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 percent. The probability distribution of the risky funds is as follows:

	Expected Return	Standard Deviation
Stock fund (S)	.20	.30
Bond fund (B)	.12	.15

The correlation between the fund returns is 0.10.

2. If you were to use only the two risky funds and will require an expected return of 14 percent, what must be the investment proportions of your portfolio? (Compare its standard deviation to that of the optimized portfolio in question 1 (=13.03%).

Answer 1

We need to find the investment proportions or weights of the stock and bond funds in the portfolio for which the expected return of the portfolio will be 14% or.14.
Suppose the weight of the stock fund be x.
Then, weight of the bond fund will be 1-x
Expected Return of stock fund=.20
Expected Return of bond fund=.12
Weight of stock fund*Expected return of stock fund + Weight of bond fund*Expected return of bond fund=Expected return of the portfolio
So, x*0.2+(1-x)*0.12=.14
=>0.2x+0.12-0.12x=.14
=>0.2x-0.12x=.14-0.12
=>0.08x=0.02
=>x=0.02/0.08=0.25
So, (1-x)=1-.25=0.75

Weight or proportion of the stock fund invested in the portfolio is .25 or 25%
Weight or proportion of the bond fund invested in the portfolio is .75 or 75%

The standard deviation of a two asset portfolio is calculated as:
[(Weight of stock fund)^2*(Standard deviation of stock fund)^2+ (Weight of bond fund)^2*(Standard deviation of bond fund)^2
+2*(Weight of stock fund)*(Weight of bond fund)*(Standard deviation of stock fund)*(Standard deviation of bond fund)*(Correlation coefficient between the funds)]^1/2

Weight or proportion of the stock fund=.25
Weight or proportion of the bond fund=.75
Standard deviation of stock fund=.30
Standard deviation of bond fund=.15
Correlation coefficient between the fund returns= 0.10
Putting these values in the formula we get:
[(.25)^2(.30)^2 + (.75)^2*(.15)^2 +2*(.25)*(.75)*(.30)*(.15)*(0.10)]^1/2
=[0.005625+0.01265625+0.0016875]^1/2
=0.01996875^1/2
=0.141310828 or 14.13% (rounded up to two decimal places)
Value of standard deviation here is 14.13%
Value of optimized portfolio in question 1 is 13.03%
So, value of standard deviation is greater than the value of the optimized portfolio.