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Investment Analysis and Portfolio Management

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					Investment Analysis and
Portfolio Management
   Lecture 6
   Gareth Myles
The Single-Index Model

    Efficient frontier
      Shows achievable risk/return combinations
      Permits selection of assets

    Can be constructed for any number of
     assets
        Given expected returns, variances and
         covariances
    Calculation is demanding in the
     information required
The Single-Index Model
  More useful if information demand can
   be reduced
  The single-index model is one way to do
   this
  Imposes a statistical model of returns
      Simplifies construction of frontier
      The model may (or may not) be accurate

    The reduced information demand is
     traded against accuracy
Portfolio Variance

    The variance of a portfolio is given by
                                        12
                      N   N        
               p   X i X j ij 
                     i 1 j 1     
  This requires the knowledge of N variances
   and N[N – 1] covariances
  But symmetry ( ij   ji ) reduces this to
   (1/2)N[N – 1] covariances
Portfolio Variance

    So N + (1/2)N[N – 1]
                         = (1/2)N[N + 1]
   pieces of information are required to compute
   the variance
  Example
        If a portfolio is composed of all FT 100 shares then
            (1/2)N[N + 1] = 5050
    This is not even an especially large portfolio
Portfolio Variance

    Where can the information come from?
        1. Data on financial performance (estimation)
        2. From analysts (whose job it is to understand
         assets)
  But brokerages are typically organized into
   market sectors such as oil, electronics,
   retailers
  This structure can inform about variances but
   not covariances between sectors
  So there is a problem of implementation
Model
  A possible solution is to relate the returns on
   assets to some underlying variable
  Let the return on asset i be modelled by
                 ri   iI   iI rI  e iI
  ri  = return on asset i, rI = return on index,
   eiI = random error
  Return is linearly related to return on the index
  This model is imposed and may not capture the
   data
Model

  Three assumptions are placed on this model
  The expected error is zero:
                       E e iI   0
    The error and the return on the index are
     uncorrelated:
                    Ee iI rI  rI   0
    The errors are uncorrelated between assets:
                     Ee iI , e kI   0
Model

  The model is         ri
   estimated using data
  Observe the return                                x

   on the market and
   the return on the                     x
                                                 x
                                                         x
                                 x
   asset                                     x
                             x       x
  Carry out linear
   regression to find
   line of best fit                                          rI
Model

    The estimated values are
                   T
                                   
                    rI , j  rI ri , j  ri   
                   j 1
           iI                                    ,  iI  ri   iI rI
                           rI , j  rI 2
                              T

                          j 1
    With E e iI   0
        The estimation process ensures the average error
         is zero
    The value of iI is the gradient of the fitted line
Model

    If the model is applied to all assets it need not
     follow that
                     Ee iI , e kI   0

  If the covariance of errors are non-zero this
   indicates the index is not the only explanatory
   factor
  Some other factor or factors is correlated with
   (or “explains”) the observed returns
Model
    Note:
                    iI covarianceof i with index
              iI  2 
                   I      varianceof index
    Note:
                            II  1
    And
                            II  0
    These observations permits a characterization
     of assets
Assets Types

  If  iI  1 then the     ri
   asset is more volatile
   (or risky) than the
   market
  This is termed an
   “aggressive” asset



                                 rI
Assets Types

  If iI  1 then the    ri
   asset is less volatile
   than the market
  This is termed a
   “defensive” asset




                               rI
Risk

  For an individual asset
  If ri   iI   iI rI  e iI then

        Eri  ri 
        2
        i
                         2



             E  iI   iI rI  e iI   iI   iI rI 
                                                        2


             E iI rI  rI   e iI 
                                        2



                
             E  rI  rI   2eiI iI rI  rI   e
                     2
                    iI
                              2                              2
                                                            iI   
Risk

    This can be written
                      
                    i
                     2       2
                            iI
                                   2
                                   I
                                       2
                                       ei

    So risk is composed of two parts:
        1. market (or systematic) risk
                            
                             2
                            iI
                                   2
                                   I
        2. unique (or unsystematic) risk
                            e2i
Return

    Portfolio return
                   N
             rp   X i ri
                  i 1
                   N
                 X i  iI   iI rI  e iI 
                  i 1
                    N            N                N
                  X i iI   X i  iI rI   X ie iI
                   i 1          i 1             i 1

                  pI   pI rI  e pI
Return

    Hence

                                  
       rp  E  pI   pI rI  e pI   pI   pI rI

  The portfolio has a value of beta
  This also determines its risk
Risk

    Portfolio variance is

           Erp  rp 
           2
           p
                                   2



                   
                E  pI   pI rI  e pI   pI   pI rI          
                                                                   2




               E      2
                         pI   rI  rI  2
                                              2 pI rI  rI e pI  e 2
                                                                        pI   
                  
                    2
                    pI
                          2
                          I
                                    2
                                    ep
Risk

    The final expression can also be written

                             2
                  N
                             2  N 2 2
            2   X i iI   I   X i  ei 
             p
                  i 1             i 1      
   Consequence: now need to only know 
                                                    2
                                                   I
   and  ei, i = 1,...,N
          2


  For example, for FT 100 need to know 101
   variances (reduced from 5050)
Diversified Portfolio
                                                 1
    A large portfolio that is evenly held X i 
                                                 N
  The non-systematic variance is
                  N
                       2 2
                                 N  1 2 2 
          ep   X i  ei       ei 
           2

                 i 1        i 1  N  
                                            
  This tends to 0 as N tends to infinity, so only
   market risk is left
Diversified Portfolio

    That is
                     
                   2
                   p
                            2
                            pI
                                 2
                                 I
                                          2
                                          ep
    tends to
                         
                        2
                        p
                                 2
                                 pI
                                      2
                                      I


   I2 is undiversifiable market risk
   ep is diversifiable risk
     2
Market Model

  A special case of the single-index model
  The index is the market
        The set of all assets that can be purchased
    The market model has two additional
     properties
        Weighted-average beta = 1
        Weighted-average alpha = 0
    Issue: how is the market defined?
        This is discussed for CAPM
Adjusting Beta

  The value of beta for an asset can be
   calculated from observed data
  This is the historic beta
  There are two reasons why this value
   might be adjusted before being used
      Sampling
      Fundamentals

				
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