Semi variance

Standard deviation treats a big gain and an equally big loss as the same amount of "risk" — it measures dispersion in both directions equally. For many investors that is not how risk actually feels: an unusually good month is not something they worry about.

Semi variance answers to that intuition. It measures dispersion using only the returns that fall below the mean — or, in its close cousin target semi variance, below some other target return the investor cares about — and simply ignores how far above the mean the good months went.

The workbook: "Semi variance measures the dispersion of the return below the mean return. Target Semi variance measures the dispersion of the return below the target return. In case of symmetrically distributed return, semi variance will be proportional to variance and provides no additional insight. Forecasting asymmetrical distribution of return is very difficult and may not be a good forecast of future risk. Due to these reasons' semi variance is not so popularly used in practice."

The workbook's own standard-deviation example, a few paragraphs earlier in the same section, shows exactly what semi variance is reacting against: a portfolio with a monthly standard deviation of 3.27% and an expected return of 2% is said to have a 66.7% probability of landing between 2% − 3.27% and 2% + 3.27% in any month — the upper and lower tails are treated identically. Semi variance would instead look only at the portion of that distribution below the 2% mean.

Illustrative figures, not from the workbook. Take a small-cap portfolio's five monthly returns: +6%, −3%, +2%, −5%, +4%, giving a mean of 0.8%. Only two months fall below the mean: −3% (a deviation of −3.8 percentage points) and −5% (a deviation of −5.8 points). Squaring and averaging just these two below-mean deviations gives a semi-variance of (3.8² + 5.8²) ÷ 2 = 24.04, for a semi-standard-deviation of ≈4.90% — higher, in this particular small sample, than the portfolio's full-sample standard deviation of ≈4.17% (computed across all five months), because the two down-months happened to be unusually large relative to the up-months. That gap between the two figures is exactly the kind of asymmetry the workbook says makes semi variance hard to forecast reliably.

A portfolio manager compares two funds with the same standard deviation of 6% monthly. Fund A's bad months are moderate and its good months are occasionally spectacular; Fund B's good and bad months are both moderate and roughly symmetric. Standard deviation alone cannot tell them apart — both funds show 6%.

Semi variance can. Fund A, whose downside months are smaller relative to its standard deviation than its upside months (the volatility is skewed toward gains), would show a lower semi variance than Fund B, whose losses and gains are more evenly sized. An investor who only fears losses — which is most investors — would reasonably prefer Fund A even though both funds report an identical standard deviation, which is precisely the blind spot semi variance is designed to catch. This is also why the Sortino ratio, which is built on semi-deviation rather than standard deviation, exists alongside the Sharpe ratio.

Chapter 20, section 20.3.1 (Total Risk and Downside Risk), introduces semi variance and target semi variance immediately after the standard-deviation discussion, and explains why the workbook itself says the measure is not popular in practice. Expect a conceptual question on the difference between total risk and downside risk, and a question on why semi variance adds little value for symmetric return distributions.

• Semi variance ≠ target semi variance. Semi variance measures dispersion below the mean; target semi variance measures dispersion below a separately chosen target return — the workbook states both definitions side by side precisely because they are often confused.
• For a symmetric distribution, semi variance adds no new information — it will simply be proportional to ordinary variance. It only matters when the return distribution is skewed.
• The workbook itself says semi variance is not popularly used in practice, because forecasting an asymmetric return distribution reliably is difficult — do not present it as the industry-standard risk measure; that remains standard deviation.
• Semi variance is a variance-type measure; its square root gives the semi standard deviation, exactly as ordinary variance relates to standard deviation.