Buffered probability of exceedance

Buffered Probability of Exceedance (bPOE) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The bPOE at a threshold, x, is equal to one minus the probability at which the Conidtional Value at Risk (CVaR) — also known as the Superquantile or Expected Shortfall (ES) — is equal to x. In other words, it is the proportion of worst-case scenarios that average to x. The figure shows the bPOE for a random variable X, at threshold x (marked in red) as the blue shaded area.

bPOE shaded in blue for the threshold, x, marked in red.

bPOE has its origins in the concept of Buffered Probability of Failure (bPOF) which was developed by Rockafellar and Royset to measure failure risk in the optimization of structures. It was further developed and defined as the inverse superquantile by Mafusalov and Uryasev. Like CVaR, bPOE is a more robust measure of tail risk, as it considers not only the probability that events/losses will exceed the threshold x, but also the magnitude of these potential events.

Formal Definition

For a random variable, $X$ the Buffered Probability Distribution ${\bar {p}}_{x}(X)$ at threshold $x\in [E[X],\sup X]$ is given by:

${\bar {p}}_{x}(X)=\min _{a\geq 0}E[a(X-x)+1]^{+}=\min _{\gamma <x}{\frac {E[X-\gamma ]^{+}}{x-\gamma }}$

where $[\cdot ]^{+}=\max\{\cdot ,0\}$.

If $X$ is continuous, then the bPOE can also be expressed as:

${\bar {p}}_{x}(X)=\{1-\alpha |{\bar {q}}_{\alpha }(X)=x\}$

where ${\bar {q}}_{\alpha }(X)$ is the CVaR of $X$ with probability $\alpha$.