Question

Suppose you are a fund manager, currently managing a diversified risky portfolio that consists of equity index fund A (40%), bond index fund B (30%), and international equity fund C (30%). The portfolio has an expected rate of return of 12% and a standard deviation of 25%. Lisa, a project manager in IBM, is one of your clients. After some discussion with her, you suggest Lisa to invest her total $800,000 personal wealth in your fund and a T-bill money market fund, which has a return of 4%. You can assume the quadratic utility function for some of the following analysis.

a) If Lisa chooses to invest 70% of a portfolio in your fund and 30% in the T-bill, what is the expected value and standard deviation of the rate of return on her portfolio?

b) Suppose that Lisa prefers to invest in your fund a certain proportion so that maximizes the expected return on the overall portfolio subject to the constraint that the overall portfolio’s standard deviation will not exceed 18%. What is the investment proportion of your fund in Lisa’s new portfolio? What is the expected rate of return on her overall portfolio?

c) From Lisa’s risk preference questionnaire, you learnt that her risk-aversion coefficient is A = 1.2. What proportions of the total investment should you suggest that she invest in your fund and the T-bill, respectively?

d) Assume Lisa still keeps her allocation as in part a), i.e., 70% in your fund and 30% in the T-bill. Few weeks later, Lisa is granted some IBM company’s stocks worth $200,000, as bonus of the year. By looking at historical return data, you provide her with the following forecast information. What’s the expected return of Lisa’s new portfolio that includes IBM? What’s the standard deviation of her new portfolio?

Answer 1

1) Return is 9.6 % and Std Dev is 17.5 %

Return	Fund	T Bill		Standard Deviation	Fund	T Bill
Expected Return	12%	4%		Expected Std Deviation	25%	0%
Weight	0.7	0.3		Weight	0.7	0.3
Individual Return	8.40%	1.20%		Individual	17.50%	0.00%
Portfolio Return	9.60%			Portfolio Std Dev	17.50%

2) Std Dev of Portfolio at 18% i.e can be achevied by 18%/25% = 0.72 since std deviation of T Bill is 0.

Return	Fund	T Bill		Standard Deviation	Fund	T Bill
Expected Return	12%	4%		Expected Std Deviation	25%	0%
Weight	0.72	0.28		Weight	0.72	0.28
Individual Return	8.64%	1.12%		Individual	18.00%	0.00%
Portfolio Return	9.76%			Portfolio Std Dev	18.00%

3)

Utility Score	=Expected Return-0.5* sd^2*A
	0.12-0.5*0.25*0.25*1.2
	0.0825

Return	Fund	T Bill
Expected Return	12%	4%
Weight	0.53	0.47
Individual Return	6.36%	1.88%
Portfolio Return	8.24%

4) Information about IBM is not available to work the answer