How is semivariance-based portfolio optimization implemented?

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AcadiFi · Accepted Answer

Semivariance optimization substitutes semicovariance for covariance in mean-variance. Semicovariance between assets i and j: S_ij = E[(r_i - μ_i)_- (r_j - μ_j)_-] using only observations where both returns are below their means. This matrix is endogenous to the portfolio because whether observation t is included depends on whether the portfolio return at t is below target. This circularity makes direct quadratic programming inapplicable—the covariance-like matrix changes with weights. Two practical approaches: 1. Estreada's Fixed-Target Method: use a fixed threshold (0 or risk-free rate) instead of portfolio-dependent mean. Semicovariance S_ij(τ) = E[(r_i - τ)_- (r_j - τ)_-] is then fixed, allowing standard QP. Results approximate true semivariance but are tractable. 2. Iterative Algorithm: start with equal weights, compute semivariance matrix from historical downside scenarios, optimize as QP with that matrix, recompute matrix with new weights, iterate to convergence. Usually converges in 5-10 iterations for stable datasets. 3. Sample-based LP: formulate as linear program with auxiliary variables z_t = max(0, τ - R_portfolio_t). Minimize Σ z_t² approximated via quadratic constraints. Handles true portfolio-dependent semivariance exactly. ```mermaid flowchart LR A[Returns] --> B[Extract Downside
observations] B --> C[Semicovariance
Matrix] C --> D[QP Optimization] D --> E[Downside Frontier] ``` Comparison to variance-based results: semivariance portfolios typically overweight low-skew low-downside-vol assets (utilities, staples) versus mean-variance. Performance during crashes often better but sacrifice upside capture. Back-tests show mean-semivariance outperforms Markowitz in fat-tailed regimes but underperforms in extended bull markets. Covariance estimation error is more severe for semivariance due to smaller effective sample.

How is semivariance-based portfolio optimization implemented?

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