Chapter 7: Optimal Risky Portfolios (Review Questions)

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 22. Compute the estimated annual risk premiums, standard deviations, and Sharpe ratios for the two portfolios. The risk premium for the S&P portfolio is: _____

(1+.05)^1-1=0.05

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 10. If you were to use only the two risky funds and still require an expected return of 14%, what would be the investment proportions of your portfolio? Compare its standard deviation to that of the optimized portfolio in Problem 9. What do you conclude? So the proportions are 25% invested in the stock fund and 75% in the bond fund. The standard deviation of this portfolio will be: ____

σP = [(0.252 900) + (0.752 225) + (2 x 0.25 x 0.75 x 45)]1/2 = 14.13% **This is considerably greater than the standard deviation of 13.04% achieved using T-bills and the optimal portfolio.**

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 24. What should be Greta's capital allocation? The resulting investment composition will be S&P: ______

0.6888 .6049 = 41.67% Hedge: 0.6888 .3951 = 27.21%. The remaining 31.11% will be invested in the risk-free asset.

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 22. Compute the estimated annual risk premiums, standard deviations, and Sharpe ratios for the two portfolios. The hedge fund Sharpe ratio is _____

10/35 = 0.2857.

15. Suppose you have a project that has a .7 chance of doubling your investment in a year and a .3 chance of halving your investment in a year. What is the standard deviation of the rate of return on this investment? Standard deviation = ____

4725^1/2 = 68.74%

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 22. Compute the estimated annual risk premiums, standard deviations, and Sharpe ratios for the two portfolios. S&P Sharpe ratio is ____

5/20 = 0.25

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 23. Assuming the correlation between the annual returns on the two portfolios is indeed zero, what would be the optimal asset allocation? The resulting Sharpe ratio is _____

6.9753/18.3731= 0.3796

15. Suppose you have a project that has a .7 chance of doubling your investment in a year and a .3 chance of halving your investment in a year. What is the standard deviation of the rate of return on this investment? Variance = _____

[0.7 × (100 − 55)2] + [0.3 × (-50 − 55)2] = 4725

The following data are for Problems 17 through 19: The correlation coefficients between several pairs of stocks are as follows: Corr(A, B) = .85; Corr(A, C) = .60; Corr(A, D) = .45. Each stock has an expected return of 8% and a standard deviation of 20%. 17. If your entire portfolio is now composed of stock A and you can add some of only one stock to your portfolio, would you choose (explain your choice): a. B b. C c. D d. Need more data 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)=0.5

The following table of compound annual returns by decade applies to Problems 20 and 21 Look at Table 20. Input the data from the table into a spreadsheet. Compute the serial correlation in decade returns for each asset class and for inflation. Also find the correlation between the returns of various asset classes. What do the data indicate?

Rearrange the table (converting rows to columns) and compute serial correlation results in the following table: Look at word document --> For 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.

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 5. Tabulate and draw the investment opportunity set of the two risky funds. Use investment proportions for the stock fund of 0% to 100% in increments of 20%.

See word document

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 6. Draw a tangent from the risk-free rate to the opportunity set. What does your graph show for the expected return and standard deviation of the optimal portfolio?

see word document

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 7. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio The proportion of the optimal risky portfolio invested in the stock fund is given by: _____

w_S=([E(r_S)-r_f]×σ_B^2-[E(r_B)-r_f]×Cov(r_S,r_B))/([E(r_S)-r_f]×σ_B^2+[E(r_B)-r_f]×σ_S^2-[E(r_S)-r_f+E(r_B)-r_f]×Cov(r_S,r_B)) =([(.20-.08)×225]-[(.12-.08)×45])/([(.20-.08)×225]+[(.12-.08)×900]-[(.20-.08+.12-.08)×45])=0.4516 w_B=1-0.4516=0.5484

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 b. What is the proportion invested in the money market fund and each of the two risky funds? Setting E(rC) = 14% we find: _____

y = 0.7884 and (1 − y) = 0.2119 (the proportion invested in the T-bill fund).

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 22. Compute the estimated annual risk premiums, standard deviations, and Sharpe ratios for the two portfolios. The 3-year risk premium for the hedge fund portfolio is _____

(1+.1)^1-1=0.1

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 8. What is the Sharpe ratio of the best feasible CAL? The reward-to-volatility ratio of the optimal CAL is: _____

(E(r_p)-r_f)/σ_p =(.1561-.08)/.1654=0.4601

3. Which of the following statements about the minimum-variance portfolio of all risky securities is valid? (Assume short sales are allowed.) Explain. a. Its variance must be lower than those of all other securities or portfolios. b. Its expected return can be lower than the risk-free rate. c. It may be the optimal risky portfolio. d. It must include all individual securities

(a) Answer (a) is valid because it provides the definition of the minimum variance portfolio.

2. When adding real estate to an asset allocation program that currently includes only stocks, bonds, and cash, which of the properties of real estate returns most affects portfolio risk? Explain. a. Standard deviation. b. Expected return. c. Covariance with returns of the other asset classes

(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.)

1. Which of the following factors reflect pure market risk for a given corporation? a. Increased short-term interest rates. b. Fire in the corporate warehouse. . Increased insurance costs. d. Death of the CEO. e. Increased labor costs

(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.

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 10. If you were to use only the two risky funds and still require an expected return of 14%, what would be the investment proportions of your portfolio? Compare its standard deviation to that of the optimized portfolio in Problem 9. What do you conclude? 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

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 22. Compute the estimated annual risk premiums, standard deviations, and Sharpe ratios for the two portfolios. The S&P 3-year standard deviation is: _____

0.2×√1=0.20 .

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 22. Compute the estimated annual risk premiums, standard deviations, and Sharpe ratios for the two portfolios. The hedge fund 3-year standard deviation is: _____

0.35×√1=0.35

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 27. Repeat Problem 24 using an annual correlation of .3 The resulting investment composition will be S&P: _____

0.5214 0.5771 = 30.09% Hedge: .5214 .4229 = 22.05%. The remaining 47.86% will be invested in the risk-free asset.

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 26. Repeat Problem 23 using an annual correlation of .3. The resulting Sharpe ratio is _____

7.11/21.33 = 0.3336.

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 4. What are the investment proportions in the minimum-variance portfolio of the two risky funds, and what are the expected value and standard deviation of its rate of return?ted value and standard deviation of its rate of return? From the standard deviations and the correlation coefficient we generate the covariance matrix [note that]: _____

Bonds Stocks Bonds 225 45 Stocks 45 900

12. Suppose that there are many stocks in the security market and that the characteristics of stocks A and B are given as follows: Stock Expected Return Standard Deviation A 10% 5% B 15 10 Correlation = −1 Suppose that it is possible to borrow at the risk-free rate, rf . What must be the value of the riskfree rate? (Hint: Think about constructing a risk-free portfolio from stocks A and B.) 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%

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 4. What are the investment proportions in the minimum-variance portfolio of the two risky funds, and what are the expected value and standard deviation of its rate of return?ted value and standard deviation of its rate of return? The minimum variance portfolio mean and standard deviation are: ______

E(rMin) = (0.1739 × .20) + (0.8261 × .12) = .1339 = 13.39% σMin = [w_S^2 σ_S^2+w_B^2 σ_B^2+2w_S w_B "Cov"(r_S,r_B)]^(1/2) = [(0.17392 900) + (0.82612 225) + (2 0.1739 0.8261 45)]1/2 = 13.92%

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 7. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio The mean and standard deviation of the optimal risky portfolio are: _____

E(rP) = (0.4516 × .20) + (0.5484 × .12) = .1561 = 15.61% σp = [(0.45162 900) + (0.54842 225) + (2 0.4516 0.5484 × 45)]1/2 = 16.54%

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 4. What are the investment proportions in the minimum-variance portfolio of the two risky funds, and what are the expected value and standard deviation of its rate of return?ted value and standard deviation of its rate of return? The parameters of the opportunity set are: _____

E(rS) = 20%, E(rB) = 12%, σS = 30%, σB = 15%, ρ = 0.10

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 b. What is the proportion invested in the money market fund and each of the two risky funds? 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: _____

E(r_C)=(1-y)×r_f+y×E(r_P)=r_f+y×[E(r_P)-r_f]=.08+y×(.1561-.08)

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 9. You require that your portfolio yield an expected return of 14%, and that it be efficient, that is, on the steepest feasible CAL. a. What is the standard deviation of your portfolio? 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: ______

E(r_C)=r_f+(E(r_p)-r_f)/σ_P σ_C=.08+0.4601σ_C **If E(rC) is equal to 14%, then the standard deviation of the portfolio is 13.04%.

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 26. Repeat Problem 23 using an annual correlation of .3. With these weights, _____

E(r_P)=0.5771×5+0.4229×10=0.0711=7.1147% σ_P=√(.5771^2×20^2+.4229^2×35^2+2×.5771×.4229×(.3×20×35))=.2133

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 23. Assuming the correlation between the annual returns on the two portfolios is indeed zero, what would be the optimal asset allocation? With these weights, E(rp) = _____

E(r_P)=0.6049×5+0.3951×10=0.0698=6.9753%

11. Stocks offer an expected rate of return of 18% with a standard deviation of 22%. Gold offers an expected return of 10% with a standard deviation of 30%. a. In light of the apparent inferiority of gold with respect to both mean return and volatility, would anyone hold gold? If so, demonstrate graphically why one would do so

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, gold will be held as a component in a portfolio, specifically, the optimal tangency portfolio.

13. True or false: Assume that expected returns and standard deviations for all securities (including the risk-free rate for borrowing and lending) are known. In this case, all investors will have the same optimal risky portfolio.

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. True or false: The standard deviation of the portfolio is always equal to the weighted average of the standard deviations of the assets in the portfolio.

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.

The following table of compound annual returns by decade applies to Problems 20 and 21 Look at Table 21. Convert the asset returns by decade presented in the table into real rates. Repeat Problem 20 for the real rates of return. The table for real rates (using the approximation of subtracting a decade's average inflation from the decade's average nominal return) is: ______

Get Table: While 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.

11. Stocks offer an expected rate of return of 18% with a standard deviation of 22%. Gold offers an expected return of 10% with a standard deviation of 30%. b. Given the data above, reanswer (a) with the additional assumption that the correlation coefficient between gold and stocks equals 1. Draw a graph illustrating why one would or would not hold gold in one's portfolio.

If the correlation between gold and stocks equals +1, then no one would hold gold. 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), these combinations would be dominated by the stock 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 sufficiently attractive to include in a portfolio.

The following data are for Problems 17 through 19: The correlation coefficients between several pairs of stocks are as follows: Corr(A, B) = .85; Corr(A, C) = .60; Corr(A, D) = .45. Each stock has an expected return of 8% and a standard deviation of 20%. 18. Would the answer to Problem 17 change for more risk-averse or risk-tolerant investors? Explain

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%.

15. Suppose you have a project that has a .7 chance of doubling your investment in a year and a .3 chance of halving your investment in a year. What is the standard deviation of the rate of return on this investment? The probability distribution is: ______

Probability Rate of Return 0.7 100% 0.3 −50

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 b. What is the proportion invested in the money market fund and each of the two risky funds? 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

The following data are for Problems 17 through 19: The correlation coefficients between several pairs of stocks are as follows: Corr(A, B) = .85; Corr(A, C) = .60; Corr(A, D) = .45. Each stock has an expected return of 8% and a standard deviation of 20%. 17. If your entire portfolio is now composed of stock A and you can add some of only one stock to your portfolio, would you choose (explain your choice): a. B b. C c. D d. Need more data

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.

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 26. Repeat Problem 23 using an annual correlation of .3. With a ρ = .3, the optimal asset allocation is _____

W_(S&P)=(5 × 35^2 - 10 × (0.3 × 20 × 35)) /(5 × 35^2 +10 × 20^2- (5 + 10) × (0.3 × 20 × 35)) = 0.5771 W_Hedge= 1 - 0.5771 = 0.4229 .

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 23. Assuming the correlation between the annual returns on the two portfolios is indeed zero, what would be the optimal asset allocation? With a ρ = 0, the optimal asset allocation is: Ws&p = ______

W_(S&P)=(5×35^2-10×(0×20×35))/(5×35^2+10×20^2-(5+10)×(0×20×35))=0.6049

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 23. Assuming the correlation between the annual returns on the two portfolios is indeed zero, what would be the optimal asset allocation? With a ρ = 0, the optimal asset allocation is: Whedge = ______

W_Hedge=1-0.6049=0.3951 .

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 25. If the correlation coefficient between annual portfolio returns is actually .3, what is the covariance between the returns? With ρ = 0.3, the annual covariance is ______

With ρ = 0.3, the annual covariance is .3×.2×.35=0.021.

The following data are for Problems 17 through 19: The correlation coefficients between several pairs of stocks are as follows: Corr(A, B) = .85; Corr(A, C) = .60; Corr(A, D) = .45. Each stock has an expected return of 8% and a standard deviation of 20%. 19. Suppose that in addition to investing in one more stock you can invest in T-bills as well. Would you change your answers to Problems 17 and 18 if the T-bill rate is 8%?

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.

15. Suppose you have a project that has a .7 chance of doubling your investment in a year and a .3 chance of halving your investment in a year. What is the standard deviation of the rate of return on this investment? Mean = ______

[0.7 × 100%] + [0.3 × (-50%)] = 55%

The following table of compound annual returns by decade applies to Problems 20 and 21 Look at Table 20. Input the data from the table into a spreadsheet. Compute the serial correlation in decade returns for each asset class and for inflation. Also find the correlation between the returns of various asset classes. What do the data indicate? 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% 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.)

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term bond fund, and the third is a money market fund that provides a safe return of 8%. The characteristics of the risky funds are as follows: Expected Return Standard Deviation Stock fund (S) 20% 30% Bond fund (B) 12 15 The correlation between the fund returns is .10 4. What are the investment proportions in the minimum-variance portfolio of the two risky funds, and what are the expected value and standard deviation of its rate of return?ted value and standard deviation of its rate of return? The minimum-variance portfolio is computed as follows: _____

wMin(S) =(σ_B^2-"Cov" (r_S,r_B))/(σ_S^2+σ_B^2-2"Cov" (r_S,r_B))=(225-45)/(900+225-(2×45))=0.1739 wMin(B) = 1 0.1739 = 0.8261

The following data are for Problems 17 through 19: The correlation coefficients between several pairs of stocks are as follows: Corr(A, B) = .85; Corr(A, C) = .60; Corr(A, D) = .45. Each stock has an expected return of 8% and a standard deviation of 20%. 17. If your entire portfolio is now composed of stock A and you can add some of only one stock to your portfolio, would you choose (explain your choice): a. B b. C c. D d. Need more data 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: ______

w_Min (I)=(σ_J^2-"Cov" (r_I,r_J))/(σ_I^2+σ_J^2-2"Cov" (r_I,r_J)) w_Min (J)=1-w_Min (I)

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 24. What should be Greta's capital allocation? Greta has a risk aversion of A=3, Therefore, she will invest ____ of her wealth in this risky portfolio

y=.06975/(3×.183 ^2)=0.6888=68.8%

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 27. Repeat Problem 24 using an annual correlation of .3 Greta has a risk aversion of A=3, Therefore, she will invest ____ of her wealth in this risky portfolio.

y=0.07115/(3×.2133^2)=0.5214=52.14%

16. Suppose that you have $1 million and the following two opportunities from which to construct a portfolio: a. Risk-free asset earning 12% per year. b. Risky asset with expected return of 30% per year and standard deviation of 40%. If you construct a portfolio with a standard deviation of 30%, what is its expected rate of return?

σP = 30 = y × σ = 40 × y Þ y = 0.75 E(rP) = 12 + 0.75(30 − 12) = 25.5%

12. Suppose that there are many stocks in the security market and that the characteristics of stocks A and B are given as follows: Stock Expected Return Standard Deviation A 10% 5% B 15 10 Correlation = −1 Suppose that it is possible to borrow at the risk-free rate, rf . What must be the value of the riskfree rate? (Hint: Think about constructing a risk-free portfolio from stocks A and B.) 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: _______

σP = Absolute value [wAσA - wBσB] 0 = 5 × wA − [10 ´ (1 - wA)] Þ wA = 0.6667

The following data are for Problems 17 through 19: The correlation coefficients between several pairs of stocks are as follows: Corr(A, B) = .85; Corr(A, C) = .60; Corr(A, D) = .45. Each stock has an expected return of 8% and a standard deviation of 20%. 17. If your entire portfolio is now composed of stock A and you can add some of only one stock to your portfolio, would you choose (explain your choice): a. B b. C c. D d. Need more data 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: ______

σ_Min (G)=[200×(1+ρ_IJ)]^(1/2) **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%.**

The following information applies to Problems 22 through 27: Greta has risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a standard deviation of 20%. The hedge fund risk premium is estimated at 10% with a standard deviation of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 23. Assuming the correlation between the annual returns on the two portfolios is indeed zero, what would be the optimal asset allocation? σ_P= _____

σ_P=√(.6049^2×20^2+.3951^2×35^2+2×.6049×.3951×(0×20×35))=.1837=18.3731%