Cornish-Fisher VaR

Variance-covariance VaR is fast because it only needs a mean and a standard deviation, but it assumes those two numbers fully describe the risk, which is exactly false whenever a portfolio has meaningful skew, an option-selling strategy with limited upside and occasional large downside, say, or fat tails beyond what a normal curve predicts. Rerunning the whole risk engine as a full Monte Carlo or historical simulation fixes that, but costs speed. Cornish-Fisher VaR offers a shortcut: keep the fast formula, but adjust the quantile itself using the data's actual skewness and kurtosis.

The analogy before any symbols

A tailor with only a client's height and weight can guess a reasonable suit size, using standard proportions built into any pattern book. But if the client is unusually broad-shouldered or long-armed, that standard-proportions guess will be visibly off. A skilled tailor does not remeasure the client from scratch; she takes two extra numbers, shoulder width and arm length, and nudges the standard pattern to fit. Cornish-Fisher does the same to the normal-distribution quantile: instead of remeasuring the whole loss distribution, it takes two extra numbers, skewness and kurtosis, and nudges the standard normal quantile to fit the actual shape better.

The mechanics

Start from the standard normal quantile $z_\alpha$ used in ordinary variance-covariance VaR ($-1.645$ at 95%, $-2.326$ at 99%). The Cornish-Fisher expansion replaces it with an adjusted quantile $z_{CF}$:

$z_{CF} = z_\alpha + \frac{1}{6}\big(z_\alpha^2 - 1\big)S + \frac{1}{24}\big(z_\alpha^3 - 3z_\alpha\big)K - \frac{1}{36}\big(2z_\alpha^3 - 5z_\alpha\big)S^2,$

where $S$ is the sample skewness and $K$ is the sample excess kurtosis (kurtosis in excess of the normal distribution's value of 3). In words: start from the ordinary normal quantile, then add a correction for skew (pulls the quantile further left if losses are skewed negative) and a correction for excess kurtosis (pulls it further left again if tails are fatter than normal), with a small final term that corrects for skew and kurtosis interacting. VaR is then computed exactly as before, just with $z_{CF}$ in place of $z_\alpha$:

$\text{VaR}_\alpha = -\big(\mu + z_{CF}\,\sigma\big).$

In words: same formula shape as plain variance-covariance VaR, but the "how many standard deviations out" number now reflects the actual lopsidedness and tail-heaviness of the return distribution, not just its width.

Picture this normal curve with its left side pulled down and out, more mass pushed into the tail, without changing its center or width, that stretched left tail is what negative skewness and excess kurtosis are doing to $z_{CF}$ relative to the plain $z_\alpha$.

Worked example: a book with negative skew

A short-volatility strategy has monthly returns with mean 0.5%, standard deviation 3%, skewness $S = -1.2$, excess kurtosis $K = 4.5$ (much fatter tails than normal). At 99% confidence, $z_{0.99} = -2.326$. Plain normal VaR: $-(0.5\% + (-2.326)(3\%)) = -(0.5\% - 6.978\%) = 6.478\%$ of capital. Now compute $z_{CF}$: the skew term is $\tfrac16\big((-2.326)^2-1\big)(-1.2) = \tfrac16(4.410)(-1.2) = -0.882$; the kurtosis term is $\tfrac1{24}\big((-2.326)^3-3(-2.326)\big)(4.5) = \tfrac1{24}(-12.59+6.978)(4.5) = \tfrac1{24}(-5.612)(4.5) \approx -1.052$; the cross term is small, roughly $-0.15$. Summing: $z_{CF} \approx -2.326 - 0.882 - 1.052 - 0.15 = -4.41$. Cornish-Fisher VaR: $-(0.5\% + (-4.41)(3\%)) = -(0.5\% - 13.23\%) = 12.73\%$ of capital, roughly double the plain normal estimate, purely from accounting for this strategy's known lopsided, fat-tailed payoff shape.

Worked example: a nearly-normal book barely changes

A diversified long-only equity book has skewness $S = -0.15$ and excess kurtosis $K = 0.4$, close to normal. At 95% confidence, $z_{0.95} = -1.645$: skew term $\tfrac16((-1.645)^2-1)(-0.15) = \tfrac16(1.706)(-0.15) \approx -0.043$; kurtosis term $\tfrac1{24}((-1.645)^3-3(-1.645))(0.4) = \tfrac1{24}(-4.45+4.935)(0.4) \approx 0.008$. The adjustment is tiny, $z_{CF} \approx -1.645 - 0.043 + 0.008 = -1.68$, moving VaR by only about 2%. Cornish-Fisher barely matters when a portfolio's return distribution is already close to normal, exactly as it should.

What this means in practice

Cornish-Fisher is popular in hedge fund and derivatives risk reporting because it needs only the first four moments of the historical return series, no simulation infrastructure, and slots directly into an existing variance-covariance pipeline. It is most valuable for exactly the books that break the normal assumption hardest: option sellers, merger arbitrage, credit strategies, anything with a "steady gains, occasional large loss" payoff shape.