Dec 26

Alternatives to Standard Deviation as Risk Measure

by David Harper, CFA, FRM, CIPM

FRM | CFA |

Joe Hamel asks about an old article of mine on www.investopedia.com called The Uses and Limits of Volatility. He says, "Your piece on standard deviation is very informative. Do you have a source for the downside of using standard deviation in finance?"

Thanks Joe! I've got a few sources and I'll list them at the end. Just to establish some terms, when we say 'volatility' we typically refer to an annualized standard deviation. The standard deviation is the square root of the variance. The variance and the standard deviation are, of course far and away, the most popular financial measures of dispersion. In some cases, it is easier to deal with a variance (e.g., the variance scales directly with time: under assumptions, a two-day variance is equal to twice a one-day variance). But since the variance is non-intuitively given by squared units, we tend to deal with the more intuitive standard deviation which is expressed in plain old units. Either way we are talking about the most common, but hardly the only, measures of dispersion of returns.

Variance is the average of squared deviations; and it is a 'moment' employed to characterize a distribution

The formula for sample variance is given by the following.

varianceProper0

Sigma denotes standard deviation; sigma squared denotes variance; m is the number of observations; u is each observation; and u-bar is the average of all of the observations. So, the variance is basically the average of the squared deviations (for a population; in this case, for a sample, it is slightly more than the average of the squared deviations).

If we are dealing with short periods (i.e., daily or less), we make two simplifying assumptions: we assume the average return is zero (again, only for really short periods!) and we divide by (m) instead of (m-1). This makes the variance really easy to remember: for short periods, the variance is the average squared return.

variance2

If we are measuring, for example, the variance of a 10-day window, we merely square each day's (periodic) return and average them. That's the variance. Take the square root for the standard deviation (a.k.a., volatility).

We employ variance (standard deviation) to characterize a distribution

We typically employ a standard deviation to characterize a probability distribution. As in, "the returns are normally distributed with mean of X and standard deviation of Y." We do not need to specify a normal distribution, by the way. Other distributions have standard deviations, too. A lognormal distribution has a standard deviation, a Poisson distribution has a variance, and so on. The dispersion measure--standard deviation or variance--is just one parameter that helps to characterize the distribution. Technically, it is a partial moment.

But note that, as soon as we take real, actual, messy returns and characterize a distribution, we inevitably start to make a necessary mistake: the actual returns won't be as clean as the distribution. They certainly won't be as "normal" as a normal distribution which is fitted to our standard deviation (i.e., they will have fatter tails). But even after we fatten up the normal, etc, our distribution is still a mere approximation of actual returns. We superimpose a mathematical, Platonic ideal on reality. Caveat Emptor.

The 'problem' is that variance (standard deviation) doesn't care about direction

The problem is that variance treats up and down the same.

In the case of say stock prices, we are talking about the standard deviation of periodic returns, not the standard deviation of stock prices. Consider a stock that starts today at $10 under two paths. The first path is growth of 10% per year over 10 years. The second path is a loss of 10% per year over 10 years. Both series have zero volatility! See the simple spreadsheet below, the sample standard deviation of stock prices under the Grow scenario is about 5; under the Sink scenario, the standard deviation of prices is about 1.85. But the volatility of periodic returns under both is zero because there is no variation in periodic returns. Both have zero volatility, but their risk is not the same.

EditGrid Spreadsheet by bt/admin.

 

Downside risk metrics

A distribution is described by moments (e.g., a normal is described by mean, the first moment, and variance, the second moment. The normal only needs two moments). So the standard deviation is a partial moment. And if we are only concerned about losses or the downside, we really want a lower partial moment. These are the set of downside risk measures. Here is a list of a few. What they have in common is they only care about dispersion for losses, or below some target return level:

  • Semivariance: instead of the average squared deviations (i.e, variance), we take an average of squared deviations only for the returns that are below the mean. So, this is a sort of variance but only on the downside
  • Downside deviation: same as semivariance, but replace a target mean with the actual mean. So, this is sort of a variance but only on the downside based on where you want to put the hurdle.
  • Shortfall risk: the percentage of periodic returns that fall below some target return. For example, if we observe 10 returns, six of them positive, four of them negative, and our target return is zero, then shortfall risk is 40% (4 negative returns/10 total ).
  • Expected downside value: because 'shortfall risk' only gives the percentage, this metric gives the expected loss conditional on a loss. That is, what's the average loss among the losses only.
  • Expected shortfall (a.k.a., expected tail loss): this is a powerful variant on value at risk (VaR). It is the average loss among a set of losses where the set of losses is a function of our confidence. For example, what is the average loss of the worst 5% of losses.

 

More information on downside risk

My sources for downside risk measures include the following. The CIPM exam (which i just passed at the Expert level) uses Chapter 10 of Investment Performance Measurement. The CFA naturally includes a discussion of this, both in the source curriculum and Quant Methods for Investment Analysis. Finally, for a more sophisticated treatment, Chapter 4 of Noel Amenc's Portfolio Theory and Performance Analysis is a fabulous chapter from a fabulous book. (more sophisticated because his focus is risk-adjusted return measures like the Sortino ratio; most of these downside risk metrics can be put to useful work in modifying the traditional risk/return metrics like the information ratio and Sharpe ratio). The expected shortfall (ES) is maybe a little more difficult to access than the others; my favorite authority here is Kevin Dowd.

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