w-MPS Risk Aversion and the CAPM

Affiliation(s)

School of Business and Economics, Wilfrid Laurier University, Waterloo, Canada.

School of Management & Fudan Finance Center, Fudan University, Shanghai, China.

School of Business and Economics, Wilfrid Laurier University, Waterloo, Canada.

School of Management & Fudan Finance Center, Fudan University, Shanghai, China.

ABSTRACT

This paper establishes general conditions for the validity of mutual
fund separation and the equilibrium CAPM. We use partial preference orders that
display weak form mean preserving spread (w-MPS) risk aversion in the sense of
Ma (2011). We derive this result without imposing any distributional
assumptions on asset returns. The results hold even when the market contains an
infinite number of securities and a continuum number of traders, and when each
investor is permitted to hold some (arbitrary) finite portfolios. A proof of
existence of equilibrium CAPM is provided for finite economies by assuming that
when preferences are constrained on the market subspace spanned by the risk
free bond, the market portfolios admit continuous utility representations.

Cite this paper

P. Boyle and C. Ma, "w-MPS Risk Aversion and the CAPM,"*Theoretical Economics Letters*, Vol. 3 No. 6, 2013, pp. 306-316. doi: 10.4236/tel.2013.36052.

P. Boyle and C. Ma, "w-MPS Risk Aversion and the CAPM,"

References

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http://dx.doi.org/10.1016/0022-0531(70)90038-4

[2] M. Rothschild and J. Stiglitz, “Increasing Risk. II: Its Economic Consequences,” Journal of Economic Theory, Vol. 3, 1971, pp. 66-84.

http://dx.doi.org/10.1016/0022-0531(71)90034-2

[3] D. Duffie, “Security Markets: Stochastic Models,” Academic Press, 1988.

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http://dx.doi.org/10.1142/p745

[5] W. Sharpe, “Capital Asset Prices: A Theory of Capital Market Equilibrium under Conditions of Risk,” Journal of Finance, Vol. 19, 1964, pp. 425-442.

[6] J. Lintner, “The Valuation of Risky Assets and the Selection of Risky Investment in Stock Portfolios and Capital Budgets,” Review of Economics and Statistics, Vol. 47, No. 1, 1965, pp. 13-37.

http://dx.doi.org/10.2307/1924119

[7] J. Mossin, “Equilibrium in a Capital Asset Market,” Econometrica, Vol. 34, No. 4, 1966, pp. 768-783.

http://dx.doi.org/10.2307/1910098

[8] G. Chamberlain, “A Characterization of the Distribution that Imply Mean-Variance Utility Functions,” Journal of Economic Theory, Vol. 29, No. 1, 1983, pp. 185-201.

http://dx.doi.org/10.1016/0022-0531(83)90129-1

[9] J. Owen and R. Rabinovitch, “On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice,” Journal of Finance, Vol. 38, No. 3, 1983, pp. 745-752.

http://dx.doi.org/10.1111/j.1540-6261.1983.tb02499.x

[10] A. Löffler, “Variance Aversion Implies μ-σ2-Criterion,” Journal of Economic Theory, Vol. 69, No. 2, 1996, pp. 532-539. http://dx.doi.org/10.1006/jeth.1996.0067

[11] R. J. Aumann, “Markets with a Continuum of Traders,” Econometrica, Vol. 32, No. 1-2, 1964, pp. 39-50.

http://dx.doi.org/10.2307/1913732

[12] R. J. Aumann, “Existence of Competitive Equilibria in Markets with a Continuum of Traders,” Econometrica, Vol. 34, No. 1, 1966, pp. 1-17.

http://dx.doi.org/10.2307/1909854

[13] D. Schmeidler, “Competitive Equilibria in Markets with a Continuum of Traders and Incomplete Preferences,” Econometrica, Vol. 37, No. 4, 1969, pp. 578-585.

http://dx.doi.org/10.2307/1910435

[14] A. Mas-Colell, “An Equilibrium Existence Theorem without Complete or Transitive Preferences,” Journal of Mathematical Economics, Vol. 1, No. 3, 1974, pp. 237-246.

http://dx.doi.org/10.1016/0304-4068(74)90015-9

[15] D. Gale and A. Mas-Colell, “An Equilibrium Existence Theorem for a General Model without Ordered Preferences,” Journal of Mathematical Economics, Vol. 2, No. 1, 1975, pp. 9-15.

http://dx.doi.org/10.1016/0304-4068(75)90009-9

[16] R. A. Dana, “Existence, Uniqueness and Determinacy of Equilibrium in CAPM with a Riskless Asset,” Journal of Mathematical Economics, Vol. 32, No. 2, 1999, pp. 167-175. http://dx.doi.org/10.1016/S0304-4068(98)00050-0

[17] C. Hara, “Equilibrium Prices of the Market Portfolio in the CAPM with Incomplete Financial Markets,” Working Paper, University of Cambridge, 2001.

[18] L. Nielsen, “Existence of Equilibrium in CAPM,” Journal of Economic Theory, Vol. 52, No. 1, 1990, pp. 223-231. http://dx.doi.org/10.1016/0022-0531(90)90076-V

[19] N. Sun and Z. Yang, “Existence of Equilibrium and Zero-Beta Pricing Formula in the Capital Asset Pricing Model with Heterogeneous Beliefs,” Annals of Economics and Finance, Vol. 4, 2003, pp. 51-71.

[20] G. Chamberlain and M. Rothschild, “Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets,” Econometrica, Vol. 50, 1983, pp. 1281-1304.

http://dx.doi.org/10.2307/1912275

[21] F. Black, “Capital Market Equilibrium with Restricted Borrowing,” Journal of Business, Vol. 45, No. 3, 1972, pp. 444-455. http://dx.doi.org/10.1086/295472

[22] H. Markowitz, “Portfolio Selection,” Journal of Finance, Vol. 7, 1952, pp. 77-99.

[23] H. Markowitz, “Portfolio Selection,” John Wiley and Sons, Inc., New York, 1959.

[24] J. Tobin, “Liquidity Preference and Behavior towards Risk,” Review of Economic Studies, Vol. 25, 1958, pp. 65-86. http://dx.doi.org/10.2307/2296205

[25] S. Ross, “Mutual Fund Separation in Financial Theory— The Separating Distributions,” Journal of Economic Theory, Vol. 17, 1978, pp. 254-286.

http://dx.doi.org/10.1016/0022-0531(78)90073-X

[26] C. F. Huang and R. Litzenberger, “Foundations for Financial Economics,” Prentice Hall, Inc., 1988.

[27] J. Berk, “Necessary and Sufficient Conditions for the CAPM,” Journal of Economic Theory, Vol. 73, 1997, pp. 245-257. http://dx.doi.org/10.1006/jeth.1996.2218

[28] P. Dybvig and S. Ross, “Arbitrage,” In: The New Palgrave: A Dictionary of Economics, The MacMillan Press Limited, 1987.

[29] C. Aliprantis and K. Border, “Infinite Dimensional Analysis,” Springer-Verlag, 1994.

http://dx.doi.org/10.1007/978-3-662-03004-2

[1] M. Rothschild and J. Stiglitz, “Increasing Risk. I: A Definition,” Journal of Economic Theory, Vol. 2, No. 3, 1970, pp. 225-243.

http://dx.doi.org/10.1016/0022-0531(70)90038-4

[2] M. Rothschild and J. Stiglitz, “Increasing Risk. II: Its Economic Consequences,” Journal of Economic Theory, Vol. 3, 1971, pp. 66-84.

http://dx.doi.org/10.1016/0022-0531(71)90034-2

[3] D. Duffie, “Security Markets: Stochastic Models,” Academic Press, 1988.

[4] C. Ma, “Advanced Asset Pricing Theory,” Imperial College Press, 2011.

http://dx.doi.org/10.1142/p745

[5] W. Sharpe, “Capital Asset Prices: A Theory of Capital Market Equilibrium under Conditions of Risk,” Journal of Finance, Vol. 19, 1964, pp. 425-442.

[6] J. Lintner, “The Valuation of Risky Assets and the Selection of Risky Investment in Stock Portfolios and Capital Budgets,” Review of Economics and Statistics, Vol. 47, No. 1, 1965, pp. 13-37.

http://dx.doi.org/10.2307/1924119

[7] J. Mossin, “Equilibrium in a Capital Asset Market,” Econometrica, Vol. 34, No. 4, 1966, pp. 768-783.

http://dx.doi.org/10.2307/1910098

[8] G. Chamberlain, “A Characterization of the Distribution that Imply Mean-Variance Utility Functions,” Journal of Economic Theory, Vol. 29, No. 1, 1983, pp. 185-201.

http://dx.doi.org/10.1016/0022-0531(83)90129-1

[9] J. Owen and R. Rabinovitch, “On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice,” Journal of Finance, Vol. 38, No. 3, 1983, pp. 745-752.

http://dx.doi.org/10.1111/j.1540-6261.1983.tb02499.x

[10] A. Löffler, “Variance Aversion Implies μ-σ2-Criterion,” Journal of Economic Theory, Vol. 69, No. 2, 1996, pp. 532-539. http://dx.doi.org/10.1006/jeth.1996.0067

[11] R. J. Aumann, “Markets with a Continuum of Traders,” Econometrica, Vol. 32, No. 1-2, 1964, pp. 39-50.

http://dx.doi.org/10.2307/1913732

[12] R. J. Aumann, “Existence of Competitive Equilibria in Markets with a Continuum of Traders,” Econometrica, Vol. 34, No. 1, 1966, pp. 1-17.

http://dx.doi.org/10.2307/1909854

[13] D. Schmeidler, “Competitive Equilibria in Markets with a Continuum of Traders and Incomplete Preferences,” Econometrica, Vol. 37, No. 4, 1969, pp. 578-585.

http://dx.doi.org/10.2307/1910435

[14] A. Mas-Colell, “An Equilibrium Existence Theorem without Complete or Transitive Preferences,” Journal of Mathematical Economics, Vol. 1, No. 3, 1974, pp. 237-246.

http://dx.doi.org/10.1016/0304-4068(74)90015-9

[15] D. Gale and A. Mas-Colell, “An Equilibrium Existence Theorem for a General Model without Ordered Preferences,” Journal of Mathematical Economics, Vol. 2, No. 1, 1975, pp. 9-15.

http://dx.doi.org/10.1016/0304-4068(75)90009-9

[16] R. A. Dana, “Existence, Uniqueness and Determinacy of Equilibrium in CAPM with a Riskless Asset,” Journal of Mathematical Economics, Vol. 32, No. 2, 1999, pp. 167-175. http://dx.doi.org/10.1016/S0304-4068(98)00050-0

[17] C. Hara, “Equilibrium Prices of the Market Portfolio in the CAPM with Incomplete Financial Markets,” Working Paper, University of Cambridge, 2001.

[18] L. Nielsen, “Existence of Equilibrium in CAPM,” Journal of Economic Theory, Vol. 52, No. 1, 1990, pp. 223-231. http://dx.doi.org/10.1016/0022-0531(90)90076-V

[19] N. Sun and Z. Yang, “Existence of Equilibrium and Zero-Beta Pricing Formula in the Capital Asset Pricing Model with Heterogeneous Beliefs,” Annals of Economics and Finance, Vol. 4, 2003, pp. 51-71.

[20] G. Chamberlain and M. Rothschild, “Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets,” Econometrica, Vol. 50, 1983, pp. 1281-1304.

http://dx.doi.org/10.2307/1912275

[21] F. Black, “Capital Market Equilibrium with Restricted Borrowing,” Journal of Business, Vol. 45, No. 3, 1972, pp. 444-455. http://dx.doi.org/10.1086/295472

[22] H. Markowitz, “Portfolio Selection,” Journal of Finance, Vol. 7, 1952, pp. 77-99.

[23] H. Markowitz, “Portfolio Selection,” John Wiley and Sons, Inc., New York, 1959.

[24] J. Tobin, “Liquidity Preference and Behavior towards Risk,” Review of Economic Studies, Vol. 25, 1958, pp. 65-86. http://dx.doi.org/10.2307/2296205

[25] S. Ross, “Mutual Fund Separation in Financial Theory— The Separating Distributions,” Journal of Economic Theory, Vol. 17, 1978, pp. 254-286.

http://dx.doi.org/10.1016/0022-0531(78)90073-X

[26] C. F. Huang and R. Litzenberger, “Foundations for Financial Economics,” Prentice Hall, Inc., 1988.

[27] J. Berk, “Necessary and Sufficient Conditions for the CAPM,” Journal of Economic Theory, Vol. 73, 1997, pp. 245-257. http://dx.doi.org/10.1006/jeth.1996.2218

[28] P. Dybvig and S. Ross, “Arbitrage,” In: The New Palgrave: A Dictionary of Economics, The MacMillan Press Limited, 1987.

[29] C. Aliprantis and K. Border, “Infinite Dimensional Analysis,” Springer-Verlag, 1994.

http://dx.doi.org/10.1007/978-3-662-03004-2