Portfolio Selection with Monotone Mean-Variance Preferences

SSRN Electronic Journal 01/2009; 19(3):487-521. DOI: 10.2139/ssrn.1148724
Source: RePEc


We develop a Savage-type model of choice under uncertainty in which agents identify uncertain prospects with subjective compound lotteries. Our theory permits issue preference; that is, agents may not be indifferent among gambles that yield the same probability distribution if they depend on different issues. Hence, we establish subjective foundations for the Anscombe-Aumann framework and other models with two different types of probabilities. We define second-order risk as risk that resolves in the first stage of the compound lottery and show that uncertainty aversion implies aversion to second-order risk which implies issue preference and behavior consistent with the Ellsberg paradox.

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    • "Nevertheless , it is well known that mean-variance functional is not monotone and this is a serious drawback. Namely, Maccheroni et al. [10] gave a simple example when an investor with mean-variance preferences may strictly prefer less to more, thus violating one of the most compelling principles of economic rationality. For this reason, they created a new class of monotone preferences that coincide with mean-variance preferences on their domain of monotonicity, but differ where mean-variance preferences fail to be monotone and are therefore not economically meaningful. "
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    ABSTRACT: This is a follow up of our previous paper - Trybu{\l}a and Zawisza \cite{TryZaw}, where we considered a modification of a monotone mean-variance functional in continuous time in stochastic factor model. In this article we address the problem of optimizing the mentioned functional in a market with a stochastic interest rate. We formulate it as a stochastic differential game problem and use Hamilton-Jacobi-Bellman-Isaacs equations to derive the optimal investment strategy and the value function.
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    • "The purpose is to describe an optimal financial strategy which an investor can follow in order to maximize his performance criterion which is a modification of the monotone mean-variance functional. Let us recall that Maccheroni et al. [13] introduce a functional given by "
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    ABSTRACT: We consider an incomplete market with a non-tradable stochastic factor and an investment problem with optimality criterion based on a functional which is a modification of a monotone mean-variance preferences. We formulate it as a stochastic differential game problem and use Hamilton Jacobi Bellman Isaacs equations to derive the optimal investment strategy and the value function. Finally, we show that our solution coincides with the solution to classical mean-variance problem with risk aversion coefficient which is dependent on stochastic factor.
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    • "This allows us to prove existence and uniqueness of an equilibrium under general assumptions by backward induction. Typical examples of translation invariant preferences are those induced by expected exponential utility, the monotone mean-variance preferences of Maccheroni et al. (2009), mean-risk type preferences where risk is measured with a convex risk measure, optimized certainty equivalentsà la Ben-Tal and Teboulle (1986, 1987) or the divergence utilities of Cherny and Kupper (2009). The assumption of translation invariant preferences is appropriate if, for instance, agents are understood as banks or insurance companies which evaluate investments in terms of expected values and risk capital, that is, buffer capital that needs to be held to make an investment acceptable from a risk management point of view. "
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    ABSTRACT: We propose a general discrete-time framework for deriving equilibrium prices of financial securities. It allows for heterogeneous agents, unspanned random endowments and convex trading constraints. We give a dual characterization of equilibria and provide general results on their existence and uniqueness. In the special case where all agents have preferences of the same type, and in equilibrium, all random endowments are replicable by trading in the financial market, we show that a one-fund theorem holds and give an explicit expression for the equilibrium pricing kernel. If the underlying noise is generated by nitely many Bernoulli random walks, the equilibrium dynamics can be described by a system of coupled backward stochastic difference equations, which in the continuous-time limit becomes a multidimensional backward stochastic differential equation. If the market is complete in equilibrium, the system of equations decouples, but if not, one needs to keep track of the prices and continuation values of all agents to solve it.
    Mathematics of Operations Research 07/2012; DOI:10.2139/ssrn.1755610 · 1.31 Impact Factor
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