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The following article is reprinted from the May/June, 2001 issue of
On the Edge, the Interactive Data Fixed Income Analytics bimonthly newsletter.
Back-to-Basics: Standard Deviation
Teri Geske Many concepts used in investment and risk management incorporate the statistic standard deviation. For example, the volatility measure used in option models is the standard deviation of the underlying on which the option is based, a Tracking Error analysis is computed from the standard deviation of expected returns (where the expected returns are a function of the volatility, or standard deviation, of risk factors that affect returns), a Value-at-Risk analysis often use standard deviation, and so on. Since this measure is so frequently encountered in finance theory and practice, this Back-to-Basics column reviews exactly what this statistic can tell us, and what its limitations are.
First, let’s recall how a standard deviation (abbreviated “s.d.” or  Intuitively, the more the individual outcomes (xi) of a random stochastic variable differ from the mean value of that variable (µx), the greater the standard deviation. Standard deviation is viewed as a measure of risk, because the more we expect any single value for a particular variable (which could be an interest rate, a currency exchange rate, the return on a portfolio or benchmark, etc.) to deviate from the mean or expected value for that variable, the more uncertain the value the variable will obtain in the future. In other words, given two sets of data with the same expected value, the data set where the individual values are tightly clustered around the mean describes a variable whose outcome is less uncertain, or not as “risky” as, the outcome of the variable that has taken on values that are widely dispersed. Most of us are familiar with the concept of a “normal” distribution, where the values obtained by a variable are symmetrically distributed around the mean, forming a pattern often referred to as a bell-shaped curve. When a variable is normally distributed, we can know everything worth knowing about its behavior based solely on its mean and standard deviation. For example, we can state that roughly two-thirds of the time, the variable will take on a value that is within one standard deviation of its mean, 95% of the time its value will be within two standard deviations (+/-) of the mean, and over 99% of the time its value will be within three standard deviations (+/-) of the mean. In other words, if a variable is normally distributed we should very rarely observe movements in the variable that exceed a “three standard deviation move”. Given this complete picture by the provided mean and standard deviation values of a normally distributed random variable, it is tempting to assume that a given variable does adhere to a normal distribution, and in fact many do. However, some distributions have “fat tails”, meaning that the variable actually takes on extreme values more often than the probability described by the mean and standard deviation of a normal distribution, and of course, a higher frequency of extreme values implies greater risk. For example, if we observed the average daily percentage change in the S&P 500 and computed its standard deviation to be 2% (using the formula shown earlier), the rules of the normal distribution would state that over 99% of the time, daily changes in the S&P 500 would be within a range of +/-6% of the average change. If subsequent changes in the S&P 500 exceeded this range more than 1% of the time, we might conclude that the true distribution of this variable was not “normal”, in which case we cannot rely on standard deviation to completely describe the range of values of the variable is expected to achieve. Sometimes, two random variables can have the same average and standard deviation, but the pattern or shape of the distribution of the individual values the variables can achieve are different. Here is an example of two such variables, X and Y:
 In this case, variable Y has a skewed distribution, meaning its values are not symmetrically distributed around its mean. The expected returns on corporate bonds, given a risk of downgrade/default, are skewed, since there is a large probability that returns will be positive, based on the bond’s coupon income, and a small probability of an extremely negative return due to a severe downgrade or actual default. A mean and standard deviation can be computed from these possible return outcomes, but these two statistics do not fully describe the low probability of a severely negative outcome compared to the high probability of moderately positive outcomes. In other words, the mean and standard deviation of such a variable does not adequately describe the true shape of the variable’s distribution. Standard deviations can be used to compute a confidence interval, i.e. the range within which the true value of an estimated parameter will lie, given a stated probability. Confidence intervals are used to express the uncertainty of values that are estimated based on a sample set of outcomes, rather than an exhaustive set of all possible outcomes. For example, although we can calculate the expected future value of a portfolio as the average of a sampling of its potential values in the future, since we have only a limited sample data set with which to compute this average we cannot know the true (in a statistical sense) average or expected value of the portfolio. However, we can say that for a given level of confidence (e.g. 90%, 95%, 99%), we know its true expected value must lie within some interval that contains our estimated value, and we can compute this interval by adding/subtracting a certain number of standard deviations to/from our estimate. In summary, the concept of standard deviation is fairly straightforward, it is intuitive as a measure of risk and is used extensively in investment and risk management analyses. However, it should be used with a clear understanding of what it does and does not say about the distribution of possible future outcomes so that we are not too surprised when those “three-standard deviation moves” occur. As always, if you have any questions on this column or suggestions for future Back-to-Basics topics, we’d like to hear from you – please contact marketing at fia.marketing@interactivedata.com. | ||
 
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) is defined and computed. It is a measure of the dispersion of a series of data around the mean of that data, calculated as the square root of the mean squared deviation: