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 tet 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 4M$
• • • •
or . assets! the inde procedure re,uires/ . ;eta ca"cu"ations/ 9i . 0ari 0ariance ance of the errors ca"cu"ations/ $(:i) 1 <pected return of the market estimate/ <(r M) 1 0ariance 0ariance of the market estimate/ M$ The tota" is $ . 5 $ n . 6 $7! then the num'er of estimates is 7= 5 $ 6 7$ n addition! the "arge num'er of estimates re,uired for the Markowitz procedure can resu"t in "arge aggregate estimation errors when imp"ementing the procedure# ¬her advantage of using an inde mode" is that the re"ationship 'etween stocks and an inde (measured 'y 9i) may 'e more sta'"e than the re"ationships 'etween stocks (measured 'y i4)#
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The disadvantage of the inde mode" arises from the mode">s assumption that return residua"s are uncorre"ated and therefore can 'e ignored# This assumption wi"" 'e incorrect if the inde emp"oyed omits a significant risk factor# or eamp"e! we know that the sing"e-inde market mode" (essentia""y the &PM) seems to omit risk-factors associated with market-cap and re"ative price (growth or va"ue stock)# n this case! the errors are correlated # Therefore the math that a""ows for the ca"cu"ate o f i$ 6 9i$M$ 5 $(:i) and i4 6 9i9 4M$ does not work# /. What "s the ,as"& trade0!! )hen de'art"ng !r$ '+re "nde#"ng "n !avr ! an a&t"ve%- $anaged 'rt!%"
The trade-off 'etween a managed portfo"io (P) and a 'enchmark inde (;) is the pro'a'i"ity of superior performance (<(r P 2 r ;)) versus certainty of additiona" management fees# . H) des the $agn"t+de ! !"r$0s'e&"!"& r"s( a!!e&t the e#tent t )h"&h an a&t"ve "nvestr )"%% ,e )"%%"ng t de'art !r$ an "nde#ed 'rt!%"
n other words/ ?i"" high firm-specific risk keep an active investor in a diversified indeed portfo"io or wou"d the investor prefer an individua" stock@ ncreased firm-specific risk ($(:i)) reduces the etent to which an active investor wi"" 'e wi""ing to depart from an indeed portfo"io to 'uy individua" stock# The tota" risk for an individua" stock is the sum of the inde risk (the market inde M) and the firm specific risk/ i$ 6 9i$M$ 5 $(:i) The tota" risk for the market inde is M$ Aeca"" that the investor is on"y compensated for incu rring inde risk (market risk) and is not compensated for incurring firm specific risk! $(:i)! since other investors can easi"y and without cost e"iminate that risk through diversification# Therefore the we""-diversified (inde-ho"ding) investor wi"" pay more for the stock! decreasing the stocks return to compensate on"y for the inde risk# The greater the firm specific risk! the sma""er the portion of total risk the ho"der of one stock is compensated# 2. Wh- d )e &a%% a%'ha a 3nn$ar(et4 ret+rn 're$"+$ Wh- are h"gh0a%'ha st&(s des"ra,%e "nvest$ents !r a&t"ve 'rt!%" $anagers W"th a%% ther 'ara$eters he%d !"#ed5 )hat )+%d ha''en t a 'rt!%"6s Shar'e rat" as the a%'has ! "ts &$'nent se&+r"t"es "n&reased
The tota" risk premium is <(r) 2 r f 6 α 5 βB<(r M) 2 r fC # Sharpe ratio 6 B<(r) 2 r f C3
2
&"pha is ca""ed a DnonmarketE return premium 'ecause it is the portion of the risk premium that is independent of market performance# t is the risk-ad4usted ecess return# &n increase in α wi"" increase <(r) 2 r f 'ut wi"" not increase ! so as a"pha increases! the Sharpe ratio increases# Since a portfo"io>s a"pha is the weighted average of the individua" securities> a"phas! ho"ding a"" other parameters fied! an increase in any one security>s a"pha increases the portfo"io>s Sharpe ratio# 7. A 'rt!%" $anage$ent rgan"*at"n ana%-*es 9 st&(s and &nstr+&ts a $ean0var"an&e e!!"&"ent 'rt!%" +s"ng n%- these 9 se&+r"t"es. a; H) $an- est"$ates ! e#'e&ted ret+rns5 var"an&es5 and &var"an&es are needed t 't"$"*e th"s 'rt!%"
n 6 += estimates of means n 6 += estimates of variances (n$ 2 n)3$ 6 1!FF= estimates of covariances 1!GH= tota" estimates ,; I! ne &+%d sa!e%- ass+$e that st&( $ar(et ret+rns &%se%- rese$,%e a s"ng%e0"nde# str+&t+re5 h) $an- est"$ates )+%d ,e needed
n a sing"e inde mode"/ ri − rf = α i + β i (r M – rf ) + e i <,uiva"ent"y! using ecess returns/ R i = α i + β i R M + e i The variance of the rate of return on each stock can 'e decomposed into the components/ The variance for each stock in re"ation to the common market factor 6 9i$M$ The variance for each stock due to firm specific events 6 $(:i) $
Cov( r i ! r j ) = β i β jσ M
n this mode"/ The num'er of parameter estimates is/ n 6 += estimates of the mean <(r i) n 6 += estimates of the sensitivity coefficient βi n 6 += estimates of the firm-specific variance $(:i) 1 estimate of the market mean <(r M ) $ σ 1 estimate of the market variance M Therefore! in tota"! 1G$ estimates# The sing"e inde mode" reduces the tota" num'er of parameter estimates from 1!GH= to 1G$# . The $ar(et "nde# has a standard dev"at"n ! //< and the r"s(0!ree rate "s 8<. The !%%)"ng are est"$ates !r t) st&(s: St&( A
Er; 1<
="r$0S'e&"!"& Standard Dev"at"n > ?@ "; 9<
9.8
3
B
18<
1./
29<
a; What are the standard dev"at"ns ! st&(s A and B
The standard deviation of each individua" stock is given 'y/ i 6 B9i$M$ 5 $(:i)CI & 6 B(=#G)$(=#$$)$ 5 (=#8=)$CI 6 8%#FGJ ; 6 B(1#$)$(=#$$)$ 5 (=#%=)$CI 6 %F#H8J (b) Suppose that we were to construct a portfolio with following proportions: Stock A = 30%, Stock B = 45% and T-bills = 20%. Compute the expected return, standard deviation, beta, and nonsystematic standard deviation of the portfolio.
E(rP ) = wAE(rA ) + wBE(rB ) + wf rf = (0.30)(0.13) + (0.45)(0.18) + (0.25)(0.08) = 14.00% 9P 6 w&9& 5 w;9; 5 wf 9 f 6 (0.30)(0.8) + (0.45)(1.2) + 0 = 0.78 p$ 6 9 p$M$ 5 $(: p) $ $ $ σ β σ P M where is the systematic component and (e P ) is the nonsystematic component.
Since the residuals (:i ) are uncorrelated, the non-systematic variance is: $(: p) 6 w&$$(:&) 5 w;$$(:;) 5 wf $$(:f ) 6 (=#8=)$(=#8=)$ 5 (=#%7)$(=#%=)$ 5 = 6 =#=%=7 The residual (non-systematic) standard deviation of the portfolio is: (: p) 6 (0.0405) 1/2 = 20.12% The total variance of the portfolio is then: $ $ $ I $ $ I p 6 B9 p M 5 (: p)C 6 B(=#FG) (=#$$) 5 =#=%=7C 6 =#$+%7 6 $+#%7J
. Cns"der the !%%)"ng t) regress"n %"nes !r st&(s A and B "n the !%%)"ng !"g+re.
4
a; Wh"&h st&( has h"gher !"r$0s'e&"!"& r"s(
The two figures depict the Security Characteristic Line (SK) for each stock# Stock & has higher firm-specific risk 'ecause the deviations of the o'servations from the SK are "arger for Stock & than for Stock ;# Leviations are measured 'y the vertica" distance 'etween each point and the "ine# ,; Wh"&h st&( has greater s-ste$at"& $ar(et; r"s(
Market risk (9) is the s"ope of the SK and measures of systematic risk# The SK for Stock ; is steeper so Stock ;>s systematic risk is greater# &; Wh"&h st&( has h"gher R /
A $ (or the s,uared corre"ation coefficient) is the ratio of the explained variance of the stock>s return to total variance# Total variance is the sum of the explained variance p"us the unexplained variance (the stock>s residua" variance)/ 2
2
2
2
β i σ M Explained Variance βi σ M Ri = = = 2 2 2 2 Total Variance σ i β i σ M + σ ( ε i ) 2
$
$
Since ep"ained variance for stock ; ( β ; σ M ) is greater than ep"ained variance for stock & (9; 9&) and residua" variance for stock ; (σ$(:; )) is sma""er! ;>s A $ is higher than Stock &>s# d; Wh"&h st&( has h"gher a%'ha
&"pha is the intercept of the SK with the vertica" ais (the e pected return ais)# Stock & has a sma"" positive a"pha# Stock ; has a negative a"pha# Stock &>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. Cns"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 ep"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 tta% 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 tet-'ook eercise 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
7
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 <ce") and averaging it with 1#=! using the weights of $38 and 138 which moves it toward 1#=# The motivation for ad4usting 'etas toward 1#= is that over time! as 'usinesses grow! their 'etas move toward the market 'eta of 1#=# &d4usted ;eta at time t 6 9 t 6 (138)(1#=) 5 ($38)9t-1 6 (138)(1#=) 5 ($38)(1#$%) 6 1#1+ ,; S+''se that -+ est"$ate the !%%)"ng regress"n des&r","ng the ev%+t"n ! ,eta ver t"$e: t > 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.
9