Question

Problem 13-18 Optimal Sharpe Portfolio Value-at-Risk (LO3, CFA6)

You are constructing a portfolio of two assets, Asset A and Asset B. The expected returns of the assets are 12 percent and 16 percent, respectively. The standard deviations of the assets are 29 percent and 37 percent, respectively. The correlation between the two assets is 0.41 and the risk-free rate is 3.4 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 2.5 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

Sharpe ratio = (Rp - rf)/σ_p

where Rp (portfolio return) = (r_A*W_A)+(r_B*W_B)

Weight of asset A W_A is calculated as:

Formula
	rA	12%
	rB	16%
	rf	3.40%
	σA	29%
	σB	37%
	ρA,B	0.41
Numerator	N	0.0062
Denominator	D	0.0130
(N/D)	WA	0.4777
1-(N/D)	WB	0.5223
(rA*WA) + (rB*WB)	Portfolio return (Rp)	0.1409

Portfolio standard deviation calculation:

σ²_p = (W_A²σ_A²) + (W_B²σ_B²) + (2W_AW_Bσ_Aσ_Bρ_A,B)

Portfolio standard deviation = ((0.4777)^2(29%)^2 + (0.5223)^2(37%)^2 + (2*0.4777*0.5223*29%*37%*0.41))^(1/2)

= 0.2802

Sharpe ratio = (0.1409 - 0.0340)/0.2802 = 0.3815

To calculate smallest expected loss:

It needs to be calculated for a confidence level of 2.5%. Z-score for 2.5% = -1.960

Prob(R <= Expected portfolio return + Z-score(σ_p)) = 2.5%

Prob(R<= 0.1409 -(1.960*0.2802)) = 2.5%

Prob(R <= -0.4082) = 2.5%