Adjusted Sharpe Ratio: Penalizing Skew and Kurtosis

Key Takeaways

• The skew kurtosis adjusted Sharpe ratio multiplies the Sharpe ratio by a penalty based on the third and fourth moments.
• Negative skewness and high excess kurtosis lower the score, since both signal worse tail risk.
• For a normal distribution with zero skew and zero excess kurtosis, the figure equals the plain Sharpe ratio.
• It was introduced by Pezier and White in 2006 to evaluate funds with non-normal returns.

What the Skew Kurtosis Adjusted Sharpe Is

The adjusted Sharpe ratio, sometimes written ASR, was introduced by Jacques Pezier and Anthony White in 2006. They wanted a single number that kept the familiar return-per-risk shape of the Sharpe ratio but accounted for the lopsided, fat-tailed returns common in hedge funds.

The standard Sharpe ratio uses only the first two moments of a return distribution, the mean and the standard deviation. It implicitly assumes returns are normal. The adjusted version brings in the third moment, skewness, and the fourth moment, kurtosis. It multiplies the Sharpe ratio by a factor that rises slightly for favorable shapes and falls for unfavorable ones.

The Intuition

Two funds can post identical Sharpe ratios while offering very different risk. One might earn its return through steady, symmetric gains. The other might sell tail risk, booking smooth profits most months and rare devastating losses. The second fund has negative skewness and fat tails, yet the plain Sharpe ratio cannot see the difference.

The adjusted Sharpe ratio adds that vision. Negative skewness, meaning the big surprises tend to be on the downside, lowers the score. Excess kurtosis, meaning fatter tails than a normal distribution, also lowers it. A fund whose returns are symmetric and thin-tailed keeps roughly its original Sharpe. A fund hiding crash risk gets marked down.

How It Works

The adjusted Sharpe ratio multiplies the plain Sharpe ratio by a polynomial penalty in skewness and kurtosis:

ASR = SR * [ 1 + (S / 6) * SR - ((K - 3) / 24) * SR^2 ]

Where:

SR = the standard Sharpe ratio
S  = skewness of returns
K  = kurtosis of returns (so K - 3 is excess kurtosis)

The middle term, (S / 6) times SR, rewards positive skewness and penalizes negative skewness. The last term, ((K - 3) / 24) times SR squared, penalizes excess kurtosis, since heavy tails carry hidden risk. When skewness is zero and kurtosis equals 3, both correction terms vanish and the adjusted Sharpe equals the plain Sharpe ratio.

The penalty grows with the size of the Sharpe ratio itself, so the adjustment matters most for strategies that already look strong on the plain measure. That is by design, because a high Sharpe built on hidden tail risk is exactly the case worth flagging.

Worked Example

Suppose a fund has a Sharpe ratio of 1.0, skewness of -0.8, and kurtosis of 6 (so excess kurtosis is 3).

Skew term: (S / 6) times SR = (-0.8 / 6) times 1.0 = -0.133.

Kurtosis term: ((K - 3) / 24) times SR^2 = (3 / 24) times 1.0 = 0.125.

ASR = 1.0 times [1 + (-0.133) - 0.125] = 1.0 times 0.742 = 0.74

The negative skew and fat tails cut a Sharpe of 1.0 down to an adjusted 0.74. A second fund with the same Sharpe but symmetric, normal returns would keep its full 1.0, revealing it as the safer choice once the tails are accounted for.

Common Mistakes

1. Using too short a sample. Skewness and kurtosis are higher moments that need many observations to estimate reliably. A year of monthly data gives noisy, unstable inputs.

2. Confusing it with the modified Sharpe ratio. The adjusted Sharpe applies a polynomial penalty to the plain Sharpe. The modified Sharpe instead swaps standard deviation for a Cornish-Fisher value at risk in the denominator. They are different formulas.

3. Forgetting kurtosis is measured as raw, not excess. The formula uses K minus 3. If your software already reports excess kurtosis, do not subtract 3 again.

4. Trusting it on assets with huge Sharpe ratios. The penalty term scales with the Sharpe squared and the polynomial can behave oddly at very high Sharpe values. Sanity-check the output.

5. Treating the adjustment as a full tail-risk model. Two correction terms cannot capture every distribution. The adjusted Sharpe is a refinement, not a complete picture of extreme risk.

Frequently Asked Questions

What is the skew kurtosis adjusted Sharpe ratio in simple terms? The skew kurtosis adjusted Sharpe ratio is the Sharpe ratio with a penalty for return distributions that have downside-heavy skew or fat tails. Worse tails mean a lower score.

How does the skew kurtosis adjusted Sharpe ratio affect investment decisions? It helps you separate two funds with the same Sharpe ratio when one hides crash risk. The fund with negative skew and fat tails scores lower, steering you toward the safer profile.

What is a real-world example of the skew kurtosis adjusted Sharpe ratio? A strategy that sells options books smooth gains and rare large losses, giving negative skew and fat tails. Its Sharpe of 1.0 might fall to 0.74 once the adjustment penalizes those tails.

How can investors use the skew kurtosis adjusted Sharpe ratio effectively? Use a sample long enough to estimate skew and kurtosis, confirm whether your software reports raw or excess kurtosis, and compare it against the plain Sharpe to see how much tail risk is hidden.

How is the adjusted Sharpe ratio different from the modified Sharpe ratio? The adjusted Sharpe multiplies the plain Sharpe by a skew and kurtosis penalty, while the modified Sharpe replaces volatility with a Cornish-Fisher value at risk in the denominator. Both target non-normal returns differently.

Sources

1. Pezier, J. & White, A. (2006). "The Relative Merits of Investable Hedge Fund Indices and of Funds of Hedge Funds in Optimal Passive Portfolios." ICMA Centre. https://core.ac.uk/download/pdf/6560807.pdf
2. PerformanceAnalytics (CRAN). "Adjusted Sharpe ratio of the return distribution." https://search.r-project.org/CRAN/refmans/PerformanceAnalytics/html/AdjustedSharpeRatio.html
3. PerformanceAnalytics (rdrr.io). "AdjustedSharpeRatio." https://rdrr.io/cran/PerformanceAnalytics/man/AdjustedSharpeRatio.html
4. South African Journal of Economic and Management Sciences. "Establishing the risk denominator in a Sharpe ratio framework." https://sajems.org/index.php/sajems/article/view/3467/2166

Disclaimer

This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.