10. A Special Case of the Zero-Beta CAPM: The ZCAPM

The material in the book is based on earlier academic studies of U.S. stock market returns by the authors in Liu et al. (2012, 2019) and Liu (2013) as well as real-world investment services in collaboration with the Teachers Retirement System of Texas (TRS). Additionally, recent papers by Kolari et al. (2022a, b) extend their previous work to a number of major industrialized countries as well as a longer U.S. sample period.

Interested readers can refer to the random matrix theory and Markowitz’s mean-variance portfolio proofs in their book.

We have simplified the notation somewhat here. The authors use \(f(\theta ) \sigma ^2_a\) instead of simply \(\sigma ^2_a\) in Eq.  (10.2), where \(f(\theta ) > 0\) is a complex function of other terms.

The following equilibrium conditions are implied by the ZCAPM:

They are observable in the sense that they can be estimated with standard mean and variance statistical measures of market characteristics.

The authors estimated the ZCAPM to compute beta and zeta risk coefficients for thousands of U.S stocks (see Sect. 10.2 for an overview of the empirical ZCAPM). In 1964 the ZCAPM was estimated for stocks which were then sorted into quintiles by their beta and zeta coefficients to form the 25 portfolios. Next, portfolios’ returns were computed in the out-of-sample month of January 1965. This process was rolled forward one month at a time to generate monthly series of beta, zeta, and one-month-ahead returns for each portfolio from January 1965 to December 2018. For each portfolio, average returns and the standard deviations of these monthly returns were computed. All portfolios were long-only with no short positions.

Originally invented by Dempster et al. (1977), EM regression is widely used in the sciences. See also studies by Jones and McLachlan (1990), McLachlan and Peel (2000), and McLachlan and Krishnan (2008), among others. Wikipedia gives an excellent discussion of the EM algorithm with literature citations.

See https://​mba.​tuck.​dartmouth.​edu/​pages/​faculty/​ken.​french/​data_​library.

This portfolio is the smallest-size/lowest-BM (growth) stock portfolio with the lowest average realized excess return. It is well known that the three-factor model has difficulty explaining this small, growth portfolio (e.g., see Fama and French (1996)).

We should mention that some researchers have found that many anomalous factors tend to disappear after their publication in finance journals (e.g., see McLean and Pontiff (2016), Linnainmaa and Roberts (2018), and others). Nonetheless, numerous anomalous factors persist over time.