Investments, tenth edition

 Figure 10.2 

Returns as a function of the systematic factor: an arbitrage 

opportunity  

0

F (Realization of

    Macro Factor)

A

B

10

8

Return (%)

  The No-Arbitrage Equation of the APT 

 We’ve seen that arbitrage activity will quickly pin the risk premium of any zero-beta well-

diversified portfolio to zero.  

4

   Setting the expression in Equation 10.8 to zero implies that 

the alpha of  any  well-diversified portfolio must also be zero. From Equation 10.5, this 

means that for any well-diversified  P,      

E(R

P

)

5 b

P

E(R

M

) 

 (10.9)   

 In other words, the risk premium (expected excess return) on portfolio  P  is the product of 

its beta and the market-index risk premium. Equation 10.9 thus establishes that the SML 

of the CAPM applies to well-diversified portfolios simply by virtue of the “no-arbitrage” 

requirement of the APT. 

 Another demonstration that the APT results in the same SML as the CAPM is more 

graphical in nature. First we show why all well-diversified portfolios with the same beta must 

have the same expected return.  Figure 10.2  plots the returns on two such portfolios,  A   and 

B,  both with betas of 1, but with differing expected returns:  E ( r  

 A 

 )  5  10% and  E ( r  

 B 

 )  5  8%.   

 Could portfolios  A  and  B  coexist with the return pattern depicted? Clearly not: No matter 

what the systematic factor turns out to be, portfolio  A  outperforms portfolio  B,  leading to 

an arbitrage opportunity. 

 If you sell short $1 million of  B  and buy $1 million of  A,  a zero-net-investment strategy, 

you would have a riskless payoff of $20,000, as follows:   

(.10

1 1.0 3 F) 3 $1 million

2(.08 1 1.0 3 F) 3 $1 million

.02

3 $1 million 5 $20,000

 

  from long position in A

  from short position in B

  net proceeds

 

Your profit is risk-free because the factor risk cancels out across the long and short 

 positions. Moreover, the strategy requires zero-net-investment. You should pursue it on an 

4

 As an exercise, show that when  a

 P 

, 0 you reverse the position of  P  in  Z,  and the arbitrage portfolio will still 

earn a riskless excess return. 

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6/21/13   3:43 PM

6/21/13   3:43 PM

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P A R T   I I I

  Equilibrium in Capital Markets

r

f

 = 4

β

6

.5

1

10

7

F

C

D

A

Risk

Premium

(With Respect

to Macro Factor)

Expected Return (%)