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 sure rate of 5.7%. The probability distributions of the two risky funds are:

  

	Expected Return	Standard Deviation
Stock fund (S)	18%	47%
Bond fund (B)	7%	41%

The correlation between the two fund returns is .0317.

  

What is the expected return for the minimum-variance portfolio of the two risky funds? (Do not round intermediate calculations. Enter your answer as a decimal number rounded to 4 decimal places.)

Answer 1

Given the following information,

	Expected return	standard deviation
Stock fund(S)	0.18	0.47
Bond fund(B)	0.07	0.41

Correlation between the two fund returns = ρ(s,b) = 0.0317

r = 5.7% = 0.057

The expected return of a minimum variance portfolio is given by the following formula,

Expected return for a two-asset portfolio = (ws*expected return of S)+(wb*expected return of B)

In order to calculate expected return for two asset portfolio, first we need to calculate ws and wb,

Where,

ws = weight of Asset S = (σb^2 - Cov(s,b))/ (σs^2+σb^2-2Cov(s,b))

wb = 1- ws

and

σ_p² = variance of the portfolio

ws = weight of Asset S

wb = weight of Asset B

σs² = variance of Asset S

σs = standard deviation of Asset S = 0.47

σb² = variance of Asset B

σb = standard deviation of Asset B = 0.41

Cov(s,b) = covariance of returns between Asset S and Asset B

and

Cov(s,b) = ρ(s,b) * σs * σb

where ρ(s,b) = correlation of returns between Asset S and Asset B

Calculating wa and ws:

Since,

Cov(s,b) = ρ(s,b) * σs * σb

Substituting the given values we get

Cov(s,b) = 0.0317*(0.47*0.41)

Cov(s,b) = 0.0317*0.1927

Cov(s,b) = 0.006109

ws = (0.41)^2 - 0.006109)/ ((0.47)^2+(0.41)^2-2*0.006109)

ws = (0.1681 - 0.006109)/ (0.2209+0.1681-2*0.006109)

ws = 0.161991/ (0.2209+0.1681-0.01222)

ws = 0.161991/ 0.37678

ws = 0.429933

So wb = 1- ws = 1-0.429933 = 0.570067

wb = 0.570067

Now substituting ws and ws in the expected return for two asset portfolio, we get

Expected return for a two-asset portfolio = (ws*expected return of S)+(wb*expected return of B)

Expected return for a two-asset portfolio = (0.429933*0.18)+(0.570067*0.07)

Expected return for a two-asset portfolio = (0.077388)+(0.039905)

Expected return for a two-asset portfolio = 0.117293 = 11.7293%

Thus the expected return for the minimum-variance portfolio of the two risky funds is 11.7293%