5. (4 Points) Assume that you can invest in two securities: Stock A and Stock Z.

Question

5. (4 Points) Assume that you can invest in two securities: Stock A and Stock Z. There are five possible outcomes next period. The probability of each outcome and the return offered by each stock in each outcome are shown in the table below. Outcome Probability 10% 20% 30% 30% 10% Stock A 30% 22% 10% 3% -20% Stock Z 10% 25% 20% -15% 5% A. What is the expected return for each security and a portfolio consisting of 30% Stock A and 70% Stock Z? B. What is the standard deviation for each security and a portfolio consisting of 30% Stock A and 70% Stock Z? C. At what portfolio weights will the standard deviation of the two-stock portfolio be minimized? What is the portfolio standard deviation using these weights? D. What is the correlation between Stock A and Stock Z? Looking at the numbers above, BRIEFLY explain why this correlation makes sense.

Explanation / Answer

Expected returns of Stock A=0.1*30%+0.2*22%+0.3*10%+0.3*3%+0.1*(-20%)=9.3%

Expected returns of Stock Z=0.1*10%+0.2*25%+0.3*20%+0.3*(-15%)+0.1*(5%)=8%

Standard Deviaiton of Stock A=sqrt(0.1*(30%-9.3%)^2+0.2*(22%-9.3%)^2+0.3*(10%-9.3%)^2+0.3*(3%-9.3%)^2+0.1*(-20%-9.3%)^2)=13.153326%

Standard Deviaiton of Stock Z=sqrt(0.1*(10%-8%)^2+0.2*(25%-8%)^2+0.3*(20%-8%)^2+0.3*(-15%-8%)^2+0.1*(5%-8%)^2)=16.1554944%

Covariance of Stock A & Z=(0.1*(10%-8%)*(30%-9.3%)+0.2*(25%-8%)*(22%-9.3%)+0.3*(20%-8%)*(10%-9.3%)+0.3*(-15%-8%)*(3%-9.3%)+0.1*(5%-8%)*(-20%-9.3%))=0.01021

Expected returns of portfolio=30%*9.3%+70%*8%=8.39%

Standard Deviaiton of portfolio=sqrt(30%^2*13.153326%^2+70%^2*16.1554944%^2+2*30%*70%*0.01021)=13.6507%

Minimum variance is at weight of Z=(13.153326%^2-0.01021)/(13.153326%^2+16.1554944%^2-2*0.01021)=0.308559
Weight of A=1-0.308559=0.691441

Portfolio standard deviaiton at minimum variance=sqrt(0.308559^2*13.153326%^2+0.691441^2*16.1554944%^2+2*0.308559*0.691441*0.01021)=13.5948%

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