A three mutual fund separation theorem (2024)

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Abstract

We analyze a one period economy with CARA preferences, and normally distributed aggregate risk. We allow an arbitrary distribution of (dogmatic) beliefs on the expected value and variance of the aggregate risk, as well as on the agent’s risk tolerance param- eter. We show that there is a representative agent whose risk tolerance, and beliefs over expected value and variance are appropriately defined weighted averages of the same objects across all agents. We show a type of 3 fund separation theorem: any efficient allocation can be decentralized with three asset: an uncontingent bond, a Lucas’s tree, and a simple variance swap –whose price is given by the SVIX index, an index closely related to VIX, given by the the price of a portfolio of out of the money puts and calls. We give simple expressions for the relative prices of these securities in terms of the representative agent’s parameters. We also give simple expressions for each agent’s holding on the Lucas’ tree and the simple variance swap as function of the agent’s differences with the representative agent’s values. The main novelty of the setup is the explicit role for a simple variance swap –or the portfolio of puts and calls– in the decentralization of efficient allocations. In equilibrium there are different positions in the the variance swap –or to the options portfolio– if and only if there is heterogeneity in the beliefs of the aggregate risk of the economy, and hence these heterogeneity maps into trade volume. To understand the role of different assumption we also study the case of arbitrary belief and utility function for a small noise. Finally, we extend the results to a multiperiod setting, for which they hold conditionally on the history for each period.

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References listed on IDEAS

1. Merton, Robert C, 1973."An Intertemporal Capital Asset Pricing Model,"Econometrica, Econometric Society, vol. 41(5), pages 867-887, September.
2. Stephen A. Ross, 2005."Mutual Fund Separation in Financial Theory—The Separating Distributions,"World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 10, pages 309-356,World Scientific Publishing Co. Pte. Ltd..
   • Ross, Stephen A., 1978."Mutual fund separation in financial theory--The separating distributions,"Journal of Economic Theory, Elsevier, vol. 17(2), pages 254-286, April.
   • Stephen A. Ross, "undated". "Mutual Fund Separation in Financial Theory - The Separating Distributions,"Rodney L. White Center for Financial Research Working Papers 1-76, Wharton School Rodney L. White Center for Financial Research.
   • Stephen A. Ross, "undated". "Mutual Fund Separation in Financial Theory - The Separating Distributions,"Rodney L. White Center for Financial Research Working Papers 01-76, Wharton School Rodney L. White Center for Financial Research.
3. Fousseni Chabi-Yo & Eric Ghysels & Eric Renault, 2008. "On Portfolio Separation Theorems with Heterogeneous Beliefs and Attitudes towards Risk,"Staff Working Papers 08-16, Bank of Canada.
4. Chen, An & Pelsser, Antoon & Vellekoop, Michel, 2011."Modeling non-monotone risk aversion using SAHARA utility functions,"Journal of Economic Theory, Elsevier, vol. 146(5), pages 2075-2092, September.
5. Paul A. Samuelson, 1970."The Fundamental Approximation Theorem of Portfolio Analysis in terms of Means, Variances and Higher Moments,"The Review of Economic Studies, Review of Economic Studies Ltd, vol. 37(4), pages 537-542.
6. Chabi-Yo, Fousseni & Leisen, Dietmar P.J. & Renault, Eric, 2014."Aggregation of preferences for skewed asset returns,"Journal of Economic Theory, Elsevier, vol. 154(C), pages 453-489.
7. Cass, David & Stiglitz, Joseph E., 1970."The structure of investor preferences and asset returns, and separability in portfolio allocation: A contribution to the pure theory of mutual funds,"Journal of Economic Theory, Elsevier, vol. 2(2), pages 122-160, June.
8. Ian Martin, 2011. "Simple Variance Swaps,"NBER Working Papers 16884, National Bureau of Economic Research, Inc.

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• NEP-UPT-2018-08-27 (Utility Models and Prospect Theory)

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