CHAPTER 06Capital Allocation to Risky Assets

.Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $50,000 or $150,000, with equal probabilities of .5. The alternative riskless investment in T-bills pays 5%. (LO 5-3)
a. If you require a risk premium of 10%, how much will you be willing to pay for the portfolio?
b. Suppose the portfolio can be purchased for the amount you found in (a). What will the expected rate of return on the portfolio be?

. 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 bills?

Please answer the following questions3 to 7 based on the following assumption: you manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 27%. The T-bill rate is 7%.
. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund.  (LO 5-3)
a. What is the expected return and standard deviation of your client’s portfolio?
b. Suppose your risky portfolio includes the following investments in the given proportions:
Stock    Given Proportions
Stock A    27%
Stock B    33%
Stock C    40%

What are the investment proportions of your client’s overall portfolio, including the position in T-bills?
c. What is the reward-to-volatility ratio ( S ) of your risky portfolio and your client’s overall portfolio?
d. Draw the CAL of your portfolio on an expected return/standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund’s CAL.
.. Suppose the same client in the previous problem decides to invest in your risky portfolio
a proportion ( y ) of his total investment budget so that his overall portfolio will have an expected rate of return of 15%.  (LO 5-3)
a. What is the proportion y?
b. What are your client’s investment proportions in your three stocks and the T-billfund?
c. What is the standard deviation of the rate of return on your client’s portfolio?

.. Suppose the same client as in the previous problem prefers to invest in your portfolio aproportion(y ) that maximizes the expected return on the overall portfolio subject to the constraint that the overall portfolio’s standard deviation will not exceed 20%.  (LO 5-3)
a. What is the investment proportion, y?
b. What is the expected rate of return on the overall portfolio?

.. You estimate that a passive portfolio invested to mimic the S&P 500 stock index yields an expected rate of return of 13% with a standard deviation of 25%. Draw the CML and your fund’s CAL on an expected return/standard deviation diagram.  (LO 5-4)
a. What is the slope of the CML?
b. Characterize in one short paragraph the advantage of your fund over the passive fund.

. Your client (see previous problem) wonders whether to switch the 70% that is invested in your fund to the passive portfolio.  (LO 5-4)
a. Explain to your client the disadvantage of the switch.
b. Show your client the maximum fee you could charge (as a percent of the investment in your fund deducted at the end of the year) that would still leave him at least as well off investing in your fund as in the passive one. (Hint: The fee will lower the slope of your client’s CAL by reducing the expected return net of the fee.)

.. You manage an equity fund with an expected risk premium of 10% and a standard deviation of 14%. The rate on Treasury bills is 6%. Your client chooses to invest $60,000 of her portfolio in your equity fund and $40,000 in a T-bill money market fund.
a. What is the expected return and standard deviation of return on your client’s portfolio?  (LO 5-3)
b. What is the reward-to-volatility ratio for the equity fund in the previous problem?

CFA Problems
1. A portfolio of nondividend-paying stocks earned a geometric mean return of 5% between January 1, 2005, and December 31, 2011. The arithmetic mean return for the same period was 6%. If the market value of the portfolio at the beginning of 2005 was $100,000, what was the market value of the portfolio at the end of 2011?  (LO 5-1)
2. Which of the following statements about the standard deviation is/are  true? A standard deviation:  (LO 5-2)
a. Is the square root of the variance?
b. Is denominated in the same units as the original data.
c. Can be a positive or a negative number.

3. Which of the following statements reflects the importance of the asset allocation decision to the investment process? The asset allocation decision:  (LO 5-3)
a. Helps the investor decide on realistic investment goals.
b. Identifies the specific securities to include in a portfolio.
c. Determines most of the portfolio’s returns and volatility over time.
d. Creates a standard by which to establish an appropriate investment time horizon.

Use the following data in answering CFA Questions 4–6.
Investment    Expected Return, E(r)    Standard Deviation, s
1    .12    .30
2    .15    .50
3    .21    .16
4    .24    .21
Investor “satisfaction” with portfolio increases with expected return and decreases with variance according to the “utility” formula:  U 5 E(r) 2 ½ As2   where A = 4.
4. Based on the formula for investor satisfaction or “utility,” which investment would you select if you were risk averse with A = 4?  (LO 5-4)
5. Based on the formula above, which investment would you select if you were risk neutral?  (LO 5-4)
6. The variable (A) in the utility formula represents the:  (LO 5-4)
a. Investor’s return requirement.
b. Investor’s aversion to risk.
c. Certainty equivalent rate of the portfolio.
d. Preference for one unit of return per four units of risk.

Use the following scenario analysis for stocks X and Y to answer CFA Questions 7 through 9.

Bear Market    Normal Market    Bull Market
Probability       .2     .5     .3
Stock  X     -20%    18%    50%
Stock Y     -15%    20%    10%
7. What are the expected returns for stocks  X and  Y ?  (LO 5-2)

8. What are the standard deviations of returns on stocks  X and  Y ?  (LO 5-2)

9. Assume that of your $10,000 portfolio, you invest $9,000 in stock X and $1,000 in stock  Y. What is the expected return on your portfolio?  (LO 5-3)

CHAPTER 06Capital Allocation to Risky Assets

.Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either $50,000 or $150,000, with equal probabilities of .5. The alternative riskless investment in T-bills pays 5%. (LO 5-3)
a. If you require a risk premium of 10%, how much will you be willing to pay for the portfolio?
b. Suppose the portfolio can be purchased for the amount you found in (a). What will the expected rate of return on the portfolio be?

. 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 bills?

Please answer the following questions3 to 7 based on the following assumption: you manage a risky portfolio with an expected rate of return of 17% and a standard deviation of 27%. The T-bill rate is 7%.
. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund.  (LO 5-3)
a. What is the expected return and standard deviation of your client’s portfolio?
b. Suppose your risky portfolio includes the following investments in the given proportions:
Stock    Given Proportions
Stock A    27%
Stock B    33%
Stock C    40%

What are the investment proportions of your client’s overall portfolio, including the position in T-bills?
c. What is the reward-to-volatility ratio ( S ) of your risky portfolio and your client’s overall portfolio?
d. Draw the CAL of your portfolio on an expected return/standard deviation diagram. What is the slope of the CAL? Show the position of your client on your fund’s CAL.
.. Suppose the same client in the previous problem decides to invest in your risky portfolio
a proportion ( y ) of his total investment budget so that his overall portfolio will have an expected rate of return of 15%.  (LO 5-3)
a. What is the proportion y?
b. What are your client’s investment proportions in your three stocks and the T-billfund?
c. What is the standard deviation of the rate of return on your client’s portfolio?

.. Suppose the same client as in the previous problem prefers to invest in your portfolio aproportion(y ) that maximizes the expected return on the overall portfolio subject to the constraint that the overall portfolio’s standard deviation will not exceed 20%.  (LO 5-3)
a. What is the investment proportion, y?
b. What is the expected rate of return on the overall portfolio?

.. You estimate that a passive portfolio invested to mimic the S&P 500 stock index yields an expected rate of return of 13% with a standard deviation of 25%. Draw the CML and your fund’s CAL on an expected return/standard deviation diagram.  (LO 5-4)
a. What is the slope of the CML?
b. Characterize in one short paragraph the advantage of your fund over the passive fund.

. Your client (see previous problem) wonders whether to switch the 70% that is invested in your fund to the passive portfolio.  (LO 5-4)
a. Explain to your client the disadvantage of the switch.
b. Show your client the maximum fee you could charge (as a percent of the investment in your fund deducted at the end of the year) that would still leave him at least as well off investing in your fund as in the passive one. (Hint: The fee will lower the slope of your client’s CAL by reducing the expected return net of the fee.)

.. You manage an equity fund with an expected risk premium of 10% and a standard deviation of 14%. The rate on Treasury bills is 6%. Your client chooses to invest $60,000 of her portfolio in your equity fund and $40,000 in a T-bill money market fund.
a. What is the expected return and standard deviation of return on your client’s portfolio?  (LO 5-3)
b. What is the reward-to-volatility ratio for the equity fund in the previous problem?

CFA Problems
1. A portfolio of nondividend-paying stocks earned a geometric mean return of 5% between January 1, 2005, and December 31, 2011. The arithmetic mean return for the same period was 6%. If the market value of the portfolio at the beginning of 2005 was $100,000, what was the market value of the portfolio at the end of 2011?  (LO 5-1)
2. Which of the following statements about the standard deviation is/are  true? A standard deviation:  (LO 5-2)
a. Is the square root of the variance?
b. Is denominated in the same units as the original data.
c. Can be a positive or a negative number.

3. Which of the following statements reflects the importance of the asset allocation decision to the investment process? The asset allocation decision:  (LO 5-3)
a. Helps the investor decide on realistic investment goals.
b. Identifies the specific securities to include in a portfolio.
c. Determines most of the portfolio’s returns and volatility over time.
d. Creates a standard by which to establish an appropriate investment time horizon.

Use the following data in answering CFA Questions 4–6.
Investment    Expected Return, E(r)    Standard Deviation, s
1    .12    .30
2    .15    .50
3    .21    .16
4    .24    .21
Investor “satisfaction” with portfolio increases with expected return and decreases with variance according to the “utility” formula:  U 5 E(r) 2 ½ As2   where A = 4.
4. Based on the formula for investor satisfaction or “utility,” which investment would you select if you were risk averse with A = 4?  (LO 5-4)
5. Based on the formula above, which investment would you select if you were risk neutral?  (LO 5-4)
6. The variable (A) in the utility formula represents the:  (LO 5-4)
a. Investor’s return requirement.
b. Investor’s aversion to risk.
c. Certainty equivalent rate of the portfolio.
d. Preference for one unit of return per four units of risk.

Use the following scenario analysis for stocks X and Y to answer CFA Questions 7 through 9.

Bear Market    Normal Market    Bull Market
Probability       .2     .5     .3
Stock  X     -20%    18%    50%
Stock Y     -15%    20%    10%
7. What are the expected returns for stocks  X and  Y ?  (LO 5-2)

8. What are the standard deviations of returns on stocks  X and  Y ?  (LO 5-2)

9. Assume that of your $10,000 portfolio, you invest $9,000 in stock X and $1,000 in stock  Y. What is the expected return on your portfolio?  (LO 5-3)