# Conditional Value at Risk

A tail risk metric, Conditional Value at Risk (CVaR) quantifies the scale of expected losses once the Value at Risk (VaR) breakpoint has been breached.
PDF version: StatFACTs ConValue

## How Is it Useful?

Conditional Value at Risk attempts to address some of the shortcomings of Value at Risk (VaR). VaR is defined as a breakpoint that is breached only under extreme conditions. However, VaR does not describe what happens beyond that breakpoint. CVaR does. It is the average of the returns that fall beyond the VaR cut-off. CVaR is a more pessimistic measure of tail risk than VaR.

## What Is a Good Number?

Like VaR, the smaller the value for CVaR the better. One would hope that the VaR breakpoint is rarely penetrated. When VaR is surpassed, one would hope it is not exceeded by a significant amount. Therefore CVaR is most useful when viewed in conjunction with VaR.

## What Are the Limitations?

Like VaR, CVaR does not represent the maximum one could potentially lose in an investment. With any investment, the maximum possible loss is 100%.

Also, CVaR is focused solely on risk. It does not account for the upside potential of an investment. Those investments with the largest downside tail risk are often those with the greatest upside potential.

## What Does the Graph Show Me?

The graph below illustrates an idealized distribution curve of returns. Most of the time an investment’s returns occur near the center or “peak” of the distribution. When markets are doing very well, the returns will fall to the far right of the curve. However, at other times the returns will fall to the left or far-left of the distribution.

The point marked as VaR represents a breakpoint that is rarely expected to be surpassed. CVaR explores what happens on those occasions when the VaR cutoff is breached. CVaR is the average of the extreme losses in the “tail” of the distribution.

## What Are Typical Values?

CVaR is best analyzed in conjunction with its close relative, VaR. VaR is a breakpoint, CVaR is what happens when VaR is breached. With large cap US stocks and investment grade US bonds, when returns fall past the VaR breakpoint, they don’t tend to exceed the VaR by a significant amount. In contrast, more volatile asset classes like small cap US stocks, emerging markets stocks, and high yield US bonds tend to exhibit CVaRs of magnitudes up to 50% greater than VaRs. In plain English this means that when things are bad, they are really bad.

Math Corner:

The very name, Conditional Value at Risk, indicates how it is calculated. CVaR values are conditional to the calculation of VaR itself. Therefore, all of the decisions that go into the calculation of VaR will also impact CVaR. The shape of the distribution, the cut-off level used, the periodicity of the data, and assumptions about volatility stochasticity all set the value known as VaR.

Once the VaR has been established, calculating CVaR is trivial. It is simply the average of those values that fall beyond the VaR:

Where p(x)dx term is the probability density of getting a return with value “x” and “c” is the cut-off point along the distribution curve where one sets the VaR breakpoint.

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