Investment%20Analysis%20and%20Portfolio%20Management

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Investment Analysis and Portfolio Management

• Lecture 3
• Gareth Myles

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FT 100 Index
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and
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Risk

• Variance
• The standard measure of risk is the variance of
  return
• or
• Its square root the standard deviation
• Sample variance
• The value obtained from past data
• Population variance
• The value from the true model of the data

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Sample Variance
General Motors Stock Price 1962-2008
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Sample Variance
Year 93-94 94-95 95-96 96-97 97-98
Return 36.0 -9.2 17.6 7.2 34.1
Year 98-99 99-00 00-01 01-02 02-03
Return -1.2 25.3 -16.6 12.7 -40.9
Return on General Motors Stock 1993-2003
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Sample Variance
Graph of return
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Sample Variance

• With T observations sample variance is
• The standard deviation is
• Both these are biased estimators
• The unbiased estimators are

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Sample Variance

• For the returns on the General Motors stock, the
  mean return is 6.5
• Using this value, the deviations from the mean
  and their squares are given by

Year 93-94 94-95 95-96 96-97 97-98
29.5 -15.7 11.1 0.7 27.6
870.25 246.49 123.21 0.49 761.76
Year 98-99 99-00 00-01 01-02 02-03
-7.7 18.8 -23.1 6.2 -47.4
59.29 353.44 533.61 38.44 2246.76
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Sample Variance

• After summing and averaging, the variance is
• The standard deviation is
• This information can be used to compare different
  securities
• A security has a mean return and a variance of
  the return

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Sample Covariance

• The covariance measures the way the returns on
  two assets vary relative to each other
• Positive the returns on the assets tend to rise
  and fall together
• Negative the returns tend to change in opposite
  directions
• Covariance has important consequences for
  portfolios

Asset Return in 2001 Return in 2002
A 10 2
B 2 10
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Sample Covariance

• Mean return on each stock 6
• Variances of the returns are
• Portfolio 1/2 of asset A and 1/2 of asset B
• Return in 2001
• Return in 2002
• Variance of return on portfolio is 0

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Sample Covariance

• The covariance of the return is
• It is always true that
• i.
• ii.

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Sample Covariance

• Example. The table provides the returns on three
  assets over three years
• Mean returns

Year 1 Year 2 Year 3
A 10 12 11
B 10 14 12
C 12 6 9
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Sample Covariance

• Covariance between A and B is
• Covariance between A and C is

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Variance-Covariance Matrix

• Covariance between B and C is
• The matrix is symmetric

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Variance-Covariance Matrix

• For the example the variance-covariance matrix is

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Population Return and Variance

• Expectations assign probabilities to outcomes
• Rolling a dice any integer between 1 and 6 with
  probability 1/6
• Outcomes and probabilities are
• 1,1/6, 2,1/6, 3,1/6, 4,1/6, 5,1/6,
  6,1/6
• Expected value average outcome if experiment
  repeated

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Population Return and Variance

• Formally M possible outcomes
• Outcome j is a value xj with probability pj
• Expected value of the random variable X is
• The sample mean is the best estimate of the
  expected value

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Population Return and Variance

• After market analysis of Esso an analyst
  determines possible returns in 2010
• The expected return on Esso stock using this data
  is
• ErEsso .2(2) .3(6) .3(9) .2(12)
• 7.3

Return 2 6 9 12
Probability 0.2 0.3 0.3 0.2
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Population Return and Variance

• The expectation can be applied to functions of X
• For the dice example applied to X2
• And to X3

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Population Return and Variance

• The expected value of the square of the deviation
  from the mean is
• This is the population variance

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Modelling Returns

• States of the world
• Provide a summary of the information about future
  return on an asset
• A way of modelling the randomness in asset
  returns
• Not intended as a practical description

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Modelling Returns

• Let there be M states of the world
• Return on an asset in state j is rj
• Probability of state j occurring is pj
• Expected return on asset i is

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Modelling Returns

• Example The temperature next year may be hot,
  warm or cold
• The return on stock in a food production company
  in each state
• If each states occurs with probability 1/3, the
  expected return on the stock is

State Hot Warm Cold
Return 10 12 18
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Portfolio Expected Return

• N assets
• M states of the world
• Return on asset i in state j is rij
• Probability of state j occurring is pj
• Xi proportion of the portfolio in asset i
• Return on the portfolio in state j

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Portfolio Expected Return

• The expected return on the portfolio
• Using returns on individual assets
• Collecting terms this is
• So

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Portfolio Expected Return

• Example Portfolio of asset A (20), asset B
  (80)
• Returns in the 5 possible states and
  probabilities are

State 1 2 3 4 5
Probability 0.1 0.2 0.4 0.1 0.2
Return on A 2 6 9 1 2
Return on B 5 1 0 4 3
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Portfolio Expected Return

• For the two assets the expected returns are
• For the portfolio the expected return is

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Population Variance and Covariance

• Population variance
• The sample variance is an estimate of this
• Population covariance
• The sample covariance is an estimate of this

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Population Variance and Covariance

• M states of the world, return in state j is rij
• Probability of state j is pj
• Population variance is
• Population standard deviation is

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Population Variance and Covariance

• Example The table details returns in five
  possible states and the probabilities
• The population variance is

State 1 2 3 4 5
Return 5 2 -1 6 3
Probability 0.1 0.2 0.4 0.1 0.2
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Portfolio Variance

• Two assets A and B
• Proportions XA and XB
• Return on the portfolio rP
• Mean return
• Portfolio variance

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Portfolio Variance

• Population covariance between A and B is
• For M states with probabilities pj

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Portfolio Variance

• The portfolio return is
• So
• Collecting terms

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Portfolio Variance

• Squaring
• Separate the expectations
• Hence

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Portfolio Variance

• Example Portfolio consisting of
• 1/3 asset A
• 2/3 asset B
• The variances/covariance are
• The portfolio variance is

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Correlation Coefficient

• The correlation coefficient is defined by
• Value satisfies
• perfect positive correlation

rB
rA
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Correlation Coefficient

• perfect negative correlation
• Variance of the return of a portfolio

rB
rA
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Correlation Coefficient

• Example Portfolio consisting of
• 1/4 asset A
• 3/4 asset B
• The variances/correlation are
• The portfolio variance is

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General Formula

• N assets, proportions Xi
• Portfolio variance is
• But so

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Effect of Diversification

• Diversification a means of reducing risk
• Consider holding N assets
• Proportions Xi 1/N
• Variance of portfolio

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Effect of Diversification

• N terms in the first summation, N N-1 in the
  second
• Gives
• Define
• Then

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Effect of Diversification

• Let N tend to infinity (extreme diversification)
• Then
• Hence
• In a well-diversified portfolio only the
  covariance between assets counts for portfolio
  variance