You have three assets X Y and Z with expected returns of 10

You have three assets X, Y and Z with expected returns of 10%, 15% and 20%, respectively. The weights of the first two assets are 50% and 70% respectively. The standard deviations of the returns for assets X and Y are 3% and 5%. The covariance of assets X with asset Y is 20%, the covariance of asset Z with asset X is -30%, the covariance of asset Z with Y is 10% , and the covariance of asset Z with Z is 36%.

Calculate the expected return and the variance of your portfolio.

Solution

Sum of weights of he asset = 1

Thus, Weight of asset Z = 1 - (0.5 + 0.7) = - 0.20 [i.e, the asset has been in short selling ]

Expected return of the portfolio

= w1r1 + w2r2 + w3r3

= (0.5 * 10%) + (0.7 * 15%) + (-0.2 * 20%)

= 11.5%

Portfolio Variace in terms of 3 assets is given by:

s_p²= w_A² s²_A + w_B² s²_B + w_C² s²_C+ 2 w_A w_B r_AB s_A s_B+ 2 w_A w_C r_AC s_A s_C+ 2 w_B w_C r_BC s_B s_C

s = std. dev

Here, Std.Dev of A = 3%, Std.Dev = 5% and

cov (Z,Z ) = var (Z) = 0.36

Std.DEV of Z= sqrt (0.36) = 0.6

Now,

Portfolio Variance = w_a²s_a² +  w_b²s_b² +  w_c²s_c² + 2w_aw_bCov(A,B) + 2w_bw_cCov(B,C) +2w_aw_cCov(A,C)

= [ 0.5² *0.03² + 0.7²*0.05² + (-0.2)² * (0.6)²  + 2 *(0.5*0.7*0.2) + 2(0.7 * (-0.2) * 0.1) + 2*(0.5 * (-0.2) * (-0.3) ]

= 0.18785

Hope this helps.

Ask if you have any doubts