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[This chapter mathematically derives the theoretical ZCAPM. To do this, we employ the geometry of the mean-variance investment parabola to show that a new CAPM (dubbed the ZCAPM) can be developed as a special case of Black’s (1972) zero-beta CAPM. As discussed in the previous chapter, Black’s zero-beta CAPM posits that an infinite number of pairs of efficient index I and orthogonal inefficient ZI portfolios on the investment parabola are possible. Each pair of portfolios can be used to proxy the market portfolio M. Here we show that the ZCAPM is based on two orthogonal portfolios denoted I∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^*$$\end{document} and ZI∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ZI^*$$\end{document} on the parabola that have the same variance of returns. Given these two unique portfolios, we utilize random matrix theory mathematics to derive expressions for their expected returns. The resultant ZCAPM is an alternative but equivalent form of Black’s zero-beta CAPM. Unlike other CAPM models, the market factors in the ZCAPM can be readily estimated using daily market returns for assets. In applications to stocks (for example), the ZCAPM requires only the mean and cross-sectional standard deviation (return dispersion) of daily returns for all stocks in the market. These values can be easily computed. In the ZCAPM, sensitivities of an individual stock or portfolio to average stock market returns and market return dispersion are measures of beta risk and zeta risk, respectively. Unlike other models that incorporate return dispersion as an asset pricing factor, zeta risk captures both positive and negative sensitivity to return dispersion on any day t. These contrary forces of cross-sectional market volatility risk are a distinctive feature of the ZCAPM due to its close connection to the mean-variance investment parabola. In our theoretical derivation of the ZCAPM, we rely heavily on Markowitz’s (1959) mean-variance investment parabola and Black’s (1972) zero-beta CAPM, which itself is a more general form of the Sharpe’s (1964) CAPM. Hence, our ZCAPM stands on the shoulders of these classical finance theories by famous authors that were awarded the Nobel Prize in Economics. In this regard, we are most indebted to Fischer Black, who passed away before receiving Nobel recognition. We duly, albeit posthumously, celebrate his valuable insights that made possible our ZCAPM.]
Published: Mar 2, 2021
Keywords: Asset pricing; Beta risk; Cross-sectional return dispersion; efficient frontier; Efficient portfolio; Fischer Black; Harry Markowitz; Mean-variance investment parabola; Random matrix theory; Return dispersion; Securities investment; Stock market; Theoretical ZCAPM; Valuation; ZCAPM; Zero-beta CAPM. Zero-beta portfolio; Zeta risk
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