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Plate 7 of 19 · Finance

Value at Risk and Expected Shortfall

About 7 minutes to read, then three questions that count toward your mastery.

Value at Risk is the loss a portfolio should exceed only rarely. Expected Shortfall asks how bad those rare days are, and a backtest asks whether the model was right.

The idea

A one-day 99% Value at Risk of $1m means losses should exceed $1m on about one day in a hundred. It is a threshold, not a worst case: it says nothing about how far beyond it the bad days go. Expected Shortfall answers that, as the average loss on the days beyond VaR, which is why bank capital rules now use it, at 97.5%.

There are three common ways to estimate them. Historical simulation replays past returns. The normal (variance–covariance) method uses only the average and volatility, and so understates risk when returns have fat tails. Monte Carlo simulates many outcomes from a fitted model. Longer horizons are often scaled by √time, which assumes one day’s return tells you nothing about the next.

A model is checked by backtesting: count the days its VaR was broken. Kupiec’s test asks whether the count fits the confidence level; Christoffersen’s test asks whether breaks bunch together, the sign of a model slow to notice stormy weather. Banks’ 99% models are graded on 250 days: 0–4 breaks is green, 5–9 yellow, 10 or more red.

The formula

VaRc = zc × σ × √h × V, ESc = φ(zc)1 − c × σ × √h × V

c
confidence level, such as 99%
zc
the normal quantile for c: 2.326 at 99%, 1.645 at 95%
σ
daily volatility of the portfolio’s returns
h, V
horizon in days, and the portfolio’s value
φ
the standard normal density

This is the normal method with an average daily return of zero, the usual simplification over short horizons.

A worked example

Plate 7

A $10m portfolio has a daily volatility of 1.2%. Using the normal method at 99%, what are its one-day VaR and Expected Shortfall, and its ten-day VaR?

  1. One-day VaR: 2.326 × 1.2% × $10m = $279k.
  2. Expected Shortfall: φ(2.326) ÷ 0.01 = 2.665, so 2.665 × 1.2% × $10m = $320k. On the worst 1% of days, the average loss is about 15% beyond VaR.
  3. Ten days by the square-root-of-time rule: $279k × √10 ≈ $279k × 3.16 = $882k.
  4. Backtest: over the last 250 days the loss beat the VaR 6 times, against 2.5 expected. That is the Basel yellow zone; Kupiec’s p-value is 0.06, so the count alone is not quite enough to reject the model at the usual 5% level.

Report $279k as the one-day 99% VaR and $320k as Expected Shortfall. Six breaks is a warning: check whether returns have fatter tails than the normal curve, and whether the breaks came in a cluster.

Try it

The normal method, with an average daily return of zero.

$10m
1.2%
99.0%
1

Size the risk

$279k99.0% one-day VaR
Expected Shortfall
$320k
As a share of the portfolio
2.79% VaR; 3.20% ES
Breaks to expect in 250 days
2.5

Far out in the tail the normal curve is at its weakest: real returns usually have fatter tails, so historical or Monte Carlo estimates often come out higher.

The common mistake

Scaling VaR by the number of days

Multiplying by 10 instead of √10 gives $2.79m for ten days, more than three times the √time answer. Risk grows with the square root of time when returns are independent, because good and bad days partly cancel.

Remember

  • A 99% one-day VaR should be broken on about one day in a hundred; it is not the most you can lose.
  • Expected Shortfall is the average loss beyond VaR. The normal method understates both when tails are fat.
  • Ten-day VaR ≈ one-day VaR × √10. Backtest the count (Kupiec) and the clustering (Christoffersen).

Check yourself

Three questions from the Learn room’s bank. Your answers update your mastery of this skill, the same as in Practise.

Use it on real numbers

Open Value at Risk, in Finance › Risk management and try the idea on your own numbers.