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      In the previous example, we were calculating the population variance and standard deviation. All of the possible outcomes for the derivative were known.

      To calculate the sample variance of a random variable X based on n observations, x₁, x₂, … , x_n, we can use the following formula:

      (3.39)

      where

      Equation 3.38 can easily be rearranged as follows (we leave the proof of this for an exercise, too):

(3.40)

      Note that variance can be nonzero only if E[X²] ≠ E[X]².

      When writing computer programs, this last version of the variance formula is often useful, since it allows you to calculate the mean and the variance in the same loop. Also, in finance it is often convenient to assume that the mean of a random variable is close to zero. For example, based on theory, we might expect the spread between two equity indexes to have a mean of zero in the long run. In this case, the variance is simply the mean of the squared returns.

      As with the mean, for a continuous random variable we can calculate the variance by integrating with the probability density function. For a continuous random variable, X, with a probability density function, f(x), the variance can be calculated as:

      (3.41)

      It is not difficult to prove that, for either a discrete or a continuous random variable, multiplying by a constant will increase the standard deviation by the same factor:

(3.42)

      In other words, if you own $10 of an equity with a standard deviation of $2, then $100 of the same equity will have a standard deviation of $20.

      Adding a constant to a random variable, however, does not alter the standard deviation or the variance:

      (3.43)

      This is because the impact on the mean is the same as the impact on any draw of the random variable, leaving the deviation from the mean unchanged. If you own a portfolio with a standard deviation of $20, and then you add $1,000 of cash to that portfolio, the standard deviation of the portfolio will still be $20.

      It is often convenient to work with variables where the mean is zero and the standard deviation is one. From the preceding section it is not difficult to prove that, given a random variable X with mean μ and standard deviation σ, we can define a second random variable Y:

      (3.44)

      such that Y will have a mean of zero and a standard deviation of one. We say that X has been standardized, or that Y is a standard random variable. In practice, if we have a data set and we want to standardize it, we first compute the sample mean and the standard deviation. Then, for each data point, we subtract the mean and divide by the standard deviation.

      The inverse transformation can also be very useful when it comes to creating computer simulations. Simulations often begin with standardized variables, which need to be transformed into variables with a specific mean and standard deviation. In this case, we simply take the output from the standardized variable, multiply by the desired standard deviation, and then add the desired mean. The order is important. Adding a constant to a random variable will not change the standard deviation, but multiplying a non-mean-zero variable by a constant will change the mean.

      Up until now we have mostly been looking at statistics that summarize one variable. In risk management, we often want to describe the relationship between two random variables. For example, is there a relationship between the returns of an equity and the returns of a market index?

      Covariance is analogous to variance, but instead of looking at the deviation from the mean of one variable, we are going to look at the relationship between the deviations of two variables:

(3.45)

      where σXY is the covariance between two random variables, X and Y, with means μ_X and μ_Y, respectively. As you can see from the definition, variance is just a special case of covariance. Variance is the covariance of a variable with itself.

      If X tends to be above μ_X when Y is above μ_Y (both deviations are positive), and X tends to be below μ_X when Y is below μ_Y (both deviations are negative), then the covariance will be positive (a positive number multiplied by a positive number is positive; likewise, for two negative numbers). If the opposite is true and the deviations tend to be of opposite sign, then the covariance will be negative. If the deviations have no discernible relationship, then the covariance will be zero.

Earlier in this chapter, we cautioned that the expectations operator is not generally multiplicative. This fact turns out to be closely related to the concept of covariance. Just as we rewrote our variance equation earlier, we can rewrite Equation 3.45 as follows:

      (3.46)

      In the special case where the covariance between X and Y is zero, the expected value of XY is equal to the expected value of X

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