Question

You are constructing a portfolio of two assets, Asset A and Asset B. The expected returns of the assets are 11 percent and 14 percent. The standard deviations of the assets are 35 percent and 43 percent. The correlation between the two assets is .53 and the risk-free rate is 3.8 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 1 percent? ( do not round intermediate calculations. (Round Sharpes ratio answer to 4 decimal places and the s-score value to 3 decimal places when calculating 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)]=	11.00%
Asset B	E[R(e)]=	14.00%
Asset A	Stdev[R(d)]=	35.00%
Asset B	Stdev[R(e)]=	43.00%
	Var[R(d)]=	0.12250
	Var[R(e)]=	0.18490
T bill	Rf=	3.80%
Correl	Corr(Re,Rd)=	0.53
Covar	Cov(Re,Rd)=	0.0798
Asset A	Therefore W(*d)=	0.4340
Asset B	W(*e)=(1-W(*d))=	0.5660
	Expected return of risky portfolio=	12.70%
	Risky portfolio std dev (answer )=	34.86%
Sharpe ratio=	(Port. Exp. Return-Risk free rate)/(Port. Std. Dev)		=(0.127-0.038)/0.3486	=0.2553
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 =12.7-Normal distribution of 0.01*34.86

Smallest expected Loss =12.7-2.3263*34.86

Smallest expected Loss =-68.39%