Semivarianz

Semivarianz, or semi-variance, is a statistical measure that focuses on downside dispersion rather than overall variability. In finance and statistics, it is used to assess downside risk by concentrating on observations that fall below a chosen threshold, typically the mean or a target level.

Definition and variants: For a set of observations x1, x2, ..., xn with mean μ, the population semivarianz is often defined as the average of squared deviations below μ: S = E[ max(0, μ − X)² ]. A common sample version is S = (1/n) ∑ max(0, μ − x_i)². A related variant uses a specified target T instead of the mean: S_T = E[ max(0, T − X)² ]. These expressions belong to the broader class of lower partial moments, specifically the second-order LPM with threshold T.

Relation to variance: Semivarianz is related to, but distinct from, variance. Variance, Var(X) = E[(X − μ)²], accounts for deviations on both sides of the mean. Semivarianz only accumulates contributions from outcomes below the threshold, ignoring upside deviations above it. As a result, semivariance is typically lower than variance for distributions with substantial upside risk and is highly sensitive to the chosen threshold.

Applications: Semivarianz is used as a downside risk measure in risk management and portfolio optimization. It guides asset selection and weighting by prioritizing the avoidance of negative returns rather than balancing all fluctuations equally. It can be used in performance evaluation, pricing, and in models that explicitly target downside protection.

Limitations: The metric depends on the selected threshold, which can be subjective. It ignores upside volatility and can be less informative for symmetric or skewed distributions where the threshold choice strongly influences results. Despite limitations, semivarianz remains a widely cited concept for assessing downside risk.