Chapter 10
Index models for
Portfolio Selection
By
Cheng Few Lee
Joseph Finnerty
John Lee
Alice C Lee
Donald Wort
1
Chapter Outline
2
•
10.1 THE SINGLE-INDEX MODEL
•
10.1.1Deriving the Single-Index Model
•
10.1.1.1 Expected Return of a Portfolio
•
10.1.1.2 Variance of a Portfolio
•
10.1.2 Portfolio Analysis and the Single-Index Model
•
10.1.3 The Market Model and Beta
•
10.2 MULTIPLE INDEXES AND THE MIM
•
10.3 SUMMARY
•
•
The essential difference between the single- and multiple-index models is the
assumption that the single-index model explains the return of a security or a
portfolio with only the market.
The multiple-index model describes portfolio returns through the use of more than
one index.
10.1 The Single-Index Model
•
•
•
Suggested by Markowitz (1959), the single-index model was fully developed by
Sharpe (1970), who assumed that the covariances could be overlooked.
The return of an individual security was tied to two factors－a random effect and
the performance of some underlying market index.
Notationally:
Rit  a i  bi R It  eit
where:
ai and bi = regression parameters for the ith firm;
R It = the return of some underlying market index at time t;
R it = the return on security I at time t; and
e it = the random effect for the ith security at time t.
(10.1)
•
•
•
Equation (10.1) makes several assumptions about the random effect term, full
details are in text book page 354.
The expected value of the return on security i is equal to the sum of the intercept,
the adjusted return on the index, and some random effect.
This can be expressed:
E  R i   E  a i  bi R It  e it 
 E  a i   E  bi R It   E  e it 
•
(10.2)
Because a i and bi are constants, andE  e it  is equal to zero:
ri  E  R i   a i  bi rI
Where rI is equal to the mean of the returns on the market index.
(10.3)
•
•
If the mean of the returns on security i equals then the variance of security i is
equal to the expected value of squared deviations from .
This translates to

2
i
 E  R it  ri 
2
 E   a i  bi R It  e it    a i  bi rI  
 E  bi  R It  rI   e it 
 b E  R It  rI
2
i
•
•

2
2
 2 bi E  e it  R It  rI    E  e it 
2
2
(10.4)
By the third assumption, the covariance of the random effect and the deviation
of the index return from its mean are zero; also, the expected value of the
squared errors is equal to the variance of the random effect.
2
2
Thus:
2
2
 b E R r E e
i
i

It
I
 bi  I   ei
2
2
2

 it 
(10.5)
•
•
It is suggested that because variations in the returns of two different securities are
not interrelated but only connected to the market index, the covariance between
the two securities can be derived from the twice-adjusted variance of the market
index (once by the b coefficient of the first security with the market and again by
the b coefficient of the second security with the market).
The investigation starts with a statement about the covariance of the two securities:
 ij  E   R it  ri   R jt  r j  


 E [(( a i  bi R It  e it )  ( a i  bi rI ))
 (( a j  b j R It  e jt )  ( a j  b j rI )]
 E   bi  R It  rI   e it    b j  R It  rI   e jt  


 bi b j E  R It  rI

2
 bi E  e jt  R It  rI
 b j E  e it  R It  rI
   E  e it e jt 

(10.6)
•
Since according to the third and fourth assumptions the last three terms of the
last summation are equal to zero:
 ij  bi b j  I
2
•
•
•
•
(10.7)
For a portfolio with a hundred securities, this translates to 5,050 calculations.
With the Sharpe single-index model, only 100 estimates are needed for the
various security-regression coefficients and only one variance calculation, the
variance of the returns on the market.
2
In addition, 100 estimates of the unsystematic risk  ei are also needed for the
single-index model.
Hence, the single-index model has dramatically reduced the input information
needed.
10.1.1 Deriving the Single-Index Model
•
So far only the Sharpe single-index model has been utilized to study the return
of a single security i as determined by its relation to the returns on a market
index.
10.1.1.1 Expected Return of a Portfolio
•
•
•
Now consider the return of a portfolio of n securities.
The return of a portfolio of n securities is the weighted summation of the
individual returns of the component securities.
Notationally:
n
E  R pt  
 x E R 
i
i
i 1
where Rpt is the rate of return for a portfolio in period t and Xi is the weight
associated with the ith security.
Rit  a i  bi R It  eit
n
R pt 
 x a
i
i
 bi R It  e it 
i 1
n

 x a
i
i
n
 e it  
 x b R 
i
i 1
n

•
(10.8a)
i 1
n
n
 x a   x b R    x e
i
i
i
i 1
i 1
n
n
i
It
i it
i 1
 x a    x b E  R 
i
i 1
•
It
Thus,
E  R pt  
•
i
i
i
i
It
(10.8b)
i 1
This equation indicates that the return of a portfolio may be decomposed into
the summation of the weighted returns peculiar to the individual securities and
the summation of the weighted adjusted return on the market index.
Thus the portfolio may be viewed as a combination of n basic securities and a
weighted adjusted return from an investment in the market index.
10.1.1.2 Variance of a Portfolio
•
To derive the variance of the portfolio  p , consider first that the mean return of
the portfolio is equal to the expected value of the return on the portfolio.
2

2
p

 E  R pt  E  R pt 

n

x
2
i
n
2
ei

i 1

2


n
  x x bb 
i
i 1
j
i
j
(10.9)
j 1
 2
 n
 n
   x i bi    x j b j   i 
 i 1
  j 1

•
2
I
n
x
2
i
 ei
2
i 1
Because the weighted sum of the coefficients is equal to the coefficient of the
portfolio , similarly . Hence, the last equation reduces to
n

2
p
b 
2
p
2
I


x i  ei
2
2
(10.10)
i 1
•
In Equation (10.10) the last term, the weighted sum of the random effect
variances, approaches zero as n increases.
Sample Problem 10.1
•
Given the following information, what should the  of the portfolio (bp) be?
n

2
p
 0.082

2
I
 0.021
x
2
i
 ei  0.04
2
i 1
Solution:
Substituting the related information into Equation (10.10):
n

2
p
2
2
i 1
bp 
2

-  x i  ei

2
p
0 .0 8 2  0 .0 4
0 .0 2 1
 2 .0
Therefore, bp = 1.414
•
Equation (10.8b) implies that the portfolio can be viewed as an investment in n
basic securities and a weighted adjusted return in the market, or
n
 x b E R 
i
i
It
i 1
•
•
•
The return on the market can be decomposed as a combination of the expected
return plus some random effect.
When this random effect is positive, the atmosphere is bullish and when it is
negative, the atmosphere is bearish.
Notationally:
R It  E  R It   e n  1, t
 a n  1  e n  1, t
•
(10.11)
This bit of algebraic maneuvering enables the weighted adjusted investment in
the market to be viewed as an investment in an artificial security, the (n+1)th of
an n-security portfolio.
•
The weight for this (n+l)th security is the sum of the n weights multiplied by
their respective related coefficients to the market index. Thus:
n
x n 1 
 xb
i
i
(10.12)
i 1
•
•
The reason for this divergence in notation resulting in the definition of the
(n+l)th security’s return and weight is that Equation (10.8) can be simplified to
yield a working model for portfolio analysis.
Substituting the last results for X n 1 and R It into Equation (10.8B) yields:
n
R pt 
 x a
i
i
 e it   x n  1  a n  1  e n  1, t 
i 1
n 1

 x a
i
i 1
i
 e it 
(10.13)
•
Because the expected value of the random-effect terms is zero, the summation
that results after the application of the expectations operator can be expressed as
E  R pt  
n
xa
i
 x n  1 E  R It  
i
i 1
•
•
•
n 1
xa
i
i
(10.14)
i 1
This yields a formula for the return of a portfolio that is easily applied to
portfolio analysis.
Before proceeding, however, it is necessary to simplify the variance formula so
that it may be used as well.
Remembering that the variance of the portfolio is the expected value of the
squared deviations from the expected market return, the last results concerning
and may be applied:
2
V ar  R p t 
  n 1
  n 1

 n 1


 E    x i  a i  e it    E   x i  a i  e it     E    x i  a i  e it   

 i 1


  i 1
  i 1
  n 1

 E    x i  a i  e it   

  i 1
n 1

i 1
2

 n 1

x i E  a i    E   x i e it 
 i 1


n 1

i 1

x i  E  a i   E  e it   

2
2
(10.15)
 n 1

 V ar   x i e it 
 i 1

•
This result follows from the assumption that the covariances are equal to zero.
Sample Problem 10.2
Givenn RIt , Rit and  as indicated in the Table 10.1 shown below and the fact
i
that   i R It  56 , using the relationship
t 1
n
R
n
it
 n i 
t 1

t 1
find the values for  i ,  i and
n
i
R It 
e
it
t 1
ei .
Table 10.1 Single Index Model
•
Solution:
n

n

R It  28, so  i 

t 1
It

R It
t 1
n
Substitute the values  R
R It
t 1
n
t 1
•
i
56
2
28
n
 64,   i R It  56
t 1
n
R
, n = 4 into the regression line
n
It
 n i 

n
i
R It 
e
it
•
and solve for  i .
•
The  i R It column in Table 10.1 is filled by simply multiplying  (  2) by the R It
column.
The ei‘s are the amounts such that Rit     i R m t  eit is an equality, so
R it   i   i R It  eit is satisfied.
From the last column of the table we know that  e  0
Therefore
64  56  0
•
t 1
t 1
t 1
n
•
it
i 1
•
•
i 
2
4
In this section, the expected return E ( R pt ) and the variance of a portfolio
in terms of the single-index model has been derived.
V ar ( R pt )
10.1.2 Portfolio Analysis and the Single-Index Model
•
The maximization procedure maximizes a linear combination of the following
two equations:
M ax E  R p  
n 1
 x E R 
i
i
or  V ar  R p 
i 1
n 1
subject to
x
•
•
(10.16a)
n
 1 and x n  1 
i
i 1
•
 n 1

  V ar   x i e it 
 i 1

 xb
i
(10.16b)
i
i 1
The first constraint is equivalent to requiring that the sum of the weights of the
component securities is equal to one.
The second constraint requires that the weight of the market index within the
portfolio returns is equal to the summation of the weighted adjustment factors of
the component securities.
Combining the above two objective functions with the two constraints yields a
Lagrangian function:




P   E  R   V ar  R      x  1      x b  x

n
p
n 1
   xi a i 
i 1
p
n 1

i 1
1

n
i
i 1

2

i
i 1
i
n 1

 n

 n

V ar  e it   1   x i  1    2   x i bi  x n  1 
 i 1

 i 1

•
•
•
•
•
When low values are exhibited, risk aversion is pronounced; when high values
are in evidence, substantial risk taking is allowed.
This notion of the level of risk aversion is best pictured in Figure 10.1.
In Figure 10.1, A represents an investor’s objective function to minimize the risk
only－therefore, the aggressiveness to return is zero.
C represents an investor’s objective function to maximize returns only－
therefore, the aggressiveness to return is infinite.
At B an investor’s attitude toward return and risk is between A and C.
Figure 10.1 Level of Risk Aversion and Investors’ Investment Attitude
•
•
•
•
•
•
Figure 10.2 provides further illustration of the approach being utilized.
What is depicted is the variation of the objective function P as the parameter
denoting risk is varied.
Points A and C are as discussed for Figure 10.1.
Lines BD and B’D’ represent the objective function when the risk-aversion
parameter is equal to one.
Note that BD instead of represents the maximization of the objective function.
In sum, different values of  generate different optimal objective functions.
Figure 10.2 Level of Risk Aversion and Object Function
•
Now consider the three-security portfolio. In this framework, the preceding
objective function expands to:
P   x1 a1   x 2 a 2   x 3 a 3   x 4 a 4  x1 V ar  e1 t 
2
 x 2 V ar  e 2 t   x 3 V ar  e 3 t   x 4 V ar  e 4 t   1 x1  1 x 2
2
2
2
(10.17)
 1 x 3  1   2 x1b1   2 x 2 b 2   2 x 3 b3   2 x 4
•
•
•
To proceed with the Lagrangian maximization it is necessary to notice that the
above equation has six unknowns — the four weights and the two Lagrangian
coefficients.
All other values are expected to be known or estimated.
To maximize, the partial derivative of the objective function is taken with respect
to each of the six variables:  P   a  2 x      b  0
 x1
P
x2
P
 x3
P
x4
P
 1
P
2
1
1
2
1
  a 2  2 x 2
1
2
2
2
1
w h ere
 1   2 b 2  0
 1  V ar  e1 t  ;
2
  a 3  2 x 3 3   1   2 b 3  0
2
  a 4  2 x 4
2
4
 2  0
 2  V ar  e 2 t  ;
2
(10.18)
 3  V ar  e 3 t  ; an d
2
 4  V ar  e 4 t  .
2
 x1  x 2  x 3  1  0
 x1 b1  x 2 b 2  x 3 b3  x 4  0
•
Equation (10.18) is a set of six equations in six unknowns when set equal to zero.
These six equations can be rewritten in matrix format:
A
  2 12

 0
 0

 0
 1

 b1
•
•
0
x
0
0
1
 2 2
0
0
1
0
 2 3
0
1
2
2
0
0
 2
1
1
0
0
b2
b3
1
0
2
4
0


b2 
b3 

1 
0 

0 
b1
 x1 
 
x
 2
 x3 
 
 x4 
 
1
 
  2 

k
   a1 


 a2


   a3 
 



a
4 

 1 


 0 
(10.19a)
Equation (10.19A) has related the individual securities to the index and has
discarded the use of numerous covariance terms. The majority of the elements of
the matrix indicated in the first matrix of Equation (10.19A) are zero.
The majority of the elements of the matrix indicated in the first matrix of Equation
(l0.19a) are zero. For three securities case, the inputs needed for Equation (10.19a)
are three alphas, three betas, and three residual variances for individual securities.
•
•
As in Chapter 8, to solve for the x column vector of variables the P matrix is
1
inverted and each side of the equation is premultiplied by A .
This yields the x column vector of variables on the left-hand side and A  1
multiplied by k, the solution column vector, on the right-hand side.
Ax  k
1
1
A Ax  A k
(10.19b)
1
Ix  A k
•
where I is the identity matrix.
•
In order to estimate a single-index type of optimal portfolio the estimates of a
market model are needed; a discussion of the market model and beta estimates is
therefore needed as well.
Sample Problem 10.3
•During the discussion of the single-index model, a method was presented for
determining optimal portfolio weights given different levels of risk aversion.
•In this section a three-security portfolio is examined in which the returns of
the securities are related to a market index.
Ri
JNJ
IBM
BA
S&P 500
0.0080
0.0050
0.0113
−0.0009
V ar ( R i )
0.0025
0.0071
0.0083
0.0021
C ov  R i , R j 
bi
0.0007
0.0006
0.0007
0.0021
0.37
1.20
1.04
1.00
Table 10.2 Single Index Model: Three Securities Case
•The securities and the index have the following observed parameter
estimates taken from actual monthly returns during the period from April
2000 to April 2010 (see Table 10-2).
•The table includes the calculation of the beta estimate for each security.
•Check these figures, remembering that beta is equal to the covariance of
security i with the market divided by the variance of the market returns.
A matrix can be developed by utilizing the set of data in the first table of this
problem and a  of 1.0 (denoting moderate risk aversion):
A
  0.0050

0


0

0

 1.0

 0.37
x

k
0
0
0
1
0.37
 0.0142
0
0
1
1.2
0
 0.0166
0
1
1.04
0
0
 0.0042
0
0
1.0
1.0
0
0
0
1.2
1.04
 1.0
0
0









 x1    0.00 80 
  

x2
 0.0050
  

 x 3    0.0113 
  

x
0.0009
 4 

    1.0 
1
  

0
  2  

When the P matrix is invented and premultiplies each side of the equation:
x

1
A k
 x1   1.7092 
  

x2
 0.6821
  

 x 3    0.0271 
  

x

0.2143
 4 

    0.0073 
1
  

  2    0.0184 
•
•
•
This solution vector shows that investment should be divided 170.92% in JNJ,
−68.21% in IBM, and −2.71% in BA.
Additionally, the weight of x 4 is the sum of the weighted adjustments as
indicated in Equation (10.16b).
The return on this portfolio is the weighted sum of the individual returns:
E Rp  
n
 x E R 
i
i
i 1
   1.7092  0.0080     0.6821  0.0050     0.027 1  0.0113  
 0.0100
•
The variance of the portfolio is the sum of the weighted variance and
covariance terms: V ar R  n n x x 
  
p
i 1
j 1
i
j
ij
•
Taking the covariance expressions:
V ar  R p   [(1.7092  0.0025)  (  0.6821  0.0071)+ (  0.0271  0.0083)
2
2
2
 2  (1.7092   0.6821  0.0007 )  2  (1.7 092   0.0271  0.0007 )
 2  (  0.6821   0.0271  0.0006)
 0.0089
•
•
•
A portfolio has been developed that is efficient within the realm of this model.
By varying the utility factor  from risk averse to more aggressive risk posture
we can develop an efficient frontier under the Sharpe single-index model.
For a fuller graph, the analyst would continue the calculations with ever higher
values of  stretching the efficient frontier.
10.1.3 The Market Model and Beta
•
Equation (10.1) defines the market model:
Rit  a i  bi R It  eit
•
•
•
•
From this market model ai, bi, Var(ei), and Var(RIt) can be estimated: all are
required for the single-index type of optimal portfolio.
So far the regression coefficient has been referred to as bi when, in fact, it is a
risk relationship of a security with the market.
Quantified as the covariance of security i with the market index divided by the
variance of the return of the market, beta is a relational coefficient of the returns
of security i as they vary with the returns of the market.
Notationally:
n  R  r

  R  r 

bi 
•
•
(10.1)

i.I
2
I


t 1

it
it
 R It
It
 rIt 
It
2

(10.20)
Of course, due to the continuous nature of security returns, the result of the above
calculation is an estimate and therefore subject to error.
This error can be quantified by the use of standard error estimates, which provide
the ability to make interval estimations for future predictions.
•
•
•
•
•
•
•
Because of the linkage between security returns and the firm’s underlying
fundamental nature, beta estimates will vary over time.
It is the job of the security analyst to decide how to modify not only the beta
estimate but also the period of time from which the sample of returns is to be
drawn.
Some popular security-evaluation techniques use a tiered growth model to
correspond to the product life-cycle theory.
Within the scope of this theory, beta estimates for the relation of that security to
the market will vary with respect to the time period chosen for the sample of
returns.
It is obvious that the returns generated by a firm in its infancy have very little
correlation with returns during the growth or maturity phase.
It is up to the analyst to judge which phase a company may be in and to adjust
the beta estimates accordingly.
This section has briefly looked over some of the available adjustment processes
for beta estimation.
10.2 Multiple Indexes and the MIM
•
•
•
•
The multi-index model (MIM) pursues the same problem as beta adjustment,
but approaches the problem from a different angle.
The MIM tackles the problem of relation of a security not only to the market by
including a market index but also to other indexes that quantify other movements.
The covariance of a security’s return with other market influences can be added
directly to the index model by quantifying the effects through the use of
additional indexes.
If the single-index model were expanded to take into account interest rates,
factory orders, and several industry-related indexes, the model would change to:
R i  a i  bi 1 I 1  bi 2 I 2 
*
•
*
*
*
*
 biM I M  c i
*
*
(10.21)
In this depiction of the MIM, I is the actual level of index j, while b ij* is the
actual responsiveness of security i to index j.
*
j
•
Assume there is a hypothetical model that deals with two indexes:
Ri  a  b I  b I  ci
*
i
•
•
•
* *
i1 1
*
i2
*
2
(10.22)
Suppose the indexes are the market index and an index of wholesale prices.
If these two indexes are correlated, the correlation may be removed from either
index.
*
*
To remove the relation between I 1 and I 2 , the coefficients of the following
equation can be derived by regression analysis:
I 2  e 0  e1 I 1  d i
*
w here
e 0 and e1  the regression coefficients;
d i  the random error term .
*
•By the assumptions of regression analysis, is uncorrelated with. Therefore:
I 2  e 0  e1 I 1  d i
*
*
which is an index of the performance of the sector index without the effect of I1(the
market removed). Defining:
*
*
ˆ
ˆ
ˆ
d i  I 2   e 0  e1 I 1 
*
an index is obtained that is uncorrelated with the market. By solving for I 2 and
substituting into Equation (10.22):
R i  a i  bi1 I 1  bi 2 I 2  bi 2 eˆ0  bi 2 eˆ1 I 1  c i
*
Rearranging
*
*
*
*
*
*
R i   a i  bi 2 eˆ0    bi1  bi 2 eˆ1  I 1  bi 2 I 2  c i
*
*
*
*
*
*
•If the first set of terms in the brackets are defined asand the second set of terms
*
*
are defined as bi1 , b i 2 = bi 2 , I 1 = I and ci=ei , the equation can be expressed:
1
R i  a i  bi1 I 1  bi 2 I 2  ei
(10.23)
in which I 1 and I 2 are totally uncorrelated: the goal has been achieved.
•As will be seen later, these simplifying calculations will make the job of
determining variance and covariance much simpler.
The expected return can be expressed:
E  R it   E  a i  bi 1 I 1t  bi 2 I 2 t  e it 
 E  a i   E  bi1 I 1t   E  bi 2 I 2 t   E  e it 
(10.24)
 a i  bi 1 I 1  bi 2 I 2
Variance can be expressed:

2
 E  R it  R i 
w h ere R i  E  R it 
2
 E [( a i  bi 1 I 1  bi 2 I 2  e it )  ( a i  bi 1 I 1  bi 2 I 2 )]
 E [( a1  a i )  bi 1 ( I 1 t  I 1 )  bi 2 ( I 2 t  I 2 )  e it ]
 E [ bi 1 ( I 1 t  I 1 )  2 bi 1bi 2 ( I 1 t  I 1 )( I 2 t  I 2 )
2
2
 bi 2 ( I 2 t  I 2 )  2 bi 1 ( I 1 t  I 1 ) e it
2
2
 2 bi 2 ( I 2 t  I 2 ) e it  e it
2
 bi 1 E ( I 1 t  I 1 )  bi1bi 2 E [( I 1 t  I 1 )( I 2 t  I 2 )]
2
2
 bi 2 E ( I 2 t  I 2 )  bi1 E [( I 1 t  I 1 ) e i ]
2
 bi 2 E [( I 2 t  I 2 ) e i ]  E ( e i )
2
2
2
But by assumption:
E ( I 1t  I 1 )   1
2
E [( I 1t  I 1 )( I 2 t  I 2 )]  0
E [( I 1t  I 1 ) ei ]  0
2
E ( I 2t  I 2 )   2
2
and
E [( I 2 t  I 2 ) ei ]  0
2
E ( e i )   ei
2
Therefore,  i  bi1 1  bi 2 2   ei
2
2
2
2
2
2
2
(10.25)
Covariance between security i and security j can be expressed:
 ij  E [( R it  R i )( R jt  R j )][ w here R i  E ( R it ) and R j  E ( R jt )
 E {[( a i  bi1 I 1  bi 2 I 2 t  e1 t )  ( a i  bi1 I 1  bi 2 I 2 )]
 [( a j  b j 1 I 1 t  b j 2 I 2 t  e jt )  ( a j  b j 1 I 1  b j 2 I 2 )]}
 E {[ bi 1 ( I 1 t  I 1 )  bi 2 ( I 2 t  I 2 )  e i ]  [ b j 1 ( I 1 t  I 1 )
 ( b j 2 ( I 2 t  I 2 )  e jt ]}
 E [ bi 1b j 1 ( I 1 t  I 1 )  bi 2 b j 2 ( I 2 t  I 2 ) ]
2
2
 bi1b j 1 1  bi 2 b j 2 2
2
2
(10.26)
10.3 SUMMARY
•
•
•
•
This chapter has discussed the essentials of single- and multiple-index models.
The theoretical underpinnings of the theories have been explored and
numerical examples have been provided.
An efficient boundary has been derived under the guidelines of the model and
the quantitative analysis of the related parameters has been studied.
It has been shown that both single-index and multi-index models can be used
to simplify the Markowitz model for portfolio section; remember, however,
that the multi-index model is much more complicated than the single-index
model.