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variance hedge ratio, we simply take the derivative of our equation for the portfolio variance with respect to h, and set it equal to zero:

      (3.57)

      You can check that this is indeed a minimum by calculating the second derivative. Substituting h* back into our original equation, we see that the smallest variance we can achieve is:

      (3.58)

      At the extremes, where ρ_AB equals –1 or +1, we can reduce the portfolio volatility to zero by buying or selling the hedge asset in proportion to the standard deviation of the assets. In between these two extremes we will always be left with some positive portfolio variance. This risk that we cannot hedge is referred to as idiosyncratic risk.

      If the two securities in the portfolio are positively correlated, then selling $h of Security B will reduce the portfolio's volatility to the minimum possible level. Sell any less and the portfolio will be underhedged. Sell any more and the portfolio will be overhedged. In risk management it is possible to have too much of a good thing. A common mistake made by portfolio managers is to overhedge with a low-correlation instrument.

      Notice that when ρ_AB equals zero (i.e., when the two securities are uncorrelated), the optimal hedge ratio is zero. You cannot hedge one security with another security if they are uncorrelated. Adding an uncorrelated security to a portfolio will always increase its volatility.

      This last statement is not an argument against diversification. If your entire portfolio consists of $100 invested in Security A and you add any amount of an uncorrelated Security B to the portfolio, the dollar standard deviation of the portfolio will increase. Alternatively, if Security A and Security B are uncorrelated and have the same standard deviation, then replacing some of Security A with Security B will decrease the dollar standard deviation of the portfolio. For example, $80 of Security A plus $20 of Security B will have a lower standard deviation than $100 of Security A, but $100 of Security A plus $20 of Security B will have a higher standard deviation – again, assuming Security A and Security B are uncorrelated and have the same standard deviation.

      Moments

      Previously, we defined the mean of a variable X as:

      It turns out that we can generalize this concept as follows:

      (3.59)

      We refer to m_k as the kth moment of X. The mean of X is also the first moment of X.

      Similarly, we can generalize the concept of variance as follows:

      (3.60)

      We refer to μ_k as the kth central moment of X. We say that the moment is central because it is central around the mean. Variance is simply the second central moment.

      While we can easily calculate any central moment, in risk management it is very rare that we are interested in anything beyond the fourth central moment.

      Skewness

      The second central moment, variance, tells us how spread-out a random variable is around the mean. The third central moment tells us how symmetrical the distribution is around the mean. Rather than working with the third central moment directly, by convention we first standardize the statistic. This standardized third central moment is known as skewness:

      (3.61)

      where σ is the standard deviation of X.

      By standardizing the central moment, it is much easier to compare two random variables. Multiplying a random variable by a constant will not change the skewness.

A random variable that is symmetrical about its mean will have zero skewness. If the skewness of the random variable is positive, we say that the random variable exhibits positive skew. Figures 3.3 and 3.4 show examples of positive and negative skewness.

FIGURE 3.3 Positive Skew

FIGURE 3.4 Negative Skew

      Skewness is a very important concept in risk management. If the distributions of returns of two investments are the same in all respects, with the same mean and standard deviation but different skews, then the investment with more negative skew is generally considered to be more risky. Historical data suggest that many financial assets exhibit negative skew.

      As with variance, the equation for skewness differs depending on whether we are calculating the population skewness or the sample skewness. For the population statistic, the skewness of a random variable X, based on n observations, x₁, x₂, … , x_n, can be calculated as:

      (3.62)

      where μ is the population mean and σ is the population standard deviation. Similar to our calculation of sample variance, if we are calculating the sample skewness, there is going to be an overlap with the calculation of the sample mean and sample standard deviation. We need to correct for that. The sample skewness can be calculated as:

      (3.63)

Based on Equation 3.40 for variance, it is tempting to guess that the formula for the third central moment can be written simply in terms of E[X³] and μ. Be careful, as the two sides of this equation are not equal:

      (3.64)

      The correct equation is:

(3.65)

      Sample Problem

      Question:

Prove that the left-hand side of Equation 3.65 is indeed equal to the right-hand side of the equation.

      Answer:

      We start by multiplying out the terms inside the expectation. This is not too difficult to do, but, as a shortcut, we could use the binomial theorem as mentioned previously:

      Next we separate the terms inside the expectations operator and move any constants, namely μ, outside the operator:

      E[X] is simply the mean, μ. For E[X²], we reorganize our equation for variance, Equation 3.40, as follows:

      Substituting these results into our equation and collecting terms, we arrive at the final equation:

      For many symmetrical continuous distributions, the mean, median, and mode all have the same value. Many continuous distributions with negative skew have a mean that is less than the median, which is less than the mode. For example, it might be that a certain

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