Conditional Value at Risk (CVaR)
factor.formula
Conditional Value at Risk (based on VaR score) formula:
Conditional Value at Risk (based on conditional expectation) formula:
in:
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The confidence level indicates that we are concerned about the probability of loss exceeding VaR, and usually takes a higher value, such as 0.05, 0.01, etc. For example, when α = 0.05, it indicates the average expected value of the portfolio loss in the worst case of 5%. Note that here α usually represents the probability of loss tail, so in the formula, α is usually a smaller value.
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The value at risk (VaR) of asset portfolio X under probability p. VaR_p(X) represents the upper limit of the loss of asset portfolio X under probability p. That is, the maximum loss of the asset portfolio when the confidence level is (1-p).
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The loss of the portfolio (or negative value of the return). It should be noted here that losses are usually defined as negative values, so when X is less than or equal to VaR, it represents a greater loss.
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The conditional expectation of the portfolio loss X when the loss is less than or equal to VaR_{\alpha}. That is, the average expected value of the loss X when the loss exceeds VaR_{α}.
factor.explanation
Conditional Value at Risk (CVaR) is the average expected value of losses when losses exceed the Value at Risk (VaR) at a given confidence level α. CVaR can measure risk more comprehensively because it not only considers the possibility of losses exceeding VaR, but also the extent of losses after exceeding VaR. Compared with VaR, CVaR has better mathematical properties, such as subadditivity, and is therefore more effective in risk management and optimization problems. CVaR is more robust than VaR, especially when dealing with non-normal distributions and fat-tailed distributions. CVaR can be seen as a supplement to VaR, which can more effectively manage extreme risks and provide decision makers with more comprehensive risk information.