In Section 7.7 we dealt with sums of random variables, under the restrictive assumption of independence. Finally, armed with covariance, we may tackle the general case.
THEOREM 8.4 (Variance of the sum/difference of two random variables) Given two random variables X and Y, the variance of their sum and difference is
This theorem is somewhat reassuring; if we take the difference of two random variables, rather than the sum, the minus does count after all, but it plays a role depending on covariance. It is instructive to consider a pair of alternative proofs of the first relationship (the second one is easily obtained from the first by using property 3 of covariance). One possibility is using the definition of variance directly:
Another possibility is to take advantage of property 4 of covariance:
Using the second proof technique, it is fairly easy to come up with the following statement.
THEOREM 8.5 (Variance of the sum of random variables) Given a collection of random variables Xi, i = 1,…, n, we obtain
We see that, in the case of mutually independent variables, covariances will be zero, and the variance of the sum does boil down to the sum of variances. A very interesting application of the theorem arises in the area of financial portfolio management.
Example 8.5 (A classical model of risk in financial portfolio management) Consider the task of an investor who must allocate her wealth to two risky financial assets. The returns of the two assets are modeled by random variables R1 and R2 respectively. Assume that the investor knows the two expected returns, μ1 and μ2; the two variances of the return, 
and 
; and the covariance, σ12. What the investor has to decide is the fraction of wealth that she should allocate to each asset; let us denote the weights of the two assets in the portfolio by w1 and w2, respectively. The two weights must add up to 1
and they cannot be negative if short-selling is ruled out (see Example 1.3). Hence, the return of the portfolio is a weighted sum of random variables:
There are numerous issues involved in portfolio decisions, but a preliminary requirement is to figure out the expected return of the portfolio and some measure of risk, in order to trade them off. It is easy to see that the expected return is
One possible measure of risk is standard deviation of portfolio return. We will see a few shortcomings of this risk measure later, but it is certainly a useful input for the decision of our investor. To find standard deviation, one has to find variance first. This is a bit trickier than the expected value:
Taking a careful look at variance, as a function of portfolio weights w1 and w2, should ring a bell: This is a quadratic form, a concept that we introduced in Section 3.8. We also know that quadratic forms can be expressed in a very compact way by using vectors and matrices. So, let us group portfolio weights into the following vector:
Let us also group variances and covariances into the following matrix:
where 
, i = 1,2. Matrix Σ is called the covariance matrix. Given property 2 of covariance, σij = σji; so Σ is a symmetric matrix. Now it is easy to check that variance of portfolio return can be expressed as
The finding of the example above can be easily generalized to a collection of n random variables grouped into the following vector:
Let us also introduce the vector of expected values and the covariance matrix
where , i = 1,…, n. If we form a linear combination
then we have
Since we know that variance is nonnegative, we may immediately deduce the following property of the covariance matrix.
PROPERTY 8.6 The covariance matrix Σ is a symmetric, positive semidefinite matrix.
This property has several implications. We may be interested in minimizing risk as measured by variance, with respect to a vector of decision variables x, subject to constraints. The objective function is the quadratic form xTΣx, which, by the property above, is a convex function.4 Furthermore, we know that a symmetric matrix has orthogonal eigenvectors.5 This can be exploited in data reduction methods such as principal component analysis.
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