# Quantitative risk measures

### Mathematics of actuarial and financial disaster

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 financial history, probably that’s the niche to worry about filling.

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

TODO: introduce risk coherence, extreme value theory, and especially expected shortfall. Discuss application to climate risk, network bandwidth and multithreading.

For the definitions that follow, $$X$$ is a random variable the payoff of a portfolio at some future time, and our quantile of interest is $$0 < \alpha < 1$$.

## Value-at-Risk

The $$\alpha-\operatorname{VaR}$$ of an asset $$X$$ with cdf $$F$$ is, up to minus signs, an estimate of the icdf. It is defined as

\begin{aligned} \operatorname{VaR}_\alpha(X) &:= \inf\{x\in\mathbb{R}:\mathbb{P}(X<-x)\leq 1-\alpha\}\\ &= \inf\{x\in\mathbb{R}:1-F(-x)\geq \alpha\}\\ \end{aligned}

## Expected shortfall

“the expected loss of portfolio value given that a loss is occurring at or above the $$q$$-quantile.”

The expected shortfall (ES) is defined

$ES_{\alpha} := \frac{1}{\alpha}\int_0^{\alpha} \operatorname{VaR}_{\gamma}(X)d\gamma\\ = -\frac{1}{\alpha}\left(E[X \ 1_{\{X \leq x_{\alpha}\}}] + x_{\alpha}(\alpha - \mathbb{P}[X \leq x_{\alpha}])\right)$

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

It is also known as the Conditional Value-at-Risk, $$\alpha-\operatorname{CVaR}.$$

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?