Conditional Value at Risk

Also seen as: CVaR, expected shortfall, ES, CVaR 95%, tail loss

How is Conditional Value at Risk calculated?

CVaR averages every loss at or beyond the VaR:

where: $L$ is the strategy's loss on one day and $\text{VaR}_{95\%}$ is the loss that only the worst 5% of days reach. In words: line up every day of the backtest from worst to best, keep the worst 5%, and average their losses. Because it is an average of the days beyond the VaR, it can never be smaller than the VaR itself.

How is CVaR different from VaR?

VaR answers "where do the bad days begin?" and CVaR answers "how bad are they on average?". VaR is blind to what happens past its threshold: a strategy whose worst days are all just past the VaR and one whose worst days include a collapse can share the same VaR. CVaR sees the difference, because the collapse pulls its average down. That sensitivity to the far tail is why CVaR is the measure risk managers prefer when losses have fat tails, and why the Minimum CVaR allocation method uses it as its objective.

How does Fincanva handle it?

• Daily, at 95%, historical. The CVaR in Performance Metrics is read straight off the backtest's daily returns and written as a positive loss: "3.0%" is an average loss of 3.0% on the worst days.
• Performance Metrics shows it in the Worst days · 95% group as Daily CVaR · historical — see what every number in Performance Metrics means.
• The Stress test sets the Daily CVaR 95% of the strategy's historical days beside the same figure under the scenario. When the worst 5% of days shrinks to a single day, the row is marked "· a single day": the VaR and the CVaR are then equal and say little — see stress test.
• A simulation computed before the row existed shows "n/a", never 0%.

What does it look like in practice?

Two strategies both read Daily VaR · historical 2.0%. The first reads Daily CVaR · historical 2.6%: past the threshold its bad days stay close to it. The second reads 4.1%: among its worst days are a few very deep ones, and they drag the average far past the threshold. The VaR alone would call them equally risky on a bad day; the CVaR shows the second one's tail is much heavier.

What counts as a good value?

Lower means shallower bad days. The gap between CVaR and VaR is often more telling than either figure: a CVaR only a little above the VaR means the tail is thin, a CVaR far above it means the worst days hide much deeper ones. Like VaR, it describes the history that was tested — a period with a crash in it carries a higher CVaR than one without — and it is not a cap on what a future day can lose.