Investments, tenth edition

   The Regression Equation of the Single-Index Model 

 Because rates of return on market indexes such as the S&P 500 can be observed, we have 

a considerable amount of past data with which to estimate systematic risk. We denote the 

market index by  M,  with excess return of  R  

 M 

   5   r  

 M 

   2   r  

 f 

  , and standard deviation of  s  

 M 

 . 

Because the index model is linear, we can estimate the sensitivity (or beta) coefficient of a 

security on the index using a single-variable linear regression. We regress the excess return 

of a security,  R  

 i 

   5   r  

 i 

   2   r  

 f 

  , on the excess return of the index,  R  

 M 

 . To estimate the regression, 

we collect a historical sample of paired observations,  R  

 i 

 ( t ) and  R  

 M 

 ( t ), where  t  denotes the 

date of each pair of observations (e.g., the excess returns on the stock and the index in a 

particular month).  

3

    The     regression  equation     is 

 

   R

i

(t)

5 a

i

1 b

i

R

M

(t)

1 e

i

(t) 

 (8.8)  

The intercept of this equation (denoted by the Greek letter alpha, or  a ) is the security’s 

expected excess return  when the market excess return is zero.  The slope coefficient,  b  

 i 

 ,  is 

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6/21/13   4:09 PM

6/21/13   4:09 PM

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P A R T   I I

  Portfolio Theory and Practice

the security beta. Beta is the security’s sensitivity to the index: It is the amount by which the 

security return tends to increase or decrease for every 1% increase or decrease in the return 

on the index.  e  

 i 

  is the zero-mean, firm-specific surprise in the security return in time   t,  

also called the    residual.