What is lower partial moment and how is it used in risk measurement?

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AcadiFi · Accepted Answer

Lower partial moment of order n around threshold tau is LPM_n(tau) = E[max(tau - R, 0)^n]. It captures only downside deviations from tau, raised to power n. Three commonly used orders: LPM_0 is shortfall probability P(R < tau), LPM_1 is expected shortfall magnitude E[tau - R | R < tau] * P(R < tau), LPM_2 is semi-variance (variance of returns below tau). For a portfolio with 12% mean return, 16% volatility, and MAR (minimum acceptable return) threshold 5%, standard deviation penalizes upside vol (hitting 35% return contributes the same as -20%). LPM_2 with tau=5% only penalizes returns below 5%. For a right-skewed return distribution, LPM_2 is much lower than variance, properly rewarding upside. The Sortino ratio = (mean - tau) / sqrt(LPM_2) uses semi-deviation in denominator, replacing Sharpe's symmetric volatility. For convex hedge fund returns, Sortino can be 2x Sharpe because upside vol is inflated in Sharpe's denominator. Kappa ratios generalize this using higher LPM orders for tail-risk emphasis. ```mermaid graph LR A[Return
distribution] --> B{Below tau?} B -->|Yes| C[Count in
LPM n] B -->|No| D[Ignore
upside] ```

What is lower partial moment and how is it used in risk measurement?

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