Chapter 5 The Mathematics of Diversification

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Chapter 5The Mathematics of Diversification
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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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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)
• 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)
• Solution The weights of the minimum variance
  portfolios in the previous 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)

• 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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Example of Variance-Covariance Matrix Computation
in Excel
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Portfolio Mathematics (Matrix Form)

• Define w as the (vertical) vector of weights on
  the different assets.
• Define the (vertical) vector of expected
  returns
• Let V be their variance-covariance matrix
• The variance of the portfolio is thus
• Portfolio optimization consists of minimizing
  this variance subject to the constraint of
  achieving a given expected return.

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Portfolio Variance in the 2-asset case

• We have
• Hence

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Covariance Between Two Portfolios (Matrix Form)

• Define w1 as the (vertical) vector of weights on
  the different assets in portfolio P1.
• Define w2 as the (vertical) vector of weights on
  the different assets in portfolio P2.
• Define the (vertical) vector of expected
  returns
• Let V be their variance-covariance matrix
• The covariance between the two portfolios is

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The Optimization Problem

• Minimize
• Subject to
• where E(Rp) is the desired (target) expected
  return on the portfolio and is a vector of
  ones and the vector is defined as

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Lagrangian Method

• Min
• Or Min
• Thus Min

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Taking Derivatives
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• The last equation solves the mean-variance
  portfolio problem. The equation gives us the
  optimal weights achieving the lowest portfolio
  variance given a desired expected portfolio
  return.
• Finally, plugging the optimal portfolio weights
  back into the variance
• gives us the efficient portfolio frontier

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Global Minimum Variance Portfolio

• In a similar fashion, we can solve for the global
  minimum variance portfolio
• The global minimum variance portfolio is the
  efficient frontier portfolio that displays the
  absolute minimum variance.

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Another Way to Derive the Mean-Variance Efficient
Portfolio Frontier

• Make use of the following property if two
  portfolios lie on the efficient frontier, any
  linear combination of these portfolios will also
  lie on the frontier. Therefore, just find two
  mean-variance efficient portfolios, and
  compute/plot the mean and standard deviation of
  various linear combinations of these portfolios.

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Some Excel Tips

• To give a name to an array (i.e., to name a
  matrix or a vector)
• Highlight the array (the numbers defining the
  matrix)
• Click on Insert, then Name, and finally
  Define and type in the desired name.

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Excel Tips (Contd)

• To compute the inverse of a matrix previously
  named (as an example) V
• Type the following formula minverse(V) and
  click ENTER.
• Re-select the cell where you just entered the
  formula, and highlight a larger area/array of the
  size that you predict the inverse matrix will
  take.
• Press F2, then CTRL SHIFT ENTER

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Excel Tips (end)

• To multiply two matrices named V and W
• Type the following formula mmult(V,W) and
  click ENTER.
• Re-select the cell where you just entered the
  formula, and highlight a larger area/array of the
  size that you predict the product matrix will
  take.
• Press F2, then CTRL SHIFT ENTER

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Single-Index Model 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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Proof
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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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Proof
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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