ࡱ> ^ `bjbj[[ Rf99V@  ttttt8,JJ```;~,HHHHHHH$OMP\HYt ;; Htt``WJ<... t`t`H. H..$DF`BM$nE,HJ<JE]P&]PXF]PtFvsT.D HH(J ]P :: CHAPTER 7: OPTIMAL RISKY PORTFOLIOS PROBLEM SETS 1. (a) and (e). Short-term rates and labor issues are factors that are common to all firms and therefore must be considered as market risk factors. The remaining three factors are unique to this corporation and are not a part of market risk. 2. (a) and (c). After real estate is added to the portfolio, there are four asset classes in the portfolio: stocks, bonds, cash, and real estate. Portfolio variance now includes a variance term for real estate returns and a covariance term for real estate returns with returns for each of the other three asset classes. Therefore, portfolio risk is affected by the variance (or standard deviation) of real estate returns and the correlation between real estate returns and returns for each of the other asset classes. (Note that the correlation between real estate returns and returns for cash is most likely zero.) 3. (a) Answer (a) is valid because it provides the definition of the minimum variance portfolio. 4. The parameters of the opportunity set are: E(rS) = 20%, E(rB) = 12%, S = 30%, B = 15%,  = 0.10 From the standard deviations and the correlation coefficient we generate the covariance matrix [note that EMBED Equation.DSMT4 ]: BondsStocksBonds 225 45Stocks 45 900The minimum-variance portfolio is computed as follows: wMin(S) = EMBED Equation.3  wMin(B) = 1 ( 0.1739 = 0.8261 The minimum variance portfolio mean and standard deviation are: E(rMin) = (0.1739 .20) + (0.8261 .12) = .1339 = 13.39% Min =  EMBED Equation.3  = [(0.17392 ( 900) + (0.82612 ( 225) + (2 ( 0.1739 ( 0.8261 ( 45)]1/2 = 13.92% 5. Proportion in Stock FundProportion in Bond FundExpected ReturnStandard Deviation 0.00% 100.00%12.00%15.00% 17.39 82.6113.3913.92minimum variance 20.00 80.0013.6013.94 40.00 60.0015.2015.70 45.16 54.8415.6116.54tangency portfolio 60.00 40.0016.8019.53 80.00 20.0018.4024.48 100.00 0.0020.0030.00Graph shown below.  EMBED Word.Picture.8  6. The above graph indicates that the optimal portfolio is the tangency portfolio with expected return approximately 15.6% and standard deviation approximately 16.5%. 7. The proportion of the optimal risky portfolio invested in the stock fund is given by:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.DSMT4  The mean and standard deviation of the optimal risky portfolio are: E(rP) = (0.4516 .20) + (0.5484 .12) = .1561 = 15.61% p = [(0.45162 ( 900) + (0.54842 ( 225) + (2 ( 0.4516 ( 0.5484 45)]1/2 = 16.54% 8. The reward-to-volatility ratio of the optimal CAL is:  EMBED Equation.3  9. a. If you require that your portfolio yield an expected return of 14%, then you can find the corresponding standard deviation from the optimal CAL. The equation for this CAL is:  EMBED Equation.3  If E(rC) is equal to 14%, then the standard deviation of the portfolio is 13.04%. b. To find the proportion invested in the T-bill fund, remember that the mean of the complete portfolio (i.e., 14%) is an average of the T-bill rate and the optimal combination of stocks and bonds (P). Let y be the proportion invested in the portfolio P. The mean of any portfolio along the optimal CAL is:  EMBED Equation.DSMT4  Setting E(rC) = 14% we find: y = 0.7884 and (1 " y) = 0.2119 (the proportion invested in the T-bill fund). To find the proportions invested in each of the funds, multiply 0.7884 times the respective proportions of stocks and bonds in the optimal risky portfolio: Proportion of stocks in complete portfolio = 0.7884 ( 0.4516 = 0.3560 Proportion of bonds in complete portfolio = 0.7884 ( 0.5484 = 0.4323 10. Using only the stock and bond funds to achieve a portfolio expected return of 14%, we must find the appropriate proportion in the stock fund (wS) and the appropriate proportion in the bond fund (wB = 1 " wS) as follows: 0.14 = 0.20 wS + 0.12 (1 " wS) = 0.12 + 0.08 wS ( wS = 0.25 So the proportions are 25% invested in the stock fund and 75% in the bond fund. The standard deviation of this portfolio will be: P = [(0.252 ( 900) + (0.752 ( 225) + (2 ( 0.25 ( 0.75 ( 45)]1/2 = 14.13% This is considerably greater than the standard deviation of 13.04% achieved using T-bills and the optimal portfolio. 11. a.  EMBED Word.Picture.8  Even though it seems that gold is dominated by stocks, gold might still be an attractive asset to hold as a part of a portfolio. If the correlation between gold and stocks is sufficiently low (or even negative), gold will be held as a component in a portfolio, specifically, the optimal tangency portfolio. If the correlation between gold and stocks equals +1, then no one would be willing to hold gold: its return is lower than stocks, its standard deviation is higher, and, with perfect correlation, it offers no diversification benefits (such as those described in part a). The optimal CAL would be composed 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), any portfolio that contains gold would be dominated by the stock portfolio.  EMBED Word.Picture.8  Of course, this situation could not be an equilibrium. As long as no one is willing to hold gold, its price will fall and its expected rate of return will increase until it became sufficiently attractive to include in a portfolio. 12. Since Stock A and Stock B are perfectly negatively correlated, a risk-free portfolio can be created and the rate of return for this portfolio, in equilibrium, will be the risk-free rate. To find the proportions of this portfolio [with the proportion wA invested in Stock A and wB = (1  wA ) invested in Stock B], set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation is: P = Absolute value [wAA ( wBB] 0 = 5 wA " [10 ( (1 wA)] ( wA = 0.6667 The expected rate of return for this risk-free portfolio is: E(r) = (0.6667 10) + (0.3333 15) = 11.667% Therefore, the risk-free rate is: 11.667% 13. 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. 14. False. The portfolio standard deviation equals the weighted average of the component-asset 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 is a weighted sum of the elements in the covariance matrix, with the products of the portfolio proportions as weights. 15. The probability distribution is: ProbabilityRate of Return0.7100%0.3"50Mean = [0.7 100%] + [0.3 (-50%)] = 55% Variance = [0.7 (100 " 55)2] + [0.3 (-50 " 55)2] = 4725 Standard deviation = 47251/2 = 68.74% 16. P = 30 = y  = 40 y ( y = 0.75 E(rP) = 12 + 0.75(30 " 12) = 25.5% 17. The correct choice is (c). Intuitively, we note that since all stocks have the same expected rate of return and standard deviation, we choose the stock that will result in lowest risk. This is the stock that has the lowest correlation with Stock A. 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 the portfolio is restricted to Stock A and one additional stock, the objective is to find G for any pair that includes Stock A, and then select the combination with the lowest variance. With two stocks, I and J, the formula for the weights in G is:  EMBED Equation.3  Since all standard deviations are equal to 20%:  EMBED Equation.DSMT4  This intuitive result is an implication of a property of any efficient frontier, namely, that the covariances of the global minimum variance portfolio with all other assets on the frontier are 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) reduces to:  EMBED Equation.DSMT4  This leads to the intuitive result that the desired addition would be the stock with the lowest correlation with Stock A, which is Stock D. The optimal portfolio is equally invested in Stock A and Stock D, and the standard deviation is 17.03%. 18. No, the answer to Problem 17 would not change, at least as long as investors are not risk lovers. Risk neutral investors would not care which portfolio they held since all portfolios have an expected return of 8%. 19. Yes, the answers to Problems 17 and 18 would change. The efficient frontier of risky assets is horizontal at 8%, so the optimal CAL runs from the risk-free rate through G. This implies risk-averse investors will just hold Treasury bills. 20. Rearrange the table (converting rows to columns) and compute serial correlation results in the following table: Nominal Rates Small Company StocksLarge Company StocksLong-Term Government Bonds Treasury Bills Inflation1920s -3.72 18.36 3.983.56 -1.001930s 7.28 -1.25 4.600.30 -2.041940s 20.63 9.11 3.590.37 5.361950s 19.01 19.41 0.251.87 2.221960s 13.72 7.84 1.143.89 2.521970s 8.75 5.90 6.636.29 7.361980s 12.46 17.60 11.509.00 5.101990s 13.84 18.20 8.605.02 2.932000s6.70-1.005.002.702.50Serial Correlation 0.34 -0.35 0.550.59 0.23For example: to compute serial correlation in decade nominal returns for large-company stocks, we set up the following two columns in an Excel spreadsheet. Then, use the Excel function CORREL to calculate the correlation for the data. DecadePrevious1930s-1.25%18.36%1940s9.11%-1.25%1950s19.41%9.11%1960s7.84%19.41%1970s5.90%7.84%1980s17.60%5.90%1990s18.20%17.60%2000s-1.00%18.20%Note that each correlation is based on only seven observations, so we cannot arrive at any statistically significant conclusions. Looking at the results, however, it appears that, with the exception of large-company stocks, there is persistent serial correlation. (This conclusion changes when we turn to real rates in the next problem.) 21. The table for real rates (using the approximation of subtracting a decades average inflation from the decades average nominal return) is: Real Rates Small Company StocksLarge Company StocksLong-Term Government Bonds Treasury Bills1920s -2.72 19.36 4.98 4.561930s 9.32 0.79 6.64 2.341940s 15.27 3.75 -1.77 -4.991950s 16.79 17.19 -1.97 -0.351960s 11.20 5.32 -1.38 1.371970s 1.39 -1.46 -0.73 -1.071980s 7.36 12.50 6.40 3.901990s 10.91 15.27 5.67 2.092000s4.20-3.52.50.2Serial Correlation 0.20 -0.38 0.37 0.00While the serial correlation in decade nominal returns seems to be positive, it appears that real rates are serially uncorrelated. The decade time series (although again too short for any definitive conclusions) suggest that real rates of return are independent from decade to decade. 22. The risk premium for the S&P portfolio is:  QUOTE    EMBED Equation.DSMT4  The 3-year risk premium for the hedge fund portfolio is  QUOTE   EMBED Equation.DSMT4  The S&P 3-year standard deviation is:  QUOTE   EMBED Equation.DSMT4  . The hedge fund 3-year standard deviation is:  EMBED Equation.DSMT4  S&P Sharpe ratio is 5/20 = 0.25 The hedge fund Sharpe ratio is 10/35 = 0.2857. 23. With a  = 0, the optimal asset allocation is  EMBED Equation.DSMT4   EMBED Equation.DSMT4  . With these weights,  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The resulting Sharpe ratio is 6.9753/18.3731= 0.3796 24. Greta has a risk aversion of A=3, Therefore, she will invest  EMBED Equation.DSMT4  of her wealth in this risky portfolio. The resulting investment composition will be S&P: 0.6888  QUOTE   .6049 = 41.67% and Hedge: 0.6888 QUOTE   .3951 = 27.21%. The remaining 31.11% will be invested in the risk-free asset. 25. With  = 0.3, the annual covariance is  QUOTE   EMBED Equation.DSMT4 . 26. With a  = .3, the optimal asset allocation is  EMBED Equation.DSMT4   EMBED Equation.DSMT4  . With these weights,  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The resulting Sharpe ratio is 7.11/21.33 = 0.3336. 27. Greta has a risk aversion of A=3, Therefore, she will invest  EMBED Equation.DSMT4  of her wealth in this risky portfolio. The resulting investment composition will be S&P: 0.5214  QUOTE   0.5771 = 30.09% and Hedge: .5214 QUOTE   .4229 = 22.05%. The remaining 47.86% will be invested in the risk-free asset. CFA PROBLEMS 1. a. Restricting the portfolio to 20 stocks, rather than 40 to 50 stocks, will increase the risk of the portfolio, but it is possible that the increase in risk will be minimal. Suppose that, for instance, the 50 stocks in a universe have the same standard deviation (() and the correlations between each pair are identical, with correlation coefficient . Then, the covariance between each pair of stocks would be 2, and the variance of an equally weighted portfolio would be:  EMBED Equation.3  The effect of the reduction in n on the second term on the right-hand side would be relatively small (since 49/50 is close to 19/20 and 2 is smaller than 2), but the denominator of the first term would be 20 instead of 50. For example, if  = 45% and  = 0.2, 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. 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  = 0.2, the increase in portfolio risk was minimal. As a practical matter, this means that Hennessy would have to spread his portfolio among many industries; concentrating on just a few industries would result in higher correlations among the included stocks. 2. 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 Hennessy to 10 instead of 20 issues would increase the risk of his portfolio by a greater amount than would a reduction in 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 1.76% increase in standard deviation resulting from giving up 10 of 20 stocks is greater than the 1.14% increase that results from giving up 30 of 50 stocks. 3. The point is well taken because the committee should be concerned with the volatility of the entire portfolio. Since Hennessys portfolio is only one of six well-diversified portfolios and is smaller than the average, the concentration in fewer issues might have a minimal effect on the diversification of the total fund. Hence, unleashing Hennessy to do stock picking may be advantageous. 4. d. Portfolio Y cannot be efficient because it is dominated by another portfolio. For example, Portfolio X has both higher expected return and lower standard deviation. 5. c. d. b. a. 9. c. 10. Since we do not have any information about expected returns, we focus exclusively on reducing variability. Stocks A and C have equal standard deviations, but the correlation of Stock B with Stock C (0.10) is less than that of Stock A with Stock B (0.90). Therefore, a portfolio composed of Stocks B and C will have lower total risk than a portfolio composed of Stocks A and B. 11. Fund D represents the single best addition to complement Stephenson's current portfolio, given his selection criteria. Fund Ds expected return (14.0 percent) has the potential to increase the portfolios return somewhat. Fund Ds relatively low correlation with his current portfolio (+0.65) indicates that Fund D will provide greater diversification benefits than any of the other alternatives except Fund B. The result of adding Fund D should be a portfolio with approximately the same expected return and somewhat lower volatility compared to the original portfolio. The other three funds have shortcomings in terms of expected return enhancement or volatility reduction through diversification. Fund A offers the potential for increasing the portfolios return but is too highly correlated to provide substantial volatility reduction benefits through diversification. Fund B provides substantial volatility reduction through diversification benefits but is expected to generate a return well below the current portfolios return. Fund C has the greatest potential to increase the portfolios return but is too highly correlated with the current portfolio to provide substantial volatility reduction benefits through diversification. 12. a. Subscript OP refers to the original portfolio, ABC to the new stock, and NP to the new portfolio. i. E(rNP) = wOP E(rOP ) + wABC E(rABC ) = (0.9 ( 0.67) + (0.1 ( 1.25) = 0.728% ii. Cov =  ( (OP ( (ABC = 0.40 ( 2.37 ( 2.95 = 2.7966 ( 2.80 iii. (NP = [wOP2 (OP2 + wABC2 (ABC2 + 2 wOP wABC (CovOP , ABC)]1/2 = [(0.9 2 ( 2.372) + (0.12 ( 2.952) + (2 ( 0.9 ( 0.1 ( 2.80)]1/2 = 2.2673% ( 2.27% b. 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