# 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
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
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

 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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