Historical Value at Risk (VaR)
Historical Value at Risk (VaR)
The loss that was only exceeded (1 − confidence) of the time in the observed returns: sort the returns from worst to best, take the one at position floor((1 − c) × n), and report its loss as a positive fraction. No distribution is assumed; the history speaks for itself.
When to use: Use with at least a few dozen periods of returns to summarise downside risk in one number: "on a bad 1-in-20 period we lose at least X%". Its weakness is that it can only report losses as bad as the sample contains.
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| VaR | Value at Risk | Loss at the chosen confidence, as a positive fraction | % |
| Returns | Periodic Returns | Sequence of periodic returns as decimals (e.g. 0.05 for 5%) | % |
| Confidence | Confidence Level | Probability the loss will not be exceeded, as a decimal (0.95 for 95%) | % |
Real-Life Examples
Example 1: 95% VaR over 20 Months
Twenty monthly returns, ranging from −8.3% to +5.1%. What is the 95% VaR?
Given
Step-by-Step
In 19 of 20 months the loss was no worse than 6.1%; only one month (the −8.3%) was worse. That single worse month is the tail VaR does not describe.
Example 2: 90% VaR on the Same Data
The same twenty monthly returns at 90% confidence.
Given
Step-by-Step
Lowering the confidence to 90% moves the cutoff two returns in from the worst, so the VaR falls to 4.4%. VaR always rises with confidence.
Frequently Asked Questions
Several quantile conventions exist and they differ slightly on small samples. SnapFi uses the simplest, the sorted return at position floor((1 − c) × n), and states it so the number can be reproduced by hand.
How bad the losses beyond the cutoff are. Two portfolios can share a 5% VaR while one has a worst month of −6% and the other −40%. Expected shortfall addresses that; VaR does not.