## Well Diversified Portfolios and the Arbitrage Pricing Theory

The arbitrage opportunity described in the previous section is further obscured by the fact that it is almost always impossible to construct a precise scenario analysis for individual stocks that would uncover an event of such straightforward mispricing.

Using the concept of well-diversified portfolios, the arbitrage pricing theory, or APT, resorts to statistical modeling to attack the problem more systematically. By showing that

6We have described pure arbitrage: the search for a costless sure profit. Practitioners often use the terms arbitrage and arbitrageurs more loosely. An arbitrageur may be a professional searching for mispriced securities in specific areas such as merger-target stocks, rather than one looking for strict (risk-free) arbitrage opportunities in the sense that no loss is possible. The search for mispriced securities is called risk arbitrage to distinguish it from pure arbitrage.

mispriced portfolios would give rise to arbitrage opportunities, the APT arrives at an expected return-beta relationship for portfolios identical to that of the CAPM. In the next section, we will compare and contrast the two theories.

In its simple form, just like the CAPM, the APT posits a single-factor security market. Thus, the excess rate of return on each security, R, = rt - r, can be represented by

where alpha, a,, and beta, (3,, are known, and where we treat RM as the single factor.

Suppose now that we construct a highly diversified portfolio with a given beta. If we use enough securities to form the portfolio, the resulting diversification will strip the portfolio of nonsystematic risk. Because such a well-diversified portfolio has for all practical purposes zero firm-specific risk, we can write its returns as

(This portfolio is risky, however, because the excess return on the index, RM, is random.)

Figure 7.11 illustrates the difference between a single security with a beta of 1.0 and a well-diversified portfolio with the same beta. For the portfolio (Panel A), all the returns plot exactly on the security characteristic line. There is no dispersion around the line, as in Panel B, because the effects of firm-specific events are eliminated by diversification. Therefore, in Equation 7.6, there is no residual term, e.

Notice that Equation 7.6 implies that if the portfolio beta is zero, then RP = aP. This implies a riskless rate of return: There is no firm-specific risk because of diversification and no factor risk because beta is zero. Remember, however, that R denotes excess returns. So the equation implies that a portfolio with a beta of zero has a riskless excess return of aP, that is, a return higher than the risk-free rate by the amount aP. But this implies that aP must equal zero, or else an immediate arbitrage opportunity opens up. For example, if aP is greater than zero, you can borrow at the risk-free rate and use the proceeds to buy the well-diversified zero-beta portfolio. You borrow risklessly at rate rf and invest risklessly at rate rf + aP, clearing the riskless differential of aP.

well-diversified portfolio

A portfolio sufficiently diversified that nonsystematic risk is negligible.

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