Investment Analysis and
Portfolio Management
Lecture 6
Gareth Myles
Announcement

There is no lecture next week (Thursday
27th February)
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


 p   X i X j ij 
 i 1 j 1

N

N
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 modeled by
I
ri   i
I
 i rI
I
 i
I
= return on index,  i
= return on asset i, rI
= random error
 Return is linearly related to return on the index
 This model is imposed and may not capture the
data
 ri
Model

Three assumptions are placed on this model
 The expected error is zero:
  0
I
E i

The error and the return on the index are
uncorrelated:

I
E i

rI  rI   0
The errors are uncorrelated between assets:


E  iI ,  kI  0
Model

The model is
ri
estimated using data
 Observe the return
on the market and
the return on the
asset
 Carry out linear
regression to find
line of best fit
x
x
x
x
x
x
x
x
rI
Example
British American Tobacco
Barclays
rBarclays 20
rBAT 10
8
10
6
0
4
-10
0
-5
-2
-4
-6

0
5
-10
2
-10
-5
0
5
10
rFT 100
-20
-30
-40
Monthly data on stock return and FTSE100
return
 Observe different scales on vertical axis
10
rFT 100
Example
20
10
0
-10
-8
-6
-4
-2
0
-10
-20
-30
-40
2
4
6
8
10
BAT
Barclays
Model

The estimated values are
 rI , j  rI ri, j  ri 
T
 iI

With

I
i
j 1
0
2


r

r
 I, j I
T
,  iI  ri   iI rI
j 1
The estimation process ensures the average error
is zero
I
 The value of  i is the gradient of the fitted line

Example
British American Tobacco
Barclays
rBarclays
rBAT 10
30
8
20
6
10
4
0
-10
2
-5
0
-10
0
-10
-5
-2
0
5
10
5
10
rFT 100
-20
rFT 100
-4
-30
-6
-40
y = 2.6642x - 0.62
R² = 0.6304
rBAT  1.4379  0.4925 rFT 100  
rBarclays  0.62  2.6642 rFT 100  
R 2  0.2183
R 2  0.63 .04
Example
30
20
10
BAT
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Barclays
Linear (BAT)
-10
-20
-30
-40
Linear (Barclays)
Model

If the model is applied to all assets it need not
follow that

I
I
cov  i ,  k

 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:
I
i
I
i  2
I
covariance of i with index

variance of index

Note:

And

These observations permits a characterization
of assets
I
I
1
I
I
0
Assets Types
If iI  1 then the
asset is more volatile
(or risky) than the
market
 This is termed an
“aggressive” asset

ri
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   iI then
  Eri  ri 
2
2
i
 E iI   iI rI   iI   iI   iI rI 
2
 E iI rI  rI    iI 
2

 E  rI  rI   2 iI iI rI  rI   
2
iI
2
2
iI

Risk

This can be written
    
2
i

2
iI
2
I
2
i
So risk is composed of two parts:
1. market (or systematic) risk
 
2
iI
2
I
2. unique (or unsystematic) risk
 2i
Return

Portfolio return
N
rp   X i ri
i 1
N
  X i  iI   iI rI   iI 
i 1
N
N
N
i 1
i 1
  X i iI   X i  iI rI   X i iI
i 1
  pI   pI rI   pI
Return

Hence


rp  E  pI   pI rI   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

E
pI
  pI rI   pI   pI   pI
2
pI
rI  rI 
   
2
pI
2
I
2
2
p
r
2
I
 2 pI rI  rI  pI  
2
pI

Risk

The final expression can also be written
2
N



2
2
2 2
 p   X i iI   I   X i  i 
 i 1

 i 1

N
Consequence: now need to only know  I
2
and  i, i = 1,...,N
 For example, for FT 100 need to know 101
variances (reduced from 5050)

2
Diversified Portfolio


1
A large portfolio that is evenly held X i 
N
The non-systematic variance is
2
N
N


 1  2
2
2 2
 p   X i  i       i 
 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
p
tends to
  
2
p
2
pI
2
I
 is undiversifiable market risk
  is diversifiable risk

2
I
2
p
Market Model

A special case of the single-index model
 The index is the market


The market model has two additional
properties



The set of all assets that can be purchased
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
