Investments, tenth edition

  The Zero-Beta Model 

 Efficient frontier portfolios have a number of interesting characteristics, independently 

derived by Merton and Roll.  

13

   Two of these are

    1.  Any portfolio that is a combination of two frontier portfolios is itself on the efficient 

frontier.  

   2.  Every portfolio on the efficient frontier, except for the global minimum-variance 

portfolio, has a “companion” portfolio on the bottom (inefficient) half of the frontier 

  

13

 Robert C. Merton, “An Analytic Derivation of the Efficient Portfolio Frontier,”  Journal of Financial and 

 Quantitative  Analysis,  1972. Richard Roll, “A Critique of the Asset Pricing Theory’s Tests: Part I: On Past and 

Potential Testability of the Theory,”  Journal of Financial Economics  4 (1977). 

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P A R T   I I I

  Equilibrium in Capital Markets

with which it is uncorrelated. Because it is uncorrelated, the companion portfolio 

is referred to as the    zero-beta  portfolio    of the efficient portfolio. If we choose 

the market portfolio  M  and its zero-beta companion portfolio  Z,  then we obtain a 

CAPM-like equation

 

   E( r

i

)

2 E(r

Z

)

5 3E(R

M

)

2 E(R

Z

)

4

 

Cov( r

i

, r

M

)

s

M

2

5 b

i

3E(r

M

)

2 E(r

Z

)

4   (9.12)  

    Equation 9.12 resembles the SML of the CAPM, except that the risk-free rate is 

replaced with the expected return on the zero-beta companion of the market-index 

portfolio.     

 Fischer Black used these properties to show that Equation 9.12 is the CAPM equation 

that results when investors face restrictions on borrowing.  

14

   In this case, at least some 

investors will choose portfolios on the high risk-premium portion of the efficient frontier. 

Put differently, investors who would otherwise wish to borrow and leverage their portfo-

lios but who find it impossible or costly will instead tilt their portfolios toward high-beta 

stocks and away from low-beta ones. As a result, prices of high beta stocks will rise, and 

their risk premiums will fall. The SML will be flatter than in the simple CAPM. You see 

from Equation 9.12 that the risk premium on the market portfolio is smaller (because the 

expected return on the zero-beta portfolio is greater than the risk-free rate) and therefore 

the reward to bearing beta risk is smaller.