6. Means and variances of collections (portfolios)

Means add, Variances DON'T.

Say I take 2 objects and combine them, as in a portfolio.

Call them X and Y, then

$$\fbox{$E( X + Y) = E(X) + E(Y).$ }$$

If we combine them in a weighted fashion, say 0.25 of the first and 0.75 of the second then

$$\fbox{$E(0.25\, X + 0.75\,Y) = 0.25\,E(X) + 0.75\,E(Y)$ .}$$

An example using Air.com and Bricks.

Recall that Air.com has a mean return of 0.55 and Bricks has a mean return of -0.35.

Consider an equally weighted portfolio, with 1/2 Air.com and 1/2 Bricks.

Then the expected return on the portfolio is

$$\fbox{$E(0.5\, X + 0.5\,Y) = 0.5\,E(X) + 0.5\,E(Y) = 0.5\times 0.55 + 0.5\times (-0.35) = 0.1$ .}$$

The formula for arbitrary weights, w₁ and w₂ is

$$\fbox{$E(w_1\, X + w_2\,Y) = w_1\,E(X) + w_2\,E(Y)$ .}$$

Variances are a bit more complex. One can show that

$$\fbox{$Var( X + Y) = Var(X) + Var(Y) + 2\,Cov(X,Y)$ ,}$$

so that variances do not add (except when ...).

If we put weights in to make a portfolio then the formula becomes

$$\fbox{$Var( w_1\,X + w_2\,Y) = w_1^2\,Var(X) + w_2^2\,Var(Y) + 2\,w_1w_2 Cov(X,Y)$ .}$$

An example using Air.com and Bricks.

Recall that Air.com has a variance of 5.5475 and Bricks has a variance of 2.5275 (you just worked this out!), and that their covariance is 0.11.

If we use a 3:1 weighted portfolio, so that Air.com has a weight of 0.75 and Bricks has a weight of 0.25, then we have

$$\fbox{$Var( 0.75\,X + 0.25\,Y) = 0.75^2\,Var(X) + 0.25^2\,Var(Y) + 2\times 0.75\times 0.25 Cov(X,Y)$ .}$$

$$\fbox{$Var( 0.75\,X + 0.25\,Y) = 0.75^2\times 5.5475 + 0.25^2... ...es 2.5275 + 2\times 0.75\times 0.25 \times 0.11 = 3.319687$ .}$$