Investments, tenth edition

   The APT and the CAPM 

 The APT is built on the foundation of well-diversified portfolios. However, we’ve seen, 

for example in  Table 10.1 , that even large portfolios may have non-negligible residual risk. 

Some indexed portfolios may have hundreds or thousands of stocks, but active portfolios 

generally cannot, as there is a limit to how many stocks can be actively analyzed in search 

of alpha. How does the APT stand up to these limitations? 

 Suppose we order all portfolios in the universe by residual risk. Think of Level 0 portfo-

lios as having zero residual risk; in other words, they are the theoretically well-diversified 

portfolios of the APT. Level 1 portfolios have very small residual risk, say up to 0.5%. 

Level 2 portfolios have yet greater residual SD, say up to 1%, and so on. 

 If the SML described by Equation 10.9 applies to all well-diversified Level 0 portfolios, 

it must at least approximate the risk premiums of Level 1 portfolios. Even more important, 

while Level 1 risk premiums may deviate slightly from Equation 10.9, such deviations 

should be unbiased, with alphas equally likely to be positive or negative. Deviations should 

be uncorrelated with beta or residual SD and should average to zero. 

 We can apply the same logic to portfolios of slightly higher Level 2 residual risk. Since 

all Level 1 portfolios are still well approximated by Equation 10.9, so must be risk pre-

miums of Level 2 portfolios, albeit with slightly less accuracy. Here too, we may take 

comfort in the lack of bias and zero average deviations from the risk premiums pre-

dicted by Equation 10.9. But still, the precision of predictions of risk premiums from 

Equation 10.9 consistently deteriorates with increasing residual risk. (One might ask why 

we don’t transform Level 2 portfolios into Level 1 or even Level 0 portfolios by further 

diversifying, but as we’ve pointed out, this may not be feasible in practice for assets with 

considerable residual risk when active portfolio size or the size of the investment universe 

is limited.) If residual risk is sufficiently high and the impediments to complete diversifica-

tion are too onerous, we cannot have full confidence in the APT and the arbitrage activities 

that underpin it. 

 Despite this shortcoming, the APT is valuable. First, recall that the CAPM requires that 

almost all investors be mean-variance optimizers. We may well suspect that they are not. 

The APT frees us of this assumption. It is sufficient that a small number of sophisticated 

arbitrageurs scour the market for arbitrage opportunities. This alone produces an SML, 

Equation 10.9, that is a good and unbiased approximation for all assets but those with 

significant residual risk. 

 Perhaps even more important is the fact that the APT is anchored by observable port-

folios such as the market index. The CAPM is not even testable because it relies on an 

unobserved, all-inclusive portfolio. The reason that the APT is not fully superior to the 

CAPM is that at the level of individual assets and high residual risk, pure arbitrage may 

be insufficient to enforce Equation 10.9. Therefore, we need to turn to the CAPM as a 

complementary theoretical construct behind equilibrium risk premiums. 

 It should be noted, however, that when we replace the unobserved market portfolio of 

the CAPM with an observed, broad index portfolio that may not be efficient, we no longer 

can be sure that the CAPM predicts risk premiums of all assets with no bias. Neither model 

therefore is free of limitations. Comparing the APT arbitrage strategy to maximization of 

the Sharpe ratio in the context of an index model may well be the more useful framework 

for analysis.  

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  Equilibrium in Capital Markets