the expected value of Y:
(3.47)
If the covariance is anything other than zero, then the two sides of this equation cannot be equal. Unless we know that the covariance between two variables is zero, we cannot assume that the expectations operator is multiplicative.
In order to calculate the covariance between two random variables, X and Y, assuming the means of both variables are known, we can use the following formula:
If the means are unknown and must also be estimated, we replace n with (n – 1):
If we replaced yi in these formulas with xi, calculating the covariance of X with itself, the resulting equations would be the same as the equations for calculating variance from the previous section.
Correlation
Closely related to the concept of covariance is correlation. To get the correlation of two variables, we simply divide their covariance by their respective standard deviations:
(3.48)
Correlation has the nice property that it varies between –1 and +1. If two variables have a correlation of +1, then we say they are perfectly correlated. If the ratio of one variable to another is always the same and positive then the two variables will be perfectly correlated.
If two variables are highly correlated, it is often the case that one variable causes the other variable, or that both variables share a common underlying driver. We will see later, though, that it is very easy for two random variables with no causal link to be highly correlated. Correlation does not prove causation. Similarly, if two variables are uncorrelated, it does not necessarily follow that they are unrelated. For example, a random variable that is symmetrical around zero and the square of that variable will have zero correlation.
Sample Problem
Question:
X is a random variable. X has an equal probability of being –1, 0, or +1. What is the correlation between X and Y if Y = X2?
Answer:
We have:
First we calculate the mean of both variables:
The covariance can be found as:
Because the covariance is zero, the correlation is also zero. There is no need to calculate the variances or standard deviations.
As forewarned, even though X and Y are clearly related, the correlation is zero.
Application: Portfolio Variance and Hedging
If we have a portfolio of securities and we wish to determine the variance of that portfolio, all we need to know is the variance of the underlying securities and their respective correlations.
For example, if we have two securities with random returns XA and XB, with means μA and μB and standard deviations σA and σB, respectively, we can calculate the variance of XA plus XB as follows:
where ρAB is the correlation between XA and XB. The proof is left as an exercise. Notice that the last term can either increase or decrease the total variance. Both standard deviations must be positive; therefore, if the correlation is positive, the overall variance will be higher compared to the case where the correlation is negative.
If the variance of both securities is equal, then Equation 3.49 simplifies to:
Now we know that the correlation can vary between –1 and +1, so, substituting into our new equation, the portfolio variance must be bound by 0 and 4σ2. If we take the square root of both sides of the equation, we see that the standard deviation is bound by 0 and 2σ. Intuitively this should make sense. If, on the one hand, we own one share of an equity with a standard deviation of $10 and then purchase another share of the same equity, then the standard deviation of our two-share portfolio must be $20 (trivially, the correlation of a random variable with itself must be one). On the other hand, if we own one share of this equity and then purchase another security that always generates the exact opposite return, the portfolio is perfectly balanced. The returns are always zero, which implies a standard deviation of zero.
In the special case where the correlation between the two securities is zero, we can further simplify our equation. For the standard deviation:
(3.51)
We can extend Equation 3.49 to any number of variables:
(3.52)
In the case where all of the Xi's are uncorrelated and all the variances are equal to σ, Equation 3.50 simplifies to:
(3.53)
This is the famous square root rule for the addition of uncorrelated variables. There are many situations in statistics in which we come across collections of random variables that are independent and have the same statistical properties. We term these variables independent and identically distributed (i.i.d). In risk management we might have a large portfolio of securities, which can be approximated as a collection of i.i.d. variables. As we will see, this i.i.d. assumption also plays an important role in estimating the uncertainty inherent in statistics derived from sampling, and in the analysis of time series. In each of these situations, we will come back to this square root rule.
By combining Equation 3.49 with Equation 3.42, we arrive at an equation for calculating the variance of a linear combination of variables. If Y is a linear combination of XA and XB, such that:
(3.54)
then, using our standard notation, we have:
Correlation is central to the problem of hedging. Using the same notation as before, imagine we have $1 of Security A, and we wish to hedge it with $h of Security B (if h is positive, we are buying the security; if h is negative, we are shorting the security). In other words, h is the hedge ratio. We introduce the random variable P for our hedged portfolio. We can easily compute the variance of the hedge portfolio using Equation 3.55:
(3.56)
As a risk manager, we might be interested to know what hedge ratio would achieve the portfolio with the least variance. To