Elton, Gruber, Brown, and Goetzmann
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions to Text Problems: Chapter 8
Chapter 8: Problem 1
Given the correlation coefficient of the returns on a pair of securities i and j, the
securities covariance can be expressed as the securities correlation coefficient
times the product of their standard deviations:
ij = ij i j
But if we assume that all pairs of securities have the same constant correlation, * ,
then the constant-correlation expression for covariance is:
CC ij = * i j
Given the assumptions of the Sharpe single-index model, the single-index models
expression for the covariance between the returns on a pair of securities is:
2
SIM ij = i j m
im jm 2
= 2
2 m
m m
= im 2i m jm 2j m m
2
m m
= im jm i j
If the assumptions of both the constant correlation and single-index model hold,
then we have CC ij = SIM ij :
* i j = im jm i j or * = im jm
This must hold for all pairs of securities, including i and j, i and k and j and k. So we
have:
* = im jm
* = im km
* = jm km
The only solution to the above set of equations is:
im = jm = km = *
Therefore, for any security i we have:
im im i m im i *
i = = = = i
m 2 2
m m m
In other words, given that all pairs of securities have the same correlation
coefficient and that the Sharpe single-index model holds, each securitys beta is
proportional to its standard deviation, where the proportion is a constant across all
*
securities equal to .
m
Elton, Gruber, Brown, and Goetzmann 8-1
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 8
Chapter 8: Problem 2
Start with a general 3-index model of the form:
Ri = ai* + bi*1 I1* + bi*2 I2* + bi*3 I3* + ci (1)
Set I1* = I1 and define an index I2 which is orthogonal to I1 as follows:
I 2* = 0 + 1 I1 + d t or I2 = dt = I2* ( 0 + 1 I1)
which gives:
I2* = 0 + 1 I1 + I2
Substituting the above expression into equation (1) and rearranging we get:
( ) ( )
Ri = ai* + bi*2 0 + bi*1 + bi*2 1 I1 + bi*2 I2 + bi*3 I3* + ci
The first term in the above equation is a constant, which we can define as a1 . The
coefficient in the second term of the above equation is also a constant, which we
can define as bi1 . We can then rewrite the above equation as:
Ri = ai + bi1 I1 + bi*2 I2 + bi*3 I3* + ci (2)
Now define an index I3 which is orthogonal to I1 and I2 as follows:
I3* = 0 + 1 I1 + 2 I2 + et or I 3 = et = I 3* ( 0 + 1 I1 + 2 I 2 )
which gives:
I 3* = 0 + 1 I1 + 2 I 2 + I 3
Substituting the above expression into equation (2) and rearranging we get:
( )
Ri = ai + bi 3 0 + bi1 + bi 3 1 I1 + bi*2 + bi*3 2 I2 + bi*3 I3 + ci
In the above equation, the first term and all the coefficients of the new orthogonal
indices are constants, so we can rewrite the equation as:
Ri = ai + bi1 I1 + bi 2 I2 + bi 3 I3 + ci
Elton, Gruber, Brown, and Goetzmann 8-2
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 8
Chapter 8: Problem 3
Recall from the earlier chapter on the single-index model that an expression for
the covariance of returns on two securities i and j is:
( ) [(
ij = i j E Rm Rm + j E ei Rm Rm + i E ej Rm Rm + E[ei ej ]
2
)] [ ( )]
The first term contains the variance of the market portfolio, the second two terms
contain the covariance of the market portfolio with the residuals and the last term
is the covariance of the residuals.
Given that one of the models assumptions is that the covariance of the market
portfolio with the residuals is zero and that, from the problem, the covariance of
the residuals equals a constant K, the derived covariance between the two
securities is:
2
ij = i j m +K
One expression for the variance of a portfolio is:
N N N
P2 = Xi2 i2 + X j Xk jk
i =1 j =1 k =1
k j
Recalling that the single-index models expression for the variance of a security is
i2 = i2 m
2
+ ei2 and substituting that expression and the derived expression for
covariance into the above equation and rearranging gives:
N N N N N N
P2 = Xi2 i2 m
2
+ Xi2 ei2 + X j Xk j k m
2
+ X j Xk K
i =1 i =1 j =1 k =1 j =1 k =1
k j k j
N N N N N
=
i =1 j =1
2
Xi X j i j m +
i =1
Xi2 ei2 + X X K
j =1 k =1
j k
k j
N
N
2 N N N
= Xi i Xi i m + Xi2 ei2 + X X K j k
i =1 i =1 i =1 j =1 k =1
k j
N N N
= 2
P
2
m +
i =1
Xi2 2
ei + K
j =1 k =1
X j Xk
k j
Elton, Gruber, Brown, and Goetzmann 8-3
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 8
Chapter 8: Problem 4
Using the result from Problem 2, we have:
Ri = ai + bi1 I1 + bi 2 I2 + bi 3 I3 + ci
Since the residual ci always has a mean of zero (by construction if necessary), we
have the following expression for expected return:
R i = a i + b i1 I1 + b i 2 I 2 + b i 3 I 3
The variance formula is:
( (
i2 = E ai + bi1 I1 + bi 2 I2 + bi 3 I3 + ci ai + bi1 I1 + bi 2 I2 + bi 3 I3
))
2
= E (b (I I ) + b (I ) ( ) )
I2 + bi 3 I3 I3 + ci
2
i1 1 1 i2 2
Carrying out the squaring, noting that the indices are all orthogonal with each
other and making the usual assumption that the residual is uncorrelated with any
index gives us:
i2 = bi21 I21 + bi22 I22 + bi23 I23 + ci2
The covariance formula is:
( (
ai + bi1 I1 + bi 2 I2 + bi 3 I3 + ci ai + bi1 I1 + bi 2 I2 + bi 3 I3
i2 = E
))
( ( ))
a j + b j1 I1 + b j 2 I2 + b j 3 I3 + c j a j + b j1 I1 + b j 2 I2 + b j 3 I3
[( ( ) ( ) ( ) ) ( ( ) ( ) (
= E bi1 I1 I1 + bi 2 I2 I2 + bi 3 I3 I3 + ci b j1 I1 I1 + b j 2 I2 I2 + b j 3 I3 I3 + c j ) )]
Carrying out the multiplication, noting that the indices are all orthogonal with
each other, making the usual assumption that the residuals are uncorrelated with
any index and assuming that the residuals are uncorrelated with each other gives
us:
ij = bi1b j1 I21 + bi 2 b j 2 I22 + bi 3 b j 3 I23
Elton, Gruber, Brown, and Goetzmann 8-4
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 8
Chapter 8: Problem 5
The formula for a security's expected return using a general two-index model is:
Ri = ai + bi1 I1 + bi 2 I 2
Using the above formula and data given in the problem, the expected return for,
e.g., security A is:
R A = a A + b A1 I1 + b A2 I 2
= 2 + 0.8 8 + 0.9 4
= 12%
Similarly:
RB = 17% ; RC = 12.6%
The two-index models formula for a securitys own variance is:
i2 = bi21 I21 + bi22 I22 + ci2
Using the above formula, the variance for, e.g., security A is:
A2 = b A2 1 I21 b A2 2 I22 + cA
2
= (0.8 ) (2 ) + (0.9) (2.5) + (2 )
2 2 2 2 2
= 2.56 + 5.0625 + 4 = 11.6225
Similarly, 2B = 16.4025, and 2C = 13.0525.
C. The two-index model's formula for the covariance of security i with security j is:
ijj = bi1b j1 I21 + bi 2 b j 2 I22
Using the above formula, the covariance of, e.g., security A with security B is:
AB = b A1bB1 I21 + b A2 bB2 I22
= (0.8 )(1.1)(2 ) + (0.9)(1.3 )(2.5)
2 2
= 3.52 + 7.3125 = 10.8325
Similarly, AC = 9.0675, and BC = 12.8975.
Elton, Gruber, Brown, and Goetzmann 8-5
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 8
Chapter 8: Problem 6
For an industry-index model, the text gives two formulas for the covariance
between securities i and k. If firms i and k are both in industry j, the covariance
between their securities' returns is given by:
2
ik = bim bkm m + bij bkj Ij2
Otherwise, if the firms are in different industries, the covariance of their securities'
returns is given by:
2
ik = bim bkm m
If only firms A and B are in the same industry, then:
2
AB = b Am bBm m + b A2 bB 2 I22
= (0.8 )(1.1)(2 ) + (0.9)(1.3 )(2.5)
2 2
= 3.52 + 7.3125 = 10.8325
The second formula should be used for the other pairs of firms:
2
AC = b Am bCm m
= (0.8 )(0.9)(2 ) = 2.88
2
2
BC = bBm bCm m
= (1.1)(0.9)(2 ) = 3.96
2
Chapter 8: Problem 7
The answers for this problem are found in the same way as the answers for problem 6,
except that now only firms B and C are in the same industry. So for firms B and C, the
covariance between their securities' returns is:
2
BC = bBm bCm m + bB2 bC2 I22
= (1.1)(0.9)(2 ) + (1.3 )(1.1)(2.5)
2 2
= 3.96 + 8.9375 = 12.8975
Elton, Gruber, Brown, and Goetzmann 8-6
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 8
The other formula should be used for the other pairs of firms:
2
AB = b Am bBm m
= (0.8 )(1.1)(2 ) = 3.52
2
2
AC = b Am bCm m
= (0.8 )(0.9)(2 ) = 2.88
2
Chapter 8: Problem 8
To answer this problem, use the procedure described in Appendix A of the text.
First, I1 is defined as being equal to I*1 , then I*2 is regressed on I1 to obtain the given
regression equation. Since dt is uncorrelated with I1 by the techniques of regression
analysis, dt is an orthogonal index to I1. So, define I2 = dt. Then express the given
regression equation as:
I*2 = 1 + 1.3 I1 + I2.
Now, substitute the above equation for I*2 into the given multi-index model and
simplify:
Ri = 2 + 1.1 I*1 + 1.2 I*2 + ci
= 2 + 1.1 I1 + 1.2 (1 + 1.3 I1 + I2) + ci
= 2 + 1.1 I1 + 1.2 + 1.56 I1 + 1.2 I2 + ci
= 3.2 + 2.66 I1 + 1.2 I2 + ci
The two-index model has now been transformed into one with orthogonal indices
I1 and I2, where I1 = I*1, and I2 = dt = I*2 - 1 - 1.3 I1.
Elton, Gruber, Brown, and Goetzmann 8-7
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 8
