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1、Mean-Preserving-Spread Risk Aversion and The CAPM Phelim P. Boyle and Chenghu Ma October 27, 2006 Abstract This paper establishes conditions under which the classical CAPM holds in equilibrium.Our derivation uses simple arguments to clarify and extend results available in the literature.We show that
2、 if agents are risk averse in the sense of mean-preserving-spread (MPS) the CAPM will necessarily hold, along with two-fund separation. We derive this re- sult without imposing any distributional assumptions on asset returns. The CAPM holds even when the market contains an infi nite number of securi
3、ties and when investors only hold fi nite portfolios. Our paper com- plements the results of Duffi e(1988) who provided an abstract derivation of the CAPM under some somewhat more technical assumptions. In addition we use simple arguments to prove the existence of equilib- rium with MPS-risk-averse
4、investors without assuming that the market is complete. Our proof does not require any additional restrictions on the asset returns, except that the co-variance matrix for the returns on the risky securities is non-singular. Keywords: CAPM equilibrium, two-fund separation, generalized effi cient por
5、tfolio, MPS-risk-aversion. JEL Classifi cation: D50, D81, G10, G11 1Introduction This paper provides general conditions for the validity of the classical CAPM as an equilibrium model in economies with a frictionless market. First, we show that, if equilibrium exists, then the asset returns must sati
6、sfy the CAPM if all investors are MPS-risk-averse (Theorem1). Second, we prove the existence Phelim P. Boyle is with the School of Business and Economics, Wilfrid Laurier University Canada. Chenghu Ma is from the Wang Yanan Institute for Studies in Economics (WISE), Xiamen University, China. This ve
7、rsion has benefi tted from discussions with Jonathan Berk, Darrell Duffi e, Juan Pedro G omez, Chiaki Hara and Zaifu Yang. We thank the referee for comments on an earlier version. Ma is grateful for a research grant from the ESRC, UK. Boyle acknowledges support from the Social Sciences and Humanitie
8、s Research Council of Canada. For correspondence please e-mail: chmaukyahoo.ca. 1 of equilibrium in the CAPM without assuming complete markets (Theorem 2).What is remarkable for the existence of equilibrium CAPM lies in the fact that, as illustrated in Section 4 below, generically, the optimal deman
9、d correspondences for MPS-risk-averse investors do not exist for an arbitrary given set of security prices. Yet, we manage to prove the existence of an equilibrium CAPM by restricting the prices to be located in a zero-measure set. Precisely, the existence proof is based on the validity of the CAPM
10、when equilibrium exists as is proved in Theorem 1. It is noted that prices satisfying the CAPM constitute a measure-zero set in a suitably defi ned topological space for security prices. Given the importance of the CAPM, it is of interest to see to what extent it holds in equilibrium and this topic
11、has been discussed in the literature. It has been known for a long time1that if all investors have mean-variance preferences, then the CAPM will necessarily hold. It is also known that mean-variance pref- erences persist when asset returns are elliptically distributed (see Chamberlain 1983, and Owen
12、 and Rabinovitch 1983). It is, therefore, of particular inter- est to explore if the CAPM holds when preferences are not necessarily in the mean-variance class (but, see Duffi e 1988, and the discussions below). Following the insights of Sharpe-Lintner-Mossin, the key observation which leads to the
13、validity of the CAPM is that (mean-variance) investors optimally choose to hold combinations of two effi cient portfolios: the risk free asset and the so-called tangent portfolio. This is known as two-fund separation theorem (see Black 1972 and Tobin 1958, and also Bottazzi, Hens and L offl er 1998
14、for recent developments). Therefore, to seek conditions for the CAPM, it is suffi cient to seek conditions under which the two-fund separation theorem holds. The fi rst eff ort in this direction was made by Cass and Stiglitz (1970). They used an expected utility framework and derived a parametric sp
15、ecifi cation of expected utility functions which were suffi cient for two-fund separation in the sense that, given the utility function, changes in wealth would not change the risky portfolio which the investor would optimally invest (if the optimal solution exists). In contrast to Chamberlain (1983
16、), and Owen and Rabinovitch (1983), Cass and Stiglitzs observation on two-fund separation was made without imposing any distributional restrictions on asset returns. As a result, it is not clear if Cass and Stiglitzs separating risky portfolios remain the same for diff erent utility functions belonging to the parametric class discovered by them. Further to Cass and Stiglitz (1970), Ross (1978) developed distributional conditions on asset returns to ensure two-fund separation with the separating
