Minimizing measures of risk by saddle point conditions.

The minimization of risk functions is becoming a very important topic due to its interesting applications in Mathematical Finance and Actuarial Mathematics. This paper addresses this issue in a general framework. Many types of risk function may be involved. A general representation theorem of risk functions is used in order to transform the initial optimization problem into an equivalent one that overcomes several mathematical caveats of risk functions. This new problem involves Banach spaces but a mean value theorem for risk measures is stated, and this simplifies the dual problem. Then, optimality is characterized by saddle point properties of a bilinear expression involving the primal and the dual variable. This characterization is significantly different if one compares it with previous literature. Furthermore, the saddle point condition very easily applies in practice. Four applications in finance and insurance are presented.Risk minimization; Saddle point condition; Actuarial and finantial aplications;

Mean-Variance-Skewness Portfolio Performance Gauging: A General Shortage Function and Dual Approach

This paper proposes a nonparametric efficiency measurement approach for the static portfolio selection problem in mean-variance-skewness space. A shortage function is defined that looks for possible increases in return and skewness and decreases in variance. Global optimality is guaranteed for the resulting optimal portfolios. We also establish a link to a proper indirect mean-variance-skewness utility function. For computational reasons, the optimal portfolios resulting from this dual approach are only locally optimal. This framework permits to differentiate between portfolio efficiency and allocative efficiency, and a convexity efficiency component related to the difference between the primal, non-convex approach and the dual, convex approach. Furthermore, in principle, information can be retrieved about the revealed risk aversion and prudence of investors. An empirical section on a small sample of assets serves as an illustration.shortage function, efficient frontier, mean-variance-skewness, portfolios, risk aversion, prudence

Portfolio selection problems in practice: a comparison between linear and quadratic optimization models

Several portfolio selection models take into account practical limitations on the number of assets to include and on their weights in the portfolio. We present here a study of the Limited Asset Markowitz (LAM), of the Limited Asset Mean Absolute Deviation (LAMAD) and of the Limited Asset Conditional Value-at-Risk (LACVaR) models, where the assets are limited with the introduction of quantity and cardinality constraints. We propose a completely new approach for solving the LAM model, based on reformulation as a Standard Quadratic Program and on some recent theoretical results. With this approach we obtain optimal solutions both for some well-known financial data sets used by several other authors, and for some unsolved large size portfolio problems. We also test our method on five new data sets involving real-world capital market indices from major stock markets. Our computational experience shows that, rather unexpectedly, it is easier to solve the quadratic LAM model with our algorithm, than to solve the linear LACVaR and LAMAD models with CPLEX, one of the best commercial codes for mixed integer linear programming (MILP) problems. Finally, on the new data sets we have also compared, using out-of-sample analysis, the performance of the portfolios obtained by the Limited Asset models with the performance provided by the unconstrained models and with that of the official capital market indices

Controlled diffusion processes

This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.Comment: Published at http://dx.doi.org/10.1214/154957805100000131 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Simplified approaches for portfolio decision analysis

Traditional choice decisions involve selecting a single, best alternative from a larger set of potential options. In contrast, portfolio decisions involve selecting the best subset of alternatives — alternatives that together maximize some measure of value to the decision maker and are within their available resources to implement. Examples include capital investment, R&D project selection, and maintenance planning. Portfolio decisions involve a combinatorial aspect that makes them more theoretically and computationally challenging than choice problems, particularly when there are interactions between alternatives. Several portfolio decision analysis methods have been developed over the years and an increasing interest has been noted in the field of portfolio decision analysis. These methods are typically called “exact” methods, but can also be called prescriptive methods. These are generally computationally-intensive algorithms that require substantial amounts of information from the decision maker, and in return yield portfolios that are provably optimal or optimal within certain bounds. These methods have proved popular for choice decisions — for example, those based on multiattribute value or utility theory. But whereas information and computational requirements for choice problems are probably manageable for the majority of diligent decision makers, it is much less clear that this is true of portfolio decisions. That is, for portfolio decisions it may be more common that decision makers do not have the time, expertise and ability to exert the effort to assess all the information required of an exact method. Heuristics are simple, psychologically plausible rules for decision making that limit the amount of information required and the computation effort needed to turn this information into decisions. Previous work has shown that people often use heuristics when confronted with traditional choice problems in unfacilitated contexts, and that these can often return good results, in the sense of selecting alternatives that are also ranked highly by exact methods. This suggests that heuristics may also be useful for portfolio decisions. Moreover, while the lower information demands made by choice problems mean that heuristics have not generally been seen as prescriptive options, the more substantial demands made by portfolio decisions make a priori case for considering their use not just descriptively, but as tools for decision aid. Very little work exists on the use of heuristics for portfolio decision making, the subject of this thesis. Durbach et al. (2020) proposed a family of portfolio selection heuristics known collectively as add-the-best. These construct portfolios by adding, at every step, the alternative that is best in a greedy sense, with different definitions of what “best” is. This thesis extends knowledge on portfolio heuristics in three main respects. Firstly, we show that people use certain of the add-the-best heuristics when selecting portfolios without facilitation, in a context where there are interactions between alternatives. We run an experiment involving actual portfolio decision making behaviour, administered to participants who had the opportunity to choose as many alternatives as they wanted, but under the constraint of a limited budget. This experiment, parts of which were reported in Durbach et al. (2020), provides the first demonstration of the use of heuristics in portfolio selections. Secondly, we use a simulation experiment to test the performance of the heuristics in two novel environments: those involving multiple criteria, and those in which interactions between projects may be positive (the value of selecting two alternatives is more than the sum of their individual values) or negative (the opposite). This extends the results in Durbach et al. (2020), who considered only environments involving a single criterion and positive interactions between alternatives. In doing so we differentiate between heuristics that guide the selection of alternatives, called selection heuristics, and heuristics for aggregating performance across criteria, which we call scoring heuristics. We combine various selection and scoring heuristics and test their performance on a range of simulated decision problems. We found that certain portfolio heuristics continued to perform well in the presence of negative interactions and multiple criteria, and that performance depended more on the approach used to build portfolios (selection heuristics) than on the method of aggregation across criteria (scoring heuristics). We also found that in these extended conditions heuristics continued to provide outcomes that were competitive with optimal models, but that heuristics that ignored interactions led to potentially poor results. Finally, we complement behavioral and simulation experimental studies with an application of both exact methods and portfolio heuristics in a real-world portfolio decision problem involving the selection of the best subset of research proposals out of a pool of proposals submitted by researchers applying for grants from a research institution. We provide a decision support system to this institution in the form of a web-based application to assist with portfolio decisions involving interactions. The decision support system implements exact methods, namely the linear-additive portfolio value model and the robust portfolio model, as well as two portfolio heuristics found to perform well in simulations

Country portfolios in open economy macro models

This paper develops a simple approximation method for computing equilibrium portfolios in dynamic general equilibrium open economy macro models. The method is widely applicable, simple to implement, and gives analytical solutions for equilibrium portfolio positions in any combination or types of asset. It can be used in models with any number of assets, whether markets are complete or incomplete, and can be applied to stochastic dynamic general equilibrium models of any dimension, so long as the model is amenable to a solution using standard approximation methods. We first illustrate the approach using a simple two-asset endowment economy model, and then show how the results extend to the case of any number of assets and general economic structure.Econometric models ; Equilibrium (Economics) - Mathematical models ; Macroeconomics - Econometric models ; Monetary policy

Set optimization - a rather short introduction

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems