Chapter 6 The Mathematics of Diversification

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Chapter 6The Mathematics of Diversification
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Outline

• Introduction
• Linear combinations
• Single-index model
• Multi-index model

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Introduction

• The reason for portfolio theory mathematics
• To show why diversification is a good idea
• To show why diversification makes sense logically

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Introduction (contd)

• Harry Markowitzs efficient portfolios
• Those portfolios providing the maximum return for
  their level of risk
• Those portfolios providing the minimum risk for a
  certain level of return

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Linear Combinations

• Introduction
• Return
• Variance

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Introduction

• A portfolios performance is the result of the
  performance of its components
• The return realized on a portfolio is a linear
  combination of the returns on the individual
  investments
• The variance of the portfolio is not a linear
  combination of component variances

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Return

• The expected return of a portfolio is a weighted
  average of the expected returns of the
  components

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Variance

• Introduction
• Two-security case
• Minimum variance portfolio
• Correlation and risk reduction
• The n-security case

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Introduction

• Understanding portfolio variance is the essence
  of understanding the mathematics of
  diversification
• The variance of a linear combination of random
  variables is not a weighted average of the
  component variances

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Introduction (contd)

• For an n-security portfolio, the portfolio
  variance is

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Two-Security Case

• For a two-security portfolio containing Stock A
  and Stock B, the variance is

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Two Security Case (contd)

• Example
• Assume the following statistics for Stock A and
  Stock B

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Two Security Case (contd)

• Example (contd)
• What is the expected return and variance of this
  two-security portfolio?

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Two Security Case (contd)

• Example (contd)
• Solution The expected return of this
  two-security portfolio is

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Two Security Case (contd)

• Example (contd)
• Solution (contd) The variance of this
  two-security portfolio is

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Minimum Variance Portfolio

• The minimum variance portfolio is the particular
  combination of securities that will result in the
  least possible variance
• Solving for the minimum variance portfolio
  requires basic calculus

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Minimum Variance Portfolio (contd)

• For a two-security minimum variance portfolio,
  the proportions invested in stocks A and B are

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Minimum Variance Portfolio (contd)

• Example (contd)
• Assume the same statistics for Stocks A and B as
  in the previous example. What are the weights of
  the minimum variance portfolio in this case?

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Minimum Variance Portfolio (contd)

• Example (contd)
• Solution The weights of the minimum variance
  portfolios in this case are

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Minimum Variance Portfolio (contd)

• Example (contd)

Weight A
Portfolio Variance
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Correlation and Risk Reduction

• Portfolio risk decreases as the correlation
  coefficient in the returns of two securities
  decreases
• Risk reduction is greatest when the securities
  are perfectly negatively correlated
• If the securities are perfectly positively
  correlated, there is no risk reduction

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The n-Security Case

• For an n-security portfolio, the variance is

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The n-Security Case (contd)

• The equation includes the correlation coefficient
  (or covariance) between all pairs of securities
  in the portfolio

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The n-Security Case (contd)

• A covariance matrix is a tabular presentation of
  the pairwise combinations of all portfolio
  components
• The required number of covariances to compute a
  portfolio variance is (n2 n)/2
• Any portfolio construction technique using the
  full covariance matrix is called a Markowitz model

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Single-Index Model

• Computational advantages
• Portfolio statistics with the single-index model

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Computational Advantages

• The single-index model compares all securities to
  a single benchmark
• An alternative to comparing a security to each of
  the others
• By observing how two independent securities
  behave relative to a third value, we learn
  something about how the securities are likely to
  behave relative to each other

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Computational Advantages (contd)

• A single index drastically reduces the number of
  computations needed to determine portfolio
  variance
• A securitys beta is an example

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Portfolio Statistics With the Single-Index Model

• Beta of a portfolio
• Variance of a portfolio

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Portfolio Statistics With the Single-Index Model
(contd)

• Variance of a portfolio component
• Covariance of two portfolio components

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Multi-Index Model

• A multi-index model considers independent
  variables other than the performance of an
  overall market index
• Of particular interest are industry effects
• Factors associated with a particular line of
  business
• E.g., the performance of grocery stores vs. steel
  companies in a recession

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Multi-Index Model (contd)

• The general form of a multi-index model