Solutions to questions in Chapter 8 except those in PS4. The minimum-variance portfolio is found by applying the formula:

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1 Solutions to questions in Chapter 8 except those in PS4 1. The parameters of the opportunity set are: E(r S ) = 20%, E(r B ) = 12%, σ S = 30%, σ B = 15%, ρ =.10 From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(r S,r B ) = ρσ S σ B ]: Bonds Stocks Bonds Stocks The minimum-variance portfolio is found by applying the formula: w Min (S) = σ 2 B Cov(r S,r B ) σ 2 = S + σ2 B 2Cov(r S,r B ) =.1739 w Min (B) = =.8261 The minimum variance portfolio mean and standard deviation are: E(r Min ) = = 13.39% σ Min = [w 2 S σ2 S + w2 B σ2 B + 2w S w B Cov(r S,r B )]1/2 = [( ) + ( ) + ( )] 1/2 = 13.92% 2. % in stocks % in bonds Exp. return Std. Dev 0.00% % % 82.61% minimum variance 20.00% 80.00% % 60.00% % 54.84% tangency portfolio 60.00% 40.00% % 20.00% % 0.00%

2 INVESTMENT OPPORTUNITY SET CML Tangency Pf Efficient frontier of risky assets Min.Var E(r) σ The graph approximates the points: Min. Variance Portf. 13.4% 13.9% Tangency Portfolio The proportion of stocks in the optimal risky portfolio is given by: w S = = [E(r S ) r f ]σ 2 B [E(r B ) r f ]Cov(r S,r B ) [E(r S ) r f ]σ 2 B + [E(r B ) r f ]σ2 S [E(r S ) r f + E(r B ) r f ]Cov(r S,r B ) (20 8)225 (12 8)45 (20 8)225 + (12 8)900 [ ]45 =.4516 w B =

3 The mean and standard deviation of the optimal risky portfolio are: E(r p ) = = 15.61% σ p = [( ) + ( ) + ( )] 1/2 = 16.54% 5. The reward-to-variability ratio of the optimal CAL is: E(r p ) r f = σ p = a. If you require your portfolio to yield a mean return of 14% you can find the corresponding standard deviation from the optimal CAL. The formula for this CAL is: E(r C ) = r f + E(r p) r f σ p σ C = σ C Setting E(r C ) equal to 14% we find that the standard deviation of the optimal portfolio is 13.04%. b. To find the proportion invested in T-bills we remember that the mean of the complete portfolio, 14%, is an average of the T-bill rate and the optimal combination of stocks and bonds, P. Let y be the proportion in this portfolio. The mean of any portfolio along the optimal CAL is: E(r C ) = (l y) r f + y E(rp) = r f + y [E(rp) r f ] = 8 + y ( ) Setting E(r C ) = 14% we find: y =.7884, and 1 y =.2116, the proportion in T-bills. To find the proportions invested in each of the funds we multiply.7884 by the proportions of the stocks and bonds in the optimal risky portfolio: Proportion of stocks in complete portfolio = =.3560 Proportion of bonds in complete portfolio = = Using only the stock and bond funds to achieve a portfolio mean of 14% we must find the appropriate proportion in the stock fund, w S, and w B = 1 w S in the bond fund. The portfolio mean will be: 14 = 20w S + 12(1 w S ) = w S w S =.25 So the proportions will be 25% in stocks and 75% in bonds. The standard deviation of this portfolio will be: σ p = ( ) 1/2 = 14.13% 8-3

4 This is considerably larger than the standard deviation of 13.04% achieved using T-bills and the optimal portfolio. 8. With no opportunity to borrow you wish to construct a portfolio with a mean of 24%. Since this exceeds the mean on stocks of 20%, you will have to go short on bonds, which have a mean of 12%, and use the proceeds to buy additional stock. The graphical representation of your risky portfolio is point Q on the following graph: OPPORTUNITY SET WITH NO RISK-FREE ASSET Q P Standard Deviation (%) Point Q is the stock/bond combination with mean of 24%. Let w S be the proportion of stocks and 1 w S be the proportion of bonds required to achieve the 24% mean. Then: 24 = 20 w S + 12 (1 w S ) = w S w S = 1.50, and 1 w S =.50 Therefore, you would have to sell short an amount of bonds equal to.50 of your total funds, and invest 1.50 times your total funds in stocks. The standard deviation of this portfolio would be: σ Q = [ (.50) (1.50) (.50) 45] 1/2 = 44.87% 8-4

5 If you were allowed to borrow at the risk-free rate of 8%, the way to achieve the target expected return of 24% would be to invest more than 100% of your funds in the optimal risky portfolio, moving out along the CAL to the right of P, up to R, on the following graph. R is the point on the optimal CAL which has the mean of 24%. Using the formula for the optimal CAL we can find the corresponding standard deviation: E(r C ) = σ C = 24 Setting E(r C ) = 24, we get: σ C = 34.78%, which is considerably less than the 44.87% standard deviation you would get without the ability to borrow at the risk-free rate of 8% INVESTMENT OPPORTUNITY SET R Q Tangency Pf, P CML Efficient frontier of risky assets Min.Var What is the portfolio composition of point R on the optimal CAL? The mean of any portfolio along this CAL is: E(r C ) = r f + y[e(r P ) r f ] where y is the proportion invested in the optimal risky portfolio P and r P is the mean of that portfolio, which is 15.61%. 24 = 8 + y( ) y =

6 This means that for every $1 of your own funds invested in portfolio P, you would borrow an additional $ and invest it also in portfolio P. 9. a Optimal CAL P Stocks Gold Standard Deviation(%) Even though gold seems dominated by stocks, it still might be an attractive asset to hold as a part of a portfolio. If the correlation between gold and stocks is sufficiently low, it will be held as an element in a portfolio -- the optimal tangency portfolio. b. If gold had a correlation coefficient with stocks of +1, it would not be held. The optimal CAL would be comprised of bills and stocks only. Since the set of risk/return combinations of stocks and gold would plot as a straight line with a negative slope (see the following graph), it would be dominated by the stocks portfolio. Of course, this situation could not persist. If no one desired gold, its price would fall and its expected rate of return would increase until it became a sufficiently attractive asset to hold. 8-6

7 Stocks 10 Gold 5 r f Standard Deviation(%) 11. False. If the borrowing and lending rates are not identical, then depending on the tastes of the individuals (that is, the shape of their indifference curves), borrowers and lenders could have different optimal risky portfolios. 12. False. The portfolio standard deviation equals the weighted average of the componentasset standard deviations only in the special case that all assets are perfectly positively correlated. Otherwise, as the formula for portfolio standard deviation shows, the portfolio standard deviation is less than the weighted average of the component-asset standard deviations. The portfolio variance will be a weighted sum of the elements in the covariance matrix, with the products of the portfolio proportions as weights. 13. The probability distribution is: Probability Rate of Return.7 100%.3 50% Mean = ( 50) = 55% Variance =.7 (100 55) ( 50 55) 2 =

8 Standard deviation = /2 = 68.74% 15. a. Restricting the portfolio to 20 stocks rather than will increase the risk of the portfolio, but possibly not by much. If, for instance, the 50 stocks in a universe had the same standard deviation, σ, and the correlations between each pair were identical with correlation coefficient ρ (so that the covariance between each pair would be ρσ2 ), the variance of an equally weighted portfolio would be (see Appendix A, equation 8A.4), σ 2 P = 1 n σ2 + n 1 n ρσ 2 The effect of the reduction in n on the second term would be relatively small (since 49/50 is close to 19/20 and ρσ 2 is smaller than σ 2 ), but the denominator of the first term would be 20 instead of 50. For example, if σ = 45% and ρ =.2, then the standard deviation with 50 stocks would be 20.91%, and would rise to 22.05% when only 20 stocks are held. Such an increase might be acceptable if the expected return is increased sufficiently. b. Hennessy could contain the increase in risk by making sure that he maintains reasonable diversification among the 20 stocks that remain in his portfolio. This entails maintaining a low correlation among the remaining stocks. For example, in part (a), with ρ =.2, the increase in portfolio risk was minimal. As a practical matter, this means that Hennessy would need to spread his portfolio among many industries; concentrating on just a few would result in higher correlation among the included stocks. 16. Risk reduction benefits from diversification are not a linear function of the number of issues in the portfolio. Rather, the incremental benefits from additional diversification are most important when you are least diversified. Restricting Hennesey to 10 instead of 20 issues would increase the risk of his portfolio by a greater amount than would reducing the size of the portfolio from 30 to 20 stocks. In our example, restricting the number of stocks to 10 will increase the standard deviation to 23.81%. The increase in standard deviation of 1.76% from giving up 10 of 20 stocks is greater than the increase of 1.14% from giving up 30 stocks when starting with The point is well taken because the committee should be concerned with the volatility of the entire portfolio. Since Hennessey's portfolio is only one of six well-diversified portfolios and i smaller than the average, the concentration in fewer issues could have a minimal effect on the diversification of the total fund. Hence, unleashing Hennessey to do stock picking may be advantageous. 18. c. Intuitively, we note that since all stocks have the same expected rate of return and standard deviation, we wish to choose the stock that will result in lowest risk. This will be the one with the lowest correlation with stock A. 8-8

9 More formally, we note that when all stocks have the same expected rate of return, the optimal portfolio for any risk averse investor is the global minimum variance portfolio (G). When restricted to A and one more stock, the objective is to find G for any pair that includes A, and select the combination with the lowest variance. With two stocks, I and J, the formula for the weights in G is: w Min (I) = σ 2 J Cov(r I,r J ) σ 2 I + σ2 J 2Cov(r I,r J ) w Min (J) = 1 w Min (I) Since all standard deviations are equal to 20%, Cov(r I,r J ) = ρσ I σ J = 400ρ, and w Min (I) = w Min (J) =.5 This intuitive result is an implication of a property of any efficient frontier, namely, that the covariance of the global minimum variance portfolio with any other asset on the frontier is identical and equal to its own variance. (Otherwise, additional diversification would further reduce the variance.) In this case, the standard deviation of G(I,J) will reduce to: σ Min (G) = [200(1 + ρ IJ )] 1/2 This leads us directly to the intuitive result that the desired addition would be the stock with the lowest correlation with A, which is D. The optimal portfolio is equally invested in A and D, and the standard deviation will be 17.03%. 19. No, at least as long as they are not risk lovers. Risk neutral investors will not care which portfolio they hold since all portfolios yield 8%. 20. No change. The efficient frontier of risky assets is horizontal at 8%, so the optimal CAL runs from the risk-free rate through G. The best G is here, again, the one with the lowest variance. The optimal complete portfolio will, as usual, depend on risk aversion. 21. d. Portfolio Y cannot be efficient because it is dominated by another portfolio. For 8-9

10 example, Portfolio X has higher expected return and lower standard deviation. 22. c. 23. d. 24. b. 25. a. 27. We know nothing about expected returns, so we focus exclusively on reducing variability. Portfolios C and A have equal standard deviations, but the correlation of Portfolio C with Portfolio B is lower than that of Portfolio A with Portfolio B, so a portfolio comprised of C and B will have lower total risk than a portfolio comprised of A and B. 8-10

11 28. Rearranging the table (converting rows to columns), and computing serial correlation results in the following table: Nominal Rates Small company Large company L.T. Gov t Bonds Intermediate Gov t Bonds T-Bills Inflation 1920s* s s s s s s s Serial Correlation For example: to compute serial correlation in decade nominal returns for large-company stocks, we set up the following two columns and use the Correlation function of the spreadsheet to find that the serial correlation is Decade Previous 1930s -1.25% 18.36% 1940s 9.11% -1.25% 1950s 19.41% 9.11% 1960s 7.84% 19.41% 1970s 5.90% 7.84% 1980s 17.60% 5.90% 1990s 18.20% 17.60% Note that each correlation is based on only seven observations, so we cannot really arrive at any statistically significant conclusion. Looking at the numbers, however, it appears that there is persistent serial correlation with the exception of large-company stocks. This conclusion changes when we turn to real rates in the next problem. 8-11

12 29. The table for real rates (using the approximation of subtracting a decade s average inflation from the decade s average nominal return) is: Small company Large company L.T. Gov t Bonds Intermediate Gov t Bonds T-Bills 1920s* s s s s s s s Serial Correlation The positive serial correlation in decade nominal returns has vanished and it appears that real rates are serially uncorrelated. The decade time series (although again too short for any definitive conclusion) suggests that real rates of return are independent from decade to decade. 8-12