P&L distributions
The central risk concept that we will employ within the backtest is that of tail risk, as defined in [3]. For a P&L distribution with cumulative distribution F (.) and density f (.), tail risk (TR) at confidence level 1-α, is related to VaR and ES (expected shortfall)
Though it is possible to define the sample estimator for tail risk at the PIT (or p-values) time series level, we choose to make a further transformation to the exponential context. There are several interconnected reasons for this but the main reason is the complicated form of the moment- and cumulant-generating functions which necessitates the use of potentially complex numerical techniques to obtain the saddle point.
In contrast, making an additional transformation to the exponential context allows us to solve for the saddle point analytically; this solution is, moreover, well-defined over the complete interval of interest for tail losses.
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