The first thing we should observe is that variance cannot be negative, as it is the expected value of a squared deviation. It is zero for a random variable that is not random at all, i.e., a constant. In doing calculations, the following identity is quite useful:
This is the analog of Eqs. (4.5) and (4.6) in descriptive statistics, and proving it is a good exercise:
When dealing with expected value, we considered Properties 6.6 and 6.7, that essentially state that the expected value of a sum of random variables is the sum of their expected values. Does this carry over to variance? The answer is “not really.” The first surprise is that, given numbers α and β, we have
We see that a number α multiplying a random variable can be “taken outside” variance, but it gets squared. This is not surprising at all, since variance is a squared deviation. We also see that β does not play any role at all. Again, this makes sense, since shifting a probability distribution does change its expected value, a measure of location, but not its dispersion. It is easy to prove this property formally using the definition of variance and the properties of expectation:
Equation (6.8) suggests that there is something intrinsically “nonlinear” in variance, since the scale factor α gets squared. By the same token, we could wonder whether the variance of a sum of random variables is just the sum of variances:
Actually, this property does not hold in general.
Example 6.8 To see why Eq. (6.9) cannot hold in general, an intuitive example is helpful. Consider a simple financial portfolio allocation problem. We should allocate $100,000 either to IBM or Microsoft stock shares. Say that 60% of our wealth is invested in IBM, and the rest in Microsoft:
The return on our investment depends on the returns of the two stock shares, which are two random variables. Denoting the two random variables by RIBM and RMS respectively, profit/loss is a random variable:
Imagine, for the sake of the argument, that μIBM = 5% and μMS = 10%. Since Microsoft has a higher expected return than IBM, why don’t we invest all of our wealth in Microsoft shares? The easy answer is that, arguably, greater expected return comes with greater risk. How can we measure risk? The answer is definitely tricky, but a first attempt could be considering standard deviation of return, which is linked to variance. Now what is the variance of P/L? We will find the complete answer later, in Section 8.3.2, but we may immediately understand that we cannot assess risk of our portfolio without considering the relationships between the two returns.
- If IBM price increases when Microsoft price falls, and vice versa, there is an offsetting mechanism that should reduce risk, and this should be reflected somehow in variance of profit/loss.
- If the two prices have nothing to do with each other, i.e., if RIBM and RMS are independent random variables, whatever this means exactly, we will not see a big reduction in risk, but maybe some diversification benefit will result.
- On the contrary, if the two returns go hand in hand, then we do not have a well-diversified portfolio.10 This should be reflected by a relative increase in overall variance with respect to the two previous cases.
Indeed, an old piece of advice suggests that we should not place all of our eggs into one basket.
The example shows that we are missing something, i.e., the mutual dependence of random variables, which we deal Still, there is something that we may state here, without any proof. Equation (6.9) holds if all of the involved variables are independent.
PROPERTY 6.10 (Variance of a sum of independent variables) Let Xi, i = 1,…, m, be independent random variables with variance 
, and let αi, i = 0,…, m be arbitrary real numbers. Let us also define random variables:
with variance 
and 
, respectively. Then
Be sure to notice that we may add up variances, but not standard deviations.
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