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Portfolio Theory and Risk Management

Course category: 
Module code: 
Dr Zaq Coelho


Students are expected to acquire the skills and knowledge necessary to apply modern risk measures and portfolio management tools to balance investment risk and return. The emphasis here is on employing the concept of diversification to manage investment in stock. A more general approach involves utility functions and the construction of portfolios using expected utility optimisation.

Learning objectives

At the end of the module you should be able to:

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


Indicative Content:

  • Mean and variance as measures of return and risk.
  • Risk and return of a portfolio of two assets, diversification. Construction of the feasible set.
  • Risk minimisation for two assets. Finding the market portfolio in two-assets market. Discussion of the separation principle (single fund theorem). Market imperfections: different rates for borrowing and lending.
  • Non-linear optimisation: Lagrange multipliers.
  • General case of many assets, risk-minimization, efficient frontier and its characterization (reduction to the two-assets case: two-fund theorem). The role of risk-free asset, Capital Market Line, market portfolio.
  • Market imperfections: no short-selling.
  • Capital Asset Pricing Model, Security Market Line, practical applications and equilibrium theory.
  • Value-at-Risk (VaR). (if possible, time allowing)
  • Expected utility maximisation. (if possible, time allowing)

Recommended texts

  • S. Benninga and B. Czaczkes, Financial Modelling, MIT Press, 1997.
  • M. Capinski and T. Zastawniak, Mathematics for Finance: An Introduction to Financial Engineering, Springer 2003.
  • E.K.P. Chong and S.H. Zak, An Introduction to Optimisation, Wiley 1996.
  • T.E. Copeland and J.F. Weston, Financial Theory and Corporate Policy, Addison Wesley 1992.
  • D. Duffie, Dynamic Asset Pricing Theory, Princeton University Press 2001.
  • E.J. Elton and M.J. Gruber, Modern Portfolio Theory and Investment Analysis, John Wiley & Sons, 1985.
  • R.A. Haugen, Modern Investment Theory, Prentice Hall, 1993.
  • D.G. Luenberger, Investment Science, Oxford University Press  1998.


Two one-hour lectures and one one-hour seminar per week for one term.


Coursework (20%) and unseen written examination (80%).

Please note that this module forms part of the MSc in Mathematical Finance; please see the relevant Handbook at for details on assessment rules, including penalties for late submission of coursework.

Elective information

This module is not available as an elective.


Basic calculus, linear algebra, probability theory and statistics.


Not available to undergraduate students who have already taken Mathematical Finance I.

Edited 7 Jan 2013 - 19:59 by admin

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