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 3%. The probability distribution of the funds is as follows:

	Expected Return	Standard Deviation
Stock Fund	20%	40%
Bond Fund	10%	15%
Risk-free	3%
Correlation	20%

1. Find the investment proportions in the minimum variance portfolio (MVP) of the two risk asset (5 points)
2. Find the expected return and standard deviation for the minimum variance portfolio (5 points)
3. What portion of your wealth should go to S and B respectively to achieve the tangent portfolio (i.e., the portfolio with the highest Sharpe ratio) (5 points)

Answer 1

Expected Return of Stock Fund (A)=	20%
standard deviation(σ A) =	40%
Expected Return of Bond fund (B)=	10%
standard deviation(σ B) =	15%
Correlation= 20% or	0.20
risk free rate	3%
covariance between Two (a &b) formula=Correlation*Std A*Std dev B
0.20*40%*15%
0.012
So Covariance is 0.012
Formula for Minimum Variance Portfolio (weight A) = ((σB)^2 - CoV AB)/((σA)^2+(σB)^2-(2*Cov. AB))
((15%)^2-(0.012))/((40%)^2+(15%)^2-(2*(0.012)))
0.06624605678
So Weight of A or Stock fund investment proportion is 0.06624605678 or	6.6246%
weight of B =1-0.06624605678 =0.9337539432	93.3754%
Expected return of Minimum variance Portfolio= (weight of A * Expected return of A) + (Weight of B * Expected retun of B)
(6.6246%*20%)+(93.3754%*10%)
0.1066246 or 10.66%
standard deviation of portfolio formula =√( (wA * σA ) ^2 + (wB * σB ) ^2 +( 2 * wA* wB*Cov AB ))
=√( (6.6246%*40%)^2 + (93.3754%*15%)^2 +(2*6.6246%*93.3754%*0.012))
=0.1476631857 or 14.77%

Weight of stock fund investment proportion is 6.62%

weight of B =1-0.06624605678 =93.37%

Expected return of MVP is 10.66%

standard deviaiton of MVP is 14.77%

c.

The tangent portfolio the portfolio with the highest Sharpe ratio is called Optimal risky portfolio. We will calculate weight of risky assets in Optimal risky portfolio

Weight of Stock fund in Optimal risky portfolio formula =(((Er A- Rf) * σB^2) - ( (Er B - Rf) * Cov AB )))/ (((Er A - Rf)*σB^2) + ( (Er B - Rf) * σA^2 )- ( (Er A - Rf +ErB-Rf)* Cov AB))

=(((20%- 3%) * (15%)^2) - ( (10% - 3%) *0.012 ))/ (((20%-3%)*(15%)^2) + ( (10%-3%) * (40%)^2 )- ( (20%-3%+10%-3%)* 0.012))

=0.2457801564 or 24.58%

So weight of Stock fund is 0.2458 or 24.58%

weight of Bond fund is 1-0.2458 =0.7542 or 75.42%