Investments, tenth edition

  Executing Arbitrage 

 

Imagine a single-factor market where the well-diversified portfolio,  

M,   represents 

the market factor,  F,  of Equation 10.1. The excess return on any security is given by 

 R  

 i 

   5   a  

 i 

   1   b  

 i  

  R  

 M 

   1   e  

 i  

 , and that of a well-diversified (therefore zero residual) portfolio,  P,   is   

 

R

P

5 a

P

1 b

P

R

M

 

 (10.4)     

 

E(R

P

)

5 a

P

1 b

P

E(R

M

) 

 (10.5)   

 Now suppose that security analysis reveals that portfolio  P  has a positive alpha.  

2

    We  also 

estimate the risk premium of the index portfolio,  M,  from macro analysis.

   

 Since neither  M  nor portfolio  P  have residual risk, the only risk to the returns of the two 

portfolios is systematic, derived from their betas on the common factor (the beta of the 

index is 1.0). Therefore, you can eliminate the risk of  P  altogether: Construct a zero-beta 

portfolio, called  Z,  from  P  and  M  by appropriately selecting weights  w  

 P 

  and  w  

 M 

   5  1  2   w  

 P 

  

on each portfolio:   

 b

Z

5 w

P

b

P

1 (1 2 w

P

)b

M

5 0

 b

M

5 1

 

 w

P

5

1

1

2 b

P

; w

M

5 1 2 w

P

5

2b

P

1

2 b

P

 

 (10.6)  

Therefore, portfolio  Z  is riskless, and its alpha is   

 

a

Z

5 w

P

a

P

1 (1 2 w

P

)a

M

5 w

P

a

P

 

 (10.7)   

 The risk premium on  Z  must be zero because the risk of  Z  is zero. If its risk premium 

were not zero, you could earn arbitrage profits. Here is how: 

 Since the beta of  Z  is zero, Equation 10.5 implies that its risk premium is just its alpha. 

Using Equation 10.7, its alpha is  w  

 P  

  a  

 P 

 ,  so   

 

E(R

Z

)

5 w

P

a

P

5

1

1

2 b

P

 

a

P

 

 (10.8)   

 You now form a zero-net-investment arbitrage portfolio: If  b  

 P 

  , 1 and the risk premium 

of  Z  is positive (implying that  Z  returns more than the risk-free rate), borrow and invest 

the proceeds in  Z . For every borrowed dollar invested in  Z,  you get a net return (i.e., net of 

paying the interest on your loan) of    

1

1

2 b

P

 

a

P

.  This is a money machine, which you would 

work as hard as you can.  

3

   Similarly if  b  

 P 

   .  1, Equation 10.8 tells us that the risk pre-

mium is negative; therefore, sell  Z  short and invest the proceeds at the risk-free rate. Once 

again, a money machine has been created. Neither situation can persist, as the large volume 

of trades from arbitrageurs pursuing these strategies will push prices until the arbitrage 

opportunity disappears (i.e., until the risk premium of portfolio Z equals zero).   

  

2

 If the portfolio alpha is negative, we can still pursue the following strategy. We would simply switch to a short 

position in  P,  which would have a positive alpha of the same absolute value as  P ’s, and a beta that is the negative 

of  P ’s. 

  

3

 The function in Equation 10.8 becomes unstable at  b  

 P 

   5  1. For values of  b  

 P 

  near 1, it becomes infinitely large 

with the sign of  a  

 P 

 . This isn’t an economic absurdity, since in that case, the sizes of your long position in  P   and 

short position in  M  will be almost identical, and the arbitrage profit you earn  per dollar invested  will be nearly 

infinite. 

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6/21/13   3:43 PM

6/21/13   3:43 PM

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  C H A P T E R  

1 0

  Arbitrage Pricing Theory and Multifactor Models of Risk and Return

333