Semi-variance: Choosing the Best Formula

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:

Classic Semi-Variance Formula

Classic Semi-Variance Formula

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 rt.  This is simply:

Min Return Threshold SV

Min Return Threshold SV

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:

Modified Semi-Variance

Modified Semi-Variance

Finally, another useful formulation is one I call “Modified Drawdown Only” (MDO) semivariance.  The name is self-explanatory… only drawdown events are counted.  SVmdo does not require ravg (r bar) nor rt.  It produces nearly identical values to SVmod 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.

Modified Drawdown-Only Semi-variance

Modified Drawdown-Only Semi-variance

Why might  SVmdo 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 SVmdo is best suited.

Pick Your Favorite Risk Measure

Personally, I slightly favor SVmdo over SVmod 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, σd, can be expected to differ by the square root of 2.  I consider this mathematically untidy.  Conversely, I consider SVmod to be the most elegant formulation.


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