Modified Sharpe: Cornish-Fisher VaR as Risk

Key Takeaways

• The modified Sharpe Cornish-Fisher VaR ratio divides excess return by a modified value at risk.
• The modified VaR uses a Cornish-Fisher expansion to adjust for skewness and excess kurtosis.
• It was introduced by Favre and Galeano in 2002 for funds with non-normal returns.
• For perfectly normal returns, the modified VaR collapses to standard VaR and the ratio behaves like the Sharpe.

What the Modified Sharpe Cornish-Fisher VaR Is

The modified Sharpe ratio was proposed by Laurent Favre and Jose-Antonio Galeano in 2002. It keeps the structure of the Sharpe ratio, return per unit of risk, but changes the definition of risk in the denominator.

Instead of standard deviation, it uses a modified value at risk, often written MVaR. Value at risk estimates the loss a portfolio is unlikely to exceed at a chosen confidence level, such as 95% or 99%. The modified version corrects that estimate for the actual shape of returns using the Cornish-Fisher expansion, so it captures the heavier downside that fat-tailed, skewed assets carry.

The Intuition

Standard deviation treats upside and downside swings as equally risky and assumes returns follow a bell curve. For hedge funds, options strategies, and credit portfolios, that assumption breaks down. Their returns often lean negative and pile up in the tails, where rare large losses live.

A plain Sharpe ratio understates the danger of these strategies because volatility alone misses the tail. The modified Sharpe puts a tail-aware risk measure in the denominator. By using a value at risk that has been stretched to reflect negative skew and excess kurtosis, it asks a sharper question: how much excess return do you earn for each unit of plausible worst-case loss?

How It Works

The modified Sharpe ratio divides excess return by the modified value at risk:

Modified Sharpe = (mean return - risk-free rate) / MVaR

The modified value at risk uses the Cornish-Fisher expansion to adjust the normal quantile for skewness and kurtosis:

z_cf = z + (z^2 - 1)*S/6 + (z^3 - 3z)*K/24 - (2z^3 - 5z)*S^2/36
MVaR = -( mean + z_cf * sigma )

Where:

z     = the standard normal quantile at the confidence level (for example -1.645 at 95%)
S     = skewness of returns
K     = excess kurtosis (kurtosis minus 3)
sigma = standard deviation of returns

When skewness and excess kurtosis are both zero, z_cf equals z and the modified VaR reduces to ordinary parametric VaR. As negative skew and fat tails grow, z_cf pushes the quantile further into the loss region, raising MVaR and lowering the ratio. The expansion is built on the first four moments of the distribution.

Worked Example

Suppose a fund has a monthly mean return of 1%, standard deviation of 4%, skewness of -0.5, excess kurtosis of 2, and a risk-free rate of 0.2%. Use a 95% confidence level, so z = -1.645.

Cornish-Fisher quantile: z_cf = -1.645 + ((-1.645)^2 - 1)(-0.5)/6 + ((-1.645)^3 - 3(-1.645))2/24 - (2(-1.645)^3 - 5*(-1.645))*(-0.5)^2/36

Working the terms gives roughly z_cf = -1.94 after the skew and kurtosis adjustments deepen the quantile.

MVaR = -(0.01 + (-1.94)*0.04) = -(0.01 - 0.0776) = 0.0676, or about 6.76%.

Modified Sharpe = (0.01 - 0.002) / 0.0676 = 0.008 / 0.0676 = 0.118

A plain Sharpe using the 4% standard deviation would be 0.008 / 0.04 = 0.20. The tail adjustment cut the score nearly in half, exposing the downside the Sharpe ratio glossed over.

Common Mistakes

1. Feeding it raw kurtosis. The Cornish-Fisher expansion uses excess kurtosis, which is kurtosis minus 3. Forgetting to subtract 3 corrupts the quantile.

2. Using it on near-normal data. When skew and excess kurtosis are tiny, the modified Sharpe barely differs from the plain Sharpe. The extra estimation noise is not worth it for symmetric assets.

3. Trusting it with extreme moments. The Cornish-Fisher expansion can misbehave when skewness and kurtosis are very large, sometimes producing a non-monotonic quantile. Check that the implied VaR is sensible.

4. Confusing modified VaR with conditional VaR. Modified VaR estimates the loss threshold at a confidence level. Conditional VaR, or expected shortfall, averages the losses beyond that threshold. They are different risk measures.

5. Ignoring the confidence level. A 95% and a 99% modified Sharpe are not comparable. Always state the confidence level alongside the figure.

Frequently Asked Questions

What is the modified Sharpe Cornish-Fisher VaR ratio in simple terms? The modified Sharpe Cornish-Fisher VaR ratio is the Sharpe ratio with tail-aware value at risk in place of standard deviation. It charges strategies for their likely worst losses, not just volatility.

How does the modified Sharpe Cornish-Fisher VaR ratio affect investment decisions? It lowers the score of funds with negative skew and fat tails, so a smooth-looking strategy that hides crash risk ranks worse. That helps you avoid funds whose plain Sharpe flatters them.

What is a real-world example of the modified Sharpe Cornish-Fisher VaR ratio? A fund with a 0.20 Sharpe but negative skew and fat tails can fall to a modified Sharpe near 0.12 once the Cornish-Fisher adjustment widens its value at risk.

How can investors use the modified Sharpe Cornish-Fisher VaR ratio effectively? Use excess kurtosis in the expansion, state the confidence level, and apply it to non-normal assets like hedge funds or option strategies where the adjustment actually matters.

How is the modified Sharpe different from the adjusted Sharpe ratio? The modified Sharpe Cornish-Fisher VaR ratio swaps standard deviation for a tail-aware value at risk in the denominator. The adjusted Sharpe instead multiplies the plain Sharpe by a skew and kurtosis penalty.

Sources

1. Favre, L. & Galeano, J.-A. (2002). "Mean-Modified Value-at-Risk Optimization with Hedge Funds." Journal of Alternative Investments. https://www.researchgate.net/publication/255594354_Mean-Modified_Value-at-Risk_Optimization_with_Hedge_Funds
2. ABC Quant. "Modified Value-at-Risk (Cornish-Fisher MVaR)." https://www.abcquant.com/resources/knowledge-base/item/152-modified-value-at-risk
3. PerformanceAnalytics. "VaR: calculate various Value at Risk (VaR) measures." RDocumentation. https://www.rdocumentation.org/packages/PerformanceAnalytics/versions/2.0.4/topics/VaR
4. PerformanceAnalytics (Braverock). "VaR.CornishFisher." http://www.braverock.com/brian/R/PerformanceAnalytics/html/VaR.CornishFisher.html

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.