Investments, tenth edition

  information  ratio    divides the alpha of the portfolio by the nonsystematic risk 

of the portfolio, called “tracking error” in the industry. It measures abnormal return 

per unit of risk that in principle could be diversified away by holding a market 

index portfolio.  

   5.   Morningstar risk-adjusted return:     MRAR(g)

5 B

1

T a

T

t

51

¢

1

1 r

t

1

1 r

ft

≤

2g

R

12

g

2 1  

     The Morningstar rating is a sort of harmonic average of excess returns, where 

 t   5  1, . . . ,   T  are monthly observations,  

9

    and   g  measures risk aversion. Higher  g  

means greater punishment for risk. For mutual funds, Morningstar uses  g   5  2, 

which is considered a reasonable coefficient for an average retail client.  

10

    MRAR 

can be interpreted as the risk-free equivalent excess return of the portfolio for an 

investor with risk aversion measured by  g .   

 Each performance measure has some appeal. But each does not necessarily provide 

consistent assessments of performance, because the risk measures used to adjust returns 

differ  substantially.   

  

7

 We place bars over  r  

 f 

  as well as  r  

 P 

  to denote the fact that because the risk-free rate may not be constant over the 

measurement period, we are taking a sample average, just as we do for  r  

 P 

 . Equivalently, we may simply compute 

sample average  excess   returns. 

  

8

 In many cases performance evaluation assumes a multifactor market. For example, when the Fama-French 

3- factor model is used, Jensen’s alpha will be:    a

P

5 r

P

2 r

f

2 b

PM 

(r

M

2 r

f

)

2 s

P

r

SMB

2 h

P

r

HML

  where  s  

 P 

  is the 

loading on the SMB portfolio and  h  

 P 

  is the loading on the HML portfolio. A multifactor version of the Treynor 

measure also exists. See footnote 13. 

  

9

 The fraction (1  1   r  

 t 

 )/(1  1   r  

 ft 

 ) is well approximated by 1 plus the excess return,  R  

 t 

 . 

  

10

 The MRAR measure is the  certainty-equivalent geometric average excess return  derived from a more sophis-

ticated utility function than the mean-variance function we used in Chapter 6. The utility function is called  con-

stant relative risk aversion (CRRA).  When investors have CRRA, their capital allocation (the fraction of the 

portfolio placed in risk-free versus risky assets) does not change with wealth. The coefficient of risk aversion is: 

 A   5  1  1   g .  When   g   5  0 (equivalently,  A   5  1), the utility function is just the geometric average of gross excess 

returns:

MRAR(0)

5 c q

T

t

51

(1

1 R

t

)

d

12

T

2 1

   

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bod61671_ch24_835-881.indd   840

7/25/13   3:13 AM

7/25/13   3:13 AM

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  C H A P T E R  

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  Portfolio Performance Evaluation 

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