How is the Conditional Value at Risk (CVaR) calculated?

Conditional Value at Risk (CVaR), also known as Expected Shortfall, is a risk measure that quantifies the potential loss beyond a certain confidence level. It provides a more comprehensive assessment of risk compared to traditional risk measures such as Value at Risk (VaR).

To calculate CVaR, the following steps are typically followed:

1. Determine the confidence level: The first step is to select the desired confidence level, which represents the probability of the loss exceeding a certain threshold. For example, if a 95% confidence level is chosen, it means that the CVaR will represent the expected loss beyond the 95th percentile.

2. Calculate the VaR: Value at Risk (VaR) is the maximum potential loss at a given confidence level. It represents the threshold beyond which the loss is expected not to exceed. VaR can be calculated using various statistical methods, such as historical simulation, parametric models, or Monte Carlo simulation.

3. Identify the tail distribution: Once the VaR is determined, the next step is to identify the tail distribution of the losses beyond the VaR threshold. This can be done by analyzing historical data or using statistical techniques to estimate the distribution parameters.

4. Calculate the expected loss beyond VaR: Using the tail distribution, the expected loss beyond the VaR threshold is calculated. This is done by integrating the tail distribution function from the VaR threshold to infinity. The integration can be performed numerically or using mathematical techniques depending on the complexity of the distribution.

5. Compute the CVaR: Finally, the CVaR is obtained by multiplying the expected loss beyond VaR by the probability of the loss exceeding the VaR threshold. This can be expressed as CVaR = (Probability of loss > VaR) * (Expected loss beyond VaR).

In summary, CVaR is calculated by determining the confidence level, calculating the VaR, identifying the tail distribution, calculating the expected loss beyond VaR, and then multiplying it by the probability of the loss exceeding the VaR threshold.