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Financial Modeling

Stutzer Performance Index: Skewness-Adjusted Sharpe & Cumulant Decay Rate

Adjust the Sharpe ratio for third-moment crash skewness using Michael Stutzer’s (2000) large-deviations information decay rate I_P and Sharpe-equivalent SQRT(2 * I_P).

Why negatively skewed carry and option-selling strategies inflate the standard Sharpe ratio

Short-put and carry strategies collect steady small gains interrupted by rare catastrophic drops—producing high mean-to-volatility Sharpe ratios right up until a left-tail blowup! Michael Stutzer (2000) ranked portfolios by maximizing the asymptotic exponential decay rate I_P at which average excess return Prob(R_bar <= 0) <= EXP(-T * I_P) shrinks toward zero as horizon T grows.

Under zero skewness (Gaussian excess returns), I_P = 0.5 * SR^2 so Stutzer’s Sharpe-equivalent index SIGN(SR) * SQRT(2 * I_P) equals the ordinary Sharpe Ratio identically! Expanding the cumulant generating function through third central moment m_3 = skew * sigma^3 gives exact saddlepoint theta* = (-sigma^2 + SQRT(sigma^4 - 2 * mu * m_3)) / m_3 and I_P = -(mu * theta* + 0.5 * sigma^2 * theta*^2 + (1 / 6) * m_3 * theta*^3), penalizing negative skewness while rewarding positive convexity.

Return Skewness (mu=2%, sigma=5%, Raw SR=0.4000)Information Rate I_PStutzer Sharpe-Equivalent
Negative Crash Skew (-1.50)0.06860.3704 (-296 bps penalty vs Sharpe)
Symmetric Gaussian Skew (0.00)0.0800 (0.5 * 0.4^2)0.4000 (Matches Sharpe identically)
Positive Trend Skew (+1.00)0.09650.4393 (+393 bps convexity bonus)

Three-Cumulant Saddlepoint Information Decay Rate I_P Formula

Return exact 0.5 * (mu / sigma)^2 when ABS(skew) < 1E-9, and evaluate the 3-cumulant saddlepoint theta* when skewness B4 is non-zero.

=LET(mu_ex, B2, sig, B3, skw, B4, sr_val, mu_ex / sig, IF(ABS(skw) < 0.000000001, ROUND(0.5 * (sr_val ^ 2), 4), LET(v_two, sig ^ 2, m_thr, skw * (sig ^ 3), disc, (v_two ^ 2) - 2 * mu_ex * m_thr, IF(disc < 0, "DOMAIN_LIMIT", LET(th_opt, (-v_two + SQRT(disc)) / m_thr, ROUND(-(mu_ex * th_opt + 0.5 * v_two * (th_opt ^ 2) + (1 / 6) * m_thr * (th_opt ^ 3)), 4))))))

Stutzer Sharpe-Equivalent Ratio SIGN(mu) * SQRT(2 * I_P)

Map unrounded I_P onto the familiar Sharpe scale via SIGN(B2) * SQRT(2 * I_P) so analysts can compare directly against raw Sharpe B2 / B3.

=LET(mu_ex, B2, sig, B3, skw, B4, sr_val, mu_ex / sig, IF(ABS(skw) < 0.000000001, ROUND(sr_val, 4), LET(v_two, sig ^ 2, m_thr, skw * (sig ^ 3), disc, (v_two ^ 2) - 2 * mu_ex * m_thr, IF(disc < 0, "DOMAIN_LIMIT", LET(th_opt, (-v_two + SQRT(disc)) / m_thr, ip_val, -(mu_ex * th_opt + 0.5 * v_two * (th_opt ^ 2) + (1 / 6) * m_thr * (th_opt ^ 3)), ROUND(SIGN(mu_ex) * SQRT(2 * MAX(0, ip_val)), 4))))))

Try the example

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Further reading