Folio Lab/Documentation

Minimum Semivariance

Minimise the variance of downside returns only — penalising losses while leaving upside volatility untouched. A natural risk measure when investors care about the asymmetry of return distributions.

Overview

Variance treats positive and negative deviations symmetrically, which penalises the "good" volatility of upside moves alongside the "bad" volatility of losses. Semivariance corrects this by measuring only the variance of returns falling below a benchmark — a natural concept introduced by Markowitz himself in 1959.

The Minimum Semivariance portfolio finds the long-only allocation that minimises this downside variance. For investors with asymmetric loss preferences (most retail investors and risk-averse institutions), semivariance better matches the risk they actually care about.

Mathematical Formulation

Let be the portfolio return at time and a benchmark (typically the periodic risk-free rate). The semivariance is:

The optimisation problem is:

Folio Lab uses pypfopt's EfficientSemivariance estimator with the periodic risk-free rate as the benchmark and CLARABEL as the solver. Min-semivariance is solved as a quadratic program with auxiliary slack variables for the negative-deviation terms.

Advantages & Limitations

Advantages

  • Asymmetric: Doesn't penalise positive volatility.
  • Distribution-aware: Naturally handles skewed returns.
  • Investor-friendly: Matches the loss-averse view that most clients hold.
  • Convex QP: Tractable and reliable to solve.

Limitations

  • Data hungry: Halves the usable observations relative to variance.
  • Benchmark choice matters: Different thresholds give different portfolios.
  • Not coherent: Lacks subadditivity in general.
  • Sample sensitive: Empirical semivariance is noisier than variance.

References

  • Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investments. John Wiley & Sons.
  • Estrada, J. (2008). "Mean-semivariance optimization: A heuristic approach." Journal of Applied Finance, 18(1), 57-72.
  • pypfopt documentation — EfficientSemivariance.