The Theory and Practice of Investment Management: Asset Allocation, Valuation, Portfolio Construction, and Strategies, Second Edition

THE “TWO BETA” TRAP

Two distinct meanings of the word “beta” are used in modern financial theory. These meanings are sufficiently alike for people to converse—some with one meaning in mind, some with the other—without realizing they are talking about two different things. The meanings are sufficiently different, however, that one can validly derive diametrically opposite conclusions depending on which one is used. The net result of all this can be like an Abbot and Costello vaudeville comedy routine with portfolio theory rather than baseball as its setting. This is what Markowitz calls the “two beta trap.”⁵⁰ Below we first review the background of the two betas and then tabulate propositions that are true for one concept and false for the other.

Beta₁₉₆₃

In Chapter 3, we discussed Sharpe’s “single-index” (or one-factor) model of covariance introduced in 1963.⁵¹ This model assumes that the returns of different securities are correlated with each other because each is dependent on some underlying systematic factor. This can be written as

(4.13)

where the expected value of u_i is zero, and u_i is uncorrelated with F and every other u_j .

Originally F was denoted by I and described as an “underlying factor, the general prosperity of the market as expressed by some index.” We have changed the notation from I to F to emphasize that r_i depends on the underlying factor rather than ...