The Living Thing / Notebooks : Quantitative risk measures (Actuarial, financial)

See also risk perception, copula models, climate change.

Actuarial bread-and-butter. The mathematical study of measuring the chances of something terrible happening. This is usually a financial risk, but can also be extreme weather conditions, earthquakes, whatever.

How do you evaluate how bad the worst cases are when deciding whether to do something? Generally this involved ignoring how good the best scenario is; Given recent financial history, probably that’s the niche to worry about filling.

How do you trade off the badness and likeliness of the bad cases? This is distinct from “statistical risk bounds”, which are the domain of statistical learning theory.

TODO: introduce risk coherence, extreme value theory, and especially expected shortfall for a non-financial audience. Discuss application to climate risk, network bandwidth and multithreading.

Expected shortfall

Noted here because I need it surprisingly often.

\(X\) is a random variable the payoff of a portfolio at some future time, and our quantile of interest is \(0 < \alpha < 1\). The expected shortfall (ES) is defined

\begin{align*} ES_{\alpha} := \frac{1}{\alpha}\int_0^{\alpha} \text{VaR}_{\gamma}(X)d\gamma\\ = -\frac{1}{\alpha}\left(E[X \ 1_{\{X \leq x_{\alpha}\}}] + x_{\alpha}(\alpha - P[X \leq x_{\alpha}])\right) \end{align*}

where VaR is Value At Risk, and where \(x_{\alpha} = \inf\{x \in \mathbb{R}: P(X \leq x) \geq \alpha\}\)

According to Wikipedia, I might care about the dual representation, \(ES_{\alpha} = \inf_{Q \in \mathcal{Q}_{\alpha}} E^Q[X]\) with \(\mathcal{Q}_{\alpha}\) the set of probability measures absolutely continuous with respect to the physical measure \(P\), such that \(\frac{dQ}{dP} \leq \alpha^{-1}\) almost surely.

Why might I care about that again?


I don’t really understand this yet, but Shige Peng just gave a talk wherein he argued that the generalised, sublinear expectation operator derived from distributional uncertainty, generate coherent risk measures. See, e.g. Peng04


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