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.

🏗 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?

## Subadditivity

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).

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. 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. 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. 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.

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.