Question

2. Suppose there are three assets with returns r1, r2, and r3 with the covariance matrix given by: 0.01 r" E(R)0.03 r3 0.09 and Var() Cov(ri, r2) Cov(ri,T3) V(R) Cov(ri,r2) Var(r2) Cov(r2,3) 0.01 0.02-0.02 0.06 0.03 (4) Cov(ri, r3) Cov(r2, Var(ra) -0.02 0.03 0.08 (a) Discuss in detail the different properties of the three assets. (b) Calculate the return and variance of a portfolio in which an investor puts 50% in asset 1 and 50% in asset 2, Hint: In matrix notation, the portfolio return is T3 (c) Calculate the return and variance of a portfolio in which an investor puts equal amounts into each of the three stocks (d) Discuss the change in the variance of the portfolio from part (b) to part (c)

Answer 1

a) Looking at the return vector E(R) and the diagonal of the variance matrix for individual asset variance, we have:

Asset 1: Expected Return = 0.01, Variance = 0.01

Asset 2: Expected Return = 0.03, Variance = 0.06

Asset 3: Expected Return = 0.09, Variance = 0.08

Hence, we can see from above that the asset with lowest variance/risk (Asset 1) is also the one with the lowest return. As the risk increases, the Expected return also increases. While Asset 3 has the highest Expected return of 0.09, it also is the most risky asset. Whereas Asset 2 is the one with moderate risk, while also giving a moderate return.

However, the risk comes down when we combine any 2 assets, as can be seen from the lower covariance terms in the Covariance matrix V(R).

b) Given, w₁ = 0.5, w₂ = 0.5, w₃ = 0 => Portfolio weight vector is: (0.5, 0.5, 0)

Hence, return of the portfolio, r_P = (w1  w2 w3) * = 0.5*0.01 + 0.5*0.03 + 0*0.09 = 0.02

0.01 0.03 0.09

Variance of the portfolio = (w1 w2 w3) * $\sum$ * (w1 w2 w3)^T

=
0.5 0.5 0

0.01 0.02 -0.02 0.02 0.06 0.03 一0.02 0.03 0.08
*
0 1 0
= 0.0275   (Multiplication of 1x3, 3x3 and 3x1 matrices)

c) Now, (w1 w2 w3) = (1/3 1/3 1/3)

Return of the portfolio =   (w1 w2 w3) * = 0.0433

0.01 0.03 0.09

Variance of the portfolio = (w1 w2 w3) * $\sum$ * (w1 w2 w3)^T  

=
1/3 1/3 1/3
*
0.01 0.02 -0.02 0.02 0.06 0.03 一0.02 0.03 0.08
* $\begin{vmatrix} 1/3\\ 1/3\\ 1/3\\ \end{vmatrix}$   

= 0.02332

d) Portfolio in (b) has a return of 0.02 and variance (risk) of 0.0275. While portfolio in (c) has return of 0.0433 and variance (risk) of 0.02332. Hence, we see that as we diversify the portfolio by additional another asset with low covariance with the first 2 assets, the variance or risk of the total portfolio comes down while the risk goes up. Hence, more diversification here helps us in improving the return while at the same time lowering the risk.