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[This chapter develops empirical models to estimate the theoretical ZCAPM. We review prior empirical studies that incorporate return dispersion (RD) in a traditional asset pricing model as well as research on asymmetric market risk. Departing from these studies, and confirming our random matrix results in Chapter 3, we use mean-variance mathematics in Markowitz to prove that the width of the mean-variance investment parabola is defined in large part by RD. This new result implies that the average market return lies on its axis of symmetry in the middle of the parabola. Using these insights about the mean-variance parabola, our empirical ZCAPM differs from traditional models in two ways: (1) beta risk is associated with average market returns rather than a proxy market portfolio and (2) zeta risk related to RD can be either positive or negative in sign. Unlike other asset pricing models, the empirical ZCAPM is a probabilistic mixture model with two components, each of which is a two-factor regression model with either positive or negative sensitivity to RD. To determine the sign of zeta risk related to RD, we estimate the empirical ZCAPM using the well-established expectation-maximization (EM) algorithm, which is an iterative method to find maximum likelihood estimates of statistical parameters. The EM algorithm assumes that, even though the sign of the zeta risk coefficient is unknown, the probability of a positive or negative sign can be estimated via latent information contained in observed data. We provide step-by-step instructions on how to estimate the ZCAPM with the EM algorithm.]
Published: Mar 2, 2021
Keywords: Asset pricing; Asymmetric market risk; Beta risk; Empirical ZCAPM; Expectation-maximization (EM) algorithm; Investment parabola; Latent variable; Macroeconomic state variable; Maximum likelihood; Mean-variance investment parabola; Return dispersion; Signal variable; Stock market; Valuation; ZCAPM; Zero-beta CAPM; Zeta risk
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