Question

You are constructing a portfolio of two assets, Asset A and Asset B. The expected returns of the assets are 8 percent and 13 percent, respectively. The standard deviations of the assets are 30 percent and 38 percent, respectively. The correlation between the two assets is 0.43 and the risk-free rate is 5.6 percent. What is the optimal Sharpe ratio in a portfolio of the two assets? What is the smallest expected loss for this portfolio over the coming year with a probability of 5 percent? (A negative value should be indicated by a minus sign. Do not round intermediate calculations. Round your Sharpe ratio answer to 4 decimal places and the z-score value to 3 decimal places when calculating your answer. Enter your smallest expected loss as a percent rounded to 2 decimal places.)

Answer 1

To find the fraction of wealth to invest in Asset A that will result in the risky portfolio with maximum Sharpe ratio
the following formula to determine the weight of Asset A 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
						Asset A	E[R(d)]=	8.00%
Asset B	E[R(e)]=	13.00%
Asset A	Stdev[R(d)]=	30.00%
Asset B	Stdev[R(e)]=	38.00%
	Var[R(d)]=	0.09000
	Var[R(e)]=	0.14440
T bill	Rf=	5.60%
Correl	Corr(Re,Rd)=	0.43
Covar	Cov(Re,Rd)=	0.0490
Asset A	Therefore W(*d)=	-0.0304
Asset B	W(*e)=(1-W(*d))=	1.0304
	Expected return of risky portfolio=	13.15%
	Risky portfolio std dev (answer )=	38.77%
Sharpe ratio=	(Port. Exp. Return-Risk free rate)/(Port. Std. Dev)		=(0.1315-0.056)/0.3877	=0.1947
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

Smallest expected Loss = Mean

-Normal distribution of probability*std dev

Smallest expected Loss =13.15-Normal distribution of 0.05*38.77

Smallest expected Loss =13.15-1.6449*38.77

Smallest expected Loss =-50.62%