Investments, tenth edition

 Figure 10.3 

An arbitrage opportunity  

infinitely large scale until the return discrepancy between the two portfolios disappears. 

Well-diversified portfolios with equal betas must have equal expected returns in market 

equilibrium, or arbitrage opportunities exist. 

 What about portfolios with different betas? Their risk premiums must be propor-

tional to beta. To see why, consider  Figure  10.3 . Suppose that the risk-free rate is 4% 

and that a well-diversified portfolio,  C,  with a beta of .5, has an expected return of 6%. 

Portfolio  C  plots below the line from the risk-free asset to portfolio  A.  Consider, there-

fore, a new portfolio,  D,  composed of half of portfolio  A  and half of the risk-free asset. 

Portfolio  D ’s beta will be (.5   3   0    1   .5   3   1.0)   5  .5, and its expected return will be 

(.5  3  4  1  .5  3  10)  5  7%. Now portfolio  D  has an equal beta but a greater expected return 

than portfolio  C.  From our analysis in the previous paragraph we know that this consti-

tutes an arbitrage opportunity. We conclude that, to preclude arbitrage opportunities, the 

expected return on all well-diversified portfolios must lie on the straight line from the 

risk-free asset in  Figure 10.3 .  

 Notice in  Figure 10.3  that risk premiums are indeed proportional to portfolio betas. The 

risk premium is depicted by the vertical arrow, which measures the distance between the 

risk-free rate and the expected return on the portfolio. As in the simple CAPM, the risk 

premium is zero for  b   5  0 and rises in direct proportion to  b .    

    10.3 

The APT, the CAPM, and the Index Model 

  Equation 10.9 raises three questions:

    1.  Does the APT also apply to less-than-well-diversified portfolios?  

   2.  Is the APT as a model of risk and return superior or inferior to the CAPM? Do we 

need both models?  

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  Arbitrage Pricing Theory and Multifactor Models of Risk and Return 

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   3.  Suppose a security analyst identifies a positive-alpha portfolio with some remaining 

residual risk. Don’t we already have a prescription for this case from the Treynor-

Black (T-B) procedure applied to the index model (Chapter 8)? Is this framework 

preferred  to  the  APT?