The portfolio selection problem is traditionally modelled by two different approaches. The first one is based on an axiomatic model of risk-averse preferences, where decision makers are assumed to possess an expected utility function and the portfolio choice consists in maximizing the expected utility over the set of feasible portfolios. The second approach, first proposed by Markowitz (1952), is very intuitive and reduces the portfolio choice to a set of two criteria, reward and risk, with possible tradeoff analysis. Usually the reward-risk model is not consistent with the first approach, even when the decision is independent from the specific form of the risk-averse expected utility function, i.e. when one investment dominates another one by second order stochastic dominance. In this paper we generalize the reward-risk model for portfolio selection. We define reward measures and risk measures by giving a set of properties these measures should satisfy. One of these properties will be the consistency with second order stochastic dominance, to obtain a link with the expected utility portfolio selection. We characterize reward and risk measures and we discuss the implication for portfolio selection.

This paper gives a survey over a common aspect of prospect theory that occurred to be of importance in a series of recent papers developed by Enrico De Giorgi, Thorsten Hens, Janos Mayer, Haim Levy, Thierry Post, Marc Oliver Rieger and Mei Wang. The common aspect of these papers is that the value function of the prospect theory of Kahneman and Tversky (1979) and similarly that of Tversky and Kahneman (1992) has to be re-modelled if one wants to apply it to portfolio selection. Instead of the piecewise power value function, a piecewise negative exponential function should be used. This functional form is still compatible with laboratory experiments but it has the following advantages over and above Kahneman and Tversky`s piecewise power function:1. The Bernoulli Paradox does not arise for lotteries with finite expected value. 2. No infinite leverage/robustness problem arises. 3. CAPM-equilibria with heterogeneous investors and prospect utility do exist. 4. It is able to simultaneously resolve the following asset pricing puzzles: the equity premium, the value and the size puzzle.

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W e consider choice over uncertain, monetary payoffs and study a general class of preferences. These preferences favor diversification, except perhaps on a subset of sufficiently disliked acts over which concentration is instead preferred. This structure encompasses a number of known models (e.g., expected utility and several variants under a concave utility function). We show that such preferences share a representation in terms of a family of measures of risk and targets. Specifically, the choice function is equivalent to selection of a maximum index level such that the risk of beating the target at that level is acceptable. This representation may help to uncover new models of choice. One that we explore in detail is the special case when the targets are bounded. This case corresponds to a type of satisficing and has descriptive relevance. Moreover, the model is amenable to large-scale optimization.