Fall 2024
Project Rules
ACTSC 445/845: Quantitative Risk Management
Purpose
The purpose of this project is for you to analyze a topic related to quantitative risk management. It can be the description of theoretical concepts with applications in QRM (for instance, quantile estimation which can be applied to VaR estimation; risk measures beyond the ones we saw; going into proofs of EVT that we skipped), a data analysis related to QRM (for instance, applying different methods going beyond the ones in the course to a dataset from the realm of finance and insurance), the description of a QRM problem along with a simulation study (e.g., change dependency in a credit risk model and compare resulting risk) or something in between
Ultimately, you should pick the topic that you are most interested in, which has applications to QRM, and has a quantitative aspect (not just a qualitative description of, for instance, the three pillar concept). A list of ideas is at the end of the document.
Project Proposal
To ensure that the topic chosen is appropriate, every group (consisting of one or two students) will submit
a project proposal. The proposal should consist of the following
The title of the project.
The name(s) and student ID(s) of the group member(s).
A 0.5 - 1 page summary of what you intend to work on in the project. This should include a brief description of the references, the methods used, the dataset used (if applicable) and the goal of your project. It should also include a reasoning as to why the chosen topic is relevant for QRM.
The project proposal will be due on November 20, 11:59PM via Crowdmark, but you are more then welcome to submit much earlier. If you are unsure if the topic chosen is appropriate or if you need help finding a topic, please do not hesitate to contact me earlier!
Project Paper
Students can work on the project either individually or in groups of up to two students.
In the former case, the project should have at least 7 pages, in the latter, at least 10 pages (+ Appendix, if applicable). Any paper should not be much longer than 20 pages, unless instructor approval was given.
If a data analysis or simulation study is performed, reproducible and well documented code should be included in the appendix.
The project paper should be a stand-alone paper: If I read it, I should understand the main idea without having to look at references. As a guideline: Your peers having taken this course should be able to understand it.
The project paper should be well written (complete, grammatically correct and coherent), well formatted (no screenshoted formulas, for instance) and well researched (proper citations, critical thought). All three aspects influence the final grade.
There should be an introduction, a main part, a conclusion, and an appendix (if applicable, can include code, additional graphics or tables,...).
You may use any R package you find, so long as you cite it.
The project paper will be due on December 9, 11:59PM via Crowdmark. Only one group member needs to upload their project paper, so long as the other group member has been added to the Crowdmark page.
Topic ideas
The following shall just give ideas for the project topic. The list of references is also anything but exhaustive: Google scholar is your friend!
a) Theoretical topics and proofs. In the course, we have skipped a number of proofs and technicalities. A project could be to explain the theory and math behind such a concept. Ideas include
The generalized inverse (aka quantile function) extends the usual notion of an inverse of a bijective function. There are similarities and differences. Furthermore, the quantile function plays a key role in simulation. This is discussed in Embrechts and Hofert (2013), for instance.
Extreme value theory. An excellent introduction to EVT is Embrechts, Klüppelberg, et al. (1999). Project ideas include giving details on proofs we have skipped or introducing aspects of EVT not discussed in the course, such as EVT for multivariate models or EVT in financial time series.
Coherent risk measures. We have motivated the four axioms of coherence, and proved that expected shortfall is coherent. There’s so much more! For instance, one can show that any coherent risk measure can be written as a generalized scenario. Or one can try to “improve” value-at-risk to repair subadditivity. An excellent reference is Artzner et al. (1999).
Copulas and multivariate distributions. There will be a number of theorems we won’t have the time to prove in the lectures!
b) Financial Time Series, see Taylor (2008). We have only slightly touched the topic of financial time series, so there are plenty of topic ideas here:
Introduction to ARMA and GARCH processes.
Multivariate Financial Time series.
Copula modelling in Financial Time Series.
For instance, you might define ARMA and GARCH processes, illustrate how they work using a simulation study and explain how to estimate parameters. Of course, you do not need to code everything by yourself - there’s a number of R packages!
c) Non-market Risk Management. This course was mainly focussing on market risk management, but there are important others: Operational risk, credit risk and more. An introduction and good references can be found, for instance, in McNeil et al. (2015). For credit risk models, Crouhy et al. (2000) give a comparative study; Carey and Hrycay (2001) use ratings to estimate default probabilities; Byström (2019) look at the future of risk management in light of technological advances, such as blockchain. Regarding operational risk management, see Pakhchanyan (2016) for a literature review or Cornalba and Giudici (2004) for some statistical models.
d) Machine Learning. Generalized linear models, and more generally machine learning techniques, are used in non-life insurance pricing (eg, car insurance). There are plenty of project topics, such as an introduction to GLM and an application to an insurance data-set which demonstrates how to price such insurance contract, see, e.g., Ohlsson and Johansson (2010). For a credit risk application, see Galindo and Tamayo (2000).
e) Estimation methods. A model in QRM is only useful if it can reasonably well fitted to data. Parameter estimation for multivariate distributions often comes with numerical challenges; see, e.g., Erik Hintz et al. (2022), E. Hintz et al. (2021), Protassov (2004), Luo and Shevchenko (2010),
Nadarajah and Kotz (2008). A project topic could be to pick a multivariate distribution and discuss estimation methods, ideally supported with a data analysis or simulation study.
f) Simulation studies. Many problems require Monte Carlo simulation to estimate risk measures, such as in Gaussian or t copula credit risk as in Glasserman and Li (2005). Project ideas include explaining these models (how do they model the loss? What is their X, its distribution and f?) and doing a simulation study (how do the answers change if I change a certain parameter and does this make sense? What happens if I change the dependency?). We have seen the Gaussian credit portfolio in the tutorial. Useful references for possible projects include Salmon (2012), Salmon (2009), Furman et al. (2016).
g) Introducing multivariate distributions. We will get to know a number of classes of multivariate distributions (in particular, normal variance mixtures and elliptical distributions). But there are more complicated multivariate models that can be used in QRM, see, e.g., Weibel et al. (2020). A project topic could be to introduce a multivariate model not discussed in the course, motivate why it would be a useful model and perform a simulation study.
References
Artzner, P., Delbaen, F., Eber, J., and Heath, D. (1999), Coherent Measures of Risk, Mathematical Finance,
9(3), 203–228.
Byström, Hans (2019), Blockchains, real-time accounting, and the future of credit risk modeling, Ledger, 4.
Carey, Mark and Hrycay, Mark (2001), Parameterizing credit risk models with rating data, Journal of banking & finance 25(1), 197–270.
Cornalba, Chiara and Giudici, Paolo (2004), Statistical models for operational risk management, Physica A: Statistical 338(1-2), 166–172.
Crouhy, Michel, Galai, Dan, and Mark, Robert (2000), A comparative analysis of current credit risk models, Journal of Banking & Finance, 24(1-2), 59–117.
Embrechts, P. and Hofert, M. (2013), A note on generalized inverses, Mathematical Methods of Operations Research, 77(3), 423–432, doi:10.1007/s00186-013-0436-7.
Embrechts, P., Klüppelberg, C., and Mikosch, T. (1999), Modelling extremal events, British actuarial journal, 5(2), 465–465.
Furman, Edward, Kuznetsov, Alexey, Su, Jianxi, and Zitikis, Ričardas (2016), Tail dependence of the Gaussian copula revisited, Insurance: Mathematics and Economics, 69, 97–103.
Galindo, Jorge and Tamayo, Pablo (2000), Credit risk assessment using statistical and machine learning: basic methodology and risk modeling applications, Computational economics, 15(1), 107–143.
Glasserman, P. and Li, J. (2005), Importance Sampling for Portfolio Credit Risk, Management Science, 51(11), 1643–1656.
Hintz, E., Hofert, M., and Lemieux, C. (2021), Normal variance mixtures: Distribution, density and parameter estimation, Computational Statistics and Data Analysis, 157C, 107175, doi:10.1016/j. csda.2021.107175.
Hintz, Erik, Hofert, Marius, and Lemieux, Christiane (2022), Computational Challenges of t and Related Copulas, Journal of Data Science, 20(1), 95–110, issn: 1680-743X, doi:10.6339/22-JDS1034.
Luo, X. and Shevchenko, P. (2010), The t copula with multiple parameters of degrees of freedom: bivariate characteristics and application to risk management, Quantitative Finance, 10(9), 1039–1054, doi:10.1080/14697680903085544.
McNeil, A., Frey, R., and Embrechts, P. (2015), Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press, doi:10.1007/s10687-017-0286-4.
Nadarajah, S. and Kotz, S. (2008), Estimation methods for the multivariate t distribution, Acta Applicandae Mathemati 102(1), 99–118.
Ohlsson, E. and Johansson, B. (2010), Non-life insurance pricing with generalized linear models, vol. 2, Springer.
Pakhchanyan, Suren (2016), Operational risk management in financial institutions: A literature review, International Journal of financial studies, 4(4), 20.
Protassov, R. (2004), EM-based maximum likelihood parameter estimation for multivariate generalized hyperbolic distributions with fixed λ , Statistics and Computing, 14(1), 67–77, doi:10.1023/B:STCO. 0000009419.12588.da.
Salmon, Felix (2009), Recipe for disaster: the formula that killed Wall Street, Wired Magazine, 17(3), 17–03.
Salmon, Felix (2012), The formula that killed Wall Street, Significance, 9(1), 16–20. Taylor, Stephen J (2008), Modelling financial time series, world scientific.
Weibel, M., Luethi, D., and Breymann, W. (2020), ghyp: Generalized Hyperbolic Distributions and Its Special Cases, R package version 1.6.1, http://CRAN.R-project.org/package=ghyp.