Apr 21, 2010
Is semi-variance a more useful measure of downside risk than standard deviation? My clients aren't worried about market surges, they're worried about market crashes.

EFF/KRF: In his classic 1959 book that defined modern portfolio theory, Markowitz considers the semi-variance as a potential measure of risk. Interest in the semi-variance fell by the wayside among academics because, at least for short holding periods (e.g., monthly), distributions of returns are rather symmetric. For symmetric distributions, the true variance and semi-variance are interchangeable, but because all the data are used to estimate the variance but only negative returns are used to estimate the semi-variance, estimates of variance are more accurate than estimates of semi-variance. For longer holding periods (e.g., a year or more), distributions of returns are right skewed, and no single measure of dispersion (e.g., the variance or the semi-variance) summarizes the overall risk of the distribution.

Let's now examine whether you really believe what you say about your client's tastes. In our (academic) terms, your statements imply that your clients are risk neutral on the upside but risk averse on the downside. If this is the case, the semi-variance, which ignores upside risk, is probably a better single measure of risk than the variance, but the conclusion is subject to the caveats above about the skewness of return distributions for longer return horizons.

Risk neutrality on the upside has a strong implication that we don't think characterizes most investors. Specifically, if such a client is faced with a choice between (i) engaging in a gamble with only positive possible payoffs, or (ii) getting the expected (mean) payoff from the gamble for certain, the client is indifferent between engaging in the gamble or taking the sure payoff, regardless of the variance of the uncertain payoffs on the gamble. If the client is indeed indifferent, you have accurately characterized the upside. But if the client says that for extreme gambles of this sort, he/she prefers to have the expected payoff for certain, the client has some amount of risk aversion on the upside.

Eugene F. Fama
The Robert R. McCormick Distinguished Service Professor of Finance at the University of Chicago Booth School of Business
Kenneth R. French
The Roth Family Distinguished Professor of Finance at the Tuck School of Business at Dartmouth College
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Eugene Fama and Ken French are members of the Board of Directors of the general partner of, and provide consulting services to Dimensional Fund Advisors LP.