CHAPTER 8: INDEXMODELS - Texas Christian Universitysbufaculty.tcu.edu/mann/_Inv II F09/Chapter 8.pdf · CHAPTER 8:INDEXMODELS, PROBLEM SETS 'I. The advantage oftheindex model, compared

CHAPTER 8: INDEX MODELS

,PROBLEM SETS

'I. The advantage of the index model, compared to the Markowitz procedure, is the vastlyreduced number of estimates required. In addition, the large number of estimates requiredfor the Markowitz procedure can result in large aggregate estimation errors whenimplementing the procedure. The disadvantage of the index model arises from the model'sassumption that return residuals are uncorrelated. This assumption will be incorrect if theindex used omits a significant risk factor.' "

2. The trade-off entailed in departing from pure indexing in favor of aq.actively managedportfolio is between the probability (or possibility) of superior performance against thecertainty of additional manag~ment fees.

3. The answer to this question can be seen from the formul~s for WO and w*. Other thingsheld equal, w ° is smaller the greater the re,sidual variance" of a canCtiaate asset for inClusionin the portfolio. Furtner, we see that regardless of beta, when WO decreases, so does w*.Therefore, other things equal, the greater the residual 'variance of an asset; the smaller itsposition in the optimal risky portfolio. That is, ,increased firm-specific risk reduces theextent to which an active investor wi~l be wil~ing to depart from an indexed portfolio.

, .4. The total risk premium equals: ex + (~ x market risk premium). We call alpha a

"nonmarket" return premium because it is the portion 'of the return premium that'isindependent of market performance.

The Sharpe ratio indicates that a higher alpha makes a security more desirable. Alpha, thenumerator of the Sharpe ratio, 'is a fixed number that is not affected by the standarddeviation of returns, the denominator of the Sharpe ratio. Hence, an increase in alphaincreases the Sharpe ratio. Since the portfolio alpha is the portfolio-weighted average ofthe securities' alphas, then, holding all other parameters fixed, an increase in a security'salpha results in an increase in the portfolio Sharpe ratio. .

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5. a. To optimize this portfolio one would need:

n = 60 estimates of means

n = 60 estimates of variances2

n - n = 1,770 estimates of covariances2

Therefore, In total: n 2 +3n = 1,890 estimates. 2

b. In a single index model: rj - rr = <Xj + ~ j (r M- rf) + e j

Equivalently, using excess returns: Rj = <Xj+ ~jRM + ej

Tp.evariance. of the rate of return on each stock can be decomposed into thecomponents.: .

(1) The variance due. to the common market factor: ~;O'~,.(2) The variance due to firm specific unanticipated events: 0'2~ej)

.The number of parameter ~stimates is:

n = 60 estimates of the me~n E(ri )

n = 60 estimates of the sensitivity coefficient ~j

n = 60 estim~tes of the firm-specific variance O'2(~i)

1.estimate of the market mean E(rM)

1 estimate of the market variance O'~

Therefore, in total, 182 estimates.

Thus, the single index model reduces the total number of required parameterestimates from 1,890 to 182: In general, the number of parameter estimates isreduced from:

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6. a. The standard deviation of each individual stock is given by:

aj =[~;a~ +a2(eJf2

Since. ~A= 0.8, ~B= 1.2, a(eA) = 30%, a(eB) = 40%, and aM= 22%, we get:

aA = (0.82 X 222+ 302)112= 34.78%

aB = (1.22 X 222 + 402 )112=:= 47.93%

b. .The expected rate of return on a portfolio is the weighted average of the expected. returns of the individual securities:

E(rp) = wAE(rA) + wBE(rB) + Wfrf

where wA, WB,and Wfare the portfolio weights for Stock A, Stock B, and T-bills,respectively. .

Substituting in the fo~ula we get:

E(rp) = (0.30 x 13) + (0.45 ~ 18) + (0.25 x 8) = 14%. .

The ~eta of a portfolio is similarly a weighted average of the betas of the individualsecurities: .

~p = WA~A+ WB~B+ W/~f

The beta for T-bills (~f ) is zero. The beta for the p~rtfolio is therefore:

~p = (0.30 x 0.8) + (0.45 x 1.2) + 0 = 0.78

The variance of this portfoli? is.:.

a; = ~;a~ +a2 (ep)

. where.~;a~ is the systematic component and a2 (ep) is the nonsystematic component.Since the residuals (ei) are uncorrelated, the non-systematic variance is:

2( ) . 2. 2( ). 2 2() 2 2(. )a ep =wAa eA. +wBa eB +w[a e[. ,

. = (0.302 X 302 ) + (0.452 X402) + (0.252 X 0) = 405 .

where a \eA) and a 2(eB) are the firm-specific (n'onsystematic) variances of Stocks Aand B, and a 2(ef),the non systematic variance of T':'bills, is zero. The residualstandard devia~ion of the portfolio is thus:

a(ep) = (405)112= 20.12%

:The total variance of the portfolio is then:

a; = (0.782 x 222) + 405 = 699.47

The standard deviation is 26.45%.

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7. a. The two figures depict the stocks' security charactGristic lines (SCL). Stock A hashigher firm-specific risk because the deviations of the observations from the SCL arelarger for Stock A than for Stock B. Deviations are measured by the vertical distanceof each observation from the SCL.

8. a.

b. Beta is the slope of the SCL, whichisthe measure of systematic risk. The SCL forStock B is steeper; hence Stock B' s systematic risk is greater. ..

c. Th~ R2 (or squared correlation coefficient) 'of the SCL is the ratio of the explainedvariance of the stock's return to total variance, and the total variance is the' sum of theexplained variance plus the unexplained yariarice (the stock's residual variance):

A2 2R2 = Pi(JM

.. ~f(J~+(J2(eJ

Since the explained variance for Stock B is greater than for Stock A (the explainedvariance is ~~(j~ ' which is greater since its beta is higher), and its residual variance(j2(eB) is smaller, its ~2 is higher than Stock A's.

d. Alpha is the intercept of the SCL with the expected return axis. Stock A has a smallpositive alpha whereas Stock B has a negativ,.r alpha; hence, Stock A's alpha is larger.

e. The correlation coefficient is simply the square root of R2, so Stock B' s correlationwith the market is higher. . . ,

Firm-specific risk is measured by the residual standard deviation~ ThuS, stock A hasmore firm-specific risk: 1O}% > 9:1% '

b. Market risk is measured by beta, the slope coefficient of the regression. A has alarger beta coefficient: 1.2 > 0:8' '. .

c. R2 measures th~ fraction of total variance of.return explained by the'market return.A's R2 is larger than B"s: 0.576 > 0.436 '-

d. Rewriting the SCL equation in terms of total return (r) rather than excess return (R):. .rA- rf = a + 13(rM-:-rr) ~ rA= a + rf (l - 13)+ 13rM

The intercept is now equal to:

a + rf(l - ~) = 1 + rf(1- 1.2)

'Since rf = 6%, the intercept would be: I ---= 1.2 = -0.2%

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9. The standard deviation of each stock can be derived from the followi!lg equation forR2:

R 2 _~;(j~ ...;.Explained variancei - (j2 - Total variance.

'"Therefore:

(j2 = ~i(j~. ~ 0.72X 20

2= 980

A Ri 0.20(jA =31.30%

For stock B:

(j2 = 1.22X 20

2= 4 800

B 0.12 ,.

(jB = 69.28%

10. The systematic risk for A is:

~i(j~ = 0.702 X 202 = 1~6

The firm-specific risk of A (the residual variance) is the difference between A's total riskand"its systematic risk: . .

980 - 196 = 784

The systematic risk for B is:

~~(j~ = 1.202x 202 = 576

B's firm-specific risk (residual variance) is:

4800 - 576 = 4224

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12. Note that the c0l!elation is the square root of R2:p=.JR2,

Cov(rA,fM) = PcrAcrM= 0.201/2 X 31.30 x 20 = 280

Cov(rB,!M) = PcrBcrM~ 0.12112X 69.28 x 20 = 480

13. .for portfolio P we can compute: ,

crp= [(0.62 x 980) + (0042 X 4-800) + (2 x 004 x 0.6 x 336]112= [1282.08]112 = 35.81 %

~p = (0.6 x 0.7)+ (004 x 1.2) = 0.90

cr2(ep) = cr~- ~~cr~ = 1282.08 - (0.902 x 400) = 958.08

Cov(rp,rM) = ~pcr~ =0.90 x 400=360

This same result can also be attained using the covariances of the individual stocks withthe market:

Cov(rp,rM) = Cov(0.6r~ + OArB,rM) = 0.6Cov(rA, rM) + OACov(rB,rM)•. = (0.6 x 280) + (004 x 480) = 360

14. ,Note that the variance of T-bills is zero, and the c~)Varianceof T-bills with any ass'etiszero. Therefore, for portfolio Q:

= [(0.52 xl,282.08)+(0.32 x400)+(2xO.5xO.3x360) r/2 = 21.55%

~Q = wp~p + W M~M= (0.5xO.90)+ (O.3xl) +0 = 0.75

cr2 (eQ) = cr~ -.:~~cr~ = 464.52 ~ (0.752 x400) == 239.52

Cov(rQ,rM) ='~Qcr~ = 0.75x400 = 300

15. a. Merrill Lynch adjusts beta by taking the sample estimate of beta and averaging itwith 1.0, using.the weights of 2/3 and 1/3, as follows:

adjusted beta = [(2/3) x i,24] + [(1/3) x 1.0] := 1.16,

b. If you use your current estimate of beta to be ~t-l :::;1.24, then

~t = 0.3 + (0.7 x 1.24) = 1.16~

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16. For Stock A: .

aA ;,.rA .:...[rr + ~A(rM-rr)] = 11- [6 +0.8(12 - 6)] = 0.2%

For stock B:

aB = 14 - [6 + 1.5(12 -6)]= -1%

Stock A would be a good addition to,a well-diversified portfolio. A short position in Stock .B may be desirable.

17. a.

Alpha (a), Expected excess returnai = fj - [rr+ ~i(rM- rr)] E(n) - rf

a = 20% - [8% + 1'.3(16% - 8%)] = 1.6% 20% - 8% = 12%A •

aB=18% - [8% + 1.8(16% - 8%)] = - 4,4% 18% - 8% = 10%

ac = 17% - [8% + 0.7(16% - 8%)] = 3,4% 17% - 8% = 9%

aD = 12% - [8% +1.0(16% - 8%)] = - 4.0% 12% - 8% = 4%

Stocks A and C have positive alphas, whereas stocks Band D have negative alphas,

The residual variances are:

cr2(eA) = 582 = 3,364

cr2(eB) = ,712 = 5,041

cr2( ec) = 602 = 3~600

cr2(eD) = 552 = 3,025

b. To construct the optimal risky portfolio, we first determine the optimal activepo~folio. ~sing the Treynor-Black techniq~e, we construct the active portfolio:

a 0.1~(e)cr\e) D:1. / ~(e)

A ...0,000476 ....... ........:::0,6144..............B -0.000873 1.1265............................................. ...-------- ....... _----_ ....... __ ...............

C 0.000944 -1.2181-----.-.- .................•.................... ................................................

D -0.001322 1.7058Total -0.000775 1.0000

Do not be concerned that the positive alpha stocks hav~ negative weights andyiceversa. We will see that the entire position in the active portfolio will be negative;returning everything to good order. . .

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With.the.se weights, the forecast for the active portfolio is:'

u = [-0.6142 x 1.6] +"[1.1265 x (- 4.4)]- [1.2181 x 3.4] + [1.7058 x (~4.0)]

=-16.90%

~ = [-0.6142 x 1.3] + [1.1265 x 1.8] -,-[1.2181 x 0.70] + [1.7058 x 1] = 2.08

The high beta (higher than any individual beta) results from the short positions in therelatively low beta stocks and the long positipns in the relatively high beta stocks.

a2(e) = [(-0.6142)2 x 3364]'+ [1.12652 x 5041] + [(-1.2181)2 x 3600] + [1.70582 x 3025]

=: 21,809.6a(e) = 147.68% .

Here, again, the levered position in stock B [with high a2(e)] overcomes thediversification effect, and results in a high residual standard deviation. The optimalrisky portfolio has a proportion w* in the active portfolio, computed as follows:

wo=u/a

2(e) . = -16.90/21,809.6=_0.05124

[E(rM)-rf]/'a~ 8/232 .

The negative position is justif~ed for the reason stated earlier.

The adjustment for beta is:

. w* = . w 0 = - 0.05124 = -0.04861+ (1- ~).w0 . 1+ (1- 2.08)(-0.05124)

. Since w* is negative, the result is a positive position in stocks with positive alphasand a negative position in stocks with negative alphas. The position in the indexportfolio is: ' '.

1 - (-0.0486) = 1.0486 .

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Compare this to the market's Sh':lrpe measure:

SM = 8/23 = 0.3478

The difference is: 0.0184

Note that the only-moderate improvement in performance results from the fact thatonly a small position is taken in the active portfolio A because of its large residualvariance.

d. To calculate the exact makeup of the complete portfolio, we first compute the meanexcess return of the optimal risky portfolio and its variance. The risky portfolio betais given by: .

~p = WM + (WA X ~A) = 1.0486 + [(-O.0~86) x 2.08] = 0.95

E(Rp) = Up + ~pE(RM) = [(-0.0486) x (-16.90%)] + (0.95 x 8%) = 8.42%

cr; = ~;(J~ + (J2(ep) = (0.95 X 23)2 +((-0.04862)x 21,809.6) = 528.94

(Jp = 23.00% .

Since A =2.8, the optimal position in this portfolio is:

y = 8.42 = 0.5685. 0.0Ix2 ..8x528.94

In contrast, with a passive strategy:

8y =-----= 0.54010.0Ix2.8x232

This is a,difference of: 0.0284

The final positions of the complete portfolio are:

1- 0.5685'=0.5685 x 1.0486 =

0.5685 x (-0.0486) x (-0.6142) =0.5685 x (-0.0486) x 1.1265 =0.5685 x (-0.0486) x (-1.2181) =0.5685 x (-0.0486) x 1.7058 =

43.l5%59.61%1.70%

-3.11%3.37%

-4.71%100.00%

[sum is subject to rounding error]

Note that M may include positive proportions of stocks A through D .

BillsM,ABCD

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18. a. If a rrianager is not allowed to sell short he will not include stocks with negativealphas in his portfolio, so he will consider only A and C:

a\e)a ala2(e)a a2( e) "La / a\e)

A 1.6 3,364 0.000476 0.3352C 3.4, 3,600 0.000944 0.6648'

0.001420. 1.0000

The forecast for. the active portfolio is:

a = (0.3352 x 1.6) + (0.6648 x 3.4) = 2.80%

f3= (0.3352 x 1.3) + (0.664.8 x 0.7) = 0.90

a2(e) = (0.33522 x 3,364)+ (0.66482 x 3,6(0) = 1,969.03. ,f": : ~ i .- , •

a(e) = 44.37%

The weight in the active portf?lio is:

w = ala2(e) = 2.80/1,969.03 = 0.0940

o E(R' ) I 2 .. 8 I 232 0,M aM .

Adjusting for beta:

w* = __ w_o__ = 0.094 . = 0.09311+ (1- f3)w° 1+ [(1- 0.90) x 0.094]

The information ratio of the active portfolio is:

A = a la(e) =2.80/44.37 == 0.0631

Hence, the square of Sharpe's measure is: •

S2 = (8/23)2 + 0.06312 = 0.1250

Therefore: S =0.3535

The market's Sharpe measure 'is: SM= 0.3478

When sh'ort sales are allowed (Problem 18), the manager's Sharpe measure is higher(0.3662). The reducti0!1 in the Sharpe measure ,is the cost of the short sale restriction.

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The characteristics' of the optimal risky portfolio are:

~p = WM + WA X ~A = ('1- 0.0931) + (0.0931 x 0.9) = 0.99. .'

E(Rp) = <Xp+~pE(RM) = (0.0931 x 2.8%) + (0.99 x 8%) = 8.18%

(J~ = ~;(J~ + (J2(ep) = (0.99x 23)2 + (0.09312 x 1,969.03) = 535.54

(Jp'= 23.14%

With A = 2.8, the optimal position in this portfolio is: .

8.18y = ------ = 0.54550.0Ix2.8x535.54

The 'final positions i.neach asset are: I

BillsMAC

1- 0.5455 =0.5455 x (1 - 0.0931) =

0.5455 x 0.0931 x 0.3352 =0.5455 x 0.0931 x 0.6648 =

45.45%49047%1'.70%3.38%'

100.00%

b. The mean and variance of the optimized complete portfolios in the unconstrained andshort-sales constrained cases, aJ;ldfor th~ passive strategy are:

UnconstrainedConstrainedPassive

E(Rc)

0.5685 x 8.42 = 4.790.5455 x 8.18 = 4.460.5401 x ~.OO= 4.3~

(J~

0.56852 x 528.94 = 170.950.54552 x 535.54 = 159.360.54012 x 529.00 = 154.31

The utility levels below are computed using the formula: E(rc) - 0.005A(J~

Unconstrained .8 + 4.79 -' (0.005 x 2.8 x 170:95) = 10.40. .

Constrained 8 + 4.46 - (0.005 x 2.8 x 159.36) = 10.23

Passive 8 + 4.32 - (0.005 x 2.8 x i54.31) = 10.16

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19., All alphas are reduced to 0.3 times their values in the original case. Therefore, the relativeweights of each security in the active portfolio are unchanged, but the alpha of the activeportfolio is only 0.3 times its previous value: 0.3 x -16.90% = -5.07%The investor will take a: smaller position in the active portfolio. The optimal risky portfoliohas a proportion w* in the active portfolio as follows:

wo= a/a2(e) ,=-5.07/21,809.6 =-0.01537E(rM -rf)/a~ 8/232

The negative position is j~stified for the reason given earlier..the adjustment for bet~ is:

.w.*=' wo = -0.01537 =-0,01511+ (1- ~)w 0 1+ [(1- 2.08) x (-0.01537)]

Since w* is negative, the result is a positive position in stocks with positive alphas and anegative position in stocks with negative alphas. The p~sition in the index portfolio is:

1 - (-0.0151) = 1.0151

To calculate Sharpe's measure for the optimal risky portfolio we compute the informationratio for the active portfolio and Sharpe's measure for the market portfolio.' Theinformation ratio of the active portfolio is 0.3 times its previous value:

A = a /a(e)= -5.07/147.68= -0.0343 and A2 =0.001i8Hence, the square of Sharpe's measure of the optimizerJ risky portfolio is:

S2 = S2M"+A2 = (8/23)2 +0.00118=; 0.1222

S = 0.3495

Compare this to the market's Sharpe measure: SM= 8/23 = 0.3478The difference is: 0.0017

Note that the reduction of the forecast alphas by a factor of 0,.3 reduced the squared.information ratio and the improyement in the squ~ed Sharpe flltio by a factor of:

20.3 = 0.09

20. If each of th~ alpha fqrecasts is doubled, then the alpha of the active portfolio will alsodouble. Other things equal, the information ratio (IR) of the active portfolio also doubles.The square of the Sharpe ratio for the optimized portfolio (S-square) equals the square ofthe Sharpe ratio for the market index (SM -square) plus the square of the information ratio.Since the information ratio has doubled,~its square quadruples. Therefore: .S-sq\lar~ = SM-square + (4 x IR) .Compared to the previous S-square, the difference is: 3IRNow you can embark on the calculations to verify this result.

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CFA PROBLEMS

1. The regression results provide quantitative measures offeturn and risk based on monthlyreturns over the five-year period. '

f3for ABC was 0.60, considerably less than the averagestock'sf3 of 1.0. This indicatesthat, when the S&P 500 rose or fell by 1 percentage point, ABC's return on average rose orfell by only 0.60 percentage point. Therefore, ABC'-s systematic risk (or market risk) waslow 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 ABCwas -3,2%. ABC's unsystematiC risk (or residual risk), as measured by cree), was '13.02%:For ABC, R2 was 0.35, indicating closeness of fit to the linear regression greater than thevalue for a typical stock. '.f3for XYZ was som~what higher, at 0.97, indicating XYZ's return pattern was very similartothe f3for the market index. Therefore, XYZ stock had average systematic risk for theperiod examined. Alpha for XYZ was positive and quite large, indicating a return ofalmost 7.3%, on average, for XYZindependent of market return. Residual risk was,.21.45%, half again as much as ABC's, indicating a wider scatter of observations aroundthe regression line for XYZ. Correspondingly, the fit of th'e regression model wasconsiderably less than that of ABC, consistent with an R2 of only 0.17.

The effects of including one or the other of these stocks in a diversified portfolio may bequite 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 twobrokerage houses may help the analyst draw inferencys for the future. The three estimatesof ABC's f3are similar, regardless of the sample period of the underlying data. The rangeof these estimates is 0.60 to 0.71, well below the market average f3of 1.0. The threeestimates of XYZ' s ~ vary significantly among the three sources, ranging as high as 1.45forthe weekly data over the most recent two years. One could infer that XYZ's f3for thefuture might be well above 1.0, meaning it might have'somewhat greater systematic riskthan was implied by the monthly regression for the five-year period. .

These stocks appear to have signifi~antly different systematic risk characteristics. If thesestocks 'are added to a diversified p0l't:fo,lio,XYZ will add.more to total volatility.

2. The R2 of the regression is: 0.702 = 0.49 .

Therefore, 51% of total variance is unexplained by the market; this is nonsystematic risk.

3. '9= 3 + f3.(l1'- 3) => f3= 0.75

4. d.

5. b.

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