Chapter 6: Risk Aversion and Capital Allocation to Risky Assets (Review Questions)
9. What must be true about the sign of the risk aversion coefficient, A, for a risk lover? Draw the indifference curve for a utility level of .05 for a risk lover. The corresponding indifference curve is ____ sloping in the graph above (see Problem 6), and is labeled Q9.
downward
4. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of .5. The alternative risk-free investment in T-bills pays 6% per year. b. Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio? If the portfolio is purchased for $118,421 and provides an expected cash inflow of $135,000, then the expected rate of return [E(r)] is as follows: _____
$118,421 × [1 + E(r)] = $135,000
4. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of .5. The alternative risk-free investment in T-bills pays 6% per year. a. If you require a risk premium of 8%, how much will you be willing to pay for the portfolio? With a risk premium of 8% over the risk-free rate of 6%, the required rate of return is 14%. Therefore, the present value of the portfolio is: _____
$135,000/1.14 = $118,421
4. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of .5. The alternative risk-free investment in T-bills pays 6% per year. c. Now suppose that you require a risk premium of 12%. What price are you willing to pay? The present value of the portfolio is now:_____
$135,000/1.18 = $114,407
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. 28. Your client ponders whether to switch the 70% that is invested in your fund to the passive portfolio b. Show him the maximum fee you could charge (as a percentage of the investment in your fund, deducted at the end of the year) that would leave him at least as well off investing in your fund as in the passive one. (Hint: The fee will lower the slope of his CAL by reducing the expected return net of the fee.) Setting these slopes equal we have: ______
(.10-f)/.28=0.20⇒ f = 0.044 = 4.4% per year
For Problems 23 through 26: Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%, that rf = 5%, and that your fund has the parameters given in Problem 21. 26. What is the largest percentage fee that a client who currently is lending (y < 1) will be willing to pay to invest in your fund? What about a client who is borrowing (y > 1)? The maximum feasible fee, denoted f, depends on the reward-to-variability ratio. For y < 1, the lending rate, 5%, is viewed as the relevant risk-free rate, and we solve for f as follows: _____
(.11-.05-f)/.15=(.13-.05)/.25 f=.06-(.15×.08)/.25=.012",or " 1.2%
4. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of .5. The alternative risk-free investment in T-bills pays 6% per year. a. If you require a risk premium of 8%, how much will you be willing to pay for the portfolio? The expected cash flow is: ____
(0.5 × $70,000) + (0.5 × 200,000) = $135,000.
1. Which of the following choices best completes the following statement? Explain. An investor with a higher degree of risk aversion, compared to one with a lower degree, will most prefer investment portfolios a. with higher risk premiums. b. that are riskier (with higher standard deviations). c. with lower Sharpe ratios. d. with higher Sharpe ratios.
(d) While a higher or lower Sharpe ratios are not an indication of an investor's tolerance for risk, any investor will always prefer investment portfolios with higher Sharpe ratios. The Sharpe ratio is simply a tool to absolutely measure the return premium earned per unit of risk.
5. Consider a portfolio that offers an expected rate of return of 12% and a standard deviation of 18%. T-bills offer a risk-free 7% rate of return. What is the maximum level of risk aversion for which the risky portfolio is still preferred to T-bills? 5. When we specify utility by U = E(r) - 0.5Aσ2, the utility level for T-bills is: ____
0.07
5. Consider a portfolio that offers an expected rate of return of 12% and a standard deviation of 18%. T-bills offer a risk-free 7% rate of return. What is the maximum level of risk aversion for which the risky portfolio is still preferred to T-bills? In order for the risky portfolio to be preferred to bills, the following must hold: _____
0.12 - 0.0162A > 0.07 Þ A < 0.05/0.0162 = 3.09
For Problems 23 through 26: Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%, that rf = 5%, and that your fund has the parameters given in Problem 21. 25. Solve Problems 23 and 24 for a client who uses your fund rather than an index fund. Therefore, y = ____
1 for 0.89 ≤ A ≤ 2.67
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. a. Explain to your client the disadvantage of the switch. Our target is: E(rC) = 11.5%. Therefore, the proportion that must be invested in my fund is determined as follows: _____
115 = .08 + .10 × y ---> y=(.115-.08)/.10=0.35
4. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of .5. The alternative risk-free investment in T-bills pays 6% per year. b. Suppose that the portfolio can be purchased for the amount you found in (a). What will be the expected rate of return on the portfolio? Therefore, E(r) = ____%. The portfolio price is set to equate the expected rate of return with the required rate of return.
14%
5. Consider a portfolio that offers an expected rate of return of 12% and a standard deviation of 18%. T-bills offer a risk-free 7% rate of return. What is the maximum level of risk aversion for which the risky portfolio is still preferred to T-bills? A must be less than _____for the risky portfolio to be preferred to bills.
3.09
4. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of .5. The alternative risk-free investment in T-bills pays 6% per year. c. Now suppose that you require a risk premium of 12%. What price are you willing to pay? If the risk premium over T-bills is now 12%, then the required return is: _____
6% + 12% = 18%
For Problems 23 through 26: Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%, that rf = 5%, and that your fund has the parameters given in Problem 21. 26. What is the largest percentage fee that a client who currently is lending (y < 1) will be willing to pay to invest in your fund? What about a client who is borrowing (y > 1)? For y > 1, the borrowing rate, 9%, is the relevant risk-free rate. Then we notice that, even without a fee, the active fund is inferior to the passive fund because: ______
= .11 - .09 - f / .15 = 0.13 < .13 - .09 / .25 = 0.16 or 16%
3. What do you think would happen to the equilibrium expected return on stocks if investors perceived higher volatility in the equity market? Relate your answer to Equation 6.7.
Assuming no change in risk tolerance, that is, an unchanged risk-aversion coefficient (A), higher perceived volatility increases the denominator of the equation for the optimal investment in the risky portfolio (Equation 6.7). The proportion invested in the risky portfolio will therefore decrease.
Use these inputs for Problems 13 through 19: You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. 17. Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 16%. b. What are your client's investment proportions in your three stocks and the T-bill fund?
Client's investment proportions: 20.0% in T-bills 0.8 × 25% = 20.0% in Stock A 0.8 × 32% = 25.6% in Stock B 0.8 × 43% = 34.4% in Stock C
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. a. Explain to your client the disadvantage of the switch. To achieve a target mean of 11.5%, we first write the mean of the complete portfolio as a function of the proportion invested in my fund (y): _____
E(rC) = .08 + y × (.18 − .08) = .08 + .10 × y
Use these inputs for Problems 13 through 19: You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. 19. Your client's degree of risk aversion is A = 3.5. b. What are the expected value and standard deviation of the rate of return on your client's optimized portfolio?
E(rC) = 0.08 + 0.10 × y* = 0.08 + (0.3644 × 0.1) = 0.1164 or 11.644% sC = 0.3644 × 28 = 10.203%
21. Consider the following information about a risky portfolio that you manage and a risk-free asset: E(rP) = 11%, σP = 15%, rf = 5%. a. Your client wants to invest a proportion of her total investment budget in your risky fund to provide an expected rate of return on her overall or complete portfolio equal to 8%. What proportion should she invest in the risky portfolio, P, and what proportion in the risk-free asset?
E(rC) = 8% = 5% + y × (11% - 5%) -> y=(.08-.05)/(.11-.05)=0.5
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. 28. Your client ponders whether to switch the 70% that is invested in your fund to the passive portfolio. a. Explain to your client the disadvantage of the switch. With 70% of his money invested in my fund's portfolio, the client's expected return is 15% per year with a standard deviation of 19.6% per year. If he shifts that money to the passive portfolio (which has an expected return of 13% and standard deviation of 25%), his overall expected return becomes: ______
E(rC) = rf + 0.7 × [E(rM) − rf] = .08 + [0.7 × (.13 - .08)] = .115, or 11.5%
Use these inputs for Problems 13 through 19: You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. 17. Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 16%. a. What is the proportion y?
E(rC) = rf + y × [E(rP) - rf] = .08 + y × (.18 - .08) If the expected return for the portfolio is 16%, then: 16% = 8% + 10% × y Þ Therefore, in order to have a portfolio with expected rate of return equal to 16%, the client must invest 80% of total funds in the risky portfolio and 20% in T-bills.
Use these inputs for Problems 13 through 19: You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. 18. Suppose that your client prefers to invest in your fund a proportion y that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio's standard deviation will not exceed 18%. b. What is the expected rate of return on the complete portfolio?
E(r_C)=.08+.1×y=.08+(0.6429×.1)=14.429%
22. Investment Management Inc. (IMI) uses the capital market line to make asset allocation recommendations. IMI derives the following forecasts: ∙ Expected return on the market portfolio: 12% ∙ Standard deviation on the market portfolio: 20% ∙ Risk-free rate: 5% Samuel Johnson seeks IMI's advice for a portfolio asset allocation. Johnson informs IMI that he wants the standard deviation of the portfolio to equal half of the standard deviation for the market portfolio. Using the capital market line, what expected return can IMI provide subject to Johnson's risk constraint? Johnson requests the portfolio standard deviation to equal one half the market portfolio standard deviation. The market portfolio , which implies . The intercept of the CML equals and the slope of the CML equals the Sharpe ratio for the market portfolio (35%). Therefore using the CML: _____
E(r_P)=r_f+(E(r_M)-r_f)/σ_M σ_P =0.05+0.35×0.10=0.085=8.5%
Use these inputs for Problems 13 through 19: You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. 13. Your client chooses to invest 70% of a portfolio in your fund and 30% in an essentially risk-free money market fund. What are the expected value and standard deviation of the rate of return on his portfolio?
Expected return = (0.7 × 18%) + (0.3 × 8%) = 15% Standard deviation = 0.7 × 28% = 19.6%
2. Which of the following statements are true? Explain. a. A lower allocation to the risky portfolio reduces the Sharpe (reward-to-volatility) ratio
False
For Problems 23 through 26: Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%, that rf = 5%, and that your fund has the parameters given in Problem 21. 25. Solve Problems 23 and 24 for a client who uses your fund rather than an index fund. For a borrowing position: ______
For a borrowing position: A<(0.11-0.09)/(0.15^2 )=0.89
4. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $70,000 or $200,000 with equal probabilities of .5. The alternative risk-free investment in T-bills pays 6% per year. d. Comparing your answers to (a) and (c), what do you conclude about the relationship between the required risk premium on a portfolio and the price at which the portfolio will sell?
For a given expected cash flow, portfolios that command greater risk premiums must sell at lower prices. The extra discount from expected value is a penalty for risk.
For Problems 23 through 26: Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%, that rf = 5%, and that your fund has the parameters given in Problem 21. 25. Solve Problems 23 and 24 for a client who uses your fund rather than an index fund. For a lending position: _____
For a lending position: A>(0.11-0.05)/(0.15^2 )=2.67
20. Look at the data in Table 6.7 on the average excess return of the U.S. equity market and the standard deviation of that excess return. Suppose that the U.S. market is your risky portfolio c. What do you conclude upon comparing your answers to (a) and (b)?
In part (b), the market risk premium is expected to be lower than in part (a) and market risk is higher. Therefore, the reward-to-volatility ratio is expected to be lower in part (b), which explains the greater proportion invested in T-bills.
Use these inputs for Problems 13 through 19: You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. 14. Suppose that your risky portfolio includes the following investments in the given proportions: Stock A 25% Stock B 32% Stock C 43% What are the investment proportions of your client's overall portfolio, including the position in T-bills?
Investment proportions: 30.0% in T-bills 0.7 × 25% = 17.5% in Stock A 0.7 × 32% = 22.4% in Stock B 0.7 × 43% = 30.1% in Stock C
For Problems 23 through 26: Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%, that rf = 5%, and that your fund has the parameters given in Problem 21. 26. What is the largest percentage fee that a client who currently is lending (y < 1) will be willing to pay to invest in your fund? What about a client who is borrowing (y > 1)? ______ will not be clients of the fund. We find that f is negative: that is, you would need to pay investors to choose your active fund. These investors desire higher risk-higher return complete portfolios and thus are in the borrowing range of the relevant CAL. In this range, the reward-to-variability ratio of the index (the passive fund) is better than that of the managed fund.
More risk tolerant investors (who are more inclined to borrow)
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. 28. Your client ponders whether to switch the 70% that is invested in your fund to the passive portfolio b. Show him the maximum fee you could charge (as a percentage of the investment in your fund, deducted at the end of the year) that would leave him at least as well off investing in your fund as in the passive one. (Hint: The fee will lower the slope of his CAL by reducing the expected return net of the fee.) The fee would reduce the reward-to-volatility ratio, i.e., the slope of the CAL. The client will be indifferent between my fund and the passive portfolio if the slope of the after-fee CAL and the CML are equal. Let f denote the fee: Slope of CALL with Fee = _____ Slope of CML (which requires no fee) = _____
Slope of CAL with fee =(.18-.08-f)/.28=(.10-f)/.28 Slope of CML (which requires no fee)=(.13-.08)/.25=0.20
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. 27. Draw the CML and your funds' CAL on an expected return-standard deviation diagram. a. What is the slope of the CML?
Slope of the CML=(.13-.08)/.25=0.20
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. 29. Consider again the client in Problem 19 with A = 3.5. b. Is the fee (percentage of the investment in your fund, deducted at the end of the year) that you can charge to make the client indifferent between your fund and the passive strategy affected by his capital allocation decision (i.e., his choice of y)?
The answer here is the same as the answer to Problem 28(b). The fee that you can charge a client is the same regardless of the asset allocation mix of the client's portfolio. You can charge a fee that will equate the reward-to-volatility ratio of your portfolio to that of your competition.
21. Consider the following information about a risky portfolio that you manage and a risk-free asset: E(rP) = 11%, σP = 15%, rf = 5%. c. Another client wants the highest return possible subject to the constraint that you limit his standard deviation to be no more than 12%. Which client is more risk averse?
The first client is more risk averse, preferring investments that have less risk as evidenced by the lower standard deviation.
6. Draw the indifference curve in the expected return-standard deviation plane corresponding to a utility level of .05 for an investor with a risk aversion coefficient of 3. (Hint: Choose several possible standard deviations, ranging from 0 to .25, and find the expected rates of return providing a utility level of .05. Then plot the expected return-standard deviation points so derived.) Points on the curve are derived by solving for E(r) in the following equation: _____
U = 0.05 = E(r) - 0.5Aσ2 = E(r) - 1.5σ2
5. Consider a portfolio that offers an expected rate of return of 12% and a standard deviation of 18%. T-bills offer a risk-free 7% rate of return. What is the maximum level of risk aversion for which the risky portfolio is still preferred to T-bills? The utility level for the risky portfolio is: _____
U = 0.12 - 0.5 × A × (0.18)2 = 0.12 - 0.0162 × A
7. Now draw the indifference curve corresponding to a utility level of .05 for an investor with risk aversion coefficient A = 4. Comparing your answer to Problem 6, what do you conclude? Repeating the analysis in Problem 6, utility is now: _____
U = E(r) - 0.5Aσ2 = E(r) - 2.0σ2 = 0.05
Use these inputs for Problems 13 through 19: You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. 15. What is the reward-to-volatility (Sharpe) ratio (S) of your risky portfolio? Your client's?
Your reward-to-volatility (Sharpe) ratio: S =(.18-.08)/.28=0.3571 Client's reward-to-volatility (Sharpe) ratio: S=(.15-.08)/.196=0.3571
7. Now draw the indifference curve corresponding to a utility level of .05 for an investor with risk aversion coefficient A = 4. Comparing your answer to Problem 6, what do you conclude? The indifference curve in Problem 7 differs from that in Problem 6 in slope. When A increases from 3 to 4, the increased risk aversion results in _____
a greater slope for the indifference curve since more expected return is needed in order to compensate for additional σ.
9. What must be true about the sign of the risk aversion coefficient, A, for a risk lover? Draw the indifference curve for a utility level of .05 for a risk lover. This amounts to ____
a negative coefficient of risk aversion.
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. 27. Draw the CML and your funds' CAL on an expected return-standard deviation diagram. b. Characterize in one short paragraph the advantage of your fund over the passive fund.
b. My fund allows an investor to achieve a higher mean for any given standard deviation than would a passive strategy, i.e., a higher expected return for any given level of risk.
8. Draw an indifference curve for a risk-neutral investor providing utility level .05. The ____ for a risk neutral investor is zero.
coefficient of risk aversion
8. Draw an indifference curve for a risk-neutral investor providing utility level .05. The _____ is a horizontal line, labeled Q8 in the graph above (see Problem 6).
corresponding indifference curve in the expected return-standard deviation plane
8. Draw an indifference curve for a risk-neutral investor providing utility level .05. Therefore, the corresponding utility is equal to the ____
portfolio's expected return.
9. What must be true about the sign of the risk aversion coefficient, A, for a risk lover? Draw the indifference curve for a utility level of .05 for a risk lover. A ____, rather than penalizing portfolio utility to account for risk, derives greater utility as variance increases.
risk lover
7. Now draw the indifference curve corresponding to a utility level of .05 for an investor with risk aversion coefficient A = 4. Comparing your answer to Problem 6, what do you conclude? The equal-utility combinations of expected return and standard deviation are presented in the table below. The indifference curve is the upward sloping line in the graph on the next page, labeled Q7 (for Question 7). --->
s s^ 2 E(r) 0.00 0.0000 0.0500 0.05 0.0025 0.0550 0.10 0.0100 0.0700 0.15 0.0225 0.0950 0.20 0.0400 0.1300 0.25 0.0625 0.1750
6. Draw the indifference curve in the expected return-standard deviation plane corresponding to a utility level of .05 for an investor with a risk aversion coefficient of 3. (Hint: Choose several possible standard deviations, ranging from 0 to .25, and find the expected rates of return providing a utility level of .05. Then plot the expected return-standard deviation points so derived.) The values of E(r), given the values of σ2, are therefore: _____
s s^2 E(r) 0.00 0.0000 0.05000 0.05 0.0025 0.05375 0.10 0.0100 0.06500 0.15 0.0225 0.08375 0.20 0.0400 0.11000 0.25 0.0625 0.14375 **The bold line in the graph on the next page (labeled Q6, for Question 6) depicts the indifference curve.**
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. a. Explain to your client the disadvantage of the switch. Therefore, the shift entails a decrease in mean from 15% to 11.5% and a decrease in standard deviation from 19.6% to 17.5%. Since both mean return and standard deviation decrease, it is not yet clear whether the move is beneficial. The disadvantage of the shift is _____
that, if the client is willing to accept a mean return on his total portfolio of 11.5%, he can achieve it with a lower standard deviation using my fund rather than the passive portfolio.
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. a. Explain to your client the disadvantage of the switch. Thus, by using my portfolio, _____ can be achieved with a standard deviation of only 9.8% as opposed to the standard deviation of 17.5% using the passive portfolio.
the same 11.5% expected return
For Problems 23 through 26: Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%, that rf = 5%, and that your fund has the parameters given in Problem 21. 24. What is the range of risk aversion for which a client will neither borrow nor lend, that is, for which y = 1? For values of risk aversion within this range, the client will neither borrow nor lend but will hold a portfolio composed only of the optimal risky portfolio: ______
y = 1 for 0.64 ≤ A ≤ 1.28
Use these inputs for Problems term-3913 through 19: You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. 19. Your client's degree of risk aversion is A = 3.5. a. What proportion, y, of the total investment should be invested in your fund?
y*= (E(r_P)-r_f)/(Aσ_P^2 )=(0.18-0.08)/(3.5×0.28^2)=0.10/0.2744=0.3644 Therefore, the client's optimal proportions are: 36.44% invested in the risky portfolio and 63.56% invested in T-bills.
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. 29. Consider again the client in Problem 19 with A = 3.5. a. If he chose to invest in the passive portfolio, what proportion, y, would he select? The formula for the optimal proportion to invest in the passive portfolio is: y*=(E(r_M)-r_f)/(Aσ_M^2 ) Substitute the following: E(rM) = 13%; rf = 8%; σM = 25%; A = 3.5: _____
y*=(0.13-0.08)/(3.5×0.25^2 )=0.2286",or " 22.86%" in the passive portfolio"
20. Look at the data in Table 6.7 on the average excess return of the U.S. equity market and the standard deviation of that excess return. Suppose that the U.S. market is your risky portfolio. b. What if you believe that the 1973-1995 period is representative? If the period 1973-1995 is assumed to be representative of future expected performance, then we use the following data to compute the fraction allocated to equity: A = 4, E(rM) − rf = 6.11%, σM = 18.34% and y* is given by: _____
y*=(E(r_M)-r_f)/(Aσ_M^2 )=0.0611/(4×0.1834^2 )=0.4541 Therefore, 45.41% of the complete portfolio should be allocated to equity and 54.59% should be allocated to T-bills.
20. Look at the data in Table 6.7 on the average excess return of the U.S. equity market and the standard deviation of that excess return. Suppose that the U.S. market is your risky portfolio. a. If your risk-aversion coefficient is A = 4 and you believe that the entire 1927-2018 period is representative of future expected performance, what fraction of your portfolio should be allocated to T-bills and what fraction to equity? If the period 1927-2018 is assumed to be representative of future expected performance, then we use the following data to compute the fraction allocated to equity: A = 4, E(rM) − rf = 8.34%, σM = 20.36% (we use the standard deviation of the risk premium from Table 6.7). Then y* is given by: ______
y*=(E(r_M)-r_f)/(Aσ_M^2 )=0.0834/(4×0.2036^2 )=0.5030 That is, 50.30% of the portfolio should be allocated to equity and 49.70% should be allocated to T-bills.
For Problems 23 through 26: Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%, that rf = 5%, and that your fund has the parameters given in Problem 21. 24. What is the range of risk aversion for which a client will neither borrow nor lend, that is, for which y = 1? For y to be greater than 1 (the investor is a borrower), A must be small enough: _____
y= (E(r_M)-r_f) / (Aσ_M^2 )>1 --> A<(0.13-0.09) / (0.25^2)=0.64
For Problems 23 through 26: Suppose that the borrowing rate that your client faces is 9%. Assume that the equity market index has an expected return of 13% and standard deviation of 25%, that rf = 5%, and that your fund has the parameters given in Problem 21. 24. What is the range of risk aversion for which a client will neither borrow nor lend, that is, for which y = 1? For y to be less than 1.0 (that the investor is a lender), risk aversion (A) must be large enough such that: _____
y=(E(r_M)-r_f)/(Aσ_M^2 )<1 --> A>(0.13-0.05)/(0.25^2 )=1.28
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. a. Explain to your client the disadvantage of the switch. The standard deviation of the complete portfolio using the passive portfolio would be: _____
σC = 0.7 × σM = 0.7 × 25% = 17.5%
Use these inputs for Problems 13 through 19: You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. 17. Suppose that your client decides to invest in your portfolio a proportion y of the total investment budget so that the overall portfolio will have an expected rate of return of 16%. c. What is the standard deviation of the rate of return on your client's portfolio?
σC = 0.8 × σP = 0.8 × 28% = 22.4%
Use these inputs for Problems 13 through 19: You manage a risky portfolio with an expected rate of return of 18% and a standard deviation of 28%. The T-bill rate is 8%. 18. Suppose that your client prefers to invest in your fund a proportion y that maximizes the expected return on the complete portfolio subject to the constraint that the complete portfolio's standard deviation will not exceed 18%. a. What is the investment proportion, y?
σC = y × 28% If your client prefers a standard deviation of at most 18%, then: y = 18/28 = 0.6429 = 64.29% invested in the risky portfolio.
For Problems 27 through 29: You estimate that a passive portfolio, for example, one invested in a risky portfolio that mimics the S&P 500 stock index, offers an expected rate of return of 13% with a standard deviation of 25%. You manage an active portfolio with expected return 18% and standard deviation 28%. The risk-free rate is 8%. a. Explain to your client the disadvantage of the switch. The standard deviation of this portfolio would be: _____
σC = y × 28% = 0.35 × 28% = 9.8%
21. Consider the following information about a risky portfolio that you manage and a risk-free asset: E(rP) = 11%, σP = 15%, rf = 5%. b. What will be the standard deviation of the rate of return on her portfolio?
σC = y × σP = 0.50 × 15% = 7.5%
