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2023 | OriginalPaper | Chapter

5. The Zero-Beta CAPM

Authors : James W. Kolari, Seppo Pynnönen

Published in: Investment Valuation and Asset Pricing

Publisher: Springer International Publishing

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Abstract

In its early years, as discussed in the last chapter, the CAPM came under question. Empirical tests of the market model found that the relationship between CAPM beta and U.S. stock returns was weaker than expected. Relaxing some of the assumptions of the CAPM, Black (1972) proposed the zero-beta CAPM. This more general form of the CAPM adds a new zero-beta portfolio factor that is uncorrelated with the market factor.

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This graph is based on Figure 4 in Sharpe’s (1973) conference presentation to the Midwest Finance Association.

See equation (40) in Black (1972, p. 454).

See studies by Roll (1977), Levy (1983), and Levy (2007).

For example, see Pulley (1981), Kallberg and Ziemba (1983), Levy (1983), Kroll et al. (1984), Green and Hollifield (1992), Jagannathan and Ma (2003), Levy and Ritov (2010), and others.

For example, assuming all betas for assets are positive, long and short positions would be needed to get a portfolio beta equal to zero. Since it is a zero-investment portfolio like other factors, we use the notation \(\lambda\) to acknowledge this fact.

For example, see Blume (1970), Friend and Blume (1970), and Black et al. (1972).

See recent work by Ferson et al. (2019) for further discussion and citations of other studies that have progressively refined these statistical tests.

In other words, when the riskless rate is available, \(\hat{\theta }^{*2}\) is the square of the slope of the tangent line from the origin to the (ex post) efficient frontier, whereas \(\theta _p^2\) is the slope of the line from the origin to portfolio p in standard deviation and excess return space.

The intuition of representation (5.12) is as follows. Since a line in the standard deviation (\(\sigma\)) mean excess return (\(\mu\)) space through the origin is of the form \(\mu = \theta \sigma\), the square of the length of the line (by Pythagoras) is \(\sigma ^2 + \theta ^2\sigma ^2 = (1 + \theta ^2)\sigma ^2\). As a result, the ratio \(\sqrt{1 + \hat{\theta }^{*2}} / \sqrt{1 + \hat{\theta }_p^2}\) in Eq. (5.12) is the ratio of the lengths at any \(\sigma > 0\) of the tangent line with slope \(\hat{\theta }^{*2}\) and the line of portfolio p with slope \(\hat{\theta }_p\). Therefore the ratio \([(1 + \hat{\theta }^{*2})\sigma ^2] / [(1 + \hat{\theta }_p^2)\sigma ^2] = (1 + \hat{\theta }^{*2}) / (1 + \hat{\theta }_p^2)\) is the ratio of the squared lengths of the lines. The closer this ratio is to the lower bound of one, the closer the portfolio p is to the efficient frontier.

https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

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Title: The Zero-Beta CAPM
Authors: James W. Kolari
Seppo Pynnönen
Publisher: Springer International Publishing
Book: Investment Valuation and Asset Pricing
Print ISBN: 978-3-031-16783-6

Electronic ISBN: 978-3-031-16784-3

Copyright Year: 2023
DOI: https://doi.org/10.1007/978-3-031-16784-3_5