Portfolio theory for the recourse certainty equivalent maximizing investor

The portfolio selection problem with one safe andn risky assets is analyzed via a new decision theoretic criterion based on the Recourse Certainty Equivalent (RCE). Fundamental results in portfolio theory, previously studied under the Expected Utility criterion (EU), such as separation theorems, comparative static analysis, and threshold values for inclusion or exclusion of risky assets in the optimal portfolio, are obtained here. In contrast to the EU model, our results for the RCE maximizing investor do not impose restrictions on either the utility function or the underlying probability laws. We also derive a dual portfolio selection problem and provide it with a concrete economic interpretation.Research partly supported by ONR Contracts N0014-81-C-0236 and N00014-82-K-0295, and NSF Grant SES-8408134 with the Center for Cybernetic Studies, The University of Texas at Austin.Partly supported by NSF Grant DDM-8896112.Partly supported by AFOSR Grant 0218-88 and NSF Grant ECS-8802239 at the University of Maryland, Baltimore Campus.     

Production decisions under joint price and production uncertainty

While production decisions in the presence of price uncertainty have been extensively studied, this is not so for the case in which the level of production is itself uncertain. In this paper,we provide a decision analysis under multiplicative production uncertainty, both with and without price uncertainty. We depict equilibrium and obtain comparative statics results with the aid of a diagram based on the difference between expected price and marginal cost. Comparative statics results are obtained for the model with production uncertainty alone and also for simultaneous price and production uncertainty (including two special cases). We first derive results based on the Arrow–Pratt coefficients of risk aversion, and then supplement these with the Ross measure of relative risk aversion, since this proves useful in the presence of multiple sources of uncertainty. We find that increases in risk (both price and production) or input prices reduce expected output. However, expected output supply is an increasing function of (expected) price only for “low” levels of risk aversion, and in general the relationship is ambiguous.     

A simple recourse model for power dispatch under uncertain demand

Optimal power dispatch under uncertainty of power demand is tackled via a stochastic programming model with simple recourse. The decision variables correspond to generation policies of a system comprising thermal units, pumped storage plants and energy contracts. The paper is a case study to test the kernel estimation method in the context of stochastic programming. Kernel estimates are used to approximate the unknown probability distribution of power demand. General stability results from stochastic programming yield the asymptotic stability of optimal solutions. Kernel estimates lead to favourable numerical properties of the recourse model (no numerical integration, the optimization problem is smooth convex and of moderate dimension). Test runs based on real-life data are reported. We compute the value of the stochastic solution for different problem instances and compare the stochastic programming solution with deterministic solutions involving adjusted demand portions.This research is supported by the Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung of the Deutsche Forschungsgemeinschaft.     

Entry and exit decisions with linear costs under uncertainty

From the viewpoint of stochastic programming, we rigorously analyse entry and exit decisions of a project which were proposed by Dixit [A. Dixit, Entry and exit decisions under uncertainty, J. Polit. Econ. 97 (1989), pp. 620–638]. In this article, instead of assuming that the costs are constant in classical research, we assume that they are linear with respect to the price of the commodity produced by the project. Under this assumption, we obtain a condition which guarantees that investing in the project is worthless; besides, the project may be terminated when the commodity price is greater than a certain value. In contrast, there are no such results provided that the costs are constant. Moreover, we provide an explicit solution of entry and exit decisions if the project is worthy to be invested in.     

Optimal initial capital induced by the optimized certainty equivalent

This paper proposes the notion of optimal initial capital (OIC) induced by the optimized certainty equivalent (OCE), as discussed in Ben-Tal and Teboulle (1986) and Ben-Tal and Teboulle (2007). It also investigates the properties of the OIC with various types of utility functions. It is shown that the OIC can be a monetary utility function (negative value of risk measure) for future payoffs with the decision maker’s concrete criteria in the background.     

A framework for analysing decisions under risk

The main objective is to present a framework for analysing decisions under risk. The nature of much information available to decision makers is vague and imprecise, be it information for human managers in organisations or for process agents in a distributed computer environment. Some approaches address the problem of uncertainty, but many of them concentrate more on representation and less on evaluation. The emphasis in this paper is on evaluation and even though the representation used is that of probability theory, other well-established formalisms can be used. The approach allows the decision maker to be as deliberately imprecise as he feels is natural and provides him with the means for expressing varying degrees of imprecision in the input sentences. The framework we present is intended to be tolerant and to provide means for evaluating decision situations using several decision rules beside the conventional maximisation of the expected utility.     

Progressing from uncertainty to risk for DEA-based decisions

We demonstrate a methodology for estimating, with a specified probability, the interval within which the true DEA efficiency of an individual Decision Making Unit occurs. With this procedure, decisions based on DEA scores are made under conditions of risk, as opposed to the current practice in which decisions are made under uncertainty. The method applies statistical Panel Data Analysis (PDA), which provides proven and powerful methodologies for diagnostic testing of residuals and estimation of confidence intervals. Using PDA, we demonstrate, with a sample of real-world data, that DEA score residuals sometimes are independent and Normally distributed, and estimate confidence intervals based on these validated assumptions. Then, using another sample of real-world data in which violations of these assumptions do occur, we demonstrate well-known PDA Generalized Least Squares statistical models that account for the violations in the estimation of confidence intervals.     

Hedging entry and exit decisions: optimizing location under exchange rate uncertainty

^** Email: alexru00{at}ms41.hinet.net^*** Email: ctlin{at}mail.yust.edu.tw The Cobb–Douglas production function with Abel's (1983,Am. Econ. Rev., 173, 228–233) model is extended herein,and real options analysis (ROA) for entry–exit decision-makingestablished utilizing Dixit's (1989b) decision model under exchangerate uncertainty. This work considers the effects of real exchangerates on strategies that determine the locations of productionby firms that are entering markets in two countries. The ROAis also adopted to evaluate the switching location between twocountries. A continuous-time model optimization problem is solvedin closed-form. This provides a useful beginning to an importantanalysis of the effects on industry of exchange rate fluctuationswhen the optimal entry (exit) trigger for transferring locationsis important for a basic global logistics model. Furthermore,a myopic solution of the optimal entry (exit) trigger, sensitivityanalysis and some characteristics of the optimal productionstrategy are sought. This paper contributes to the problem ofchoice of foreign production strategy.     

A model for strategic planning under uncertainty

Summary In this paper the multi-period strategic planning problem for a consumer sumer product manufacturing chain is considered. Our discussion is focused on investment decisions which, are economically optimal over the whole planning horizonT, while meeting customer demands and conforming to technological requirements. In strategic planning, time and uncertainty play important roles. The uncertainties in the model are due to different levels of forecast demands, cost estimates and equipment behaviour. The main aim of this paper is to develop and analyse a multiperiod stochastic model representing the entire manufacturing chain, from the acquisitions of raw material to the delivering of final products. The resulting optimization problem is computationally intractable because of the enormous, and sometimes unrealistic, number of scenarios that must be considered in order to identify the optimal planning strategy. We propose two different solution approaches; firstly, we apply a scenario risk analysis giving the related results of experiments on a particular real data set. We then describe and investigate an Integer Stochastic Programming formulation of the problem and propose, as a solution technique, a variation of Benders decomposition method, namely theL-shaped method.     

A general framework for cooperation under uncertainty

In this paper, we introduce a general framework for situations with decision making under uncertainty and cooperation possibilities. This framework is based upon a two stage stochastic programming approach. We show that under relatively mild assumptions the associated cooperative games are totally balanced. Finally, we consider several example situations.     

Restricted recourse strategies for bounding the expected network recourse function

This paper presents bounds for the expected recourse function for stochastic programs with network recourse. Cyclic recourse, a concept introduced by Wallace [18], allows the approximation of the recourse problem by restricting the optimal flows on a set of cycles and by augmenting the original network to induce separability. We introduce a new procedure that uses again a set of cycles but does not approximate the problem; instead it solves it heuristically without altering the original network or requiring separability. The method produces tighter bounds and is computationally feasible for large networks. Numerical experiments with selected networks illustrate the effectiveness of the approach.     

A decision model for berth allocation under uncertainty

This paper studies the berth allocation problem (BAP) under uncertain arrival time or operation time of vessels. It does not only concern the proactive strategy to develop an initial schedule that incorporates a degree of anticipation of uncertainty during the schedule’s execution, but also studies the reactive recovery strategy which adjusts the initial schedule to handle realistic scenarios with minimum penalty cost of deviating from the initial schedule. A two-stage decision model is developed for the BAP under uncertainties. Moreover, a meta-heuristic approach is proposed for solving the above problem in large-scale realistic environments. Numerical experiments are conducted to validate the effectiveness and efficiency of the proposed method.     

Measuring production with random inputs and outputs using DEA and certainty equivalent

This study evaluates production operations with inputs/outputs under random influences. We introduce a measurement of efficiency using utility function families. Applying Data Envelopment Analysis (DEA) and the certainty equivalent, the proposed measurement is capable of accommodating various risk attitudes of evaluators.     

A concept of equilibrium for a game under uncertainty

In this paper, we introduce a new concept of equilibrium for an n-person game involving undetermined parameters when the players know the domain where these parameters can vary but completely ignore their behaviour. We call it ZS-equilibrium. Then we give some of its properties and sufficient conditions for its existence.     

Approximation of expected returns and optimal decisions under uncertainty using mean and mean absolute deviation

We consider economic decision problems under uncertainty consisting of choosing an optimal decisionX, so as to maximize to expected value of an objective function depending on a stochastic parameterp. The paper establishes an optimal policy intervalX _A X ¹ X _B, where the boundsX _A,X _B are given in terms of simple parameters of the distribution ofp, in particular the mean, and the mean absolute deviationd=E ¦p– ¦. The convexity assumptions needed to establish such bounds are shown to hold naturally in some classical problems of production under uncertainty.

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Zusammenfassung Wir betrachten wirtschaftliche Entscheidungsprobleme mit Unsicherheit, in denen eine optimale EntscheidungX so getroffen werden soll, daß der Erwartungswert einer Zielfunktion, abhängig von einem stochastischen Parameterp, maximiert werden soll. In dieser Arbeit wird ein IntervallX _A X ¹ X _B für die optimale Politik angegeben, wobei die SchrankenX _A,X _B durch einfache Größen der Verteilung vonp ausgedrückt werden, im besonderen durch den Mittelwert und die mittlere absolute Abweichungd=E ¦p– ¦. Ferner wird gezeigt, daß die für die Herleitung der Schranken benötigten Konvexitätsannahmen in natürlicher Weise für einige klassische Produktionsprobleme mit Unsicherheit gelten.

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Supported by BARD Project No. I-10-79 and by Technion VPR Fund-Lawrence Deutsch Research Fund.     

Scoring decisions in the context of economic uncertainty

We consider methods for incorporating forecasts of future economic conditions into acquisition decisions for scored retail credit and loan portfolios. We suppose that a portfolio manager is faced with two possible future economic scenarios, each characterised by a known probability of occurrence and by known performance functions that give expected profit and volume. We suppose further that he must choose in advance the scoring strategy and score cutoffs to optimise performance. We show that, despite the uncertainty of performance induced by economic conditions, every efficient policy consists of a single cutoff, provided the expected profit and volume performance curves in each scenario are concave. If these curves are not concave, efficient operating points can be characterised as cutoffs on a redefined score. In cases in which two scorecards are available, we show that it may be advantageous to randomly choose the scorecard to be employed, and we provide methods for selecting efficient operating points. Discussion is limited to cases with two scorecards and two economic scenarios, but our approach and results generalise to more scorecards and more economic scenarios.     

SOCRATES: A system for scheduling hydroelectric generation under uncertainty

The Pacific Gas and Electric Company, the largest investor-owned energy utility in the United States, obtains a significant fraction of its electric energy and capacity from hydrogeneration. Although hydro provides valuable flexibility, it is subject to usage limits and must be carefully scheduled. In addition, the amount of energy available from hydro varies widely from year to year, depending on precipitation and streamflows. Optimal scheduling of hydrogeneration, in coordination with other energy sources, is a stochastic problem of practical significance to PG&E. SOCRATES is a system for the optimal scheduling of PG&E's various energy sources over a one- to two-year horizon. This paper concentrates on the component of SOCRATES that schedules hydro. The core is a stochastic optimization model, solved using Benders decomposition. Additional components are streamflow forecasting models and a database containing hydrological information. The stochastic hydro scheduling module of SOCRATES is undergoing testing in the user's environment, and we expect PG&E hydrologists and hydro schedulers to place progressively more reliance upon it.     

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method.