5. Percentiles, quantiles and Value at Risk

Level 2: Distributions and why markets are not "normal"

Two portfolios with the same Value at Risk. The exact same figure on the risk report.

In its worst case, one loses 30,000 euros. The other, 400,000.

The most frequently cited measure in risk management is, by design, incapable of telling them apart. And understanding why starts with understanding what a percentile is.

Cut-off points: percentiles, quantiles, quartiles

A percentile is a cut-off point inside a distribution. The p-th percentile is the value below which exactly p percent of the observations lie.

Two concrete readings:

• The 50th percentile is what you already know as the median: the value leaving half the data below and half above.
• The 10th percentile of a stock's daily returns is the value such that 10% of historical days had a return equal to or worse than it, while the remaining 90% did better.

Quantiles are simply the same idea expressed at a different resolution:

• Quartiles split the data into four equal parts: the 25th, 50th and 75th percentiles.
• Deciles split it into ten equal parts: the 10th, 20th, 30th percentiles and so on.

Percentile, quantile and quartile are, at bottom, the same cut-off point tool applied at different levels of granularity.

How an empirical percentile is computed

Computing an empirical percentile is straightforward if you have the historical data. You sort all the observed returns from lowest to highest and find the value that leaves p percent of the observations below it.

For example, if you have 1,000 daily returns sorted from worst to best, the 5th percentile is roughly the 50th return on that sorted list, because 50 is 5% of 1,000.

If that value turns out to be -2.3%, you know two things: 5% of historical days had a loss of 2.3% or worse, and the remaining 95% did better than that.

VaR is a percentile

This percentile idea is exactly what lies behind Value at Risk, or VaR. It is one of the most frequently cited risk management concepts in finance, and one you probably already heard mentioned in the market theory course without going into its exact statistical definition.

The one-day 95% VaR of a portfolio is, quite simply, the 5th percentile of that portfolio's one-day loss distribution. That is, the loss threshold that is only expected to be exceeded — for the actual loss to be even worse — on 5% of days.

Put numbers on it. If the 95% VaR of a one million euro portfolio is 23,000 euros, the correct reading is this: 'under normal market conditions, the portfolio is not expected to lose more than 23,000 euros in a day on 95% of occasions'.

Put another way: that loss is expected to be exceeded, on average, on one day out of every twenty.

Historical or parametric

There are two common ways of computing VaR. It is worth telling them apart, because they do not always give the same answer.

Historical VaR is computed exactly as in the example above. You sort the actual returns, or losses, observed in the past and take the corresponding percentile directly from that data, without assuming any theoretical distribution.

Parametric VaR, by contrast, assumes returns follow a specific distribution — almost always the normal — and computes the percentile using that theoretical distribution's formula.

Under normality, the 5th percentile on the left sits roughly 1.65 standard deviations below the mean.

Take a portfolio with a daily mean of 0.03% and a daily standard deviation of 1.4%. Its 95% parametric VaR would be roughly 0.03% - 1.65 × 1.4% = -2.28%. That is, an expected loss of 2.28% in the worst 5% of days.

Fat tails and an optimistic VaR

This is where the link with module 3 becomes direct.

If real returns have fatter tails than the normal, parametric VaR computed under the normality assumption will systematically underestimate the true risk of extreme losses. It is built on a curve that assigns less probability to extreme events than actually exists.

Historical VaR, by using the observed data directly, captures that fat-tailed reality better. But with one condition: provided the historical sample used has included relevant stress periods.

If you only use data from a calm period, historical VaR will mislead you too, simply for lack of extreme events in the sample.

What VaR does not see

VaR also has a structural limitation worth keeping in mind at all times. It tells you the loss threshold at a given percentile, but it tells you absolutely nothing about how bad the loss can get once that threshold is breached.

Return to the two portfolios from the opening. Both have exactly the same 95% VaR, 23,000 euros. And yet:

• One of them may lose 'only' 30,000 euros in its worst case within that remaining 5%.
• The other may lose 400,000 euros in a truly extreme tail event.

VaR is, by design, blind to that difference. It only looks at the cut-off point, not at what lies beyond it.

Expected shortfall: looking inside the tail

To address that limitation there is a related measure called expected shortfall, or conditional expected loss, sometimes also called CVaR.

Instead of giving only the threshold, it computes the average of the losses inside that worst tranche: for example, the average of all losses between the 0th and the 5th percentile.

This measure does distinguish between the two portfolios in the previous section, because it does not stop at the cut-off point but looks inside the tail.

You do not need to master its calculation in detail at this level. But you should know it exists precisely because it answers the question VaR leaves hanging: 'fine, there is a 5% probability of losing more than X, but if that happens, how much could I really end up losing?'.