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

To find the fraction of wealth to invest in stock fund that will result in the risky portfolio with minimum variance
the following formula to determine the weight of stock fund in risky portfolio should be used
w(*d)= ((Stdev[R(e)])^2-Stdev[R(e)]*Stdev[R(d)]*Corr(Re,Rd))/((Stdev[R(e)])^2+(Stdev[R(d)])^2-Stdev[R(e)]*Stdev[R(d)]*Corr(Re,Rd))
	Where
stock fund	E[R(d)]=	20.00%
bond fund	E[R(e)]=	10.00%
stock fund	Stdev[R(d)]=	40.00%
bond fund	Stdev[R(e)]=	15.00%
	Var[R(d)]=	0.16000
	Var[R(e)]=	0.02250
T bill	Rf=	3.00%
Correl	Corr(Re,Rd)=	0.2
Covar	Cov(Re,Rd)=	0.0120
stock fund	Therefore W(*d) (answer a)=	0.0662
bond fund	W(*e)=(1-W(*d)) (answer a)=	0.9338
	Expected return of risky portfolio (answer b)=	10.66%
	Risky portfolio std dev (answer Risky portfolio std dev)=	14.77%
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

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)]=	20.00%
bond fund	E[R(e)]=	10.00%
stock fund	Stdev[R(d)]=	40.00%
bond fund	Stdev[R(e)]=	15.00%
	Var[R(d)]=	0.16000
	Var[R(e)]=	0.02250
T bill	Rf=	3.00%
Correl	Corr(Re,Rd)=	0.2
Covar	Cov(Re,Rd)=	0.0120
stock fund	Therefore W(*d) (answer c)=	0.2458
bond fund	W(*e)=(1-W(*d)) (answer c)=	0.7542
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