You are constructing a portfolio of two assets, Asset A and Asset B. The expecte

Question

You are constructing a portfolio of two assets, Asset A and Asset B. The expected returns of the assets are 10 percent and 16 percent, respectively. The standard deviations of the assets are 27 percent and 35 percent, respectively. The correlation between the two assets is .37 and the risk-free rate is 5.4 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 1 percent? (Negative value should be indicated by a minus sign. Do not round intermediate calculations. Round your Sharpe ratio answer to 4 decimal places and Probability answer to 2 decimal places. Omit the "%" sign in your response.)

Explanation / Answer

Answer:

The parameters which are provided in the question are as under:

Expected return of Asset-A = E(RA ) = 10%,  

Expected return of Asset-B = E(RB )= 16%,  

Standard deviation of Asset-A = A = 27%,

Standard deviation Asset-B = B= 35%,

Correlation between the fund returns = P= 0.37

From the standard deviation and correlation coefficient we generate the covariance matrix:

Asset-A

Asset-B

Asset-A

27*27= 729

27*35*0.37 = 349.65

Asset-B

27*35*0.37 = 349.65

35*35= 1225

As the portfolio is a minimum variance portfolio so the portfolio weights can be found by using this formula:

Wmin(A) ={ B2 – Cov(A, B)} / {A2+B2- 2 Cov(A B)}

              = {1225 – 349.65} / {729+1225 - 2*349.65} = 875.35 / 1254.7 = 0.698

Therefore,   Wmin(B) = 1- 0.698 = 0.302

Therefore the expected return on the portfolio:

E(P) = 0.698 x 10% + 0.302 x 16% = 0.118 or 11.8%

Standard deviation of the portfolio:

Now the Sharpe ratio is:

Rf = 5.4%

S(R) = (E(P) – Rf )/ P   = (11.8% - 5.4%)/24.8% = 0.2581

Now,

the 1% loss level is 2.33 standard deviations below the mean:

Pr[RP<0.118 – 2.33 x 0.248] = 0.01

which simplifies to

Pr[RP<-0.45984] = 0.01

We can expect a loss of -45.984 percent or worse over the next year with a 1 percent probability.

Asset-A

Asset-B

Asset-A

27*27= 729

27*35*0.37 = 349.65

Asset-B

27*35*0.37 = 349.65

35*35= 1225

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