# PART A Capital Allocation to Risky Assets

PART A Capital Allocation to Risky Assets

[endnoteRef:1]. 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) [1: ]

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?

[endnoteRef:2]. 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? [2: ]

## PART A Capital Allocation to Risky Assets

Please answer questions 3 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%.

[endnoteRef:3]. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. (LO 5-3) [3: ]

a. What is your client’s portfolio’s expected return and standard deviation?

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.

[endnoteRef:4]. Suppose the same client in the previous problem decides to invest in your risky portfolio [4: ]

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-bill fund?

c. What is the standard deviation of the rate of return on your client’s portfolio?

[endnoteRef:5]. Suppose the same client as in the previous problem prefers to invest in your portfolio a proportion (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) [5: ]

a. What is the investment proportion, y?

b. What is the expected rate of return on the overall portfolio?

[endnoteRef:6]. Suppose the same client as in the previous problem has the Utility Function: *U = E(r) – ½ A* σ2. The client has Coefficient of risk aversion A equal to 4. The client prefers to invest a proportion (y) that is the optimal capital allocation in your portfolio. [6: ]

a. What is the investment proportion, y*?

b. What is the expected rate of return on the overall portfolio?

c. What is the standard deviation of the rate of return on your client’s portfolio?

[endnoteRef:7]. 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) [7: ]

a. What is the slope of the CML?

b. Characterize in one short paragraph the advantage of your fund over the passive fund.

[endnoteRef:8]. Your client (see previous problem) wonders whether to switch the 70% that is invested in your fund to the passive portfolio. (LO 5-4) [8: ]

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.)

[endnoteRef:9]. 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) [9: ]

b. What is the reward-to-volatility ratio for the equity fund in the previous problem?

[endnoteRef:10]. 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%. 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? b. What is the expected value and standard deviation of the rate of return on your client’s optimized portfolio? [10: ]

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 = E(r) – ½ A* σ2 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)

**PART B Two Risky Portfolios**

[endnoteRef:11]. Suppose that there are many stocks in the security market and that the characteristics of stocks A and B are given as follows: [11: ]

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 risk-free rate? (Hint: Think about constructing a risk-free portfolio from stocks A and B. )

[endnoteRef:12]. A portfolio’s expected return is 12%, its standard deviation is 20%, and the risk-free rate is 4%. Which of the following would make for the greatest increase in the portfolio’s Sharpe ratio? (LO 6-3) [12: ]

a. An increase of 1% in expected return.

b. A decrease of 1% in the risk-free rate.

c. A decrease of 1% in its standard deviation.

The following data apply to problems 3-7

A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a sure rate of 5.5%. The probability distributions of the risky funds are:

Expected Return | Standard Deviation | |

Stock fund (S) | 15% | 32% |

Bond fund (B) | 9 | 23 |

The correlation between the fund returns is .15.

[endnoteRef:13]. 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%. What expected return and standard deviation does your graph show for the minimum-variance portfolio? (LO 6-2) [13: ]

[endnoteRef:14]. 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 risky portfolio? (LO 6-3) [14: ]

[endnoteRef:15]. What is the reward-to-volatility ratio of the best feasible CAL? (LO 6-3) [15: ]

[endnoteRef:16]. Suppose now that your portfolio must yield an expected return of 12% and be efficient, that is, on the best feasible CAL. (LO 6-4) [16: ]

a. What is the standard deviation of your portfolio?

b. What is the proportion invested in the T-bill fund and each of the two risky funds?

[endnoteRef:17].. If you were to use only the two risky funds and still require an expected return of 12%, what would be the investment proportions of your portfolio? Compare its standard deviation to that of the optimal portfolio in the previous problem. What do you conclude? (LO 6-4) [17: ]

[endnoteRef:18]. Suppose that many stocks are traded in the market and that it is possible to borrow at the risk-free rate, rf. The characteristics of two of the stocks are as follows: [18: ]

Stock | Expected Return | Standard Deviation |

A | 8% | 40% |

B | 13 | 60 |

Correlation =-1 |

Could the equilibrium rf be greater than 10%? ( Hint: Can a particular stock portfolio be substituted for the risk-free asset?) (LO 6-3)

PART C: Explain the underlisted.

· the concept of risk aversion and utility – (MO 1)

· how to allocate funds between a risky asset and a risk free asset – (MO 1)

· Sharpe Ratio – (MO 1,2)

· Optimal Capital Allocation – (MO 1)

· standard deviation and return for one and two security portfolios – (MO 1,2)

· the minimum variance combinations of two securities – (MO 2)

· covariance and correlation coefficients – (MO 2)

· the importance of diversification