Folio Lab/Documentation

Minimum Semivariance

Minimise the variance of below-target returns only. Semivariance is the downside-only counterpart of variance and was discussed by Markowitz himself (1959) as arguably a more economically sensible risk measure than full variance, since investors do not regret upside.

Overview

Variance is symmetric: it penalises upside and downside dispersion equally. Markowitz (1959) acknowledged this is awkward from a behavioural standpoint and proposed semivariance - the variance of returns below a target, usually the mean or zero - as an alternative. Hogan and Warren (1972) gave the first practical optimisation algorithm; Estrada (2008) proposed a heuristic approach; and the modern treatment (Bawa and Lindenberg, 1977; Sortino and van der Meer, 1991) connects semivariance to lower-partial moments and downside-CAPM.

FolioLab implements minimum semivariance via PyPortfolioOpt's EfficientSemivariance. The optimisation is a convex quadratic-programming problem in a slack-variable reformulation that handles the asymmetry of the objective cleanly. The scenarios are the daily returns of the run window, and the target is the daily risk-free rate derived from the run's annual rate. The solver order is CLARABEL, then SCS, then ECOS. Per-name weight bounds, asset exclusions, and group sum limits such as sector caps enter the same program as extra rows.

The method is the natural answer when the investor's loss aversion is asymmetric - for example, when the mandate is to avoid losing capital but upside is unconstrained. It is also a useful complement to mean-variance on heavy-tailed equity universes where the downside dispersion materially exceeds the upside dispersion.

Mathematical Formulation

Notation

  • - vector of asset returns at time
  • - target return (often zero or the risk-free rate; here, the daily risk-free rate)
  • - negative-part operator
  • - portfolio weights, summing to 1, long-only

Semivariance

Only periods where the portfolio return falls below contribute to the sum. This is the empirical lower-partial second moment with target ; setting gives mean-conditional semivariance, setting gives loss-only semivariance.

Slack-variable reformulation

The non-smooth negative-part operator is handled by introducing per-period non-negative slack variables :

The constraint together with makes at the optimum, so the objective coincides with semivariance. The result is a convex QP in that off-the-shelf solvers dispatch quickly.

Mean-semivariance frontier

In the same way that the mean-variance frontier is parameterised by a target return, a mean-semivariance frontier is parameterised by a target return with the semivariance objective. Markowitz (1959) and Estrada (2008) show that the resulting frontier is well-defined. By construction it is dominated in standard-deviation space by the mean-variance frontier.

Connection to the Sortino ratio

The Sortino ratio (Sortino and van der Meer, 1991) is the excess-return-per- unit-of-downside-deviation analogue of the Sharpe ratio:. Maximising the Sortino ratio is the asymmetric analogue of maximising the Sharpe ratio. The minimum-semivariance portfolio is the global minimum of the denominator of the Sortino objective.

Advantages & Limitations

Advantages

  • Asymmetric risk: Penalises downside only, matching investor preferences.
  • Convex QP: Tractable with off-the-shelf solvers.
  • Discussed by Markowitz: Markowitz (1959) considered semivariance as a risk measure that can be preferable to variance when return distributions are asymmetric.
  • Compatible: Slots into mean-semivariance frontiers and Sortino-maximisation programmes.

Limitations

  • Sample noise: Below-target observations are a fraction of the sample, so the estimator is noisier than full variance.
  • Target choice matters: Different can give meaningfully different portfolios.
  • Path-independent: Like variance, ignores drawdown sequencing.
  • Less established than CVaR: Tail-risk frameworks have largely converged to CVaR / EVaR for extreme-loss control.

References

  • Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investments. John Wiley & Sons.
  • Hogan, W. W., & Warren, J. M. (1972). "Computation of the Efficient Boundary in the E-S Portfolio Selection Model." Journal of Financial and Quantitative Analysis, 7(4), 1881-1896.
  • Bawa, V. S., & Lindenberg, E. B. (1977). "Capital Market Equilibrium in a Mean-Lower Partial Moment Framework." Journal of Financial Economics, 5(2), 189-200.
  • Estrada, J. (2008). "Mean-Semivariance Optimization: A Heuristic Approach." Journal of Applied Finance, 18(1), 57-72.
  • Sortino, F. A., & van der Meer, R. (1991). "Downside Risk." The Journal of Portfolio Management, 17(4), 27-31.
  • Fishburn, P. C. (1977). "Mean-Risk Analysis with Risk Associated with Below-Target Returns." American Economic Review, 67(2), 116-126.
  • Palomar, D. P. (2025). Portfolio Optimization: Theory and Application. Cambridge University Press, Chapter 10 (Portfolios with Alternative Risk Measures), Section 10.3 (Downside Risk Portfolios).
  • PyPortfolioOpt documentation - EfficientSemivariance.

Not investment advice. Past performance is not indicative of future results.