A three mutual fund separation theorem

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.