What it means
CVaR builds on value at risk, which states a loss level that will not be exceeded with a given probability over a given period. The weakness of value at risk is that it says nothing about what happens beyond that point, so two portfolios with the same value at risk can have very different disaster potential.
CVaR fixes this by averaging all the losses worse than the value at risk threshold. If the value at risk at 95% confidence is a loss of $420,000, the CVaR is the average of every outcome in that worst 5%, which will always be equal to or larger than the value at risk figure.
Businesses and regulators care because tail risk is what causes failures. Banking regulation shifted towards expected shortfall precisely because averaging the tail discourages strategies that look safe on a value at risk measure while carrying rare but catastrophic losses.
Calculating it requires a distribution of possible outcomes, usually from historical returns or from a simulation of thousands of scenarios. The outcomes are sorted from worst to best, the cut off for the chosen confidence level is identified, and the losses beyond it are averaged.
The nuance worth remembering is that CVaR is only as good as the scenarios behind it. If the historical window contains no serious crisis, or the simulation assumes returns are more well behaved than they really are, the tail will be understated no matter how sophisticated the arithmetic looks.
In practice
Real-world examples.
Example
An asset manager compares two funds that both show a 95% value at risk of $1.5m. The CVaR figures are $1.8m and $3.4m, revealing that the second fund holds positions with rare but severe downside, and the investment committee reduces its allocation to that fund.
Example
A commodities trading firm sets its overnight position limits using CVaR rather than value at risk, because a single adverse weather event can move prices far beyond the ordinary daily range. The change cuts the maximum permitted position in one volatile contract by roughly a third.
Example
An insurer uses CVaR at 99% to size the catastrophe reinsurance it buys. Averaging the worst 1% of modelled loss years gives an expected shortfall of $86m, so cover is purchased above the point where the balance sheet could absorb the hit.
Think of it
“CVaR is the abbreviation for Conditional Value at Risk-average loss in the worst cases.
Formula
Calculation
CVaR at confidence level c = Average of all losses that exceed the value at risk at confidence level c
A treasury team runs 100 equally likely one day scenarios on a $10,000,000 portfolio and sorts the results from worst to best. At 95% confidence, the tail is the worst 5 outcomes:
Loss 1: $660,000
Loss 2: $560,000
Loss 3: $500,000
Loss 4: $460,000
Loss 5: $420,000
Value at risk at 95% = $420,000, the smallest loss in the tail, which is 4.2% of the portfolio.
Sum of tail losses = $660,000 + $560,000 + $500,000 + $460,000 + $420,000 = $2,600,000
CVaR at 95% = $2,600,000 / 5 = $520,000
So the portfolio's expected shortfall is $520,000, or 5.2% of value, which is $100,000 worse than the value at risk figure alone would suggest. The treasurer sets the daily risk limit against the $520,000 number because it reflects what a bad day actually costs.Case study
Seen in the real world.
This is an illustrative and fictional case. Alderway Capital, an invented boutique investment firm, ran a strategy that sold insurance style option contracts and collected steady premiums. Its 95% value at risk was reassuringly small at around $900,000 on a $60m book, and the risk report had been green for eleven straight quarters.
A new risk officer recalculated the same book using CVaR and found expected shortfall at 95% of $4.1m, more than four times the value at risk. The reason was that the strategy produced many small gains and a handful of severe losses, exactly the shape that value at risk is blind to. The fictional board initially resisted the finding because nothing had actually gone wrong.
Alderway compromised by keeping the strategy but adding a limit expressed in CVaR terms and buying protective options on the far tail, at a cost of roughly 8% of the strategy's expected annual return. When a sharp market move arrived the following year, the realised loss was around $3.6m instead of the $12m the unhedged position would have produced, which in this illustrative story was the difference between a bad quarter and a closed fund.
Watch out
Common mistakes.
- Reading CVaR as a worst case loss. It is the average of the tail, not the maximum, and losses well beyond the CVaR figure remain entirely possible.
- Using a short or calm historical window. A dataset that excludes any real stress will produce a comfortable tail estimate that has never been tested against a crisis.
- Comparing CVaR numbers calculated at different confidence levels or horizons. A 99% ten day figure and a 95% one day figure are not comparable, so the parameters must be stated every time.
Questions
People also ask.
How is CVaR different from value at risk?
Value at risk is the threshold where the worst outcomes begin, while CVaR is the average size of the losses once that threshold is crossed.
Is CVaR always larger than value at risk?
Yes, for the same confidence level and horizon, because it averages outcomes that are all at least as bad as the value at risk point.
Do smaller businesses need it?
Rarely in the formal sense, though the underlying question of how bad a bad outcome really gets is useful for anyone managing currency exposure, commodity prices or concentrated credit risk.
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