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?

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. See, e.g. Peng04

Refs

AcTa02
Acerbi, C., & Tasche, D. (2002) Expected Shortfall: A Natural Coherent Alternative to Value at Risk. Economic Notes, 31(2), 379–388. DOI.
BCHM00
Briand, P., Coquet, F., Hu, Y., Mémin, J., & Peng, S. (2000) A Converse Comparison Theorem for BSDEs and Related Properties of g-Expectation. Electronic Communications in Probability, 5(0), 101–117. DOI.
CaWa08
Cai, Z., & Wang, X. (2008) Nonparametric estimation of conditional VaR and expected shortfall. Journal of Econometrics, 147(1), 120–130. DOI.
Chen08
Chen, S. X.(2008) Nonparametric Estimation of Expected Shortfall. Journal of Financial Econometrics, 6(1), 87–107. DOI.
ChCD05
Chen, Z., Chen, T., & Davison, M. (2005) Choquet expectation and Peng’s g -expectation. The Annals of Probability, 33(3), 1179–1199. DOI.
FeWW12
Feng, Z.-H., Wei, Y.-M., & Wang, K. (2012) Estimating risk for the carbon market via extreme value theory: An empirical analysis of the EU ETS. Applied Energy, 99, 97–108. DOI.
Fish77
Fishburn, P. C.(1977) Mean-Risk Analysis with Risk Associated with Below-Target Returns. American Economic Review, 67(2), 116–126.
HoOl16
Hofert, M., & Oldford, W. (2016) Visualizing Dependence in High-Dimensional Data: An Application to S&P 500 Constituent Data. arXiv:1609.09429 [Stat].
HBCK14
Hudson, P., Botzen, W. W., Czajkowski, J., & Kreibich, H. (2014) Risk Selection and Moral Hazard in Natural Disaster Insurance Markets: Empirical evidence from Germany and the United States.
KaMe58
Kaplan, E. L., & Meier, P. (1958) Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53(282), 457–481. DOI.
KaGa81
Kaplan, S., & Garrick, B. J.(1981) On the quantitative definition of risk. Risk Analysis, 1(1), 11–27.
LaVa03
Landsman, Z. M., & Valdez, E. A.(2003) Tail conditional expectations for elliptical distributions. North American Actuarial Journal, 7(4), 55–71. DOI.
Marc02
Marcelo G. Cruz. (2002) Modeling, measuring and hedging operational risk. . Chichester: Chichester : Wiley
McFE05
McNeil, A. J., Frey, R., & Embrechts, P. (2005) Quantitative risk management : concepts, techniques and tools. . Princeton: Princeton Univ. Press
Mosc04
Moscadelli, M. (2004) The Modelling of Operational Risk: Experience with the Analysis of the Data Collected by the Basel Committee (SSRN Scholarly Paper No. ID 557214). . Rochester, NY: Social Science Research Network
OuKM13
Ou-Yang, C., Kunreuther, H., & Michel-Kerjan, E. (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), 521–546. DOI.
Peng04
Peng, S. (2004) Nonlinear Expectations, Nonlinear Evaluations and Risk Measures. In Stochastic Methods in Finance (pp. 165–253). Springer Berlin Heidelberg DOI.
RaPM05
Rakhlin, A., Panchenko, D., & Mukherjee, S. (2005) Risk bounds for mixture density estimation. ESAIM: Probability and Statistics, 9, 220–229. DOI.
Rosa06
Rosazza Gianin, E. (2006) Risk measures via -expectations. Insurance: Mathematics and Economics, 39(1), 19–34. DOI.
Rous90
Roussas, G. 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. DOI.
Scai04a
Scaillet, O. (2004a) Nonparametric Estimation and Sensitivity Analysis of Expected Shortfall. Mathematical Finance, 14(1), 115–129. DOI.
Scai04b
Scaillet, Olivier. (2004b) Nonparametric Estimation of Conditional Expected Shortfall (FAME Research Paper Series). . International Center for Financial Asset Management and Engineering
SGVV13
Scandroglio, G., Gori, A., Vaccaro, E., & Voudouris, V. (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. DOI.
SCFK05
Sluijs, J. P. van der, Craye, M., Funtowicz, S. O., Kloprogge, P., Ravetz, J. R., & Risbey, J. (2005) Combining Quantitative and Qualitative Measures of Uncertainty in Model-Based Environmental Assessment: The NUSAP System. Risk Analysis, 25, 481–492. DOI.
SyTa00
Sy, J. P., & Taylor, J. M. G.(2000) Estimation in a Cox Proportional Hazards Cure Model. Biometrics, 56(1), 227–236. DOI.
Tayl08a
Taylor, J. W.(2008a) Estimating Value at Risk and Expected Shortfall Using Expectiles. Journal of Financial Econometrics, 6(2), 231–252. DOI.
Tayl08b
Taylor, J. W.(2008b) Using Exponentially Weighted Quantile Regression to Estimate Value at Risk and Expected Shortfall. Journal of Financial Econometrics, 6(3), 382–406. DOI.
VODG06
Vázquez, A., Oliveira, J. G., Dezsö, Z., Goh, K.-I., Kondor, I., & Barabási, A.-L. (2006) Modeling bursts and heavy tails in human dynamics. Physical Review E, 73(3), 036127. DOI.