii t px

exit 1

− probability that a subject in state i at time x stays in state i continuously until time x + t

ii t px

exit 2

≤ t pii x

ANNUITIES

THIELE’S DIFFERENTIAL EQN

Pay 1 per year continuously while in state j: Z ∞ a ¯ij e−δt t pij x = x dt

d V (i) dt t

0

(i)

= δt t V (i) − Bt n   X (ij) − µij + t V (j) − t V (i) x+t St j=0,j6=i

If payable at the start of the year: ∞ X a ¨ij v k k pij x = x k=0

Euler’s method: t−h V

active 0

(i)

(i)

= t V (i) (1 − δt h) + hBt n   X (ij) +h µij + t V (j) − t V (i) x+t St j=0,j6=i

INSURANCE

DISCRETE MARKOV CHAINS P(t) is the transition matrix at time t ij t px

= ij entry of

ii t px

ii ii = pii x · px+1 · · · px+t−1

P(x)

·

P(x+1)

exit m

· · · P(x+t−1)

0j t px

CONTINUOUS MARKOV CHAINS µij x ii t px


= 1 − t p00 x  R  = exp − 0t µ0• x+s ds

h→0

Probability of transition from one state to another depends on the model.

KOLMOGOROV’S FORWARD EQNS  X  ik kj ij jk d pij = t px µx+t − t px µx+t dt t x k=0,k6=j d pij dt t x

PERMANENT DISABILITY MODEL

Healthy 0

µ01 x

µ02 x

Disabled 1

=

prob “move − into j” at time x + t

Euler’s method turns a continuous Markov Chain into a discrete chain with time increments of h and transition “probabilities” ( hµij i 6= j ij x+t p = h x+t 1 − hµi• x+t i = j

=

Rt

00 0 s px

j6=k

If all transition forces are constant and no reentry into a state 1 a ¯ii x = i• µ +δ For the multiple decrement model with constant transitions: µ0j ¯0j A x = 0• µ +δ

RESERVES tV

(i)

− rate of payment of benefit while the policyholder is in state i

(ij)

− lump sum benefit payable instantaneously at time t on transition from state i to state j

St

PREMIUMS APV of benefit APV of annuity

− reserve at duration t for a subject in state i at that time

(i)

Bt

benefit premium = 01 t px

prob “move out of j” at time x + t

µ12 x Dead 2

0

CONSTANT FORCE AND NO RE-ENTRY

ij h px

for i 6= j h  R  t i• = exp − 0 µx+s ds

= lim

00 t px

µ0j x µ0• x

Pay 1 on each future transfer into k: Z ∞ X ij jk ¯ik = A e−δt t px µx+t dt x

11 µ01 x+s t−s px+s ds

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2012 The Infinite Actuary

H. Multiple Decrement Models MULTIPLE DECREMENT MODEL

FORCE OF DECREMENT - 2 (τ )

exit 1

µx+t = (j) µx+t

exit 2 active 0

=

(τ ) d q dt t x (τ ) t px

(j)

− prob. of decrement due to any cause m X (τ ) (j) t qx = t qx (j)

Ax =

(τ )

j v k k−1 px qx+k−1

FORCES OF DECREMENT - 1

(τ )

(1)

(2)

=

(τ ) 0 s px

Rt

cause 1

(m)

(1)

qx

(2)

(j)

· µx+s ds

Pr(J = j | T = t) =

µx+t

qx

alive 0

(j) t qx

(j)

=

(j) ¯x A =

µx

(τ )

µx

R∞ 0

exit 1 µ01 x+t exit 2 active 0

cause 3

(τ )

(j)

v t t px µx+t dt

CONSTANT TRANSITION FORCES   (j) (τ ) µ(j) t qx = (τ ) 1 − t px µ     (j) (τ ) t (j) t qx 1 − px t qx = (τ ) µ(j) +δ 1 a ¯x = (τ ) µ +δ   (j) (j) (τ ) µ ¯ A 1 − n Ex = (τ ) 1 x:n

=

µ(τ )

µ

0 is redundant so we use: (j) µx+t

1 2

X

(i)

dx

Constant Force and UDDMDT:   (j) (τ ) (τ ) qx /qx p0 (j) x = px UDDAST: For 2 UDDASTs use the midpoint:   (1) qx = q 0 (1) 1 − 12 q 0 (2) x x

q 0 (1) x

alive 0

For 3 or more UDDASTs use integration: 0 (j) tp x

(j)

· µx+t = q 0 (j) x - factor out of integral

For 3 UDDASTs (if you like memorizing):     (1) 0 (3) + 1 q 0 (2) q 0 (3) qx = q 0 (1) 1 − 12 q 0 (2) x +q x x x 3 x Hybrids: cause 1

Draw a picture. For 2 competing UDDASTs use the midpoint of the interval.

q 0 (2) x

alive 0

ASSET SHARE Recursion: [k AS + G(1 − ck ) − ek ] (1 + i) (d)

0 (2) 0 (m) = t p0 (1) x · tp x · · · tp x

(w)

(τ )

= qx+k + qx+k · k+1 CV + px+k · k+1 AS q 0 (3) x

alive 0

(τ ) t px

cause 2

+δ

ABSOLUTE RATE OF DECREMENT   R t (j) 0 (j) 0 (j) t q x = 1 − t p x = 1 − exp − 0 µx+s ds exit m

−

t qx

µ02 x+t

µ0m x+t

dx (τ ) `x

(3)

qx

1 − t px

(τ )

cause 2

(τ ) µx+t

If all forces are a constant multiple of the total force for all t  

¯(j) A x

=

q 0 (j) x =

Rates of Decrement

k=1

µ0j x+t

Probabilities of Decrement

i6=j

j=1 ∞ X

Super secret approximation:

(j) d q dt t x (τ ) t px

(τ )

qx

Each cause is independent of the other causes.

(j)

(j)

qx − prob. of decrement due to cause #j

MDT vs. AST

µx+t = µx+t + µx+t + · · · µx+t  R  (τ ) t (τ ) t px = exp − 0 µx+s ds (j) t qx

exit m

ABSOLUTE RATE OF DECREMENT

cause 3

Notice how in the rate of decrement “worlds” there is only one cause of decrement. For the rates of decrement we have 3 tables and for the multiple decrement “world” we have one table with 3 competing decrements. 0 (j) tq x

n-th Asset Share using Asset Share Creation: (d)

(w)

n−1 X

G(1 − ck ) − ek − vqx+k − vqx+k · k+1 CV

k=0

(τ ) n−k Ex+k

For additional premium of ∆G: ∆AS =

n−1 X

∆G(1 − ck )

k=0

(τ ) n−k Ex+k

(j)

≥ t qx

=

n−1 X

∆G(1 − ck )k px v k

k=0

(τ ) n Ex

(τ )

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2012 The Infinite Actuary

I. Multiple Life Models JOINT-LIFE STATUS

LAST-SURVIVOR STATUS

CONTINGENT PROBABILITIES

REVERSIONARY ANNUITY

Txy = min[Tx , Ty ]

Txy = max[Tx , Ty ]

1 n q xy

(y) gets money after (x) dies:

Failure on the first death.

Txy = Tx + Ty − Txy

t qxy

t pxy

− probability first death of (x) and (y) occurs within t years

that for (xy) = that for (x) + that for (y) −that for (xy)

− probability that (x) will attain x + t and (y) will attain y + t

t qxy

t pxy

= t px · t py

t qxy

= 1 − t pxy

(work with p’s)

fxy (t) = t pxy · µx+t:y+t µx+t:y+t = µx+t + µy+t Z n n qxy = t pxy µx+t:y+t dt ◦

0 ∞

Z

exy =

t pxy 0 ∞ X

exy =

dt

k pxy

k=1

0

¯xy = A

t qxy

= t qx · t qy

t pxy

− probability at least one of (x) or (y) survives for t years

(work the the q’s)

t pxy

= 1 − t qxy

t pxy

= t px + t py − t pxy

∞

v

Other versions: a ¯x|y:n = a ¯y:n − a ¯xy:n a ¯x:n |y = a ¯y − a ¯xy:n

0 2 n q xy

In other words: a ¯u|v = a ¯v − a ¯uv

n

Z =

t py

µy+t · t qx dt

= n q xy1 + n q xy2

1 n q xy

= n q xy2 + n qx · n py

Probability (x) dies more than n years after death of (y):

= n+1 qxy − n qxy

n| qxy

= n| qx + n| qx − n| qxy

◦

◦

2 · ∞ q x+n:y

n px

◦

t

t pxy

µx+t:y+t dt

a ¯xy = a ¯x + a ¯y − a ¯xy

CONTINGENT INSURANCE Z ∞ ¯1 = A v t · t px µx+t · t py dt xy

¯x + A ¯y − A ¯xy ¯xy = A A 1 Axy:n

=

1 Ax:n

+

1 Ay:n

−

−d

0

1 Axy:n c

¯ 2= A xy

¯xy = 1 − δ¯ A axy 1 a ¨ xy

− probability that (y) dies after (x) and before n years from now

If (x) and (y) independent: Z n 1 n q xy = t px µx+t · t py dt

n qy

n| qxy

0

Pxy =

s

0

fxy (t) = t px µx+t + t py µy+t − t pxy µx+t:y+t

◦

0 2 n q xy

a ¯x|y = a ¯y − a ¯xy

COMMON SHOCK Find total total total

the total force for each status: force on (x) = µ∗x + λ force on (y) = µ∗y + λ force on (xy) = µ∗x + µ∗y + λ

Pr[(x) dies first] =

µ∗ x ∗ µ∗ x +µy +λ

Pr[(y) dies first] =

µ∗ y ∗ +λ µ∗ +µ x y

exy = ex + ey − exy

exy = pxy (1 + ex+1:y+1 ) Z ∞ a ¯xy = v t t pxy dt Z

− probability that last death happens within t years

1 n q xy

− probability (x) dies before (y) and before n years from now  Z n Z ∞ = fxy (s, t) dt ds

∞

Z

v t · t py µy+t · t qx dt

0

EXACTLY ONE STATUS

¯ 2= A xy

∞

Z

Pr[Tx = Ty ] = ¯x = A

µ∗ x +λ µ∗ x +λ+δ

¯y = A

µ∗ y +λ µ∗ y +λ+δ

λ ∗ µ∗ x +µy +λ

¯y+t dt v t · t px µx+t · t py A

0 [1] t pxy

− probability exactly 1 of (x) and (y) live t years

¯y = A ¯ 1 +A ¯ 2 A xy xy

[1] t pxy

= t px + t py − 2 t pxy

SBP for a payment of 1 at the death of (x) if he dies more than n years after the death of (y):

[1]

¯x + a ¯y − 2¯ axy a ¯xy = a

n Ex

¯ 2 ·A x+n:y

CONSTANT FORCE µx +µy µx +µy +δ

¯xy = A a ¯xy = 1 n q xy

1 µx +µy +δ

=

¯1 = A xy

µx q µx +µy n xy µx µx +µy +δ

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2012 The Infinite Actuary

DE MOIVRE’S LAW ◦

exx =

ω−x 3

◦

◦

◦

exy = y−x px · eyy +y−x qx · ey 2 n q xy

=

¯1 = A xy

1 q 2 n xy

a ¯y:ω−x ω−x

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2012 The Infinite Actuary

J. Other Topics PENSION MATHEMATICS

PROFIT SIGNATURE

Replacement ratio: pension income in year after retirement R= salary in the year before retirement

( Πt =

Profit(0) t−1 px · Profit(t)

t=0 t>0

where t−1 px is the probability that policy is in force at beginning of year t.

Salary Scale: sy salary received in year of age y to y + 1 = sx salary received in year of age x tp x + 1

v(t) − current market price on a t year zerocoupon bond that pays $1 at time t.

Solve for j such that: n X Πt vjt = 0 t=0

− think of it as the discount function. yt − t year spot rate of interest v(t) =

1 (1 + yt )t

PROFIT MARGIN

f (t, t+k) − forward rate from time t to t + k

NPV Pa ¨x:n

a ¨(x)y =

k px

v(k)

k=0

A risk is diversifiable if: q P Var[ N i=1 Xi ] lim =0 N →∞ N

DISCOUNTED PAYBACK PERIOD Find the smallest m such that: m X Πt vrt ≥ 0

UNIVERSAL LIFE - Notation

− Expenses Incurred + Interest Earned on Premium less Expenses

ECt − expense charge for year t

The profit for year 0 is the negative of the expenses incurred at time 0. Annual Profit = Previous Rsv accumulated with interest + Premium Collected

− Expected Cost of Benefits

Type A:

Annual Profit =

ADBt = F A − AVt DBt = F A

Previous Rsv (usually AV) + Premium Collected − Expenses Incurred

ADBt = F A DBt = F A + AVt

+ Interest Earned on Prev Rsv plus Premium less Expenses − Expected Cost of Benefits

1 ∗ CoIt = ADBt Ax+t−1:1 = ADBt · vq qx+t−1

− Expected Surrender Benefits − Expected Cost of Ending Rsv

CoItA = ∗ [F A−(AVt−1 +Pt −ECt −CoItA )(1+ict )]vq qx+t−1

∗ [F A−(AVt−1 +Pt −ECt )(1+ict )]vq qx+t−1 ∗ 1−vq qx+t−1 (1+ic t)

Type B: CoItB

= FA ·

∗ vq qx+t−1

CORRIDOR FACTORS To qualify as life insurance the death benefit must be at least a certain multiple (γt for year t) of the account value. ADBtc = (γt − 1)AVt

t=0

F A − face amount AVt − account value at end of year t CVt − cash value at end of year t DBt − death benefit for year t ADBt − additional death benefit for year t CoIt − cost of insurance for year t SCt − surrender charge for year t ict − credited interest rate in year t It − credited interest in year t vq − discount factor used in CoI calc ∗ qx+t−1 −mort. rate used in CoI calc for year t

ANNUAL PROFIT

The profit for year 0 is the negative of the expenses incurred at time 0.

CoItA =

k

0 ∞ X

DBt = AVt + ADBt

Type A:

Profit Margin =

v(t) (1 + f (t, t + k)) = v(t + k) Z ∞ ¯ y= A(x) v(t) t px µx+t dt

UNIVERSAL LIFE - Annual Profit

Type B: INTERNAL RATE OF RETURN

INTEREST RATE RISK

UNIVERSAL LIFE - COI

UNIVERSAL LIFE - Roll Forward

ADBtf = F A − AVt

Account Value Roll Forward:

ADBt = max[ADBtc , ADBtf ]

Starting AV (AVt−1 )

∗ CoIt = ADBt · vq qx+t−1

+ Premium (Pt ) − Expense Charge (ECt ) − Mortality Charge (CoIt ) + Credited Interest (It ) = Ending Account Value (AVt ) AVtA =

∗ (AVt−1 +Pt −ECt −F A vq qx+t−1 )(1+ict ) ∗ 1−vq qx+t−1 (1+ic t)

AVtB = same as numerator for AVtA

CVt = max[AVt − SCt , 0]

− Expected Cost of Ending Rsv

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2012 The Infinite Actuary