Investments, tenth edition

  The Information Ratio 

 Equations 8.20 and 8.21 yield the optimal position in the active portfolio once we know its 

alpha, beta, and residual variance. With    w

A

*

   in  the  active  portfolio  and     1

2 w

A

*

   invested  in 

the index portfolio, we can compute the expected return, standard deviation, and Sharpe 

ratio of the optimal risky portfolio. The Sharpe ratio of an optimally constructed risky port-

folio will exceed that of the index portfolio (the passive strategy). The exact relationship is

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6/21/13   4:10 PM

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  C H A P T E R  

8

 Index 

Models 

275

 

   S

P

2

5 S

M

2

1 B

a

A

s(e

A

)

R

2

 

 (8.22)   

 Equation 8.22 shows us that the contribution of the active portfolio (when held in its opti-

mal weight,    w

A

*

 ) to the Sharpe ratio of the overall risky portfolio is determined by the ratio 

of its alpha to its residual standard deviation. This important ratio is called the     information