## 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 their investments (an example of market risk), a borrower defaulting on a loan (credit risk), or unexpected natural disasters (operational risk). Furthermore, these risks come from many sources, and many of them are likely to be strongly linked. In this course, we look at how to quantify risk using probability theory. We will look at market risk and credit risk in detail, with an emphasis on the role that dependence plays in exacerbating or mitigating risk.

#### Learning outcomes

On completion of this unit, successful students will be able to:

• Apply convex optimisation theory to solve unseen portfolio optimisation problems, and discuss the consequences of these solutions.
• Solve decision-making problems using utility theory, and interpret theoretical aspects and concrete situations in terms of risk aversion.
• Evaluate risk from loss distributions using probabilistic and statistical tools.
• Model financial situations displaying dependence, compare copula and joint distribution approaches and interpret the results of this modelling.
• Develop and apply credit risk models.

#### Assessment methods

• Other - 10%
• Written exam - 90%

#### Assessment Further Information

Coursework- 10%

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

#### Syllabus

• Portfolio management based on mean-variance approach; convex optimisation
• Utility theory; risk aversion
• Capital asset pricing model; efficient market hypothesis
• Risk measures, including Value at Risk and Tail Value at Risk; properties of risk measures
• Multivariate models for market risk: aggregate risk; loss distributions; multivariate and normal variance mixture distributions
• Copulas: joint distributions; independence and perfect dependence; Archimedean copulas; dependence measures
• Credit risk management: components of credit risk model; scoring and Merton models; mixture models; dependence between default events

McNeil, A., Frey, R. and Embrechts, P. Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton 2005.

Nelsen, R. B. An introduction to copulas. 2nd ed., Springer, 2006.

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

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

#### Feedback methods

You will receive direct feedback on homework which you hand in, via marking of the work. Your marked coursework will include comments indicating where you performed well and where you could improve, which I hope you will be able to use in order to focus your revision. Please speak to me during tutorials, or come to my office hour, if you would like more personal feedback on your understanding or on a specific problem.

#### Study hours

• Independent study hours - 0 hours

#### Teaching staff

Alexander Watson - Unit coordinator