Expected Shortfall (CVaR) - Measuring the Depth of the Tail

By EC Assets Research Team, Risk Research · Published 20 May 2026 · Updated 14 October 2026

Expected Shortfall: Expected shortfall, or conditional value-at-risk (CVaR), is the average loss in the cases where value-at-risk is breached - it measures how bad losses are inside the tail, not just where the tail begins. Unlike VaR it is a coherent risk measure, and it is the basis of Basel FRTB market-risk capital.

What Expected Shortfall Measures

Expected shortfall, also called conditional value at risk (CVaR) or expected tail loss, answers the question that value at risk leaves open: if things go badly, how bad is bad? VaR gives a threshold, the loss not exceeded on 99 days out of 100. Expected shortfall gives the average loss on the remaining day: the mean of the tail beyond VaR.

The distinction matters because the tail is where portfolios fail. Two books with the same VaR can lose a little beyond it or a great deal; VaR cannot tell them apart, expected shortfall can. And unlike VaR, expected shortfall is a coherent risk measure in the sense of Artzner and co-authors (1999): the risk of a combined portfolio can never exceed the sum of its parts, so diversification never appears to add risk.

How It Works

$$ES_\alpha = E[L \mid L \ge \text{VaR}\alpha] = \frac{1}{1-\alpha}\int\alpha^1 \text{VaR}_u , du$$

Expected shortfall is the average of all VaR figures beyond the chosen confidence level. For a normally distributed loss with mean μ and standard deviation σ it has a closed form:

$$ES_\alpha = \mu + \sigma,\frac{\varphi(z_\alpha)}{1-\alpha}$$

where z is the normal quantile and φ the normal density. With the mean set to zero the multiples of volatility are:

Under normality, 97.5 percent expected shortfall sits almost exactly at 99 percent VaR, which is why regulators chose that pairing.

In practice the normal formula is a starting point at most. Expected shortfall is usually estimated from a historical or simulated loss distribution, averaging the losses beyond the empirical quantile.

Worked Example

A 100 million book has daily volatility of 1.2 percent. Compare a normal distribution with a Student t distribution with four degrees of freedom, scaled to the same volatility, a common stand-in for the fat tails of daily returns:

Same volatility, same 97.5 percent VaR; the fat-tailed book's expected shortfall is about a fifth higher.

A risk report built on 97.5 percent VaR would show the two books as identical. Expected shortfall shows that the second loses 3.39 million on an average bad day against 2.81 million for the first. At 99 percent the gap widens to 4.43 million against 3.20 million, because fat tails matter more the further out one looks.

When It Applies (and Limitations)

It rests on few observations. A year of daily data holds about 250 days, so a 97.5 percent expected shortfall averages roughly six losses. One extra crisis day can move the estimate sharply. Longer windows and stressed calibration periods help; false precision does not.

It is hard to backtest on its own. Gneiting (2011) showed that expected shortfall is not elicitable: unlike VaR, no scoring function rewards a correct forecast of it in isolation. Fissler and Ziegel (2016) showed it can be evaluated jointly with VaR, which is why regulatory backtesting still counts VaR exceptions.

It assumes positions can be exited. A one-day measure says nothing about a position that takes weeks to unwind. The trading-book rules therefore scale expected shortfall by liquidity horizons from 10 to 120 days depending on the risk factor.

It does not see the path. A one-day figure describes the average bad day, not a run of them. Losses in crises cluster: five consecutive days at the 97.5 percent expected shortfall would cost a 100 million normal book about 14 million, a sequence that a single-day measure never shows. Multi-day horizons and drawdown measures cover what it misses.

It inherits the model's blind spots. An expected shortfall estimated on a calm window is as optimistic as a VaR estimated on the same window. Scenarios outside the data belong to stress testing.

Why It Matters for Institutional Investors

It is now the regulatory standard. The Basel Committee's Fundamental Review of the Trading Book replaced 99 percent VaR with 97.5 percent expected shortfall as the basis for market-risk capital, calibrated to a period of stress. The VaR calculator shows how the normal quantiles behind both measures relate.

It is the honest measure for short-optionality strategies. Covered calls, short strangles and variance-selling programmes earn small, steady gains and concentrate their losses in a deep, rare tail. VaR understates that tail risk; expected shortfall puts a number on it.

It makes risk budgets add up. Because it is coherent, expected shortfall can be decomposed consistently into contributions by position or manager, and a portfolio's total never exceeds the sum of its parts. Risk budgets built on it reward genuine diversification instead of occasionally punishing it.

It changes the conversation with a committee. "On our worst day in forty we lose 2.8 million on average" is a statement a board can relate to capital and liquidity, in a way that a percentile alone is not.

References

1. Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent Measures of Risk. Mathematical Finance, 9(3), 203-228.
2. Acerbi, C., & Tasche, D. (2002). On the Coherence of Expected Shortfall. Journal of Banking & Finance, 26(7), 1487-1503.
3. Rockafellar, R. T., & Uryasev, S. (2000). Optimization of Conditional Value-at-Risk. Journal of Risk, 2(3), 21-41.
4. Gneiting, T. (2011). Making and Evaluating Point Forecasts. Journal of the American Statistical Association, 106(494), 746-762.
5. Fissler, T., & Ziegel, J. F. (2016). Higher Order Elicitability and Osband's Principle. Annals of Statistics, 44(4), 1680-1707.
6. Basel Committee on Banking Supervision. (2019). Minimum Capital Requirements for Market Risk. Bank for International Settlements.