Abstract
We trace the Treynor development of the CAPM extending the work of markowitz in portfolio selection. The Treynor work illuminates issues regarding the estimation of security beta, the role played by i diosyncratic risk, and the optimal composition of efficient portfolios. It offer a systematic way in which managers might insert their evolving items of market rates and associated risk.
I wish to thank John B. Guerard Jr. for helpful comments and stimulation.
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Notes
- 1.
How does one explain that only the mean and variance of returns and not other moments play a role? One can justify this by an implicit assumption that the probability distribution of returns belongs to a family of distributions described by only two parameters, or that the expected utility function is of such a form that it depends only on the mean and variance of the relevant distribution.
- 2.
It should be noted that Markowitz did not actually solve for γ; rather his version focused only on risky assets and imposed non-negativity constraints on the elements of γ. Thus what he derived from the first order conditions were rules for inclusion in and/or exclusion from (of securities) in an optimal portfolio.
- 3.
From the point of view of computation, entering the constraint as \(k^{2} =\gamma ^{{\prime}}\Sigma \gamma\) simplifies operations, but makes the Lagrange multiplier harder to interpret in terms of common usage in finance; if, however, we enter the constraint as \(k = (\gamma ^{{\prime}}\Sigma \gamma )^{1/2}\), we complicate the computations somewhat, we do not change the nature of the solution, but we can interpret the Lagrange multiplier in terms of common usage comfortably. We should also bear in mind that if risk is defined in terms of the standard deviation rather than the variance, a certain intuitive appeal is lost. For example, it is often said that security returns are subject to two risks, market risk and idiosyncratic risk. If we also say, as we typically do, that market risk is independent of idiosyncratic risk, then we have the following situation: denote the market risk by the variance of a certain random variable, say σ mar 2 and the idiosyncratic risk by the variance σ idio 2 then the risk of the security return is the sum σ mar 2 +σ idio 2. On the other hand, if we define risk in terms of the standard deviation, then the two risks are not additive, i.e. the risk of the security is not σ mar +σ idio but \(\sqrt{\sigma _{\mathrm{mar } }^{2 } +\sigma _{ \mathrm{idi0 } }^{2}}\), which is smaller, when we use as usual the positive square root. This problem occurs whenever there is aggregation of independent risks.
- 4.
I say ‘somewhat dynamic’ because we still operate within what used to be called a ‘certainty equivalent’ environment, in that the underlying randomness is not fully embraced as in option price theory.
- 5.
The intellectual history of the evolution of CAPM is detailed in the excellent and comprehensive paper by French (2003), which details inter alia the important but largely unacknowledged role payed by the unpublished paper Treynor (1962). We cite Lintner (1965a) in the cite both Lintner paper of 1965 in his capital market development.
- 6.
See Dhrymes (2013, pp. 202–203).
- 7.
See Dhrymes (2013, pp. 46–47).
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Dhrymes, P.J. (2017). Portfolio Theory: Origins, Markowitz and CAPM Based Selection. In: Guerard, Jr., J. (eds) Portfolio Construction, Measurement, and Efficiency. Springer, Cham. https://doi.org/10.1007/978-3-319-33976-4_2
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