Investment Management Tutorial

1. Investment Management Tutorial October 10, 2008 James Kozyra

2. You are faced with the probability distribution on the stock market index fund given below. Suppose the price of a put option on a share of the index fund with an exercise price of $110 and maturity of one year is $12. The current share price is $100 and a cash dividend of $4 per share is expected to be paid during the year. Chapter 5 Problem 16

3. What is the probability distribution of the HPR on the put option Chapter 5 Problem 16

4. What is the probability distribution of the HPR on a portfolio consisting of one share of the index fund and a put option? The cost of one share and a put is $112 ($110 + $12). Chapter 5 Problem 16

5. Chapter 5 Problem 16

6. Two investment advisors are comparing performance. One averaged a 19% rate of return and the other a 16% rate of return. However, the beta of the first investor was 1.5, whereas the that of the second was 1. a) Can you tell which investor was a better predictor of individual stocks (aside from the issue of general movements in the market)? Chapter 8 Problem 14

7. a) To determine which investor was the better predictor we look at their abnormal return, which is the ex-post alpha. This means that the abnormal return is the difference between the actual return and the return predicted by the SML. Without the parameters of the equation (risk-free rate and market rate of return) we cannot determine which investor was more accurate. Chapter 8 Problem 14

8. b) If the t-bill rate were 6% and the market return during the period were 14%, which investor would be the superior stock selector? Investor 1 = 19 � [6 + 1.5(14-6)] = 19 � 18 = 1% Investor 2 = 16 � [6 + 1(14-6)] = 16 � 14 = 2% Chapter 8 Problem 14

9. c) What if the t-bill rate were 3% and the market return were 15%? Investor 1 = 19 � [3 + 1.5(15-3)] = 19 � 21 = -2% Investor 2 = 16 � [3 + 1(15-3)] = 16 � 15 = 1% Chapter 8 Problem 14

10. First case � the second investor has the larger abnormal return and appears to be the superior stock selector. He appears to have done a better job finding underpriced stocks. Second case � the second investor again is the superior stock selector. The first investor�s predictions appear to be worthless. Chapter 8 Problem 14

11. 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%. A passive portfolio has an expected rate of return of 13% and a standard deviation of 25%. Your client ponders whether to switch the 70% that he has invested in your risky portfolio to the passive portfolio. Chapter 6 Problem 21

12. a) Explain to your client the disadvantages of the shift. Current Portfolio E(r) = (.3 x 8%) + (.7 x 18%) = 15% Std Dev = .7 x 28% = 19.6% Portfolio after the shift E(r) = (.3 x 8%) + (.7 x 13%) = 11.5% Std Dev = .7 x 25% = 17.5% Chapter 6 Problem 21

13. a) Therefore, the shift entails a decline in the expected return from 14% to 11.5% and a decline in the standard deviation from 19.6% to 17.5%. The disadvantage is that the client could achieve an 11.5% expected return in my portfolio, with a lower standard deviation. Chapter 6 Problem 21

14. We first must write the mean of the complete portfolio as a function of the proportions invested in my portfolio, y: E(r) = 8 + y(18-8) E(r) = 8 + 10y Given the target 11.5% return, the proportion that must be invested in the fund is determined as follows: 11.5 = 8 + 10y Chapter 6 Problem 21

15. 11.5 = 8 + 10y 10y = 11.5 � 8 10y = (11.5 � 8) / 10 y = 0.35 The standard deviation of the portfolio would thus be 9.8% (.35 x 28%). Achieving the 11.5% return can be done with a standard deviation of 9.8% in my portfolio as opposed to 17.5% in the passive portfolio. Chapter 6 Problem 21

16. b) Show your client the maximum fee you could charge that would still leave him/her at least as well off investing in your fund. (Hint: the fee will lower the slope of the CAL by reducing E(r) net of the fee. The fee would reduce the reward-to-variability ratio (CAL slope). Clients will be indifferent if the slope of the after-fee CAL and the CML are equal. Chapter 6 Problem 21

17. Let f denote the fee Slope of the CAL with the fee = (18 � 8 � f) / 28 = (10 � f) / 28 Slope of the CML (no fee required) = (13 � 8) / 25 = .20 We must set the slopes to be equal. Chapter 6 Problem 21

18. (10 � f) / 28 = 0.20 (10 � f) = 28 x 0.20 (10 � f) = 5.6 f = 10 � 5.6 f = 4.4 Therefore the maximum fee that can be charged to make the client indifferent between portfolios is 4.4% per year. Chapter 6 Problem 21

19. You are considering investing $1,000 in a T-bill that pays 0.05 and a risky portfolio, P, constructed with two risky securities, X and Y. The weights of X and Y in P are 0.6 and 0.4, respectively. X has an expected rate of return of 0.14 and variance of 0.01, and Y has an expected rate of return of 0.10 and a variance of 0.0081. Portfolio Selection Problem

20. If you want to form a portfolio with an expected rate of return of 10%, what percentages of your money must you invest in the t-bill, X, and Y respectively if you keep X and Y in the same proportions to each other as in portfolio P (60:40). E(r) Portfolio = (0.6 x 14%) + (0.4 x 10%) E(r) T-bill = w x 5% Portfolio Selection Problem

21. Portfolio Selection Problem

22. Portfolio Selection Problem

23. What would be the dollar value of your position in X, Y, and the t-bills, respectively if you decide to hold a portfolio that has an expected outcome of $1,120 HPR = ($1,120 - $1,000) / $1,000 HPR = 12% Portfolio Selection Problem

24. Portfolio Selection Problem

25. Portfolio Selection Problem