Quantitative risk measurement

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. BTW, this is distinct from the “risk” in “statistical risk bounds”, which is the domain of statistical learning theory.

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? This is about the risk measures themselves. For a useful class of distributions to use in this context it might be worth considering extreme value theory.

🏗 introduce risk coherence, and Discuss application to climate risk, and scheduling.

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 \(\alpha-\operatorname{VaR}\) 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?

Subadditivity/coherence

TBC

G-expectation

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, although it is not immediately obvious to me how this works. See, e.g. Peng (2004).

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.

Rosenblatt transform

Mentioned here 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.

Acerbi, Carlo, and Dirk Tasche. 2002. “Expected Shortfall: A Natural Coherent Alternative to Value at Risk.” Economic Notes 31 (2): 379–88. https://doi.org/10.1111/1468-0300.00091.

Briand, Philippe, François Coquet, Ying Hu, Jean Mémin, and Shige Peng. 2000. “A Converse Comparison Theorem for BSDEs and Related Properties of G-Expectation.” Electronic Communications in Probability 5 (0): 101–17. https://doi.org/10.1214/ECP.v5-1025.

Cai, Zongwu, and Xian Wang. 2008. “Nonparametric Estimation of Conditional VaR and Expected Shortfall.” Journal of Econometrics, Econometric modelling in finance and risk management: An overview, 147 (1): 120–30. https://doi.org/10.1016/j.jeconom.2008.09.005.

Charpentier, Arthur, and Emmanuel Flachaire. 2019. “Pareto Models for Risk Management.” December 25, 2019. http://arxiv.org/abs/1912.11736.

Chen, Song Xi. 2008. “Nonparametric Estimation of Expected Shortfall.” Journal of Financial Econometrics 6 (1): 87–107. https://doi.org/10.1093/jjfinec/nbm019.

Chen, Zengjing, Tao Chen, and Matt Davison. 2005. “Choquet Expectation and Peng’s G -Expectation.” The Annals of Probability 33 (3): 1179–99. https://doi.org/10.1214/009117904000001053.

Feng, Zhen-Hua, Yi-Ming Wei, and Kai Wang. 2012. “Estimating Risk for the Carbon Market via Extreme Value Theory: An Empirical Analysis of the EU ETS.” Applied Energy 99 (November): 97–108. https://doi.org/10.1016/j.apenergy.2012.01.070.

Fishburn, Peter C. 1977. “Mean-Risk Analysis with Risk Associated with Below-Target Returns.” American Economic Review 67 (2): 116–26.

Hofert, Marius, and Wayne Oldford. 2016. “Visualizing Dependence in High-Dimensional Data: An Application to S&P 500 Constituent Data.” September 29, 2016. http://arxiv.org/abs/1609.09429.

Hubbard, Douglas W. 2014. How to Measure Anything: Finding the Value of Intangibles in Business. 3 edition. Hoboken, New Jersey: Wiley.

Hudson, Paul, WJ Wouter Botzen, Jeffrey Czajkowski, and Heidi Kreibich. 2014. “Risk Selection and Moral Hazard in Natural Disaster Insurance Markets: Empirical Evidence from Germany and the United States.”

Kaplan, E. L., and Paul Meier. 1958. “Nonparametric Estimation from Incomplete Observations.” Journal of the American Statistical Association 53 (282): 457–81. https://doi.org/10.1080/01621459.1958.10501452.

Kaplan, Stanley, and B. John Garrick. 1981. “On the Quantitative Definition of Risk.” Risk Analysis 1 (1): 11–27. http://onlinelibrary.wiley.com/doi/10.1111/j.1539-6924.1981.tb01350.x/full.

Marcelo G. Cruz. 2002. Modeling, Measuring and Hedging Operational Risk. Chichester: Chichester : Wiley.

McNeil, Alexander J, Rüdiger Frey, and Paul Embrechts. 2005. Quantitative Risk Management : Concepts, Techniques and Tools. Princeton: Princeton Univ. Press.

Moscadelli, Marco. 2004. “The Modelling of Operational Risk: Experience with the Analysis of the Data Collected by the Basel Committee.” SSRN Scholarly Paper ID 557214. Rochester, NY: Social Science Research Network. http://papers.ssrn.com/abstract=557214.

Ou-Yang, Chieh, Howard Kunreuther, and Erwann Michel-Kerjan. 2013. “An Economic Analysis of Climate Adaptations to Hurricane Risk in St. Lucia.” The Geneva Papers on Risk and Insurance - Issues and Practice 38 (3): 521–46. https://doi.org/10.1057/gpp.2013.18.

Peng, Shige. 2004. “Nonlinear Expectations, Nonlinear Evaluations and Risk Measures.” In Stochastic Methods in Finance, 165–253. Lecture Notes in Mathematics 1856. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-44644-6_4.

———. 2007. “G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Itô Type.” In Stochastic Analysis and Applications, edited by Fred Espen Benth, Giulia Di Nunno, Tom Lindstrøm, Bernt Øksendal, and Tusheng Zhang, 541–67. Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-70847-6_25.

———. 2010. “Nonlinear Expectations and Stochastic Calculus Under Uncertainty.” February 24, 2010. http://arxiv.org/abs/1002.4546.

Peng, ShiGe. 2009. “Survey on Normal Distributions, Central Limit Theorem, Brownian Motion and the Related Stochastic Calculus Under Sublinear Expectations.” Science in China Series A: Mathematics 52 (7): 1391–1411. https://doi.org/10.1007/s11425-009-0121-8.

Pesenti, Silvana M., Alberto Bettini, Pietro Millossovich, and Andreas Tsanakas. 2020. “Scenario Weights for Importance Measurement (SWIM) – an R Package for Sensitivity Analysis.” SSRN Scholarly Paper ID 3515274. Rochester, NY: Social Science Research Network. https://doi.org/10.2139/ssrn.3515274.

Pesenti, Silvana M., Pietro Millossovich, and Andreas Tsanakas. 2019. “Reverse Sensitivity Testing: What Does It Take to Break the Model?” European Journal of Operational Research 274 (2): 654–70. https://doi.org/10.1016/j.ejor.2018.10.003.

Rakhlin, Alexander, Dmitry Panchenko, and Sayan Mukherjee. 2005. “Risk Bounds for Mixture Density Estimation.” ESAIM: Probability and Statistics 9 (June): 220–29. https://doi.org/10.1051/ps:2005011.

Rosazza Gianin, Emanuela. 2006. “Risk Measures via -Expectations.” Insurance: Mathematics and Economics 39 (1): 19–34. https://doi.org/10.1016/j.insmatheco.2006.01.002.

Roussas, George G. 1990. “Asymptotic Normality of the Kernel Estimate Under Dependence Conditions: Application to Hazard Rate.” Journal of Statistical Planning and Inference 25 (1): 81–104. https://doi.org/10.1016/0378-3758(90)90008-I.

Scaillet, O. 2004. “Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall.” Mathematical Finance 14 (1): 115–29. https://doi.org/10.1111/j.0960-1627.2004.00184.x.

Scandroglio, Giacomo, Andrea Gori, Emiliano Vaccaro, and Vlasios Voudouris. 2013. “Estimating VaR and ES of the Spot Price of Oil Using Futures-Varying Centiles.” International Journal of Financial Engineering and Risk Management 1 (1): 6–19. https://doi.org/10.1504/IJFERM.2013.053713.

Sluijs, Jeroen P van der, Matthieu Craye, Silvio O Funtowicz, Penny Kloprogge, Jerome R Ravetz, and James Risbey. 2005. “Combining Quantitative and Qualitative Measures of Uncertainty in Model-Based Environmental Assessment: The NUSAP System.” Risk Analysis 25: 481–92. https://doi.org/10.1111/j.1539-6924.2005.00604.x.

Sy, Judy P., and Jeremy M. G. Taylor. 2000. “Estimation in a Cox Proportional Hazards Cure Model.” Biometrics 56 (1): 227–36. https://doi.org/10.1111/j.0006-341X.2000.00227.x.

Taylor, James W. 2008a. “Estimating Value at Risk and Expected Shortfall Using Expectiles.” Journal of Financial Econometrics 6 (2): 231–52. https://doi.org/10.1093/jjfinec/nbn001.

———. 2008b. “Using Exponentially Weighted Quantile Regression to Estimate Value at Risk and Expected Shortfall.” Journal of Financial Econometrics 6 (3): 382–406. https://doi.org/10.1093/jjfinec/nbn007.

Vázquez, Alexei, João Gama Oliveira, Zoltán Dezsö, Kwang-Il Goh, Imre Kondor, and Albert-László Barabási. 2006. “Modeling Bursts and Heavy Tails in Human Dynamics.” Physical Review E 73 (3): 036127. https://doi.org/10.1103/PhysRevE.73.036127.