How does the Cornish-Fisher expansion adjust VaR for non-normality, and when should you use it?

The Cornish-Fisher (CF) expansion is a pragmatic adjustment to the normal VaR that accounts for skewness and excess kurtosis without requiring a full distributional fit. It's a middle ground between naive normal VaR and computationally intensive methods like EVT.

The Formula:

z_CF = z + (z^2 - 1)/6 x S + (z^3 - 3z)/24 x K - (2z^3 - 5z)/36 x S^2

Where:

• z = standard normal quantile (e.g., -2.326 for 99% VaR)
• S = skewness of the return distribution
• K = excess kurtosis of the return distribution

Then: VaR_CF = mu - z_CF x sigma

Worked Example — Havenbrook Capital equity portfolio:

Daily returns: mu = 0.03%, sigma = 1.2%, S = -0.7 (negative skew), K = 4.5 (fat tails)

Standard 99% VaR (normal):

VaR = 0.03% - (-2.326) x 1.2% = 0.03% + 2.79% = 2.82%

Cornish-Fisher adjustment:

z = -2.326

z^2 = 5.410, z^3 = -12.584

Term 1 (skewness): (5.410 - 1)/6 x (-0.7) = 4.410/6 x (-0.7) = -0.5145

Term 2 (kurtosis): (-12.584 + 6.978)/24 x 4.5 = -5.606/24 x 4.5 = -1.051

Term 3 (skew^2): -(2(-12.584) - 5(-2.326))/36 x 0.49 = -(-25.168 + 11.630)/36 x 0.49 = -(-13.538/36) x 0.49 = 0.184

z_CF = -2.326 - 0.5145 - 1.051 + 0.184 = -3.708

VaR_CF = 0.03% - (-3.708) x 1.2% = 0.03% + 4.45% = 4.48%

The CF adjustment increases VaR from 2.82% to 4.48% — a 59% increase due to fat tails and negative skewness.

When CF Works Well:

• Moderate departures from normality (|S| < 2, K < 8)
• Confidence levels up to 99.5%
• As a quick analytical adjustment to parametric VaR
• When full distributional fitting is impractical

When CF Breaks Down:

• Extreme kurtosis (K > 10): Higher-order terms become important
• Very high confidence levels (99.9%+): The expansion diverges
• Bimodal distributions: CF assumes unimodal
• Can produce non-monotonic quantile functions (VaR increases then decreases with confidence)

FRM Exam Tips:

• Know the formula and be able to compute z_CF with given S and K
• Negative skewness makes z_CF more negative (larger VaR)
• Positive excess kurtosis also makes z_CF more negative (fatter tails)
• CF is a quick shortcut — for serious tail risk, use EVT

Practice Cornish-Fisher calculations in our FRM Part II question bank.