Investments, tenth edition

  Risk and Covariance in the Single-Index Model 

 

Remember that one of the problems with the Markowitz model is the overwhelming 

 number of parameter estimates required to implement it. Now we will see that the index 

model simplification vastly reduces the number of parameters that must be estimated. 

Equation 8.8 yields the systematic and firm-specific components of the overall risk of each 

security, and the covariance between any pair of securities. Both variances and covariances 

are determined by the security betas and the properties of the market index:

     Total  risk

5 Systematic risk 1 Firm-specific risk

 s

i

2

5 b

i

2

s

M

2

1 s

2

(e

i

)

 Covariance

5 Product of betas 3 Market-index risk

  

Cov(r

i

, r

j

)

5 b

i

b

j

s

M

2

 (8.10)

 Correlation

5 Product of correlations with the market index

 Corr(r

i

, r

j

)

5

b

i

b

j

s

M

2

s

i

s

j

5

b

i

s

M

2

b

j

s

M

2

s

i

s

M

s

j

s

M

5 Corr(r

i

, r

M

)

3 Corr(r

j

, r

M

)    

 Equations 8.9 and 8.10 imply that the set of parameter estimates needed for the single-

index model consists of only  a ,  b , and  s ( e ) for the individual securities, plus the risk pre-

mium and variance of the market index. 

 

 

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6/21/13   4:09 PM

6/21/13   4:09 PM

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  C H A P T E R  

8

 Index 

Models 

261