What is Conditional Value at Risk (CVaR)?

Conditional Value at Risk (CVaR), also known as Expected Shortfall, is a risk assessment metric that provides the average of losses that exceed the Value at Risk (VaR) threshold, offering a deeper insight into tail risks. While VaR identifies the maximum loss within a confidence interval, CVaR measures the expected loss when this threshold is breached, giving a more comprehensive view of potential extreme losses in a portfolio.

CVaR Calculation: CVaR is calculated by taking the average of the losses that exceed the VaR level at a given confidence interval. The formula for CVaR is:

CVaR = E[Loss | Loss > VaR] Here, E[Loss] represents the expected loss, and the condition Loss > VaR denotes that the losses are greater than the VaR threshold.

Understanding Conditional Value at Risk

CVaR enhances risk management by focusing on the tail end of the loss distribution:
  • Tail Risk Measurement: CVaR captures the risk of extreme losses beyond the VaR estimate, offering a clearer understanding of tail risk, especially in highly volatile markets or during financial crises.
  • Improved Risk Assessment: By considering the average of the worst losses, CVaR provides a more conservative estimate of risk than VaR, which only looks at the threshold value.
  • Example: If the 1-day VaR for a portfolio is $100,000 at a 95% confidence level, and the average of the losses beyond this $100,000 threshold is $150,000, the 1-day CVaR would be $150,000.

Example Calculation

Suppose you are managing a portfolio with a 1-day VaR of $50,000 at a 99% confidence level. If the average loss on days when losses exceed $50,000 is $80,000, the 1-day CVaR is $80,000. This means that on the worst 1% of trading days, the average loss is expected to be $80,000.

CVaR provides a clearer understanding of the severity of worst-case losses and is useful for managing tail risks that VaR might underestimate.

Evaluation

CVaR is a more robust risk measure than VaR, particularly in environments where extreme losses can occur. It is particularly useful for managing portfolios with non-normal distributions or assets that exhibit heavy tails. However, CVaR requires more computational effort and data than VaR and can be more sensitive to assumptions about market conditions. Nevertheless, it is a crucial metric for comprehensive risk management.