Risk measures and probability distortion

Value at Risk (VaR) at level $\alpha$ is the $\alpha$-quantile of a loss distribution: the smallest loss $x$ such that $P(X\le x)\ge \alpha$. It answers "how bad can it get, $\alpha\%$ of the time?" but famously ignores everything beyond the quantile — two portfolios with identical $\text{VaR}_{99\%}$ can have wildly different tail severity. It is also not subadditive in general, meaning diversification can appear to increase risk under VaR, a serious flaw for capital aggregation.

Tail VaR / Expected Shortfall (TVaR) fixes this: it is the average loss given that the loss exceeds the VaR threshold, $\text{TVaR}_\alpha(X)=E[X\mid X> \text{VaR}_\alpha(X)]$ (for continuous $X$), equivalently the average of all quantiles from $\alpha$ to 1. TVaR is coherent in the Artzner–Delbaen–Eber–Heath sense: it satisfies monotonicity, subadditivity, positive homogeneity, and translation invariance — the four axioms that make a risk measure behave sensibly under diversification and scaling, and the reason regulatory and rating-agency capital frameworks (Solvency II internal models, many economic capital frameworks) favour it over VaR.

Distortion and the Choquet integral. Both VaR and TVaR are special cases of a more general construction: replace the probability measure $P$ with a distorted measure $g\circ P$ for a non-decreasing function $g:[0,1]\to[0,1]$ with $g(0)=0,g(1)=1$, and integrate using the Choquet integral — sort outcomes ascending and weight each by the increment of $g$ applied to the cumulative probability, rather than the raw probability itself:

$D_g(X) = \int_0^\infty g\big(P(X>x)\big)\,dx \quad (\text{for } X\ge 0),\qquad\text{discretely: weight}_i = g(F_i)-g(F_{i-1}).$

If $g(t)=t$, no distortion occurs and $D_g(X)=E[X]$ exactly — this is why ADE's Choquet layer collapses to plain expectation when $\lambda=0$ (Wang) or $\kappa=1$ (PH), and it is the identity the verification suite checks directly.

The Wang transform $g_\lambda(t) = \Phi(\Phi^{-1}(t)+\lambda)$, for $\lambda>0$, shifts probability weight toward the tail: it upweights bad outcomes exactly as a risk-averse premium principle should, and it recovers TVaR in the limit and prices consistent with CAPM in a Normal setting — the Wang (1996, 2000) premium principle used in reinsurance pricing. The proportional-hazards (PH) transform $g_\kappa(t)=t^{1/\kappa}$, $\kappa\ge1$, is an equally common alternative distortion with a similar tail-loading effect and a direct link to Yaari's dual theory of risk.

Spectral risk measures generalise further: any weighted average of quantiles with a non-negative, non-decreasing weight ("risk spectrum") that integrates to 1 is coherent; TVaR is the special case with weight $\frac{1}{1-\alpha}$ above $\alpha$ and 0 below.

ADE's role. The distortion parameters ($\lambda$ or $\kappa$) are exactly the $g$ in ADE's Choquet layer $D_{g,u,p}(a)$. Setting a large $\lambda$ and $\rho\to\infty$ recovers a TVaR-like, tail-weighted valuation of each action side by side with plain EMV — letting a single engine show "what changes if I evaluate this the way a rating agency evaluates tail risk, instead of on average."

Limits. Distortion parameters, like utility parameters, must be chosen and disclosed — they are not estimated from data alone, and different stakeholders (regulator, policyholder, shareholder) may reasonably prefer different $\lambda$. ADE's sensitivity panel reports exactly where the recommended action would flip as $\lambda$ varies.