Investments, tenth edition

  Summary of Optimization Procedure 

 Once security analysis is complete, the optimal risky portfolio is formed from the index-

model estimates of security and market index parameters using these steps:

    1.  Compute the initial position of each security in the active portfolio as 

   w

i

0

5 a

i

/s

2

(e

i

).   

   2.  Scale those initial positions to force portfolio weights to sum to 1 by dividing by 

their sum, that is,    w

i

5

w

i

0

a

n

i

51

w

i

0

.   

   3.  Compute the alpha of the active portfolio:    a

A

5 a

n

i

51

w

i

a

i

.   

   4.  Compute the residual variance of the active portfolio:    s

2

(e

A

)

5 a

n

i

51

w

i

2

s

2

(e

i

).   

 5.  Compute the initial position in the active portfolio: w

e

E R

A

A

A

M

M

0

2

(

)

(

)

.

2

   6.  Compute the beta of the active portfolio:    b

A

5 a

n

i

51

w

i

b

i

.   

   7.  Adjust the initial position in the active portfolio:    w

A

*

5

w

A

0

1

1 (1 2 b

A

)w

A

0

.   

   8.  Note: the optimal risky portfolio now has weights:    w

M

*

5 1 2 w

A

*

; w

i

*

5 w

A

*

w

i

.   

   9.  Calculate the risk premium of the optimal risky portfolio from the risk 

premium of the index portfolio and the alpha of the active portfolio: 

   E(R

P

)

5 (w

M

*

1 w

A

*

b

A

)E(R

M

)

1 w

A

*

a

A

.  Notice that the beta of the risky portfolio 

is     w

M

*

1 w

A

*

b

A

  because the beta of the index portfolio is 1.  

   10.  Compute the variance of the optimal risky portfolio from the  variance 

of the index portfolio and the residual variance of the active portfolio: 

   s

P

2

5 (w

M

*

1 w

A

*

b

A

)

2

s

M

2

1 3w

A

*

s(e

A

)

4

2

.      

  An Example 

 We can illustrate the implementation of the index model by constructing an optimal port-

folio from the S&P 500 index and the six stocks for which we analyzed risk parameters in 

Section 8.3. 

 This example entails only six analyzed stocks, but by virtue of selecting three  pairs   of 

firms from the same industry with relatively high residual correlations, we put the index 

model to a severe test. This is because the model ignores the correlation between residuals 

when producing estimates for the covariance matrix. Therefore, comparison of results from 

the index model with the full-blown covariance (Markowitz) model should be instructive.