The Cornish-Fisher quantile in three correction terms, worked on a credit fund, with the grid that shows where it helps and the range where it breaks.
Take the normal quantile z, add three correction terms for skewness and excess kurtosis, and use the adjusted quantile in place of z in the usual VaR formula. On an illustrative 840M fund with a daily volatility of 0.3706 per cent, skew of -0.8 and excess kurtosis of 4 move the 99 per cent quantile from 2.3263 to 3.6089 standard deviations, and the one-day VaR from 7,242,456 to 11,235,320, 55.1 per cent higher. At 95 per cent the same correction adds only 8.2 per cent.
Worked in full in Financial Risk Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The fund is Halverton, the illustrative credit fund of Financial Risk Management, whose normal one-day VaR at 99 per cent is 7,242,456; the variance-covariance method behind that figure is worked in how to calculate parametric VaR for a portfolio. The skewness and kurtosis are illustrative: the kind of shape daily returns on a credit book show, with more large down days than up days and fatter tails than a normal. In practice they are estimated from the fund's own return history.
| Input | Value |
|---|---|
| Net assets | 840,000,000 |
| Daily volatility | 0.3706% |
| One daily standard deviation, in money | 3,113,230 |
| Skewness, S | -0.8 |
| Excess kurtosis, K (kurtosis 7.0) | 4.0 |
| Confidence | 99% |
z = NORM.S.INV(0.01) = −2.3263
Normal VaR = 2.3263 × 3,113,230 = 7,242,456, 0.862 per cent of net assets
zCF = z + (z2 − 1) × S / 6 + (z3 − 3z) × K / 24 − (2z3 − 5z) × S2 / 36
z2 − 1 = 4.4119; z3 − 3z = −5.6109; 2z3 − 5z = −13.5482
Skew term: 4.4119 × (−0.8) / 6 = −0.5883
Kurtosis term: −5.6109 × 4 / 24 = −0.9352
Skew-squared term: −(−13.5482) × 0.64 / 36 = +0.2409
zCF = −2.3263 − 0.5883 − 0.9352 + 0.2409 = −3.6089
Excel: =z+(z^2-1)*S/6+(z^3-3*z)*K/24-(2*z^3-5*z)*S^2/36
Use the excess kurtosis, which is kurtosis minus 3; Excel's KURT() already returns it, and SKEW() returns S.
Modified VaR = −zCF × daily volatility × net assets
= 3.6089 × 3,113,230 = 11,235,320, 1.338 per cent of net assets
In money, the negative skew adds 1,831,366 to the normal VaR, the fat tails add 2,911,340, and the second-order skew term gives back 749,841. The quantile is 1.551 times the normal one. The mean return is left out, as it usually is at a one-day horizon; if you include it, subtract it from the loss as in the normal case.
| Confidence | Normal z | Cornish-Fisher z | Normal VaR | Modified VaR | Uplift |
|---|---|---|---|---|---|
| 95.0% | -1.6449 | -1.7795 | 5,120,807 | 5,540,041 | 8.2% |
| 97.5% | -1.9600 | -2.5202 | 6,101,818 | 7,846,004 | 28.6% |
| 99.0% | -2.3263 | -3.6089 | 7,242,456 | 11,235,320 | 55.1% |
| 99.5% | -2.5758 | -4.5089 | 8,019,149 | 14,037,345 | 75.0% |
The kurtosis term is multiplied by z3 − 3z, which is small near the centre and changes sign at about 1.73 standard deviations. At 95 per cent it is +0.0807: fat tails slightly reduce VaR there, and only the negative skew pushes it up. At 99.5 per cent the kurtosis term is −1.5605 and the uplift is 75.0 per cent. A fund that reports 95 per cent VaR on a normal assumption is barely misstating anything; at 99 per cent the modified figure is more than half as large again as the normal one.
| Skew | Excess kurtosis 0 | 2 | 4 | 8 |
|---|---|---|---|---|
| 0.0 | 7,242,456 (1.000) | 8,698,125 (1.201) | 10,153,795 (1.402) | 13,065,135 (1.804) |
| -0.4 | 7,970,678 (1.101) | 9,426,348 (1.302) | 10,882,018 (1.503) | 13,793,358 (1.905) |
| -0.8 | 8,323,981 (1.149) | 9,779,650 (1.350) | 11,235,320 (1.551) | 14,146,660 (1.953) |
| -1.2 | 8,302,363 (1.146) | 9,758,032 (1.347) | 11,213,702 (1.548) | 14,125,042 (1.950) |
Kurtosis does most of the work: each two points of excess kurtosis add about 1.46M at 99 per cent. Skew helps less than expected, and the bottom row shows why: from -0.8 to -1.2 the VaR falls, because the S2 term, which reduces the quantile, grows faster than the linear skew term. More negative skew giving less risk is a property of the expansion, not of the fund.
Using the formula outside its range. The Cornish-Fisher expansion is a polynomial approximation, and with large skewness or kurtosis it describes no valid distribution: the adjusted quantile stops rising as the confidence rises. With skew of -1.2 and no excess kurtosis, the expansion is already not monotonic over the tail. With skew of -2 and excess kurtosis of 2, the 99 per cent quantile is 2.7592 standard deviations, a VaR of 8,590,036, below what milder skew gives; the 99.5 per cent quantile is 2.8675, and at 99.9 per cent it falls back to 2.7857. A VaR that shrinks as the confidence rises describes no real distribution. Before reporting modified VaR, check that zCF rises steadily as z goes further into the tail. If it does not, use historical simulation, a fitted fat-tailed distribution, or Monte Carlo VaR with Student-t draws.
Sample skewness and kurtosis are noisy, and a single crisis day can move them a long way. A modified VaR that jumps after one bad day is often measuring the estimate, not the fund.
The parametric construction this adjusts, from sleeve volatilities to 7,242,456, is live in the free workbooks for this book.
Normal VaR uses only volatility; modified VaR also uses the skewness and kurtosis of returns through the Cornish-Fisher expansion. On an illustrative fund with a daily volatility of 0.3706 per cent, skew of -0.8 and excess kurtosis of 4, the 99 per cent quantile moves from 2.3263 to 3.6089 standard deviations, and the one-day VaR on 840M rises from 7,242,456 to 11,235,320.
When skewness or kurtosis are large enough that the adjusted quantile stops rising as the confidence level rises, so the implied distribution is not valid. With skew of -2 and excess kurtosis of 2 the quantile is 2.7592 standard deviations at 99 per cent and 2.8675 at 99.5 per cent, then falls back to 2.7857 at 99.9 per cent. Check monotonicity before using the number.
No. In the Cornish-Fisher expansion the kurtosis term is multiplied by z cubed minus 3z, which changes sign near 1.73 standard deviations. With excess kurtosis of 4 it adds 0.9352 standard deviations at 99 per cent but subtracts 0.0807 at 95 per cent. Fat tails move risk into the far tail and out of the shoulders.
This article is one calculation from Financial Risk Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
Get the book on Amazon →Free companion files
Also on Amazon UK · Amazon Germany · Amazon France · Amazon Canada
Reading guide: corporate finance, valuation and markets → · All 1340 articles →
If this book helped, or didn’t, a few lines on Amazon are worth more than they look: they are what the next reader goes on. Write a review. The workbook stays free either way.