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Notes & exercises MATH37542 Tools for Risk Management
Preface
This “Quarto book” contains all notes, exercises and solutions for the course MATH37542 Tools for Risk Management at the University of Manchester. For course details & other resources, see the course page at Canvas.
Current document version: 0.6
Version 0.6, 29/04/2026: mistakes in computations of LGD in sols of Exercises 9.2&9.3 fixed, plus typo in expression for E[X] in Example 10.6 fixed
Version 0.5, 28/02/2026: added Chapter 8 and some small updates to Chapter 7
Version 0.4, 15/02/2026: added Chapter 7
Version 0.3, 08/02/2026: added Chapters 6, 9, 10
Version 0.2, 31/01/2026: initial release 2025/26, Chapters 1–5
About this course
Pretty much any business activity comes with (financial, or quantifiable as such) risks: potential losses of uncertain size at a (nearby1) future date. Finance and insurance are maybe obvious, but just about any business project in any sector comes with some or a lot of financial uncertainty. Obviously proper risk management by qualified and talented risk managers (i.e. you guys, in your fancy suits and things!) is then a crucial part of making any business tick.
1 So that we don’t have to worry about the “time value of money” i.e. the fact that £10 today is worth more than £10 in say a year from now etc. That’s not an actual sacrifice — it’s trivial to include a “time value of money” consideration in anything that we’ll be doing if you wish
The toolbox of a risk manager is filled with Maths, and in particular Probability Theory (because “loss of uncertain size” cries out for modelling by random variables of course!) and Stats (because any probabilistic model needs to be validated and calibrated, typically using past data). There is a dedicated and active field called Quantitative Risk Management.
In this course, you embark upon a hopefully exciting adventure as student risk manager, on a quest to understand, analyse and make strategic decisions about the risks that you are facing. We model such a risk as a (non-negative2) random variable, say \(X\), with a known distribution (as you’ll remember: describing, roughly speaking, what values \(X\) can take with what probabilities). The flow for you as risk manager is then as follows:
2 We don’t always impose this constraint, much theory is developed for general \(\R\)-valued random variables e.g. and sometimes also in the risk management context it may well make sense to allow negative values
Knowing that you are facing \(X\) i.e. that step 2 will happen, you do your analysis, decision making etc. (so during this course, we’re always positioning ourselves here in this flow).
\(X\) takes its value \(X(\omega) \in [0,\infty)\) (if non-negative) or \(X(\omega) \in \R\) (if more generally \(\R\)-valued) and this \(X(\omega)\) is the amount of money you actually end up losing.
Recall the general Probability Theory framework that is always present when we’re working with random variables: there is some experiment with outcome space \(\Omega\) and a random variable \(X\) which is nothing but a mapping from \(\Omega\) to \([0,\infty)\) (if non-negative) or \(\R\). When we say “\(X\) takes its value \(X(\omega)\)” then this is just shorthand for: the experiment is executed, an outcome \(\omega \in \Omega\) is observed and plugged into the mapping \(X\) to obtain the non-negative/real number \(X(\omega)\). We just don’t need to care about these technical ingredients in this course, and hence we simply don’t pay attention to them. See Random variables in the appendix if you’d like a slightly more elaborate reminder of this.
You may of course very well have to deal with multiple risks i.e. multiple random variables in the above flow, potentailly also with complicated dependencies!
Putting ourselves in step 1 of the above flow in this course, we’ll develop some new concepts, but also heavily use Probablity Theory you already know. The Stats side of things (which you’d for instance typically use to derive suitable distributions for the random variables involved, based on past data) we’ll mostly leave for what it is — there are plenty of Stats courses where you can pick up these techniques. We’ll also do some mild R coding. I hope that you’ll enjoy the maths, and that the motivations, examples and exercises also give you enough of a sense of just how useful this stuff can be out there in the real world!
For your convenience (and mine, really!), in addition to the ten chapters with contents for the eleven teaching weeks (minus one week for the in-class test) there are two appendices:
- In Appendix A — A quick reminder of Probability basics you can find an overview (a very condensed review is maybe a better description) of all the basic Probability tools & techniques you already know from earlier Prob courses, we’ll refer to it throughout to help you have a source available if you need a quick reminder.
- In Appendix B — A list of well known probability distributions you can find a list of some of the most well known probability distributions and some of their properties, also for easy reference. This list will also be attached to the exam.
About the organisation and format of these notes
Every chapter has a section with exercises, their solutions will become available (underneath the exercises) immediately after the tutorial in which we discussed them (refresh the relevant page if needs be).
This material comes only in this online format, no pdf’s or other formats I’m afraid. If this is annoying to you because it disturbs your usual workflow then I can at this point only apologise. You could consider using the “Print to pdf” functionality of your web browser to still produce your own pdf’s — from my quick testing it seems that Chrome/Chromium does the most satisfactory job with this. Further there are annotation tools that can handle web based materials (if you like using one and your favourite one can’t), see e.g. this and this thread on Reddit for some inspiration.
The use of R in these course materials A nice advantage of online materials: we will be using a bit of R coding throughout the course (nothing too heavy/serious), and this online format allows me to include interactive code blocks throughout the materials, looking as follows:
Listing 1: Example of an interactive code block You can run the code by hitting the (indeed) ‘Run code’ button; freely adjust the code and run again; and at any time go back to the code I initially wrote by clicking ‘Start Over’. I’d really encourage you to play around with the bits of code that we encounter as it’s a brilliant way to explore and learn! Good to be aware of:
- the code is executed on your own device (for the nerds among you: via the beauty that is WebAssembly) and so it’s all fully private for you!
- when you adjust the code in such a block and you close your browser (tab), then your code is automatically saved in your browser so that if at some later point you revisit the page (with the same browser) then you’ll see your own code back again.
The use of R in assessment The assessment for this course consists of an in-class test and an exam. The in-class test will focus on the use of R (the exam won’t use any R). Hence it is important to study the R parts of the materials as well as we go forward! For the in-class test you will be using R Studio (as you’re used to from earlier Stats courses as well).
At the moment (January 2026), this material is still a work in progress — updates will follow as we make our way through the semester!
Acknowledgements & further/other reading
The following textbooks in particular gave a lot of inspiration for these notes, and if you like this stuff enough that you’re keen for more then I can really recommend having a look at them:
- McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative risk management: concepts, techniques and tools-revised edition. Princeton university press.
- Kaas, R., Goovaerts, M., Dhaene, J., & Denuit, M. (2008). Modern actuarial risk theory: using R. Second edition. Springer Berlin Heidelberg.
- Glasserman, P. (2004). Monte Carlo methods in financial engineering. Springer New York.
Further, the final two chapters are in particular based on a set of course notes by the brilliant Robert Gaunt, Alex Watson and Sebastian Herrmann (ex-UoM).
Notice any issues/mistakes?
Obviously all this comes with the qualifier that any mistakes/typos/issues/unclarities/… are completely my own responsibility. If you catch any then I’d be very grateful if you would let me know (email address by clicking on my name at the top of this page), thank you!