Introduction to Portfolio Optimisation

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Introduction to Portfolio Optimisation

• CF901 Lecture 1

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Portfolio Optimisation

• Invest a sum of money in N different assets in
  order to make a profit.
• The assets are typically financial instruments,
  eg stocks.
• Decision problem
• Choose proportion of initial investment for each
  asset i

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Simple Optimisation Problem?
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Problems

• Return is not a deterministic function
• Return is not a continuous function
• Very many decision variables
• ... (plus many others)

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Uncertainty
The standard tool for reasoning about random
variables is the notion of expected value.

• The return of each asset ri is random variable
  with

We can simply maximise expected return?
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St Petersburg Paradox

• Flip a fair-coin until it lands tails.
• Number of flips is k
• You win 2k
• How much would you pay to bet?

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Risk

• Expected value is not the whole story..
• Previous examples did not take into account the
  risk of the outcome.
• Q How can we take into risk?

A We need to take into account the distribution
of values. Financial theory uses variance of
returns from an asset to summarise the risk
factor
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Risk management through diversification

• The expected return to a portfolio is the
  weighted average of the expected returns of the
  assets composing the portfolio. The same result
  is not generally true for the variance the
  variance of a portfolio is generally smaller than
  the weighted average of the variances of
  individual asset returns corresponding to this
  portfolio. Therein lies the gain from
  diversification.

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(c) D. Maringer
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Modern Portfolio Theory

• Suite of models based on mean-variance analysis.
• Selecting optimal trade offs between expected
  return and risk (variance)
• Reduce variance by selecting stocks that are
  negatively correlated (next lecture)
• In the early part of the course we will review
  standard the models Markowitz Tobin
  frameworks.

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Traditional Optimisation

• Linear Programming
• Quadratic Programming
• Dynamic Programming
• Gradient Search
• Various limitations
• Often assume continuous solution parameters
• Deterministic can get stuck in local optima

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Large search space

• Assume that return from each asset is certain

and that investment decision is binary
Standard continuous optimization methods do not
apply. Brute-force approach to problem requires
enumeration of 2N possible solutions!
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Local versus Global optima
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Heuristic Optimisation

• Start off with arbitrary initial solution(s)
• Repeat
• Produce new solutions from existing solutions
  using a generation rule
• Estimate quality of solutions
• Replace bad solutions with good solutions
  (replacement rule)
• Until
• Good enough solution
• No improvement in solutions
• Out of computing resource(s)

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Simulated Annealing

• Solutions
• atoms in a metal undergoing controlled cooling
• Quality of solution
• energy of atom
• Generation rule
• Random walk through search space (heat)
• Replacement rule
• New solutions replace old ones

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• Goal Find the lowest valley in a terrain.
• Approach A bouncing ball.
• Process
• 1.At the beginning allow the ball to make high
  bounces.
• 2.Slowly decreases the maximum bounce.

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Genetic Algorithms

• Solutions
• represented as bit-strings (genomes)
• Quality of solution
• fitness
• Generation rule
• Mutation
• Cross-over / Recombination
• Replacement rule
• Genomes compete only the fittest reproduce

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Course Outline

• Initially assume that returns and variances are
  known
• Mean-variance analysis models (traditional
  optimisation)
• Heuristic Methods (computational optimisation)
• Methods for estimating and forecasting returns
• Time series analysis

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Covariance
Y is large when X is large
Y is small when X is large
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Correlation
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Variance of a portfolio
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Reducing risk through increasing diversification
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Parameter estimation

• Thus far we have assumed that the mean and
  variance of the return of an asset are given.
• More realistically we might only have access to a
  limited sample of returns.

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• 90.10 113.40 102.90 114.79 111.38
• µ 106.61

93.55 87.08 99.27 96.70 91.57 µ 93.55
102.94 86.64 107.14
116.24 93.08 108.58 112.54
84.06 85.59 105.71 96.00
106.90 108.16 107.12
112.90 106.69 111.91 87.98
99.80 98.43 µ 101.92
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The portfolio optimization problem
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Recommended Texts

• Dietmar Maringer, Portfolio Management with
  Heuristic Optimization, Springer 2005.
• A. E. Eiben and J.E. Smith Introduction to
  Evolutionary Computing, Springer 2007.
• Paulo Brandimarte Numerical Methods in Finance
  and Economics A MATLAB-Based Introduction ,
  Wiley 2006
• Chris Brooks, Introductory Econometrics for
  Finance, CUP 2002.