## Quantitative Risk Management

 Unit code: MATH69551 Credit Rating: 15 Unit level: Level 6 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

None

#### Aims

To provide a fundamental understanding of the mathematical approach to risk management in general, and market and credit risk in particular, with a focus on the modelling of dependencies in aggregate risk.

#### Overview

Everybody operating in the financial system needs to consider the possibility that they will make losses, whether this risk comes from changes in the value of this investments (an example of market risk), a borrower defaulting on a loan (credit risk), or unexpected natural disasters (operational risk). In this course, we look at how to quantify risk using probability theory. We will consider market risk and credit risk in detail considering a variety of models, and using the tecnique of copulas to model dependencies between events.

#### Learning outcomes

After following this course, students should be able to:

- Perform portfolio optimisation calculations, using mean-variance theory and with general utility functions.

- Explain and apply the capital asset pricing model.

- Calculate loss distributions and risk measures, including those associated with normal mixture assumptions.

- State Sklar's theorem; find copulas from joint distrubutions, and construct dependant distrubutions via copulas; discuss wasy to select an appropriate copula.

- Describe types of credit risk model, and apply the Merton mdoel and Markov chain credit rating models for individual default events.

- Describe ways to model dependence between individual defaults.

#### Assessment methods

• Other - 10%
• Written exam - 90%

#### Assessment Further Information

Coursework- 10%

Written exam at the end of the semester, 3 hours-90%

#### Syllabus

- Mean-variance portfolio theory.

- Utility theory; risk aversion.

- Capital asset pricing model; discussion of the efficient market hypothesis.

- Risk measures, such as Value at Risk and Tail Value at Risk; properties of risk measures.

- Multivariate models for market risk: aggregate risk, loss distrubutions; multivariate normal distributions and normal variance mixture distributions.

- Copulas: definition; Sklars theorem; Archimedean copulas; dependene

- Credit risk management; types of model; Merton model; Markov chain model for credit ratings; exposure upon default; dependence between default events.

McNeil, A., Frey, R., and Embrechts, P.Quantitative Risk Managemnt: Concepts, Techniques and Tools.

Princeton University Press, Princeton 2006.

Ruppert, D., Statistics and Data Anaysis for Financial Engineering. Springer 2011.

Ingersoll, Jonathan E. Theory of Financial Decision Making. Rowman & Littlefield, 1987.

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Independent study hours - 0 hours

#### Teaching staff

Alexander Watson - Unit coordinator