Value at Risk (VaR) is a widely used risk measure in quantitative finance that measures the maximum loss a portfolio is expected to incur over a specified time horizon and confidence level. The VaR measure refers to the amount of money that a portfolio would stand to lose at a given level of confidence within a given time frame. VaR tells you how much you could lose, but it does not tell you anything about the magnitude of the loss that may exceed VaR.
Conditional Value at Risk (CVaR), on the other hand, is a more comprehensive risk measure that measures the expected value of all losses that exceed a given VaR threshold. In other words, it measures the average loss that an investor could experience beyond the VaR level, under the condition that the loss does exceed the VaR level.
CVaR is sometimes referred to as the expected tail loss or expected shortfall, as it measures the average size of the loss that occurs beyond the VaR threshold. CVaR considers all possible losses that may have occurred beyond the VaR level, and then calculates the average loss for those losses. It therefore provides a more complete picture of the worst-case losses that an investor may experience.
For example, let us suppose that you have invested into a portfolio of stocks and you have calculated the VaR of the portfolio to be $10 million, with a 95% confidence interval over a month. This means that there is a 5% chance that you will lose more than $10 million in a given month. Now, you are interested in knowing the expected size of loss in the event that the actual loss exceeds the VaR level. This is where CVaR comes into the picture. Let us suppose that the CVaR for the portfolio is $15 million. This means that if, hypothetically, you lose more than $10 million, the average size of your loss would be $15 million.
In summary, CVaR is a risk measure that provides a more comprehensive view of the downside risk associated with a portfolio than VaR alone, by taking into account the expected size of the loss beyond the VaR level.