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Week 3: Solutions to homework problems BKM chapter 5 5. Suppose your expectations regarding the stock market are as follows:

Use Equations 5.10–5.12 to compute the mean and standard deviation of the HPR on stocks. Answer: Using Equation 5.10, we can calculate the mean of the HPR as: E(r) = �

฀฀

p(s) r(s) = (0.3 × 0.44) + (0.4 × 0.14) + [0.3 × (–0.16)] = 0.14 or 14%

฀฀=1

Using Equation 5.11, we can calculate the variance as: Var(r) = σ2 = �

฀฀ p(s) ฀฀=1

[ r(s) – E(r)]2

= [0.3 × (0.44 – 0.14)2] + [0.4 × (0.14 – 0.14)2] + [0.3 × (–0.16 – 0.14)2] = 0.054 SD(r) = σ = �Var(r) = √0.054 = 0.2324 or 23.24% [Standard Deviation]

FINC3017, INVESTMENTS AND PORTFOLIO MANAGEMENT

6. The stock of Business Adventures sells for $40 a share. Its likely dividend payout and end-of-year price depend on the state of the economy by the end of the year as follows:

a. Calculate the expected holding-period return and standard deviation of the holdingperiod return. All three scenarios are equally likely. Answer: We use the below equation to calculate the holding period return of each scenario: HPR =

Ending Price － Beginning Price + Cash Dividend Beginning Price

The holding period returns for the three scenarios are: Boom:

(50 – 40 + 2)/40 = 0.30 = 30%

Normal:

(43 – 40 + 1)/40 = 0.10 = 10%

Recession: (34 – 40 + 0.50)/40 = –0.1375 = –13.75% E(HPR) = �

฀฀

p(s) r(s)

฀฀=1

= [(1/3) × 0.30] + [(1/3) × 0.10] + [(1/3) × (–0.1375)] = 0.0875 or 8.75% Var(HPR) = �

฀฀ p(s) ฀฀=1

[r(s) – E(r)]2

= [(1/3) × (0.30 – 0.0875)2] + [(1/3) × (0.10 – 0.0875)2] + [(1/3) (–0.1375 – 0.0875)2] = 0.031979 SD(r) = σ = �Var(r) =

319.79 = 0.1788 or 17.88%

b. Calculate the expected return and standard deviation of a portfolio invested half in Business Adventures and half in Treasury bills. The return on bills is 4%. Answer: E(r) = (0.5 × 8.75%) + (0.5 × 4%) = 6.375% σ = 0.5 × 17.88% = 8.94%

FINC3017, INVESTMENTS AND PORTFOLIO MANAGEMENT

8. a. Suppose you forecast that the standard deviation of the market return will be 20% in the coming year. If the measure of risk aversion in Equation 5.16 is A = 4, what would be a reasonable guess for the expected market risk premium? Answer: Given that A = 4 and the projected standard deviation of the market return = 20%, we can use the below equation to solve for the expected market risk premium: A=4=

Average(rM)－ rf Sample σM

2

=

Average(rM)－ rf (20%)2

E(rM) – rf = AσM2 = 4 × (0.20)2 = 0.16 or 16% b. What value of A is consistent with a risk premium of 9%? Answer: Solve E(rM) – rf = 0.09 = AσM2 = A × (0.20)2 , we can get A = 0.09/0.04 = 2.25 c. What will happen to the risk premium if investors become more risk tolerant? Answer: Increased risk tolerance means decreased risk aversion (A), which results in a decline in risk premiums. For Problems 12–16, assume that 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%. 12. Your client chooses to invest 70% of a portfolio in your fund and 30% in a T-bill money market fund. a. What is the expected return and standard deviation of your client’s portfolio? Answer: Allocating 70% of the capital in the risky portfolio P, and 30% in risk-free asset, the client has an expected return on the complete portfolio calculated by adding up the expected return of the risky proportion (y) and the expected return of the proportion (1 - y) of the risk-free investment: E(rC) = y × E(rP) + (1 – y) × rf = (0.7 × 0.17) + (0.3 × 0.07) = 0.14 or 14% per year The standard deviation of the portfolio equals the standard deviation of the risky fund times the fraction of the complete portfolio invested in the risky fund:

σC = y ×σP = 0.7 × 0.27 = 0.189 or 18.9% per year

FINC3017, INVESTMENTS AND PORTFOLIO MANAGEMENT

b. Suppose your risky portfolio includes the following investments in the given proportions:

What are the investment proportions of each stock in your client’s overall portfolio, including the position in T-bills? Answer: The investment proportions of the client’s overall portfolio can be calculated by the proportion of risky portfolio in the complete portfolio times the proportion allocated in each stock.

Security T-Bills Stock A Stock B Stock C

0.7 × 27% = 0.7 × 33% = 0.7 × 40% =

Investment Proportions 30.0% 18.9% 23.1% 28.0%

c. What is the Sharpe ratio (S) of your risky portfolio and your client’s overall portfolio? Answer: We calculate the reward-to-variability ratio (Sharpe ratio) using Equation 5.17. For the risky portfolio: S=

=

Portfolio Risk Premium Standard Deviation of Portfolio Excess Return

E(rP) － rf

σP

=

0.17 － 0.07 = 0.3704 0.27

For the client’s overall portfolio: S=

E(rC) － rf

σC

=

0.14 － 0.07 = 0.3704 0.189

FINC3017, INVESTMENTS AND PORTFOLIO MANAGEMENT

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. Answer:

E (r )

% P

17

CA CAL L( slope=.3704)

14

c lien lientt 7

σ 18.9

27

%

13. 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%. a. What is the proportion y? Answer: E(rC) = y × E(rP) + (1 – y) × rf = y × 0.17 + (1 – y) × 0.07 = 0.15 or 15% per year Solving for y, we get y =

0.15 － 0.07 = 0.8 0.10

Therefore, in order to achieve an expected rate of return of 15%, the client must invest 80% of total funds in the risky portfolio and 20% in T-bills.

FINC3017, INVESTMENTS AND PORTFOLIO MANAGEMENT

b. What are your client’s investment proportions in your three stocks and in T-bills? Answer: The investment proportions of the client’s overall portfolio can be calculated by the proportion of risky asset in the whole portfolio times the proportion allocated in each stock. Security T-Bills Stock A Stock B Stock C

0.8 × 27% = 0.8 × 33% = 0.8 × 40% =

Investment Proportions 20.0% 21.6% 26.4% 32.0%

c. What is the standard deviation of the rate of return on your client’s portfolio? Answer: The standard deviation of the complete portfolio is the standard deviation of the risky portfolio times the fraction of the portfolio invested in the risky asset:

σC = y ×σP = 0.8 × 0.27 = 0.216 or 21.6% per year 14. 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%. a. What is the investment proportion, y? Answer: Standard deviation of the complete portfolio = σC = y × 0.27 If the client wants the standard deviation to be equal or less than 20%, then: y = (0.20/0.27) = 0.7407 = 74.07% He should invest, at most, 74.07% in the risky fund. b. What is the expected rate of return on the overall portfolio? Answer: E(rC) = rf + y × [E(rP) – rf] = 0.07 + 0.7407 × 0.10 = 0.1441 or 14.41%

FINC3017, INVESTMENTS AND PORTFOLIO MANAGEMENT

CFA problems Use the following data in answering CFA Questions 4–6.

Suppose investor “satisfaction” with a portfolio increases with expected return and decreases with variance according to the following “utility” formula: U = E(r) − ½ Aσ2 where A denotes the investor’s risk aversion. 4. Based on the formula for investor satisfaction or “utility,” which investment would you select if you were risk averse with A = 4? Answer: Investment 3. For each portfolio: Utility = E(r) – (0.5 × 4 × σ2) Investment 1 2 3 4

E(r) 0.12 0.15 0.21 0.24

σ 0.30 0.50 0.16 0.21

Utility -0.0600 -0.3500 0.1588 0.1518

We choose the portfolio with the highest utility value. 5. Based on the formula above, which investment would you select if you were risk neutral, with A = 0? Answer: Investment 4. When an investor is risk neutral, A = 0 so that the portfolio with the highest utility is the portfolio with the highest expected return. 6. The variable (A) in the utility formula represents the: a. Investor’s return requirement. b. Is higher when the investor demands a greater risk premium as compensation for a given increase in the variance of returns. c. Preference for one unit of return per four units of risk. Answer: b. Investor’s aversion to risk.

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