Chapter 8 Fin Questions

Stock A E(r)20 beta 1.3 STDV 58 Stock B E(r)18 beta 1.8 STDV 71 Stock C E(r)17 beta 0.7 STDV 60 Stock D E(r)12 beta 1.0 STDV 55 ================== T-billsE(r) 8 STDV0 Passive equity portfolio E(r) 16 STDV23 Calculate expected excess returns, alpha values, and residual variances for these stocks.

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 betas of the individual securities:

βP = wA × βA + wB × βB + wf × β f

Suppose that we were to construct a portfolio with proportions: Stock A: .30 Stock B: .45 T-bills: .25 computer non-systematic variance

= (0.30^2 × 30^2 ) + (0.45^2 × 40^2 ) + (0.25^2 × 0) = 405 where σ2(eA ) and σ2(eB ) 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(eP ) = (405)^1/2 = 20.12%

RA = 3% +.7RM + eA RB = -2% +1.2RM +eB Qm = 20%; R-squareA = .20; R-squareB = .12 Break down the variance of each stock to the systematic and firm-specific components For B

Beta B^2 x stdv m^2 = 1.20 ^2 x 20 ^2 = 576 B's firm-specific risk (residual variance) is: 4800 - 576 = 4224 STDV B^2 = ( 1.2^2 x 20^2 / 0.12 ) = 4800

A stock recently has been estimated to have a beta of 1.24: What will a beta book compute as the "adjusted beta" of this stock?

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

What is the covariance between each stock and the market index?

Cov(ri, rm ) = corr x stdv i x stdv m

Consider the two (excess return) index model regression results for A and B: RA = 1% + 1.2 RM R -square .576 Residual standard deviation = 10.3% ====== RB - 2% + .8 RM R -square .436 Residual standard deviation 9.1% Which stock has more firm-specific risk?

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

A stock recently has been estimated to have a beta of 1.24: Suppose that you estimate the following regression describing the evolution of beta over time: beta t= .3 + .7beta t-1 What would be your predicted beta for next year?

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

A portfolio management organization analyzes 60 stocks and constructs a mean-variance efficient portfolio using only these 60 securities. If one could safely assume that stock market returns closely resemble a single-index structure, how many estimates would be needed?

In a single index model: ri - rf = α i + β i (r M - rf ) + 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: The variance due to the common market factor Bi^2stdvm^2 The variance due to firm specific unanticipated events STDV^2(ei) In this model Cov(ri,rj) =BiBjSTDV The number of parameter estimates is: n = 60 estimates of the mean E(ri ) n = 60 estimates of the sensitivity coefficient β i n = 60 estimates of the firm-specific variance σ2(ei ) 1 estimate of the market mean E(rM ) 1 estimate of the market variance 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: (n^2 +3n / 2) to (3n+2)

Consider the two (excess return) index model regression results for A and B: RA = 1% + 1.2 RM R -square .576 Residual standard deviation = 10.3% ====== RB - 2% + .8 RM R -square .436 Residual standard deviation 9.1% Which has greater market risk?

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

RA = 3% +.7RM + eA RB = -2% +1.2RM +eB Qm = 20%; R-squareA = .20; R-squareB = .12 What is the covariance between each stock and the market index?

Note that the correlation is the square root of R^2 corr = sqrt (r^2) Cov(ra, rm ) = corr x stdv a x stdv m = .20 ^1/2 x 31.30 x 20 = 280 Cov(ra, rm ) = corr x stdv a x stdv m = 0.12^1/2 x 69.28 x 20 = 480

Consider the two (excess return) index model regression results for A and B: RA = 1% + 1.2 RM R -square .576 Residual standard deviation = 10.3% ====== RB - 2% + .8 RM R -square .436 Residual standard deviation 9.1% For which stock does market movement explain a greater fraction of return variability?

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

Consider the two (excess return) index model regression results for A and B: RA = 1% + 1.2 RM R -square .576 Residual standard deviation = 10.3% ====== RB - 2% + .8 RM R -square .436 Residual standard deviation 9.1% If r f were constant at 6% and the regression had been run using total rather than excess returns, what would have been the regression intercept for stock A?

Rewriting the SCL equation in terms of total return (r) rather than excess return (R): ra-rf = alpha + beta x (rm -rf) ---> ra = alpah + rf x (1-beta) +beta x rm The intercept is now equal to: alpaha +rf X (1-beta) =1% +rf x (1-1.2) Since rf = 6%, the intercept would be: 1% + 6% (1-1.2) = 1% -1.2% = - .2%

RA = 3% +.7RM + eA RB = -2% +1.2RM +eB Qm = 20%; R-squareA = .20; R-squareB = .12 What is the standard deviation For B

STDV B^2 = Beta b^2 stdvm^2 / Rb^2 STDV B^2 = ( 1.2^2 x 20^2 / 0.12 ) = 4800 STDV B = 69.28 %

standard deviation of each individual stock

STDV i = [ Beta i^2 x Stdv m ^2 + STDV^2 x (ei)] ^1/2

The total variance of the portfolio

STDV p^2 = (0.78^2 x 22^2 ) + 405 = 699.47 405 = non systematic variance 0.78 is beta

Suppose that we were to construct a portfolio with proportions: Stock A: .30 Stock B: .45 T-bills: .25 variance of this portfolio is:

STDVp^2 = Beta p^2 x STDVm^2 +STDV^2 (ep)

The following are estimates for two stocks A E(r)13% beta 0.8 firm specific STDV 30% B E(r)18 beta 1.2 firm specific STDV 40% The market index has a standard deviations of 22% and the risk-free rate is 8%. What are the standard deviations of stocks A and B?

Since βA = 0.8, βB = 1.2, σ(eA ) = 30%, σ(eB ) = 40%, and σM = 22%, we get: σA = (0.82 × 222 + 302 )^1/2 = 34.78% σB = (1.22 × 222 + 402 )^1/2 = 47.93%

How does the magnitude of firm-specific risk affect the extent to which an active investor will be willing to depart from an indexed portfolio?

The answer to this question can be seen from the formulas for w 0 (equation 8.20) and w* (equation 8.21). Other things held equal, w 0 is smaller the greater the residual variance of a candidate asset for inclusion in the portfolio. Further, we see that regardless of beta, when w 0 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.

Based on current dividend yields and expected growth rates, the expected rates of return on stocks A and B are 11% and 14%, respectively. The beta of stock A is .8, while that of stock B is 1.5. The T-bill rate is currently 6%, while the expected rate of return on the S&P 500 index is 12%. The standard deviation of stock A is 10% annually, while that of stock B is 11%. If you currently hold a passive index portfolio, would you choose to add either of these stocks to your holdings?

Stock a alpha a= ra -[rf +beta ax (rm -rf)] =.11 - [.06 +0.8 x (.12 - .06)] = 0.2% alpha B= rb - [rf +beta b x (rm -rf )] = .14 - [.06+1.5 x (.12-.06 )] = -1% Stock A would be a good addition to a well-diversified portfolio. A short position in Stock B may be desirable.

What are the advantages of the index model compared to the Markowitz procedure for obtaining an efficiently diversified portfolio? What are its disadvantages?

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.

Suppose that we were to construct a portfolio with proportions: Stock A: .30 Stock B: .45 T-bills: .25 compute beta

The beta of a portfolio is similarly a weighted average of the betas of the individual securities: βP = wA × βA + wB × βB + wf × β f βP = (0.30 × 0.8) + (0.45 × 1.2) + (0.25 × 0.0) = 0.78

RA = 3% +.7RM + eA RB = -2% +1.2RM +eB Qm = 20%; R-squareA = .20; R-squareB = .12 What are the covariance and correlation coefficient between the two stocks?

The covariance between the returns of A and B is (since the residuals are assumed to be uncorrelated): Cov (ra, rb ) beta a x beta b stdv m^2 = 0.70 x 1.20 x 400 = 336 The correlation coefficient between the returns of A and B is: corr AB = Cov (ra, rb)/ stdv A x stdv b 336/ 31.30 x 69.28 = 0.155

Suppose that we were to construct a portfolio with proportions: Stock A: .30 Stock B: .45 T-bills: .25 Compute the expected return,

The expected rate of return on a portfolio is the weighted average of the expected returns of the individual securities: E(rP ) = wA × E(rA ) + wB × E(rB ) + wf × rf E(rP ) = (0.30 × 13%) + (0.45×18%)+(0.25×8%) = 14%

RA = 3% +.7RM + eA RB = -2% +1.2RM +eB Qm = 20%; R-squareA = .20; R-squareB = .12 What is the standard deviation For A

The standard deviation of each stock can be derived from the following equation for R2: Ri^2 = (Beta i^2 STDVm^2 ) / STDV i^2 = Explained variance / total variance STDV A^2 = Beta a^2 stdvm^2 / Ra^2 = (0.7^2 x 20^2 / .20 ) = 980 STDV a = 31.30 %

RA = 3% +.7RM + eA RB = -2% +1.2RM +eB Qm = 20%; R-squareA = .20; R-squareB = .12 Break down the variance of each stock to the systematic and firm-specific components For A

The systematic risk for A is: betaA^2 x stdv m^2 = 0.70^2 x 20^2 = 196 The firm-specific risk of A (the residual variance) is the difference between A's total risk and its systematic risk: 980 - 196 = 784 980 got from this STDV A^2 = Beta a^2 stdvm^2 / Ra^2 = (0.7^2 x 20^2 / .20 ) = 980

Why do we call alpha a "nonmarket" return premium? Why are high-alpha stocks desirable investments for active portfolio managers? With all other parameters held fixed, what would happen to a portfolio's Sharpe ratio as the alpha of its component securities increased?

The total risk premium equals: alpha + (beta × 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.

What is the basic trade-off when departing from pure indexing in favor of an actively managed portfolio?

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.

RA = 3% +.7RM + eA RB = -2% +1.2RM +eB Qm = 20%; R-squareA = .20; R-squareB = .12 For portfolio P with investment proportions of .60 in A and .40 in B, rework Problems 9, 10, and 12.

This same result can also be attained using the covariances of the individual stocks with the market: Cov(rP,rM ) = Cov(0.6rA + 0.4rB, rM ) = 0.6 × Cov(rA, rM ) + 0.4 × Cov(rB,rM ) = (0.6 × 280) + (0.4 × 480) = 360

A portfolio management organization analyzes 60 stocks and constructs a mean-variance efficient portfolio using only these 60 securities. How many estimates of expected returns, variances, and covariances are needed to optimize this portfolio

To optimize this portfolio one would need: n = 60 estimates of means n = 60 estimates of variances n^2-n / 2 = 1770 estimates of covariances n^2+3n / 2 = 1890 estimates

The correlation coefficient between the returns of A and B is:

corr AB = Cov (ra, rb)/ stdv A x stdv b = answer

Stock A E(r)20 beta 1.3 STDV 58 Stock B E(r)18 beta 1.8 STDV 71 Stock C E(r)17 beta 0.7 STDV 60 Stock D E(r)12 beta 1.0 STDV 55 ================== T-billsE(r) 8 STDV0 Passive equity portfolio E(r) 16 STDV23 The residual variances are:

stdv^2(eA ) = 58^2 = 3,364 stdv^2(eB) = 71^2 = 5,041 stdv^2(eC) = 60^2 = 3,600 stdv^2(eD) = 55^2 = 3,025