Unlike variance, there a several different formulas for semivariance (SV). If you are a college student looking to get the “right” answer on test or quiz, the formula you are looking for is most likely:

The question-mark-colon syntax simply means if the expression before the “?” is true then the term before the “:” is used, otherwise the term after the “:” is used. So** a?b:c** simply means chose

*b*if

*a*is true, else chose

*c*. This syntax is widely used in computer science, but less often in the math department. However, I find it more concise than other formulations.

Another common semivariance formula involves comparing returns to a required minimum threshold r_{t}. This is simply:

Classic mean-return semivariance should not be directly compared to mean-return variance. However a slight modification makes direct comparison more meaningful. In general approximately half of mean-adjusted returns are positive and half are negative (exactly zero is a relatively rare event and has no impact to either formula). While mean-variance always has n terms, semi-variance only uses a subset which is typically of size n/2. Thus including a factor of 2 in the formula makes intuitive sense:

Finally, another useful formulation is one I call “Modified Drawdown Only” (MDO) semivariance. The name is self-explanatory… only drawdown events are counted. SV_{mdo} does not require r_{avg} (r bar) nor r_{t}. It produces nearly identical values to SV_{mod} for rapid sampling (say for anything more frequent than daily data). For high-speed trading it also has the advantage of not requiring all of the return data *a priori*, meaning it can be computed as each return data point becomes available, rather than retrospectively.

Why might SV_{mdo }be useful in high-speed trading? One use may be in put/call option pricing arbitrage strategies. Black–Scholes, to my knowledge, makes no distinction between “up-side” and “down-side” variance, and simply uses plain variance. [Please shout a comment at me if I am mistaken!] However if put and call options are “correctly” priced according to Black–Scholes, but the data shows a pattern of, say, greater downside variance than normal variance on the underlying security, put options may be undervalued. This is just an off-the-cuff example, but it illustrates a potential situation for which SV_{mdo} is best suited.

### Pick Your Favorite Risk Measure

Personally, I slightly favor SV_{mdo }over SV_{mod} for computational reasons. They are often quite similar in practice, especially when used to rank risk profiles of a set of candidate portfolios. (The fact that both are anagrams of each other is deliberate.)

I realize that the inclusion of the factor 2 is really just a semantic choice. Since V and (classic) SV, amortized over many data sets, are expected to differ by a factor of 2, standard deviation, σ, and semideviation, σ_{d}, can be expected to differ by the square root of 2. I consider this mathematically untidy. Conversely, I consider SV_{mod} to be the most elegant formulation.