Question

The expected returns on the VBINX and PRWCX funds are 17% and 8% with standard deviations of 60% and 10%, respectively. The correlation between these funds is 0.1, and the yield on 90-day T-Bills is 2%. If you wish to construct the optimal risky portfolio on the investment opportunity set comprised of these two assets, how should you allocate your investment between these funds? IF using Excel, please show all steps.

Given the weights you computed in question 1, what is the (a) expected return, (b) risk, and (c) Sharpe ratio for the RISKY portfolio comprised of your two risky funds?

Given the information from questions 1 and 2, what would be the (a) expected return, (b) risk, and (c) Sharpe ratio of your COMPLETE portfolio, given you invest 70% in the optimal risky portfolio and the remainder in risk-free T-bills.

Answer 1

The Optimal Risky portfolio weights for a two security (A and B) portfolio is given by

and W_B= 1 - W_A

These weights correspond to the point on the Capital allocation line which is tangent from the Risk free Asset point (0,RFR)

Let A be the VBINX fund and B be the PRWCX fund

Here, E(R_A) = 17%, stdev(A) = 60%

E(R_B) = 8%, stdev(B) = 10%

R_f= 2% , Correlation coefficient = 0.1

So, W_A = [(0.17-0.02)*0.10^2 - (0.08-0.02)*0.60*0.10*0.1] / [(0.17-0.02)*0.10^2 +(0.08-0.02)*0.60^2 - (0.17-0.02+ 0.08-0.02)*0.60*0.10*0.1]

    =0.00114/0.02184

=0.052198 =5.22%

and W_B = 1-0.0522=0.9478 =94.78%

This is the optimal portfolio weights i.e. Investment should be 5.2% in VBINX fund and 94.78% in PRWCX fund

a) The return of a portfolio is the weighted return of the two stocks

So Return of this portfolio = 0.0522 * 17% +0.9478 *8% = 8.47%

b) The standard deviation of a portfolio is given by

Where Wi is the weight of the security i,

is the standard deviation of returns of security i.

and is the correlation coefficient between returns of security i and security j

So, standard deviation of portfolio =sqrt (0.0522^2*0.60^2+0.9478^2*0.10^2+2*0.0522*0.9478*0.60*0.10*0.1)

=sqrt(0.010558)

=0.102751 =10.275%

c)  Sharpe Ratio = (portfolio return -Risk free rate)/portfolio standard deviation

=(0.0847-0.02)/0.10275 = 0.62965 or 0.63

if 70% is invested in the optimal risky portfolio and the remainder in risk-free T-bills.For the complete portfolio

a) Expected Return =0.7*0.0847+0.3*0.02 = 0.065288 or 6.53%

b) Expected Standard Deviation = 0.7* 0.10275 = 0.07192 or 7.19%

c) Sharpe ratio = (0.0653-0.02)/0.0719 = 0.63