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 5%. The probability distribution of the risky funds is as follows:

Expected Return Standard Deviation

Stock Funds (S) 17% 38%

Bond Funds (B) 13 18

The correlation between the fund returns is 0.12. What is the Sharpe ratio of the best feasible CAL?

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
w(*d)= ((E[Rd]-Rf)*Var(Re)-(E[Re]-Rf)*Cov(Re,Rd))/((E[Rd]-Rf)*Var(Re)+(E[Re]-Rf)*Var(Rd)-(E[Rd]+E[Re]-2*Rf)*Cov(Re,Rd)
							Where
						Stock fund	E[R(d)]=	17.00%
bond fund	E[R(e)]=	13.00%
Stock fund	Stdev[R(d)]=	38.00%
bond fund	Stdev[R(e)]=	18.00%
	Var[R(d)]=	0.14440
	Var[R(e)]=	0.03240
T bill	Rf=	5.00%
Correl	Corr(Re,Rd)=	0.12
Covar	Cov(Re,Rd)=	0.0082
Stock fund	Therefore W(*d) =	0.2342
bond fund	W(*e)=(1-W(*d))=	0.7658
	Expected return of risky portfolio =	13.94%
	Risky portfolio std dev =	17.28%
Sharpe ratio=	(Port. Exp. Return-Risk free rate)/(Port. Std. Dev)		=(0.1394-0.05)/0.1728	=0.5174
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