انتخاب پرتفولیو پاداش-خطر و سلطه اتفاقی

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 a 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 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.

Markowitz (1952) introduced an intuitive model of return and risk for portfolio selection. This model is useful to guide one's intuition, and because of its simplicity it is also commonly used in practical finance decisions. Markowitz (1952) proposed to model return and risk in term of mean and variance, but he also suggested other measures of risk as for example the semivariance (Markowitz, 1959). The advantage of using the variance for describing the risk component of a portfolio, is principally due to the simplicity of the computation, but from the point of view of risk measurement the variance is not a satisfactory measure. First, the variance is a symmetric measure and “penalizes” gains and losses in the same way. Second, the variance is inappropriate to describe the risk of low probability events, as for example the default risk. Third, the mean–variance approach is not consistent with second-order stochastic dominance and thus with the expected utility approach for portfolio selection. This is well illustrated by the (μ,σ)-Paradox (see Copeland and Weston, 1998, Chapter 4.G). As already suggested by Markowitz (1959), Ogryczak and Ruszczynski (1997) also proposed semivariance models, where the reward–risk approach is maintained, but the choice of semivariance instead of variance makes the model consistent with second-order stochastic dominance. They also extend the consistency concept to higher order stochastic dominance by defining a more general central semideviation measure. Other risk measures have been proposed for portfolio selection, as for example Value-at-Risk (Jorion, 1997, Duffie and Pan, 1997) or Expected-Shortfall (Acerbi and Tasche, 2002, Bertsimas et al., 2004). The latter one is consistent with second-order stochastic dominance, as illustrated by Bertsimas et al. (2004), who introduced Expected-Shortfall exactly because its consistency with second-order stochastic dominance. Value-at-risk is widely used in practice, but it is only consistent with respect to first-order stochastic dominance (see Hürlimann, 2002a). Moreover, it has been shown by Artzner et al. (1997), that value-at-risk fails in controlling the risk of large losses with small probability, since it only considers the probability of certain losses to occur, but not the magnitude of these losses. Moreover, value-at-risk usually does not satisfy the subadditivity property, which ensures – if satisfied – a reasonable behavior of the risk measure when adding two positions. Artzner et al. (1999), being concerned with banking regulations, have proposed an axiomatic approach to the definition of a risk measure. They presented a set of four properties for measures of risk and they called measures satisfying these properties, coherent risk measures. Moreover, they show that a coherent risk measure can be still characterized by a non-empty set of scenarios (called generalized scenario), such that “any coherent risk measure arises as the supremum of the expected negative of final net worth for some collection of probability measures on the states of the world” ( Artzner et al., 1999, Section 4.1). Unfortunately, coherent risk measures are usually not consistent with second-order stochastic dominance. Pflug (1998) considered various classes of risk measures and gave the general properties for these classes. Generalizing the approach of Ogryczak and Ruszczynski (1997) he introduced expectation–dispersion risk measures and showed that under some conditions they are consistent with stochastic dominance, but usually not coherent. In this paper we proceed analogously to Artzner et al. (1999), i.e. for portfolio selection we give and justify a set of properties for measures of reward and measures of risk. We give a characterization of these measures and we present some examples. Future researches will be devoted to the portfolio selection problem and to the question posed by Jaschke and Küchler (2001) about a possible extension of the Markowitz theory to the reward–risk framework. As already pointed out, we would like to define a reward–risk framework for portfolio selection which is consistent (in some sense) with utility expectation. This is attained by imposing some specific axioms defining reward and risk measures. We give a brief overview of our axioms. A reward measure should be linear on the space of random variables. If this is not the case, it would be possible to increase (or decrease) the reward of a portfolio by just splitting it in two or more positions. Moreover, a reward measure should satisfy a risk-free condition, which states that the reward of a risk-free asset must be identical with the certain payoff (or change in value of the portfolio) provided by the risk-free asset. Finally, to ensure the link with the utility expectation approach, we impose a consistency of the reward measure with second-order stochastic dominance. This last axiom implies that whenever a risk is preferred to another one by all rational, risk-averse expected utility maximizers, then this risk should provide an higher reward than the dominated risk. Analogously, a risk measure is defined by the following axioms. First, the convexity, which ensures the diversification effect. In fact, if the risk measure were not convex, then it would be possible to reduce the risk of some portfolio by splitting the portfolio in two or more single positions. Second, a risk measure must be invariant under addition of a risk-free position. In a reward–risk framework, the contribution of the risk-free asset to the portfolio should be captured by the reward measure, since a risk-free asset gives a certain reward without adding risk. Third, the principle “no investment, no risk” leads to the very natural axiom that a zero position have zero risk. Finally, following the same argument introduced above for reward measures, we impose the consistency of risk measures with second-order stochastic dominance, i.e. a portfolio net payoff which is preferred to another one by all rational, risk-averse expected utility maximizers, should also have less risk. The paper is organized as follows: in Section 2 we present the portfolio selection problem and some standard results from decision theory, that offer an overview on the portfolio decision problem and serve for defining the optimization setup in our reward–risk approach. In Section 3, the definitions of reward and risk measures are given, with some examples. In Section 4, we give our main results, we characterize reward measures and we present a possible simple characterization of risk measures. Section 5 concludes. Technical results and proofs are presented in the Appendix.

In this paper we have considered the reward–risk approach for portfolio selection and we have denned reward and risk measures through a set of axioms, following the approach proposed by Artzner et al. (1999). To maintain a link between our reward–risk framework and expected utility decision theory we have imposed the isotonicity axioms for both the reward and the risk measure. The consequences of this axiom is that a reward–risk decision maker is consistent with the expected utility maximizer, at least for all the situations where the rational, risk averse expected utility decision does not depend on the shape of the utility index, i.e. when one risk dominates another risk by second-order stochastic dominance. For a finite state of the world, we have shown that on the space of random variables there exists only one reward measure, which is given by expectation under the physical probability. For risk measures we obtain a nice characterization through Choquet integral, which can be interpreted as a “weighted mean”, where the weights are non-increasing in the outcomes. We do not see any reason for suggesting one particular risk measure in the class of measure we have characterized. The choice of the “risk aversion” function g should depend on the risk characteristics of the investor.