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?

	Expected Returns	Standard Deviation	Correlation with the current portfolio
IBM	15%	30%	0.4

e) Continue with question d). Now Lisa decides to choose one of the following: i) keep all IBM stocks in her current portfolio; ii) sell all IBM stocks and invest the $200,000 in the previous held portfolio (70% your fund and 30% in the T-bill). Which one will you recommend? Please explain why.

Answer 1

a) We have

A fund with expected return of 12% and standard deviation of 25%

and T bill with return of 4% and standard deviation of 0

Any portfolio formed by combining the above two assets will have expected return equal to weighted average return of the two stocks and standard deviation equal to weight of risky stock times standard deviation of risky stock

So Expected return = 70% of Fund's return + 30% of T bill's return

= 0.7*12%+ 0.3*4%

= 9.6%

and Standard Deviation of Portfolio = 70%of Fund's standard deviation

=0.7*25%  

= 17.5%

b) For the portfolio to have a maximum standard deviation of 18%

weight in funds = 18%/25% = 0.72 = 72%

So, 72% should be invested in fund and 28 % in T bills

So, Expected Return = 0.72*12%+0.28*4% = 9.76%

c) Her Utility Score = Return on risky asset - variance of risky aset * Risk Aversion coefficient / 2

= 0.12- 0.25² * 1.2/2

=0.12-0.0375

=0.0825 = 8.25%

So, the weights suggested w_f and w_t = (1-w_f)to be invested in fund and T-bill are

w_f* 12%+ (1-w_f) * 4% = 8.25%

  w_f = 4.25%/8% = 0.53125 = 53.13%

and w_t = (1-w_f) = 46.87%

d)

After allottment of IBM stocks

New portfolio = $800000 + $200000 = $1,000,000

weight of IBM stocks  = $200,000/$1,000,000 = 20% with an expected return of 15%and a standard deviation of 30%

Weight of earlier portfolio = 80% with an expected return of 9.6% and a standard deviation of 17.5%

So, Return of new portfolio = weighted average return of constituent securities/portfolios

= 20% of IBM's expected return + 80% of earlier portflio's return

= 0.2*15%+ 0.8*9.6%

=10.68%

We know that standard deviation of a two stock portfolio can be calculated as

where w₁ and w₂ are the weights of the two stocks

and are the standard deviation of the two stocks and is the correlation coefficient between the two stocks

Hence

Standard deviation of portfolio = sqrt ( 0.2²*0.3² + 0.8²*0.175² + 2*0.2*0.8* 0.3*0.175 * 0.4)

=sqrt (0.02992)

=0.172974

= 17.30%

e) As keeping the IBM stocks has increased the return from 9.6% to 10.68% whereas it has decreased the risk from 17.5% to 17.3% , it is advisable to keep all the IBM stocks.

This has happened because of the low correlation coefficient between IBM stocks and the current portfolio which gives the benefit of diversification to the portfolio.