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投资学第10版习题答案08

投资学第10版习题答案08
投资学第10版习题答案08

CHAPTER 8: INDEX MODELS

PROBLEM SETS

1. The advantage of the index model, compared to the Markowitz procedure, is the

vastly reduced number of estimates required. In addition, the large number of

estimates required for the Markowitz procedure can result in large aggregate

estimation errors when implementing the procedure. The disadvantage of the index model arises from the model’s assumption that return residuals are uncorrelated.

This assumption will be incorrect if the index used omits a significant risk factor.

2. The trade-off entailed in departing from pure indexing in favor of an actively

managed portfolio is between the probability (or the possibility) of superior

performance against the certainty of additional management fees.

3. The answer to this question can be seen from the formulas for w0 (equation 8.20)

and w* (equation 8.21). Other things held equal, w0 is smaller the greater the

residual variance of a candidate asset for inclusion in the portfolio. Further, we see that regardless of beta, when w0 decreases, so does w*. Therefore, other things

equal, the greater the residual variance of an asset, the smaller its position in the optimal risky portfolio. That is, increased firm-specific risk reduces the extent to which an active investor will be willing to depart from an indexed portfolio.

4. The total risk premium equals: α + (β × Market risk premium). We call alpha a

nonmarket return premium because it is the portion of the return premium that is independent of market performance.

The Sharpe ratio indicates that a higher alpha makes a security more desirable.

Alpha, the numerator of the Sharpe ratio, is a fixed number that is not affected by the standard deviation of returns, the denominator of the Sharpe ratio. Hence, an increase in alpha increases the Sharpe ratio. Since the portfolio alpha is the

portfolio-weighted average of the securities’ alphas, then, holding all other

parameters fixed, an increase in a security’s alpha results in an increase in the

portfolio Sharpe ratio.

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

n = 60 estimates of means

n = 60 estimates of variances

770,12

2=-n n estimates of covariances Therefore, in total: 890,12

32=+n n estimates

b. In a single index model: r i - r f = α i + β i (r M – r f ) + e i

Equivalently, using excess returns: R i = α i + β i R M + e i

The variance of the rate of return can be decomposed into the components:

(l) The variance due to the common market factor: 22M i σβ

(2) The variance due to firm specific unanticipated events: )(σ2i e

In this model: σββ),(Cov j i j i r r =

The number of parameter estimates is:

n = 60 estimates of the mean E (r i )

n = 60 estimates of the sensitivity coefficient β i

n = 60 estimates of the firm-specific variance σ2(e i )

1 estimate of the market mean E (r M )

1 estimate of the market variance 2M σ

Therefore, in total, 182 estimates.

The single index model reduces the total number of required estimates from 1,890 to 182. In general, the number of parameter estimates is reduced from:

)23( to 232+???

? ??+n n n

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

2/1222)](β[σi M i i e σσ+=

Since βA = 0.8, βB = 1.2, σ(e A ) = 30%, σ(e B ) = 40%, and σM = 22%, we get:

σA = (0.82 × 222 + 302 )1/2 = 34.78%

σB = (1.22 × 222 + 402 )1/2 = 47.93%

b. The expected rate of return on a portfolio is the weighted average of the

expected returns of the individual securities:

E(r P ) = w A × E(r A ) + w B × E(r B ) + w f × r f

E(r P ) = (0.30 × 13%) + (0.45 × 18%) + (0.25 × 8%) = 14%

The beta of a portfolio is similarly a weighted average of the betas of the

individual securities:

βP = w A × βA + w B × βB + w f × β f

βP = (0.30 × 0.8) + (0.45 × 1.2) + (0.25 × 0.0) = 0.78

The variance of this portfolio is:

)(σβσ2222P M

P P e +=σ where 22σβM

P is the systematic component and )(2P e σis the nonsystematic component. Since the residuals (e i ) are uncorrelated, the nonsystematic

variance is:

222222

2()()()()P A A B B f f e w e w e w e σσσσ=?+?+?

= (0.302 × 302 ) + (0.452 × 402 ) + (0.252 × 0) = 405

where σ2(e A ) and σ2(e B ) are the firm-specific (nonsystematic) variances of

Stocks A and B, and σ2(e f ), the nonsystematic variance of T-bills, is zero. The

residual standard deviation of the portfolio is thus:

σ(e P ) = (405)1/2 = 20.12%

The total variance of the portfolio is then:

47.699

405)2278.0(σ222=+?=P The total standard deviation is 26.45%.

7. a. The two figures depict the stocks’ security characteri stic lines (SCL). Stock A

has higher firm-specific risk because the deviations of the observations from

the SCL are larger for Stock A than for Stock B. Deviations are measured by

the vertical distance of each observation from the SCL.

b. Beta is the slope of the SCL, which is the measure of systematic risk. The SCL

for Stock B is steeper; hence Stock B’s systematic risk is greater.

c. The R 2 (or squared correlation coefficient) of the SCL is the ratio of the

explained variance of the stock’s return to total variance, and the total

variance is the sum of the explained variance plus the unexplained variance

(the stock’s residual variance):

)(σσβσβ222222

i M i M i e R += Since the explained variance for Stock B is greater than for Stock A (the explained variance is 22σβM B , which is greater since its beta is higher), and its residual variance 2()B e σ is smaller, its R 2 is higher than Stock A’s.

d.

Alpha is the intercept of the SCL with the expected return axis. Stock A has a small positive alpha whereas Stock B has a negative alpha; hence, Stock A’s alpha is larger.

e.

The correlation coefficient is simply the square root of R 2, so Stock B’s correlation with the market is higher.

8.

a.

Firm-specific risk is measured by the residual standard deviation. Thus, stock A has more firm-specific risk: 10.3% > 9.1%

b.

Market risk is measured by beta, the slope coefficient of the regression. A has a larger beta coefficient: 1.2 > 0.8

c.

R 2 measures the fraction of total variance of return explained by the market return. A’s R 2 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 ): ()(1)A f M f A f M

r r r r r r r αβαββ-=+?-?

=+?-+?

The intercept is now equal to: (1)1%(1 1.2)f f r r αβ+?-=+?-

Since r f = 6%, the intercept would be: 1%6%(1 1.2)1% 1.2%0.2%+-=-=-

9. The standard deviation of each stock can be derived from the following

equation for R 2:

==2222σσβi M i i

R Explained variance Total variance Therefore:

%

30.31σ98020.0207.0σβσ2

22222==?==A A M A A

R For stock B:

%28.69σ800,412

.0202.1σ2

22==?=B B

10. The systematic risk for A is:

22220.7020196A M βσ?=?=

The firm-specific risk of A (the residual variance) is the difference between

A’s total risk and its systematic risk:

980 – 196 = 784

The systematic risk for B is:

22221.2020576B M βσ?=?=

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

4,800 – 576 = 4,224

11. The covariance between the returns of A and B is (since the residuals are assumed

to be uncorrelated):

33640020.170.0σββ)(Cov 2

=??==M B A B A ,r r The correlation coefficient between the returns of A and B is:

155.028

.6930.31336σσ),(Cov ρ=?==

B A B A AB r r

12. Note that the correlation is the square root of R 2:2ρR =

1/2,1/2,()0.2031.3020280

()0.1269.2820480A M A M B M B M Cov r r Cov r r ρσσρσσ==??===??=

13. For portfolio P we can compute:

σP = [(0.62 × 980) + (0.42 × 4800) + (2 × 0.4 × 0.6 × 336)]1/2 = [1282.08]1/2 = 35.81% βP = (0.6 × 0.7) + (0.4 × 1.2) = 0.90

958.08400)(0.901282.08

σβσ)(σ22222=?-=-=M P P P e Cov(r P ,r M ) = βP 2σM =0.90 ×

400=360 This same result can also be attained using the covariances of the individual stocks with the market:

Cov(r P ,r M ) = Cov(0.6r A + 0.4r B , r M ) = 0.6 × Cov(r A , r M ) + 0.4 × Cov(r B ,r M )

= (0.6 × 280) + (0.4 × 480) = 360

14. Note that the variance of T-bills is zero, and the covariance of T-bills with any asset

is zero. Therefore, for portfolio Q:

[][]%

55.21)3603.05.02()4003.0()08.282,15.0(),(Cov 2σσσ2/1222

/12

222=???+?+?=???++=M P M P M M P P Q r r w w w w (0.50.90)(0.31)(0.200)0.75Q P P M M w w βββ=+=?+?+?=

52.239)40075.0(52.464σβσ)(σ22222=?-=-=M Q Q Q e

30040075.0σβ),(Cov 2

=?==M Q M Q r r

15. a. Beta Books adjusts beta by taking the sample estimate of beta and averaging it

with 1.0, using the weights of 2/3 and 1/3, as follows:

adjusted beta = [(2/3) × 1.24] + [(1/3) × 1.0] = 1.16

b.

If you use your current estimate of beta to be βt –1 = 1.24, then βt = 0.3 + (0.7 × 1.24) = 1.168

16. For Stock A:

[()].11[.060.8(.12.06)]0.2%A A f A M f r r r r αβ=-+?-=-+?-=

For stock B:

[()].14[.06 1.5(.12.06)]1%B B f B M f r r r r αβ=-+?-=-+?-=-

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

17. a.

Alpha (α)

Expected excess return αi = r i – [r f + βi × (r M – r f ) ]

E (r i ) – r f αA = 20% – [8% + 1.3 × (16% – 8%)] = 1.6% 20% – 8% = 12% αB = 18% – [8% + 1.8 × (16% – 8%)] = – 4.4% 18% – 8% = 10% αC = 17% – [8% + 0.7 × (16% – 8%)] = 3.4% 17% – 8% = 9% αD = 12% – [8% + 1.0 × (16% – 8%)] = – 4.0% 12% – 8% = 4%

Stocks A and C have positive alphas, whereas stocks B and D have

negative alphas.

The residual variances are:

σ2(e A ) = 582 = 3,364

σ2(e B ) = 712 = 5,041

σ2(e C ) = 602 = 3,600

σ2(e D ) = 552 = 3,025

b. To construct the optimal risky portfolio, we first determine the optimal active

portfolio. Using the Treynor-Black technique, we construct the active portfolio: 0.000476 –0.6142 Be unconcerned with the negative weights of the positive α stocks —the entire

active position will be negative, returning everything to good order.

With these weights, the forecast for the active portfolio is:

α = [–0.6142 × 1.6] + [1.1265 × (– 4.4)] – [1.2181 × 3.4] + [1.7058 × (– 4.0)]

= –16.90%

β = [–0.6142 × 1.3] + [1.1265 × 1.8] – [1.2181 × 0.70] + [1.7058 × 1] = 2.08

The high beta (higher than any individual beta) results from the short positions

in the relatively low beta stocks and the long positions in the relatively high

beta stocks.

σ2(e ) = [(–0.6142)2×3364] + [1.12652×5041] + [(–1.2181)2×3600] + [1.70582×3025]

= 21,809.6

σ (e ) = 147.68%

The levered position in B [with high σ2(e )] overcomes the diversification

effect and results in a high residual standard deviation. The optimal risky

portfolio has a proportion w * in the active portfolio, computed as follows:

2022/()

.1690/21,809.60.05124[()]/.08/23

M f M e w E r r ασσ-===-- The negative position is justified for the reason stated earlier.

The adjustment for beta is:

0486.0)

05124.0)(08.21(105124.0)β1(1*00-=--+-=-+=w w w Since w * is negative, the result is a positive position in stocks with positive

alphas and a negative position in stocks with negative alphas. The position in

the index portfolio is:

1 – (–0.0486) = 1.0486

c. To calculate the Sharpe ratio for the optimal risky portfolio, we compute the

information ra tio for the active portfolio and Sharpe’s measure for the market

portfolio. The information ratio for the active portfolio is computed as follows:

A = ()

e ασ = –16.90/147.68 = –0.1144 A 2 = 0.0131 Hence, the square o

f the Sharpe ratio (S) of the optimized risky portfolio is:

1341.00131.02382222=+??

? ??=+=A S S M S = 0.3662

Compare this to the market’s Sharpe ratio:

S M = 8/23 = 0.3478 → A difference of: 0.0184

The only moderate improvement in performance results from only a small

position taken in the active portfolio A because of its large residual variance.

d. To calculate the makeup of the complete portfolio, first compute the beta, the

mean excess return, and the variance of the optimal risky portfolio:

βP = w M + (w A × βA ) = 1.0486 + [(–0.0486) ? 2.08] = 0.95

E (R P ) = αP + βP E (R M ) = [(–0.0486) ? (–16.90%)] + (0.95 × 8%) = 8.42%

()

94.5286.809,21)0486.0()2395.0()(σσβσ222222=?-+?=+=P M P P e %00.23σ=P

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

5685.094

.5288.201.042.8=??=y In contrast, with a passive strategy: 5401.0238.201.082=??=

y →A difference of: 0.0284 The final positions are (M may include some of stocks A through D):

Bills 1 – 0.5685 = 43.15%

M 0.5685 ? l.0486 = 59.61

A 0.5685 ? (–0.0486) ? (–0.6142) = 1.70

B 0.5685 ? (–0.0486) ? 1.1265 = – 3.11

C 0.5685 ? (–0.0486) ? (–1.2181) = 3.37

D 0.5685 ? (–0.0486) ? 1.7058 = – 4.71

(subject to rounding error) 100.00%

18. a. I f a manager is not allowed to sell short, he will not include stocks with negative alphas in his portfolio, so he will consider only A and C:

The forecast for the active portfolio is:

α = (0.3352 × 1.6) + (0.6648 × 3.4) = 2.80%

β = (0.3352 × 1.3) + (0.6648 × 0.7) = 0.90

σ2(e) = (0.33522 × 3,364) + (0.66482 × 3,600) = 1,969.03

σ(e) = 44.37%

The weight in the active portfolio is:

0940.023/803.969,1/80.2σ/)()(σ/α2220===M

M R E e w Adjusting for beta:

0931.0]

094.0)90.01[(1094.0w )1(1w *w 00=?-+=β-+= The information ratio of the active portfolio is:

2.800.0631()44.37

A e ασ=== Hence, the square of the Sharpe ratio is: 22

280.06310.125023S ??=+= ??? Therefore: S = 0.3535

The market’s Sharpe ratio is: S M = 0.3478

When short sales are allowed (Problem 17), the manager’s Sharpe ratio is

higher (0.3662). The reduction in the Sharpe ratio is the cost of the short sale

restriction.

The characteristics of the optimal risky portfolio are:

222222(10.0931)(0.09310.9)0.99

()()(0.0931 2.8%)(0.998%)8.18%

()(0.9923)(0.09311969.03)535.54

23.14%P M A A P P P M P P M P P w w E R E R e ββαβσβσσσ=+?=-+?==+?=?+?==?+=?+?== With A = 2.8, the optimal position in this portfolio is:

5455.054

.5358.201.018.8y =??= The final positions in each asset are:

Bills

1 – 0.5455 = 45.45% M

0.5455 ? (1 - 0.0931) = 49.47 A

0.5455 ? 0.0931 ? 0.3352 = 1.70 C

0.5455 ? 0.0931 ? 0.6648 = 3.38

100.00

b. The mean and variance of the optimized complete portfolios in the

unconstrained and short-sales constrained cases, and for the passive strategy are:

E (R C )

2σC Unconstrained

0.5685 × 8.42% = 4.79 0.5685 × 528.94 = 170.95 Constrained

0.5455 × 8.18% = 4.46 0.54552 × 535.54 = 159.36 Passive 0.5401 × 8.00% = 4.32 0.54012 × 529.00 = 154.31 The utility levels below are computed using the formula: 2σ005.0)(C C A r E -

Unconstrained

8% + 4.79% – (0.005 × 2.8 × 170.95) = 10.40% Constrained

8% + 4.46% – (0.005 × 2.8 × 159.36) = 10.23% Passive 8% + 4.32% – (0.005 × 2.8 × 154.31) = 10.16%

19. All alphas are reduced to 0.3 times their values in the original case. Therefore, the

relative weights of each security in the active portfolio are unchanged, but the alpha of the active portfolio is only 0.3 times its previous value: 0.3 × -16.90% = -5.07% The investor will take a smaller position in the active portfolio. The optimal risky portfolio has a proportion w * in the active portfolio as follows:

2022/()0.0507/21,809.60.01537()/0.08/23

M f M e w E r r ασσ-===-- The negative position is justified for the reason given earlier. The adjustment for beta is:

0151.0)]

01537.0()08.21[(101537.0)β1(1*00-=-?-+-=-+=w w w Since w * is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with negative alphas. The position in the index portfolio is: 1 – (–0.0151) = 1.0151

To calculate the Sharpe ratio for the optimal risky portfolio we compute the information ratio for the active portfolio and the Sharpe ratio for the market portfolio. The

information ratio of the active portfolio is 0.3 times its previous value: A = 5.07()147.68

e ασ-== –0.0343 and A 2 =0.00118 Hence, the square o

f the Sharpe ratio of the optimized risky portfolio is:

S 2 = S 2M + A 2 = (8%/23%)2 + 0.00118 = 0.1222

S = 0.3495

Compare this to the market’s Sharpe ratio: S M =

8%23%

= 0.3478 The 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 improvement in the squared Sharpe ratio by a factor of: 0.32 = 0.09

20. If each of the alpha forecasts is doubled, then the alpha of the active portfolio will

also double. 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 of the 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 -square = SM -square + (4 × IR )

Compared to the previous S -square, the difference is: 3IR

Now you can embark on the calculations to verify this result.

CFA PROBLEMS

1. The regression results provide quantitative measures of return and risk based on

monthly returns over the five-year period.

βfor 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, R2 was 0.35, indicating closeness of fit to the linear regression greater than the value for a typical stock.

βfor 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 7.3%, on average, for XYZ independent of market return.

Residual risk was 21.45%, half again as much as ABC’s, ind icating 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 R2 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: 0.702 = 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.

投资学10版习题答案CH18

CHAPTER 18: EQUITY VALUATION MODELS PROBLEM SETS 1. Theoretically, dividend discount models can be used to value the stock of rapidly growing companies that do not currently pay dividends; in this scenario, we would be valuing expected dividends in the relatively more distant future. However, as a practical matter, such estimates of payments to be made in the more distant future are notoriously inaccurate, rendering dividend discount models problematic for valuation of such companies; free cash flow models are more likely to be appropriate. At the other extreme, one would be more likely to choose a dividend discount model to value a mature firm paying a relatively stable dividend. 2. It is most important to use multistage dividend discount models when valuing companies with temporarily high growth rates. These companies tend to be companies in the early phases of their life cycles, when they have numerous opportunities for reinvestment, resulting in relatively rapid growth and relatively low dividends (or, in many cases, no dividends at all). As these firms mature, attractive investment opportunities are less numerous so that growth rates slow. 3. The intrinsic value of a share of stock is the individual investor’s assessment of the true worth of the stock. The market capitalization rate is the market consensus for the required rate of return for the stock. If the intrinsic value of the stock is equal to its price, then the market capitalization rate is equal to the expected rate of return. On the other hand, if the individual investor believes the stock is underpriced (i.e., intrinsic value > price), then that investor’s expected rate of return is greater than the market capitalization rate. 4. First estimate the amount of each of the next two dividends and the terminal value. The current value is the sum of the present value of these cash flows, discounted at 8.5%. 5. The required return is 9%. $1.22(1.05) 0.05.09,or 9% $32.03 k ? =+= 6. The Gordon DDM uses the dividend for period (t+1) which would be 1.05.

投资学期末试题及答案C卷

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