Investments, tenth edition

ratio.    It measures the extra return we can obtain from security  analysis  compared to the 

firm-specific risk we incur when we over- or underweight securities relative to the passive 

market index. Equation 8.22 therefore implies that to maximize the overall Sharpe ratio, 

we must maximize the information ratio of the active portfolio. 

 It turns out that the information ratio of the active portfolio will be maximized if we 

invest in each security in proportion to its ratio of  a  

 i 

 / s  

2

 ( e  

 i 

 ). Scaling this ratio so that the 

total position in the active portfolio adds up to    w

A

*

,  the weight in each security is

 

   w

i

*

5 w

A

*

 

a

i

s

2

(e

i

)

a

n

i

51

a

i

s

2

(e

i

)

 

 (8.23)  

With this set of weights, the contribution of each security to the information ratio of the 

active portfolio is the square of its  own  information ratio, that is,

 

   

B

a

A

s(e

A

)

R

2

5 a

n

i

51

B

a

i

s(e

i

)

R

2

 

 (8.24)   

 The model thus reveals the central role of the information ratio in efficiently taking 

advantage of security analysis. The positive contribution of a security to the portfolio is 

made by its addition to the nonmarket risk premium (its alpha). Its negative impact is to 

increase the portfolio variance through its firm-specific risk (residual variance). 

 In contrast to alpha, the market (systematic) component of the risk premium,  b  

 i 

  E ( R  

 M 

 ), 

is offset by the security’s nondiversifiable (market) risk,    b

i

2

s

M

2

,  and both are driven by the 

same beta. This trade-off is not unique to any security, as any security with the same beta 

makes the same balanced contribution to both risk and return. Put differently, the beta of a 

security is neither vice nor virtue. It is a property that simultaneously affects the risk  and  

risk premium of a security. Hence we are concerned only with the aggregate beta of the 

active portfolio, rather than the beta of each individual security. 

 We see from Equation 8.23 that if a security’s alpha is negative, the security will assume 

a short position in the optimal risky portfolio. If short positions are prohibited, a negative-

alpha security would simply be taken out of the optimization program and assigned a port-

folio weight of zero. As the number of securities with nonzero alpha values (or the number 

with positive alphas if short positions are prohibited) increases, the active portfolio will 

itself be better diversified and its weight in the overall risky portfolio will increase at the 

expense of the passive index portfolio. 

 Finally, we note that the index portfolio is an efficient portfolio only if all alpha values 

are zero. This makes intuitive sense. Unless security analysis reveals that a security has a 

nonzero alpha, including it in the active portfolio would make the portfolio less attractive. 

In addition to the security’s systematic risk, which is compensated for by the market risk 

premium (through beta), the security would add its firm-specific risk to portfolio variance. 

With a zero alpha, however, the latter is not compensated by an addition to the nonmarket 

risk premium. Hence, if all securities have zero alphas, the optimal weight in the active 

portfolio will be zero, and the weight in the index portfolio will be 1. However, when 

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6/21/13   4:10 PM

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276 

P A R T   I I

  Portfolio Theory and Practice

security analysis uncovers securities with nonmarket risk premiums (nonzero alphas), the 

index portfolio is no longer efficient.