What VaR and CVaR measure
Value at Risk (VaR) at 95% is the loss you should not exceed on 95% of days (or whatever period you use). It is a threshold: it says nothing about how bad the other 5% can get.
Conditional Value at Risk (CVaR), also called Expected Shortfall, is the average loss on the days worse than VaR. It answers the question VaR ignores: when things go wrong, how wrong on average? CVaR is always at least as large as VaR.
CVaR formula
For a confidence level α (for example 0.95) and a loss L:
VaRα = the α-quantile of L
CVaRα = E[ L | L ≥ VaRα ]
If returns are normal with mean μ and volatility σ, both have closed forms, where zα is the standard normal quantile and φ the normal density:
VaRα = zα σ − μ
CVaRα = σ φ(zα) / (1 − α) − μ
At 95%, z = 1.645 and φ(z) / (1 − α) = 2.063, so under normality CVaR is about 25% larger than VaR. At 99% the factors are 2.326 and 2.665.
How to calculate CVaR
- Historical method. Sort your past returns from worst to best and take the worst (1 − α) share: with 1,000 daily returns at 95%, the worst 50. VaR is the smallest loss in that group; CVaR is their average. No distribution assumption, but it only knows the past.
- Parametric method. Estimate μ and σ, then apply the formulas above. Fast and smooth, but the normal distribution understates fat tails, so real-world CVaR is usually higher.
- Monte Carlo method. Simulate thousands of scenarios from a model (with correlations, fat tails or regime switches), then apply the historical method to the simulated losses. See our guide to Monte Carlo simulation for portfolios.
This calculator uses the convention that the tail holds the worst ⌊n(1 − α)⌋ observations (at least one); VaR is the smallest loss in that tail and CVaR its average. Other conventions interpolate between observations and give slightly different numbers on small samples.
VaR vs CVaR
| VaR | CVaR (Expected Shortfall) | |
|---|---|---|
| Question answered | What loss is exceeded only (1 − α) of the time? | What is the average loss when that happens? |
| Sees tail severity | No | Yes |
| Coherent risk measure | No (can penalise diversification) | Yes (subadditive) |
| Optimisation | Non-convex, hard to optimise | Convex, solvable as a linear programme |
| Regulation | Basel II market risk | Basel III / FRTB market risk (97.5% ES) |
Because CVaR is convex, portfolios can be built to minimise it directly. That is the idea behind mean-CVaR optimisation, covered in The CVaR Approach to Portfolio Optimization, and available in our portfolio optimizer.
Frequently asked questions
What does CVaR stand for?
Conditional Value at Risk. It is the same measure as Expected Shortfall (ES) and is sometimes called Average Value at Risk or Tail VaR.
Is a higher CVaR better or worse?
Worse. CVaR is a loss, so a higher number means a heavier expected loss in bad periods.
Why is my historical CVaR much higher than the parametric one?
Real returns have fatter tails than the normal distribution. A few crash days pull the historical tail average down far more than a normal model expects, and that gap is a useful warning sign in itself.
Which confidence level should I use?
95% is common for portfolio analysis, 99% for daily trading risk limits, and 97.5% is the level used for Expected Shortfall under the Basel FRTB rules.