BKM Ch 08 Answers w CFA.docx

CHAPTER 8: INDEX MODELS 1. What are the advantages advantages ! the the "nde# $de% $de% &$'ared &$'ared t t the Mar()"t* Mar()"t* 'r&ed+r 'r&ed+ree !r ,ta"n"ng an e!!"&"ent%- d"vers"!"ed 'rt!%" What are "ts d"sadvantages

The Markowitz procedure and the Separation Property is what we used to first compute w1* to determine P* (given the risk-free rates and the risk and return characteristics of the risky assets) and then compute y* to determine * (using the risk-free rate! the risk and return of the optima" risky  portfo"io and risk aversion)# The Separation Property refers to separating the choice of the weights of the risky assets assets from the choice of the the weight of the risk-free risk-free asset# See page $1%# &"though we ta"ked a'out it in in "ecture! we did not do a comp"ete comp"ete nde Mode" ptimization# The  procedure is descri'ed in the tet starting on page $+1# The advantage of an inde mode" optimization! compared to the Markowitz procedure! is the reduced num'er of estimates re,uired# or . assets! the Markowitz procedure re,uires/ 0ariance ance (or standard deviation) ca"cu"ations/ i$ •  . 0ari • (.$ 2 .)3$ ovariance ca"cu"ations/ i4 The tota" is . 5 (. $ 2 .)3$ n . 6 $7! then the num'er of estimates is $7 5 ($7$ -$7)3$ 6 8$7 or the inde mode"! we can compare each asset asset to the inde# So so"ve for/ $ $ $ $ i  6 9i M  5  (:i) i4 6 9i9 4M$

• • • •

or . assets! the inde procedure re,uires/  . ;eta ca"cu"ations/ 9i  . 0ari 0ariance ance of the errors ca"cu"ations/ $(:i) 1 s a"pha is "arger# e; Wh"&h st&( has h"gher &rre%at"n )"th the $ar(et

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The corre"ation coefficient is the s,uare root of A $! so Stock ;>s corre"ation with the market is higher# 8. Cns"der the t) e#&ess ret+rn; "nde# $de% regress"n res+%ts !r A and B: R A > 1<  1./ R M R / > 9.7 Res"d+a% Stdev > ?@ A; > 19.<

R B > 0/<  9.8 R M R/ > 9.2   Res"d+a% Stdev > ?@B;> .1<

a; Wh"&h st&( has h"gher !"r$0s'e&"!"& r"s(

irm-specific risk is measured 'y residua" standard deviation 6 (:i) (:&)  (:;) 1=#8J  H#1J

,; Wh"&h st&( has greater s-ste$at"& $ar(et; r"s( 

Market risk is measured 'y 9# 9&  9; 1#$  =#G &; Wh"&h st&( has h"gher R /

A $ measures the fraction of tota" variance of return ep"ained 'y the market return# &>s A $  ;>s A $ =#7F+  =#%8+ d; I! r!  )ere &nstant at < and the regress"n had ,een r+n +s"ng tta% rather than e#&ess ret+rns5 )hat )+%d have ,een the regress"n "nter&e't !r st&( A

Part (d) is designed to showing the alpha will be wrong if you do not subtract the risk-free before doing the regression. Rewrite the SCL equation in terms of total return (r) rather than excess return (R): r & 2 r f   6 N 5 9(r M 2 r f ) r & 6 N 5 9(r M 2 r f ) 5 r f  r & 6 N 5 9r M 2 9r f   5 r f  r & 6 N 5 r f   2 9r f   5 9r M r & 6 N 5 r f (1 − 9) 5 9r  M r & 6 BN 5 r f (1 − 9)C 5 9r  M The slope is still βbut the intercept (in brackets) is equal to: α + rf (1 − β) N 5 r f (1 − 9) 6 1 5 =#=+(1 2 1#$) 6 -=#$ 6

Fse the !%%)"ng data !r Pr,%e$s  thr+gh 12. S+''se that the "nde# $de% !r st&(s A and B "s est"$ated !r$ e#&ess ret+rns )"th the !%%)"ng res+%t R A > <  9. R M  eA R B > 0/<  1./ R M  eB ?M > /9< R /A > 9./9 R /B > 9.1/ . What "s the standard dev"at"n ! ea&h st&(

 .ote that this is a tet-'ook eercise to understand the re"ationship 'etween A $ and i# f we are a'"e to compute the va"ues given a'ove! we cou"d compute the standard devation of the stock>s returns# The standard deviation of each stock can 'e derived from the fo""owing e,uation for A $/ 2 2 2 2  β i σ  M   Explained Variance  βi σ  M  2 2  Ri = = ⟹ σ  = i 2 2 Total Variance σ i  R i 2

2

σ  A =

2

2

 β A σ  M 

=

2

 R A

( 0.7 ) ( 0.20 )

2

0.20

=0.0980

& 6 81#8=J 2

2

σ B=

2

 β B σ  M  2

 R B

2

=

( 1.2 ) ( 0.20 ) 0.12

2

=0.4800

; 6 +H#$GJ 19. Brea( d)n the var"an&e ! ea&h st&( t the s-ste$at"& and !"r$0s'e&"!"& &$'nents

The systematic risk for & is/

2

2

 β A σ  M 

( 0.7 ) ( 0.20 ) 2

2

=

0.0196

=

The firm-specific risk of & (the residua" variance) is the difference 'etween &>s tota" variance (ca"cu"ated in ,uestion H) and its systematic variance/ 2

2

2

σ  A − β A σ  M =0.0980 =0.0196= 0.0784 The systematic risk for ; is/

2

2

 β B σ  M 

=

( 1.2 ) ( 0.20 ) 2

2 =

0.0576

The firm-specific risk of ; (the residua" variance) is the difference 'etween ;>s tota" variance (ca"cu"ated in ,uestion H) and its systematic variance/ 2

2

2

σ B− β B σ  M =0.4800 =0.0576 =0.4224 11. What are the &var"an&e and &rre%at"n &e!!"&"ent ,et)een the t) st&(s

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2

σ  M 

( 0.20 )

=

2

0.0400

=

Covar ( r A ,r B ) =σ  AB= β A β B σ  M = ( 0.70 ) ( 1.20 ) ( 0.0400 ) =0.0336 2

Correl ( r A , r B )= ρ AB =

σ  AB σ  A σ B

=

0.0336

( 0.3130 ) ( 0.6928 )

= 0.155

1/. What "s the &var"an&e ,et)een ea&h st&( and the $ar(et "nde#

Aeca"" that corre"ation 'etween the stocks and the market is the s,uare root of regression A $# ?e are given the two A $ va"ues a'ove# ov(r &!r M ) 6&M 6 O&M&M 6 (=#$=)13$(=#818=)(=#$=) 6 =#=$G= ov(r ;!r M ) 6 ;M 6 O;M;M 6 (=#1$13$)(=#+H$G)(=#$=) 6 =#=%G=

17. A st&( re&ent%- has ,een est"$ated t have a ,eta ! 1./2: a; What )"%% a ,eta ,( &$'+te as the 3adG+sted ,eta4 ! th"s st&(

 .ote that we did not cover ad4usted 'etas in c"ass! 'ut it is fair"y simp"e and  wanted to give you a ,uick "ook at what is sometimes ca""ed the DMerri""-Kynch ad4usted ;etas#E ;etas are ad4usted  'y taking the samp"e estimate of 'eta (the 'eta ca"cu"ated in 9.  9.t01. What would be your predicted beta for next year?

9t 6 (=#8)(1#=) 5 (=#F)1#$% 6 1#1+G

CFA PROBLEMS

1. The regression results provide quantitative measures of return and risk based on monthly returns over the five-year period.

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β for ABC was 0.60, considerably less than the average stock’s β of 1.0. This indicates that, when the S&P 500 rose or fell by 1 percentage point, ABC’s return on average rose or fell by only 0.60 percentage point. Therefore, ABC’s systematic risk (or market risk) was low relative to the typical value for stocks. ABC’s alpha (the intercept of the regression) was –3.2%, indicating that when the market return was 0%, the average return on ABC was –3.2%. ABC’s unsystematic risk (or residual risk), as measured by σ(e), was 13.02%. For ABC, R 2 was 0.35, indicating closeness of fit to the linear regression greater than the value for a typical stock. β for XYZ was somewhat higher, at 0.97, indicating XYZ’s return pattern was very similar to the β for the market index. Therefore, XYZ stock had average systematic risk for the period examined. Alpha for XYZ was positive and quite large, indicating a return of almost 7.3%, on average, for XYZ independent of market return. Residual risk was 21.45%, half again as much as ABC’s, indicating a wider scatter of observations around the regression line for XYZ. Correspondingly, the fit of the regression model was considerably less than that of ABC, consistent with an R 2 of only 0.17. The effects of including one or the other of these stocks in a diversified portfolio may be quite different. If it can be assumed that both stocks’ betas will remain stable over time, then there is a large difference in systematic risk level. The betas obtained from the two brokerage houses may help the analyst draw inferences for the future. The three estimates of ABC’s β are similar, regardless of the sample period of the underlying data. The range of these estimates is 0.60 to 0.71, well below the market average β of 1.0. The three estimates of XYZ’s β vary significantly among the three sources, ranging as high as 1.45 for the weekly data over the most recent two years. One could infer that XYZ’s β for the future might be well above 1.0, meaning it might have somewhat greater systematic risk than was implied by the monthly regression for the five-year period. These stocks appear to have significantly different systematic risk characteristics. If these stocks are added to a diversified portfolio, XYZ will add more to total volatility. 2. The R2 of the regression is: 0.70 2 = 0.49 Therefore, 51% of total variance is unexplained by the market; this is nonsystematic risk. 3.

9 = 3 + β (11 − 3) ⇒ β = 0.75

4.

d.

5.

b.

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