What is Conditional Value at Risk (CVaR)?

Conditional Value at Risk (CVaR), also called expected shortfall, measures the average loss in the worst cases. Where Value at Risk (VaR) says "losses should exceed €1,645 on only 5% of days", CVaR answers the natural next question: when they do, how bad are they on average?

Because it looks into the tail of the distribution, CVaR captures the risk of severe losses that VaR ignores. Bank regulators have moved from VaR to expected shortfall for measuring market risk for this reason.

The formula

For a confidence level $\alpha$ such as 95%, CVaR is the expected loss $L$ given that it is at least as large as the VaR:

\text{CVaR}_{\alpha} = E\left[\, L \mid L \ge \text{VaR}_{\alpha} \,\right]

With historical data it is simply the average of the worst outcomes: at 95%, the average loss on the worst 5% of days.

Under a normal distribution it has a closed form:

\text{CVaR}_{\alpha} = V \, \sigma \, \frac{\varphi(z_{\alpha})}{1 - \alpha}

where $V$ is the portfolio's value, $\sigma$ its volatility over the horizon and $\varphi$ the standard normal density. At 95%, the factor $\varphi(z_{\alpha})/(1 - \alpha)$ is about 2.06, compared with 1.645 for VaR.

A simple illustration

Over 100 trading days, a portfolio worth €100,000 has these five worst daily losses: €1,700, €1,900, €2,200, €3,000 and €5,200.

The 95% VaR is the threshold of the worst 5%: about €1,700.
The 95% CVaR is the average of those five days:

\text{CVaR}_{95\%} = \frac{1{,}700 + 1{,}900 + 2{,}200 + 3{,}000 + 5{,}200}{5} = 2{,}800

A second portfolio could have exactly the same VaR but worst days between €1,700 and €2,000: the same threshold, and a far less dangerous tail.

Why it matters

It sees the tail. Crashes, not ordinary bad days, do most of the lasting damage to a portfolio. CVaR is sensitive to them; VaR is not.
It rewards diversification consistently. Combining two portfolios never gives a CVaR above the sum of their separate CVaRs, a property VaR does not always have.
It is closer to how losses are experienced: in a bad period, how much could this cost on average?

Limits

Few observations. At 95% only one day in twenty feeds the average, and at 99% only one in a hundred. Estimates from short histories are unstable.
Still backward-looking. A tail event that is not in the data cannot show up in the estimate, which is why stress tests with hypothetical scenarios complement it.

A worked example

Worked through on a sample portfolio. The figures below are that portfolio’s, not yours.

What is my conditional Value at Risk?

I don't have a computed conditional Value at Risk (CVaR) for your portfolio in the loaded data.

What CVaR is: CVaR (also called expected shortfall) estimates the average loss in the worst q% of cases (e.g., the average loss beyond the 95% VaR) over a chosen time horizon and confidence level.

What I can already use from your data: your portfolio value is 138.987,47 € and the available volatility measures are 18,14% (long-term) and 8,26% (recent window). To produce CVaR I need a chosen confidence level (for example 95%), a time horizon (for example 1 month or 1 year), and a return model or historical return series (historical, parametric or Monte Carlo).

If you want, I can compute CVaR now using default assumptions (I would use your recent return history and a 95% confidence level over your chosen horizon) — tell me which confidence level and horizon you prefer, or ask me to use standard defaults and I'll run it.

See these numbers for your own portfolio

Floreo works out every figure on this page from your own holdings — returns, risk, allocation, currencies — and the assistant explains them the way this page does. Import from your broker, or try it on the sample portfolio first.

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