Why a normal distribution understates your VaR
1 min readRiskModels
A 99% VaR computed under normality and under a Student-t with four degrees of freedom can differ by more than a percentage point of the portfolio. That gap is a policy decision, not a detail.
Most risk reports still assume returns are normal, because the formula is short and the backtest usually passes. The problem shows up in the tail, exactly where the number is supposed to be useful.
The size of the gap
At 20% annual volatility and a ten-day horizon, the 99% VaR is 9.27% of the portfolio under a normal distribution and 10.56% under a Student-t with four degrees of freedom rescaled to the same variance. The Expected Shortfall gap is wider still.
In Expected Shortfall, the average loss once the threshold is crossed, the gap is wider: 10.62% under normality against 14.71% under the Student-t. These figures come from a theoretical example with the same assumptions, not from a real portfolio, but the order of magnitude shows up in practice whenever returns have heavy tails.
What to do about it
- State the distributional assumption in the methodology document, not in a footnote of the code.
- Report Expected Shortfall next to VaR: it is the number that describes what happens once the threshold is crossed.
- Backtest with exception counts and a green/amber/red reading, and record the result with every model release.
You can move the parameters yourself in the simulator on the Model Pool page.