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Portfolio Theory & Risk Management - MAT00032M

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  • Department: Mathematics
  • Module co-ordinator: Dr. Zaq Coelho
  • Credit value: 10 credits
  • Credit level: M
  • Academic year of delivery: 2022-23
    • See module specification for other years: 2021-22

Module will run

Occurrence Teaching period
A Autumn Term 2022-23
O1 Autumn Term 2022-23

Module aims

Students are expected to acquire the skills and knowledge necessary to apply risk measures and management tools and to use portfolio theory to manage and balance investment risk and return. The main emphasis here is on employing the concept of diversification for managing risky investments. A more general approach involves utility functions and the construction of portfolios using expected utility optimisation. Portfolios of various derivative securities are powerful tools for risk management. Sophisticated needs of fund managers can be addressed by designing these portfolios. Students also need to become familiar with the Capital Asset Pricing model along with its strengths and disadvantages.

Module learning outcomes

By the end of this module students should

  1. recognize methods of measuring risk, understand the relationships between them and their relevance for particular applications;
  2. understand the concept of diversification and be able to employ it to design and manage a portfolio of risky securities;
  3. understand the theoretical background of optimization schemes and be able to implement them to solve practical investment problems;
  4. be able to design a portfolio of financial instruments to meet the needs of fund managers.
  5. (time allowing) understand the advantages and disadvantages of Value at Risk (VaR), a widely accepted measure of risk; be able to compute VaR in practical applications;


Module content

Indicative Content:

  1. Mean and variance as a measure of return and risk.
  2. Risk and return of a portfolio of two assets, diversification. Construction of the feasible set, proof that the feasible set forms a hyperbola.
  3. Risk minimising for two assets. Indifference curves, optimization of portfolio selection based on individual preferences. Inclusion of risk free asset.
  4. Stochastic dominance relation - risk preferences.
  5. Finding the market portfolio in two-asset market. Discussion of the separation principle (single fund theorem).
  6. General case of many assets, risk-minimisation, efficient frontier and its characterisation (reduction to the two-asset case: Two-fund theorem).
  7. The role of risk-free asset - Capital Market Line, separation theorem, market portfolio.
  8. Capital Asset Pricing Model (theorem on linear dependence of cost of capital on beta - Security Market Line), practical applications - equilibrium theory.
  9. Utility Theory. Portfolio management based on maximising expected utility. Equivalence of solvability of optimization problem to the no-arbitrage principle.
  10. Value at Risk (VaR), definition, its significance, methods of computation. Managing VaR by means of financial engineering.



Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Portfolio Theory & Risk Management
2 hours 100

Special assessment rules



Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Portfolio Theory & Risk Management
2 hours 100

Module feedback

Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

  1. D.G. Luenberger, Investment Science, Oxford University Press 1998.
  2. E. Elton, M. Gruber, S. Brown, W. Goetzmann: Modern Portfolio Theory and Investment Analysis, 7th Edition Wiley 2007.
  3. M. Capinski and T. Zastawniak, Mathematics for Finance: An Introduction to Financial Engineering, 2nd edition, Springer 2010.
  4. S. Benninga and B. Czaczkes, Financial Modelling, MIT Press, 1997.
  5. E.K.P. Chong and S.H. Zak, An Introduction to Optimisation, Wiley 1996.
  6. T.E. Copeland and J.F. Weston, Financial Theory and Corporate Policy, Addison Wesley 1992.
  7. D. Duffie, Dynamic Asset Pricing Theory, Princeton University Press 2001.
  8. R.A. Haugen, Modern Investment Theory, Prentice Hall, 1993.

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.