Investments, tenth edition

  The Expected Return–Beta Relationship 

 Because   E  ( e  

 i 

 )  5  0, if we take the expected value of  E ( R  

 i 

 ) in Equation 8.8, we obtain the 

expected return–beta relationship of the single-index model:

 

   E(R

i

)

5 a

i

1 b

i

E(R

M

) 

 (8.9)   

 The second term in Equation 8.9 tells us that part of a security’s risk premium is due to the 

risk premium of the index. The market risk premium is multiplied by the relative sensitiv-

ity, or beta, of the individual security. We call this the  systematic  risk premium because it 

derives from the risk premium that characterizes the entire market, which proxies for the 

condition of the full economy or economic system. 

 The remainder of the risk premium is given by the first term in the equation,  a .  Alpha  is 

a  nonmarket  premium. For example,  a  may be large if you think a security is underpriced 

and therefore offers an attractive expected return. Later on, we will see that when secu-

rity prices are in equilibrium, such attractive opportunities ought to be competed away, in 

which case  a  will be driven to zero. But for now, let’s assume that each security analyst 

comes up with his or her own estimates of alpha. If managers believe that they can do a 

superior job of security analysis, then they will be confident in their ability to find stocks 

with nonzero values of alpha. 

 We will see shortly that the index model decomposition of an individual security’s risk 

premium to market and nonmarket components greatly clarifies and simplifies the opera-

tion of macroeconomic and security analysis within an investment company.