Consider two different stocks, A and B, with expected returns A = 17.0% and ?B= 14.9% and with standard deviations A = 12.4% and ?B = 10.0% for those returns. The two assets have a correlation, p, of -0.7. 

1. What is the covariance of the returns of stocks A and B? (use the population correlation to compute the covariance) Generally, will constructing a portfolio from these two stocks reduce or increase the risk compared to the individual stocks?
2. What is the expected return and standard deviation of a portfolio made up of stocks A and B which is 20% stock A (the remainder stock B)? 
3. What is the expected return and standard deviation of a portfolio made up of stocks A and B which is 50% stock A?
4. What is the expected return and standard deviation of a portfolio made up of stocks A and B which is 80% stock A?
5. Which of the portfolios above, either 2, 3, 4, offers the best combination of risk and return?

 

1. Note that,

Cov(A,B)/(SD(A)*SD(B)) = Corr(A,B)

=> Cov(A,B) = SD(A)*SD(B)*Corr(A,B)  

= 0.124*0.1*(-0.7)

= -0.00868 or -0.868%

As the covariance is negative so constructing a portfolio from these two stocks will reduce the risk compared to the individual stocks.

Now to answer the following questions we would use two properties of mean and variance.

We know for two random variables A and B and two constants m and n,

E(mA+nB) = mE(A)+nE(B)

V(mA+nB) = m²V(A)+n²V(B)+2mnCov(A,B)

2. In this case, m = 0.2 and n = 0.8

Portfolio = 0.2A+0.8B

Expected return = E(Portfolio) = 0.2*E(A)+0.8*E(B) = 0.2*0.17+0.8*0.149 = 0.1532 or 15.32%.

Variance of return = V(Portfolio) =  0.2²*V(A)+0.8²*V(B)+2*0.2*0.8*Cov(A,B)

= 0.2²*0.124²+0.8²*0.1²+2*0.2*0.8*(-0.00868)

= 0.00423744

Standard deviation of return = √V(Portfolio) =  √0.00423744 = 0.065095622 or 6.51%

3. In this case, m = 0.5 and n = 0.5

Portfolio = 0.5A+0.5B

Expected return = E(Portfolio) = 0.5*E(A)+0.5*E(B) = 0.5*0.17+0.5*0.149 = 0.1595 or 15.95%.

Variance of return = V(Portfolio) =  0.5²*V(A)+0.5²*V(B)+2*0.5*0.5*Cov(A,B)

= 0.5²*0.124²+0.5²*0.1²+2*0.5*0.5*(-0.00868)

= 0.002004

Standard deviation of return = √V(Portfolio) =  √0.002004 = 0.044766059 or 4.48%

4. In this case, m = 0.8 and n = 0.2

Portfolio = 0.8A+0.2B

Expected return = E(Portfolio) = 0.8*E(A)+0.2*E(B) = 0.8*0.17+0.2*0.149 = 0.1658 or 16.58%.

Variance of return = V(Portfolio) =  0.8²*V(A)+0.2²*V(B)+2*0.8*0.2*Cov(A,B)

= 0.8²*0.124²+0.2²*0.1²+2*0.8*0.2*(-0.00868)

= 0.00746304

Standard deviation of return = √V(Portfolio) =  √0.00746304 = 0.086388888 or 8.64%

5) Option 2 is inferior to option 2 in both aspect. In terms of return the option 4 is best but the risk (standard deviation) of option 4 is also highest. So I would suggest that option 3 is giving best combination overall. As it has moderate return and lower risk.