Value at risk and conditional VaR: how much you could lose in a bad case

In depth7 min read · updated 1 October 2026 · 6 sources

Value at risk (VaR) puts a dollar figure on the risk of loss: with 95% probability, you will not lose more than this amount in a month. The conditional VaR (also called expected shortfall) answers the more uncomfortable question: and if you do, how much on average?

How to calculate value at risk (parametric)

VaR always has three ingredients: a confidence level (usually 95% or 99%), a time horizon (one day, one month, one year) and an assumption about the distribution of returns. The simplest is the normal distribution. Then the expected return and volatility are all you need.

Conditional VaR: the average of the bad cases

VaR says nothing about how bad the remaining 5% of cases turn out. The conditional VaR is the average loss in exactly those cases. Under a normal distribution, it comes to 7.75% in the example, i.e. $7,751.

The difference is more than a technicality. Two portfolios can have the same VaR but very different losses beyond it: a strategy that includes sold options loses rarely, but then a lot. Its VaR looks harmless; its CVaR does not.

VaR of different portfolios compared

For choosing your equity allocation, an annual VaR is more intuitive than a monthly one. Under a normal distribution and with roughly typical assumptions for return and volatility, the figures for $100,000 over one year are:

The table describes an averagely bad year among many, not the most severe crisis. In real crashes, equities fall considerably further than the normal distribution suggests at 95%. It is nevertheless useful as a comparison: moving from 60/40 to pure equities increases the expected loss in a bad year by around two thirds.

Historical instead of parametric

Real returns have more extreme swings than the normal distribution. The historical VaR is more robust: sort all the monthly returns of a long backtest and read off the value at the 5% edge. With 240 months, that is the twelfth-worst month, and the historical CVaR is the average of the twelve worst.

A third method is Monte Carlo simulation: you generate many possible return paths from a model (for example with fat tails, or from resampled historical blocks) and read the VaR and CVaR off the simulated distribution. It is flexible, but only as good as the model behind it; see Monte Carlo simulation.

For advanced readers: skewness, kurtosis and time

Between the normal distribution and the historical method lies the Cornish-Fisher VaR. It corrects the quantile of the normal distribution using the skewness S and the excess kurtosis K of the returns. This is a series expansion that goes back to Cornish and Fisher (1938):

For longer horizons, VaR is often scaled up with the square root of time. For twelve months, the monthly VaR in the example becomes −(12 × 0.5 − 1.645 × 4 × √12) = 16.79%. This assumes independent monthly returns. Because bad months cluster in crises, the scaling tends to be too optimistic; moreover, it says nothing about losses along the way, which the maximum drawdown shows.

Fat tails are not a theoretical problem: Cont (2001) counts them among the stylized facts of all financial markets. A VaR based on the normal distribution is therefore regularly too low at the 99% edge.

What the research shows: VaR is not a coherent risk measure

Artzner et al. (1999) formulated four properties that a sensible risk measure should satisfy. The most important for investors is subadditivity: the risk of a mixed portfolio must not be greater than the sum of the individual risks. Diversification must not do harm. VaR violates this rule.

Expected shortfall does not have this problem; Acerbi and Tasche (2002) showed that it is coherent. It can also be optimized well:

Rockafellar and Uryasev (2000) showed how CVaR can be minimized using linear programming. This is the basis of the “Min. CVaR” optimization. Banking supervisors, too, have switched from VaR to expected shortfall for market risk in the trading book. Jorion (2007) gives a detailed account of the methods, how to test them after the fact (backtesting) and their limits.

Limits

• VaR is not a maximum loss. In crises it is regularly exceeded, often by a wide margin.
• It depends on the period: a VaR from calm years underestimates the risk. A look at the maximum drawdown complements it.
• For longer horizons, VaR grows roughly with the square root of time, as long as returns are independent.
• It describes one period, not the path: three moderately bad months in a row do not show up in the monthly VaR as one large loss.
• At 99%, you need many years of data: with 20 years of monthly returns, only two or three values lie beyond the 1% quantile.

Using VaR and CVaR sensibly

• As a comparison, not a promise: two portfolios can be compared well using the same method and the same period; neither figure is any good as a limit on the possible loss.
• Think in dollars: convert percentages to your portfolio. A CVaR of 24% on $300,000 is $72,000 in a bad year. That is a number you should know in advance.
• Several methods side by side: if historical and parametric VaR are far apart, returns are clearly not normally distributed. Then the historical value and the CVaR deserve more weight.

How Portfolio Lab calculates it

The Backtest shows the historical VaR 95% per period: the negative 5% quantile of all period (monthly or daily) returns, linearly interpolated between neighboring values. The CVaR 95% is the average loss of all periods with a return below this quantile. Neither is annualized. Optimization offers the objective “Min. CVaR (95%)”: it solves the Rockafellar-Uryasev linear program over the historical returns and finds the weights with the smallest average loss in the worst 5% of periods. More in the metrics methodology.

This article explains a method; it is not investment or tax advice. Worked examples are simplified and rounded. All terms in the glossary.