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Buffered probability of exceedance

Buffered probability of exceedance (bPOE) is a function of a random variable used in statistics and risk management, including financial risk. The bPOE is the probability of a tail with known mean value . The figure shows the bPOE at threshold as the blue shaded area. Therefore, by definition, bPOE is equal to one minus the confidence level at which the Conditional Value at Risk (CVaR) is equal to . bPOE is similar to the probability of exceedance of the threshold , but the tail is defined by its mean rather than the lowest point of the tail.

Formal definition
There are two slightly different definitions of bPOE, so called Lower bPOE and Upper bPOE. For a random variable, X the Lower bPOE, \bar{p}_x(X), at threshold x \in [E[X], \sup X ] is given by: \bar{p}_x (X) = \min_{a \geq 0} E[ a(X-x) +1 ]^+ = \min_{\gamma where [\cdot]^+ = \max\{\cdot , 0\}. bPOE can be expressed as the inverse function of CVaR: \bar{p}_x (X) = \{ 1 - \alpha | \bar{q}_\alpha (X) = x \} , where \bar{q}_\alpha (X) is the CVaR of X with confidence level \alpha. == References ==
Source: Wikipedia ↗
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