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In this paper we show how to formulate and solve robust portfolio selection problems. The objective of these robust formulations is to systematically combat the sensitivity of the optimal portfolio to statistical and modeling errors in the estimates of the relevant market parameters. We introduce "uncertainty structures" for the market parameters and show that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second-order cone programs and, therefore, the computational effort required to solve them is comparable to that required for solving convex quadratic programs. Moreover, we show that these uncertainty structures correspond to confidence regions associated with the statistical procedures used to estimate the market parameters. We demonstrate a simple recipe for efficiently computing robust portfolios given raw market data and a desired level of confidence.

Abstract. Mean-variance portfolio analysis provided the first quantitative treatment of the tradeoff between profit and risk. We describe in detail the interplay between objective and constraints in a number of single-period variants, including semivariance models. Particular emphasis is laid on avoiding the penalization of overperformance. The results are then used as building blocks in the development and theoretical analysis of multiperiod models based on scenario trees. A key property is the possibility of removing surplus money in future decisions, yielding approximate downside risk minimization.

Consider a set of users sharing a network node under an allocation scheme that provides each user with a fixed minimum and a random extra amount of bandwidth and buffer. Allocations and prices are adjusted to adapt to resource availability and user demands. Equilibrium is achieved when all users optimize their utility and demand equals supply for non-free resources. We analyze two models of user behavior. We show that at equilibrium expected return on purchasing variable resources can be higher than that on fixed resources. Thus users must balance the marginal increase in utility due to higher return on variable resources and the marginal decrease in utility due to their variability. For the first user model we further show that at equilibrium where such tradeoff is optimized all users hold strictly positive amounts of variable bandwidth and buffer. For the second model we show that if both variable bandwidth and buffer are scarce then at equilibrium every user either holds both variab...

Abstract — Consider a set of connections sharing a network node under an allocation scheme that pro-vides each connection with a fixed minimum and a random extra amount of bandwidth and buffer. Allocations and prices are adjusted to adapt to re-source availability y and user demands. We consider two scenarios of user behavior. In the first scenario a connection purchases an allocation to maximize its expected utility in such a way that the resource cost of the new allocation, and hence its connec-tion charge, remains the same as that for the old allocation. In the second scenario this budget con-straint is relaxed and a connection tries to maxi-mize its benefit, expected utility minus the resource cost. Equilibrium is achieved when all connections ach]eve their optimality and demand equals sup-ply for non-free resources. We show that at equi-librium expected return on purchasing variable re-sources can be higher than that on fixed resources. Thus connections must balance the marginal in-crease in utility due to higher return on variable resources and the marginal decrease in utility due to their variability. We further show that in equi-librium where such tradeoff is optimized all con-nections hold strictly positive amounts of variable bandwidth and bufer. I.

We propose a novel model of resource sharing schemes that provide each user with a fixed minimum and a random extra amount of bandwidth and buffer. Allocations and prices are adjusted to adapt to resource availability and user demands. At equilibrium, if it exists, all users optimize their utility and resource demand equals supply, i.e., the marginal increase in user utility due to higher return on variable resources is balanced by the marginal decrease in utility due to their variability. We show how an equilibrium might be approached using a simple price adjustment rule that does not require any knowledge on the part of the network about user utilities. We further show that at equilibrium every user holds strictly positive amounts of variable bandwidth and variable buffer, and in the same ratio. We characterize the equilibrium prices to lie in a hyperplane that can be computed by the network without having to know user utilities. We illustrate with an example how this characterizatio...

This paper attempts to give an overview of the pricing of risks in a pure exchange economy, where trade takes place at time zero and where uncertainty is revealed at time one. An economic equilibrium model under uncertainty is formulated, where conditions characterizing a Pareto optimal exchange equilib-rium are derived. We present two sets of sufficient conditions for the existence of an equilibrium, and demonstrate how equilibria can be characterized through several examples. Uniqueness of equilibrium is also discussed. Special attention is given to the principal components that the premiums in a reinsurance market must depend upon. We also apply the general theory to the risk exchange problem between a policyholder and an insurer, and in particular we compute market premiums of the resulting optimal contracts. It is emphasized throughout how the formulation of a competitive equilib-rium, rather than merely a general risk exchange formulation, is of particular interest in deriving a well-defined and unique set of equilibrium premiums in an insurance market. The theory is put into a framework which is fruitful for extensions beyond the one-period case. KEYWORDS Reinsurance market; competitive equilibrium; uniqueness of premiums; Pareto optimality; risk exchange; private insurance; CAPM; risk tolerance; complete markets. I.

The paper deals with the issues associated with identification of stocks generating abnormal returns. Following the findings of a finance theory regarding portfolio tilting, a set of price-related stocks' attributes was analyzed. The analysis was conducted with the help of rough sets methodology which allows to distinguish "important" attributes for problem description, and to generate decision rules which can be later used to predict stocks' performance. Validity of the approach was tested on the Toronto Stock Exchange data. Keywords: rough sets, decision rules, reducts, portfolio tilting, anomalies theory About the Authors Roman Slowinski is Professor of Decision and Computer Sciences and Head of the Laboratory of Intelligent Decision Support Systems, Institute of Computer Science, Poznan University of Technology, Poznan, Poland. Robert Susmaga is on the research and teaching staff of the Institute of Computer Science at Poznan University of Technology, Poznan, Poland. Wojtek Mic...

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paper represents a major extension of a section in Chapter 2 of my Ph.D. dissertation submitted to the University of California at Berkeley, where a two-asset equilibrium was developed. I am very grateful to Navneet Arora