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 5.5%. The probability distributions of the risky funds are:

	Expected Return		Standard Deviation
Stock fund (S)	15	%	32	%
Bond fund (B)	9	%	23	%

The correlation between the fund returns is 0.15.

Suppose now that your portfolio must yield an expected return of 12% and be efficient, that is, on the best feasible CAL.

a. What is the standard deviation of your portfolio?

b-1. What is the proportion invested in the T-bill fund?

b-2. What is the proportion invested in each of the two risky funds?

Answer 1

To find the fraction of wealth to invest in Stock fund that will result in the risky portfolio with maximum Sharpe ratio the following formula to determine the weight of Stock fund in risky portfolio should be used

	Where
Stock fund	E[R(d)]=	15.00%
Bond fund	E[R(e)]=	9.00%
Stock fund	Stdev[R(d)]=	32.00%
Bond fund	Stdev[R(e)]=	23.00%
	Var[R(d)]=	0.10240
	Var[R(e)]=	0.05290
T bill	Rf=	5.50%
Correl	Corr(Re,Rd)=	0.15
Covar	Cov(Re,Rd)=	0.0110
Stock fund	Therefore W(*d)=	0.6466
Bond fund	W(*e)=(1-W(*d))=	0.3534
	Expected return of risky portfolio=	12.88%
	Risky portfolio std dev=	23.34%
Sharpe ratio=	(Port. Exp. Return-Risk free rate)/(Port. Std. Dev)		=(0.1288-0.055)/0.2334	=0.3162
Where
Var = std dev^2
Covariance = Correlation* Std dev (r)*Std dev (d)
Expected return of the risky portfolio = E[R(d)]*W(*d)+E[R(e)]*W(*e)
Risky portfolio standard deviation =( w2A*σ2(RA)+w2B*σ2(RB)+2*(wA)*(wB)*Cor(RA,RB)*σ(RA)*σ(RB))^0.5
Desired return = tbill return*proportion invested in tbill+risky portfolio return *proportion invested in risky portfolio
= tbill return*proportion invested in tbill+risky portfolio return *(1-proportion invested in tbill)
0.09=0.055*Proportion invested in Tbill+0.1288*(1-Proportion invested in Tbill)
Proportion invested in Tbill (answer b-1) = (0.1288-0.09)/(0.1288-0.055)
=52.57%
proportion invested in risky portfolio = 1-proportion invested in tbill
=47.43%
Proportion invested in Bond fund (answer b-2) =proportion invested in risky portfolio *weight of Bond fund
=16.76%
Proportion invested in Stock fund (answer b-2) =proportion invested in risky portfolio *weight of Stock fund
=30.67%
std dev of portfolio (answer a) = std of risky portfolio*proportion invested in risky portfolio
0.4743*0.2334=11.07%