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Key Takeaways
- At confidence c, expected shortfall averages the worst 1 − c fraction of losses, including the required share of outcomes tied at the VaR boundary.
- CVaR is always at least as large as VaR at the same confidence level, because it averages the deep-tail losses that VaR only bounds.
- VaR gives the selected loss quantile. Expected shortfall gives the mean of the worst selected fraction.
- Expected shortfall is subadditive: the combined figure is at most the sum of the separate figures at the same confidence level.
- Sort scenarios by loss, retain the worst required probability mass, and divide their weighted loss by that mass.
What is conditional value at risk (CVaR)?
Conditional value at risk (CVaR), also called expected shortfall, is the average loss in the worst selected fraction of a loss distribution. At confidence c, it averages the worst 1 − c fraction of outcomes, so 95% CVaR is the mean loss across the worst 5%. Value at risk (VaR) identifies the boundary of that tail, and expected shortfall measures the average loss within it.
How is CVaR calculated?
Let L be loss and q be VaR at confidence c. Expected shortfall is q + E[max(L − q, 0)] / (1 − c). This formula assigns the correct probability weight at the boundary.
For a normal portfolio return with mean mu and standard deviation sigma, dollar expected shortfall is V × [sigma × phi(z) / (1 − c) − mu], where z = Phi⁻¹(c).
Formula
ES(c) = q + E[max(L − q, 0)] / (1 − c), where q = VaR(c). For normal return R with mean μ and standard deviation σ: ES(c) = V × [σ × φ(z) / (1 − c) − μ], z = Φ⁻¹(c).- L is loss in currency units and c is between 0 and 1. q is the smallest loss threshold with cumulative probability at least c.
- Phi is the standard normal cumulative function and phi is its density. z is the c quantile, 1.6449 to four decimals for c = 95%.
- The general formula handles probability tied at the threshold. The normal-return formula converts the tail return average into currency using portfolio value V.
CVaR example: normal returns at 95% confidence
A $1,000,000 portfolio has normally distributed monthly return with mean 0% and standard deviation 5 percentage points. Calculate 95% monthly VaR and expected shortfall.
- 95% VaR = 1.644854 × 5% × $1,000,000 = $82,243
- Normal expected-shortfall multiplier = 0.1031356 / 0.05 = 2.06271
- 95% expected shortfall = 2.06271 × 5% × $1,000,000 = $103,136
The 95th-percentile loss is $82,243. The worst 5% of monthly outcomes average a $103,136 loss, which is $20,893 above the threshold.
The difference is $20,893. The chart locates the −8.22% return threshold and the −10.31% mean return within that tail.
CVaR from scenarios with tied losses
Sort scenario losses from largest to smallest and retain the probability weight needed to fill the tail. If the boundary contains tied losses, use the share needed to complete that weight.
Consider 100 equally likely outcomes: 96 losses of $0 and four losses of $1,000. The 95% VaR is $0. The five worst outcomes contain four $1,000 losses and one $0 loss, so 95% expected shortfall is $800.
Averaging all outcomes at or above $0 gives $40. Averaging only outcomes strictly above $0 gives $1,000. Neither operation selects the required five outcomes in this example.
CVaR in a portfolio: subadditivity and risk allocation
Expected shortfall is subadditive: the figure for combined losses is at most the sum of the separate figures. This is an inequality, not a rule that separate expected-shortfall figures add exactly to portfolio expected shortfall.
Use risk contributions calculated from the combined portfolio when allocating its total. The Basel market-risk internal-models framework uses expected shortfall at a one-tailed 97.5% confidence level.
How traders use CVaR for position sizing
Reprice each position in the selected market scenarios, aggregate portfolio losses, and calculate the selected tail average. Compare that result with the portfolio’s risk budget before resizing the trade. For option positions, the Options Profit Calculator reprices each structure across price, date and volatility, so every tail scenario comes with the structure’s actual dollar loss.
Limitations of CVaR
- Fit the loss distribution to the real tails. A normal-return assumption understates tail losses when real returns have fat tails, and historical scenarios contain only the moves that happened in the sample.
- Use enough scenarios to fill the tail. With 100 equally likely scenarios at 95% confidence, five outcomes fill the tail, so a single scenario can move the figure sharply.
- The horizon, confidence level and repricing method all change the number, so figures compare only on the same settings.
- Size for losses beyond the average. CVaR is the tail average, not the worst case, and individual losses in the tail can be larger than it.
- Stress trading and liquidity during the horizon separately. A scenario set usually holds positions fixed over the horizon, so it leaves out trades made during that horizon and liquidity that can change during stress.
Bottom line
Expected shortfall answers the question VaR leaves open: how large losses are, on average, once the tail is reached. Use it to compare tail severity across portfolios on the same horizon and confidence level, and to budget risk from scenarios that reprice every position.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- CFA Institute: Measuring and Managing Market Risk
- Federal Reserve: Financial Stability Monitoring (FEDS Note)
How entries are sourced, calculated and updated: editorial methodology.