Question

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 rate of 8%. The probability distribution of the risky funds is as follows:

	Expected Return			Standard Deviation
Stock fund (S)		18	%		35	%
Bond fund (B)		15			20

The correlation between the fund returns is 0.12.

What are the investment proportions in the minimum-variance portfolio of the two risky funds?

Portfolio Invested in the Stock :

Portfolio Invested in the Bond:

What is the expected value and standard deviation of its rate of return?

Expected Return:

Standard Deviation:

Answer 1

The formula for minimum risk weights in a two stock portfolio is

So, W_S = (0.20²-0.35*0.20 *0.12) / ( 0.20²+0.35² -2*0.20*0.35 *0.12)

= 0.0316/0.1457 = 0.216884 or 21.69%

and W_B = 1- W_S = 1-0.216884 =0.783116=78.31%

So, portfolio invested in stock is 0.2169 and portfolio invested in Bond is 0.7831

b) The Expected value of rate of return of the portfolio is the weighted average return of its constituents

R_p = W_S *R_S +W_B *R_B  = 0.2169* 0.18 + 0.7831 *0.15 = 0.1565 or 15.65%

The standard deviation of a portfolio is given by

=sqrt (0.2169^2*0.35^2+0.7831^2*0.20^2+2 *0.35*0.20*0.2169*0.7831*0.12)

= sqrt (0.033146)

=0.182062=18.21%

=0.1821

The Expected Return is 0.1565 and the standard deviation is 0.1821