CVaRExpected

CVaRExpected, also known as Conditional Value at Risk (CVaR) or Expected Shortfall, is a risk measure used in finance and statistics to assess the expected loss given that a certain threshold of loss has been exceeded. Unlike Value at Risk (VaR), which provides a single value indicating the maximum loss at a given confidence level, CVaRExpected considers the entire distribution of losses beyond the VaR threshold. This makes it a more comprehensive measure of downside risk.

The CVaRExpected is calculated as the expected value of the losses that exceed the VaR threshold. Mathematically, if X represents the random variable for losses and VaR(α) is the Value at Risk at confidence level α, then CVaRExpected(α) is defined as:

CVaRExpected(α) = E[X | X > VaR(α)]

where E[X | X > VaR(α)] is the conditional expectation of X given that X is greater than VaR(α).

CVaRExpected is particularly useful in portfolio management and risk assessment because it provides a more accurate representation of potential losses in the tail of the loss distribution. It is often used in regulatory frameworks and by financial institutions to ensure that they are adequately prepared for extreme market events.