Quantitative risk measurement

1 Value-at-Risk

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

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

$$\begin{array}{rl}{\mathrm{VaR}}_{\alpha }(X)& :=inf\{x\in \mathbb{R}:\mathbb{P}(X<-x)\le 1-\alpha \}\\ & =inf\{x\in \mathbb{R}:1-F(-x)\ge \alpha \}\end{array}$$

2 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

$$\begin{gathered} E{S}_{\alpha }:=\frac{1}{\alpha }{\int }_0^{\alpha }{\mathrm{VaR}}_{\gamma }(X)d\gamma \\ =-\frac{1}{\alpha }(E[X{1}_{\{X\le x_{\alpha }\}}]+x_{\alpha }(\alpha -\mathbb{P}[X\le x_{\alpha }])) \end{gathered}$$

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

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

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

…Why might I care about that again?

3 Subadditivity/coherence

TBC

4 G-expectation

I don’t understand this yet, but Shige Peng just gave a talk wherein he argued that the generalised, sublinear expectation operator derived from distributional uncertainty generates coherent risk measures, although it is not immediately obvious to me how this works. See, e.g. Shige Peng (2004).

5 Sensitivity to parameters of risk measures

SWIM is a method for analysis of sensitivity for parameter assumptions in risk measures. (R package) I might use this article as a point of entry to this field if I need it.

An efficient sensitivity analysis for stochastic models based on Monte Carlo samples. Provides weights on simulated scenarios from a stochastic model, such that stressed random variables fulfil given probabilistic constraints (e.g. specified values for risk measures), under the new scenario weights. Scenario weights are selected by constrained minimisation of the relative entropy to the baseline model.

6 Rosenblatt transform

Mentioned mnemonically because it seems to arise in QRM all the time. The Rosenblatt transform is the one that takes a vector random variate with a known (continuous) joint distribution and transforms it to a uniform distribution over the unit hypercube.

7 Knightian risk

See black swans.

8 Incoming

• I. Risk Management Foundations - Machine Learning for Financial Risk Management with Python [Book]
• QRM 4-3: A Bestiary of Tails (video)

9 References