Question

Suppose you have some money to invest—for simplicity, $1—and you are planning to put a fraction w into a stock market mutual fund and the rest, 1 - w, into a bond mutual fund. Suppose that $1 invested in a stock fund yields Rs after 1 year and that $1 invested in a bond fund yields Rb, suppose that Rs is random with mean 0.07 (7%) and standard deviation 0.06, and suppose that Rb is random with mean 0.04 (4%) and standard deviation 0.04. The correlation between Rs and Rb is 0.22. If you place a fraction w of your money in the stock fund and the rest, 1 - w, in the bond fund, then the return on your investment is R=WRs + (1 - w)Rb. Suppose that w = 0.45. Compute the mean and standard deviation of R. The mean is - (Round your response to three decimal places.) The standard deviation is . (Round your response to three decimal places.) Suppose that w = 0.67. Compute the mean and standard deviation of R. The mean is I. (Round your response to three decimal places.) The standard deviation is . (Round your response to three decimal places.) What value of w makes the mean of R as large as possible? w=maximizes u. (Round your response to two decimal places.) What is the standard deviation of R for this value of w? o for this value of w. (Round your response to two decimal places.) What is the value of w that minimizes the standard deviation of R? W = minimizes the standard deviation of R. (Round your response to two decimal places.)

Answer 1

The Mean and Variance of R are given by:

Mean=w*0.07+(1-w)*.04

S.D²=w^2*0.06²+(1-w)²*0.04²+2*w*(1-w)*(0.06*.04*.22)

where 0.06×0.04×0.22=cov(RS, RB)

(1) Where w=0.45

μ=0.054

σ=0.038

(2) Where w=0.67

μ=0.060

σ=0.045

(3)The value of w which maximises the mean of R is 1

Mean = 1*.07+0*.04=0.07

(4) Standard Deviation of R for this value = 1²*0.06²+0²*.04²+2*1*0(0.06*.04*.22)

=0.06

(5)The value of w which minimises the S.D of R is 0

i.e. =0²*.06²+1²*0.04²+2*0*1*(0.06*.04*.22)

=.04