How do you compute parametric VaR when returns are non-normal? Is there a Cornish-Fisher adjustment?

Question

AcadiFi · Accepted Answer

The Cornish-Fisher expansion modifies the standard normal quantile to account for skewness and excess kurtosis, producing a more accurate VaR estimate without abandoning the parametric approach entirely.

## Standard Parametric VaR (Normal)

**VaR_alpha = -(mu + z_alpha x sigma)**

At 99% confidence: z_0.01 = -2.326
VaR_99% = -(mu - 2.326 x sigma)

This assumes returns are perfectly normal — zero skewness, kurtosis = 3.

## Cornish-Fisher Adjusted Quantile

Replace z_alpha with an adjusted quantile z_CF:

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

Where:
- z = standard normal quantile (e.g., -2.326 for 99%)
- S = skewness of the return distribution
- K = excess kurtosis (kurtosis - 3)

Then: **VaR_CF = -(mu + z_CF x sigma)**

## Worked Example

Crestmont Capital estimates the following for their portfolio daily returns:
- Mean (mu) = 0.03%
- Std dev (sigma) = 1.50%
- Skewness (S) = -0.60 (negative, left tail is heavier)
- Excess kurtosis (K) = 3.50 (fat tails)

**Standard normal VaR (99%):**
VaR_normal = -(0.03% + (-2.326) x 1.50%) = -(0.03% - 3.489%) = **3.46%**

**Cornish-Fisher adjusted z:**
z = -2.326
z_CF = -2.326 + ((-2.326)^2 - 1) x (-0.60)/6 + ((-2.326)^3 - 3(-2.326)) x 3.50/24 - (2(-2.326)^3 - 5(-2.326)) x (-0.60)^2/36

Term 1: (-2.326)
Term 2: (5.41 - 1) x (-0.10) = -0.441
Term 3: (-12.58 + 6.978) x 0.1458 = -5.602 x 0.1458 = -0.817
Term 4: -(2(-12.58) - 5(-2.326)) x 0.36/36 = -(-25.16 + 11.63) x 0.01 = -(-13.53) x 0.01 = +0.135

z_CF = -2.326 - 0.441 - 0.817 + 0.135 = **-3.449**

**Cornish-Fisher VaR (99%):**
VaR_CF = -(0.03% + (-3.449) x 1.50%) = -(0.03% - 5.174%) = **5.14%**

**The CF adjustment increases the VaR from 3.46% to 5.14% — a 49% increase!** This shows how much the normal assumption underestimates risk when returns are negatively skewed with fat tails.

```mermaid
flowchart LR
    NORMAL["Normal VaR
z = -2.326
VaR = 3.46%"] --> CF["Cornish-Fisher
Adjusted z = -3.449
VaR = 5.14%"]
    CF --> DIFF["49% Higher
Captures skew + kurtosis"]
    NEG_SKEW["Negative Skew
S = -0.60"] --> CF
    FAT_TAILS["Excess Kurtosis
K = 3.50"] --> CF
```

## When to Use Cornish-Fisher

- Quick parametric adjustment when you have skewness/kurtosis estimates
- More accurate than normal VaR but less computationally expensive than full simulation
- Works best when departures from normality are moderate; extreme non-normality may require EVT or simulation

Practice VaR methodology in our FRM question bank.

How do you compute parametric VaR when returns are non-normal? Is there a Cornish-Fisher adjustment?

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