The Empirical ZCAPM

As discussed in the previous chapter, the theoretical ZCAPM by Kolari, Liu, and Huang (KLH) (A new model of capital asset prices: Theory and Evidence. Palgrave Macmillan, Cham, Switzerland, 2021) contains beta risk related to average market returns and zeta risk associated with the cross-sectional return dispersion of all assets in the market. Zeta risk takes into account sensitivity to market return dispersion that can be positive or negative in sign. That is, asset returns can increase or decrease depending on the sign of zeta risk. In the real world, the positive or negative direction of zeta risk is unknown. KLH provide an empirical form of the ZCAPM that solves the problem of the direction of zeta risk. Using the expectation-maximization (EM) algorithm, an estimate of the probability that zeta risk is positive or negative for asset can be obtained from historical daily returns within (for example) the past year. Based on this probability, a mixture of two-factor models can be specified. One model corresponds to a positive zeta risk, and the second model has negative zeta riskNegative zeta risk. Which model is operative for an asset depends on the probability estimate that gives the direction of zeta risk. It turns out that the EM algorithm is well-known in the hard sciences and widely used in regression analyses. In finance it has been utilized but is not widely known. Even so, it is commonly used in our everyday lives. For example, weather mapping technology that forecasts the direction that a storm will move in the next hour often utilizes the EM algorithm. In this chapter, we give an overview of the empirical ZCAPM. For readers seeking details of the procedure, we recommend referring to KLH’s book. As we will see, with some simple modifications, the theoretical ZCAPM can be adapted to readily available empirical data in the financial markets. To justify the use of the empirical ZCAPM for the creation of efficient investment portfolios in forthcoming chapters, we review some earlier tests of the model. These analyses compare the empirical ZCAPM to other well-known asset pricing models discussed in Chapter 3 . The results are convincing. In standard out-of-sample cross-sectional regression tests, the empirical ZCAPM has very strong goodness-of-fitGoodness-of-fit as measured by $$R^2$$ R 2 estimates. Also, zeta risk is highly significant in these cross-sectional tests; indeed, zeta risk dominates all of the prominent factors in the literature in terms of being significantly priced in the cross section of average stock returns. These and other results suggest that the empirical ZCAPM is a valid asset pricing model. The fact that the empirical ZCAPM, which is a CAPM theory-based model, consistently outperforms multifactor modelsMultifactor models (at times by large margins) is quite a reversal in the field of asset pricing. Multifactor models have gained popularity over time due to better explaining stock returns than the market model version of the CAPM. However, the ZCAPM is a CAPM model that outperforms the multifactor models. Interestingly, KLH have argued that long/short, zero-investment factors in multifactor models are actually rough measures of cross-sectional return dispersion. As such, they are akin to the ZCAPM that incorporates total return dispersion as a new asset pricing factor. In effect, total return dispersion subsumes all of the multifactorsMultifactor which become redundant as they capture only different slices of return dispersion. Paradoxically, multifactor models are cousins of the ZCAPM so to speak, and by implication linked to the CAPM. Rather than the CAPM being dead, it is alive and well in the form of the ZCAPM. Also, through an unexpected twist, the CAPM is even alive to some extent in the form of multifactor models!