Finding the Covariance of a portfolio given the weights of two assets, with their probability and return % for each asset.

Question

I am given the details below:

The following Table reflects data relating to two assets X and Y, which are present in a portfolio in the ratio 60:40

Probability | Return on X(%) | Return on Y(%)

    0.2            -10            -1.40
    0.4             25            -2.25
    0.3             25            -2.00 
    0.1             10            -10.00

I had to state what type of portfolio it was and the answer was that it was a nil covariance portfolio.

The calculation to prove this was written as: Cov = (0.2x-25x1.38)+(0.4x10x0.53)+(0.3x5x0.78)+(0.1x-5x-7.22)=0

The first value in the bracket is the probability, and the third is the return of Y minus the expected return of Y but I think the middle number is incorrect.

Can anyone confirm?

Answer 1

The mean

$E[X] = \Sigma P(Scenario_1) X_i\\ E[Y] = \Sigma P(Scenario_1) Y_i$

$\bar X = (0.2)(-10) + (0.4)(25) + (0.3)(25) + (0.1)(10) = 16.5\\ \bar Y = -2.78$

Each asset has a variance.

$\sigma_X^2 = \Sigma P(Scenario_i)(X_i - \bar X)\\ \sigma_Y^2 = \Sigma P(Scenario_i)(Y_i - \bar Y)$

$\sigma_X^2 = 195.3\\ \sigma_Y^2 = 5.9$

There is a covariance between the two assets.

$\text{Cov}_{X,Y}=\Sigma P(Scenario_i)(X_i - \bar X)(Y_i - \bar Y)$

$\text{Cov}_{X,Y} = 1.17$

And the portfolio with weights $\omega_x, \omega_y$ will have a variance.

$\sigma_P^2=\omega_X^2\sigma_X^2 + \omega_Y^2\sigma_Y^2 + 2\omega_X\omega_Y\text{Cov}_{X,Y} = 71.8$

And expected return $E[R] = 8.8$