Risk management for mathematical optimization under uncertainty

We present a general multistage stochastic mixed 0-1 problem where the uncertainty appears everywhere in the objective function, constraints matrix and right-hand-side. The uncertainty is represented by a scenario tree that can be a symmetric or a nonsymmetric one. The stochastic model is converted in a mixed 0-1 Deterministic Equivalent Model in compact representation. Due to the difficulty of the problem, the solution offered by the stochastic model has been traditionally obtained by optimizing the objective function expected value (i.e., mean) over the scenarios, usually, along a time horizon. This approach (so named risk neutral) has the inconvenience of providing a solution that ignores the variance of the objective value of the scenarios and, so, the occurrence of scenarios with an objective value below the expected one. Alternatively, we present several approaches for risk averse management, namely, a scenario immunization strategy, the optimization of the well known Value-at-Risk (VaR) and several variants of the Conditional Value-at-Risk strategies, the optimization of the expected mean minus the weighted probability of having a "bad" scenario to occur for the given solution provided by the model, the optimization of the objective function expected value subject to stochastic dominance constraints (SDC) for a set of profiles given by the pairs of threshold objective values and either bounds on the probability of not reaching the thresholds or the expected shortfall over them, and the optimization of a mixture of the VaR and SDC strategies.

Income risk of EU coal-fired power plants after Kyoto

Coal-fired power plants may enjoy a significant advantage relative to gas plants in terms of cheaper fuel cost. Still, this advantage may erode or even turn into disadvantage depending on CO2 emission allowance price. This price will presumably rise in both the Kyoto Protocol commitment period (2008-2012) and the first post-Kyoto years. Thus, in a carbon-constrained environment, coal plants face financial risks arising in their profit margins, which in turn hinge on their so-called "clean dark spread". These risks are further reinforced when the price of the output electricity is determined by natural gas-fired plants' marginal costs, which differ from coal plants' costs. We aim to assess the risks in coal plants' margins. We adopt parameter values estimated from empirical data. These in turn are derived from natural gas and electricity markets alongside the EU ETS market where emission allowances are traded. Monte Carlo simulation allows to compute the expected value and risk profile of coal-based electricity generation. We focus on the clean dark spread in both time periods under different future scenarios in the allowance market. Specifically, bottom 5% and 10% percentiles are derived. According to our results, certain future paths of the allowance price may impose significant risks on the clean dark spread obtained by coal plants.

The Relationship between Risk and Expected Return in Europe

Revised: 2006-07

Asset pricing and systematic liquidity risk: an empirical investigation of the Spanish stock market

Systematic liquidity shocks should affect the optimal behavior of agents in financial markets. Indeed, fluctuations in various measures of liquidity are significantly correlated across common stocks. Accordingly, this paper empirically analyzes whether Spanish average returns vary cross-sectionally with betas estimated relative to two competing liquidity risk factors. The first one, proposed by Pastor and Stambaugh (2002), is associated with the strength of volume-related return reversals. Our marketwide liquidity factor is defined as the difference between returns highly sensitive to changes in the relative bid-ask spread and returns with low sensitivities to those changes. Our empirical results show that neither of these proxies for systematic liquidity risk seems to be priced in the Spanish stock market. Further international evidence is deserved.

Two-Stage Stochastic Optimization. An Application in the Third Sector

[en] It is known that most of the problems applied in the real life present uncertainty. In the rst part of the dissertation, basic concepts and properties of the Stochastic Programming have been introduced to the reader, also known as Optimization under Uncertainty. Moreover, since stochastic programs are complex to compute, we have presented some other models such as wait-and-wee, expected value and the expected result of using expected value. The expected value of perfect information and the value of stochastic solution measures quantify how worthy the Stochastic Programming is, with respect to the other models. In the second part, it has been designed and implemented with the modeller GAMS and the optimizer CPLEX an application that optimizes the distribution of non-perishable products, guaranteeing some nutritional requirements with minimum cost. It has been developed within Hazia project, managed by Sortarazi association and associated with Food Bank of Biscay and Basic Social Services of several districts of Biscay.

Option-Implied Preferences Adjustments and Risk-Neutral Density Forecasts

The main objective of this paper is to analyse the value of information contained in prices of options on the IBEX 35 index at the Spanish Stock Exchange Market. The forward looking information is extracted using implied risk-neutral density functions estimated by a mixture of two-lognormals and three alternative risk-adjustments: the classic power and exponential utility functions and a habit-based specification that allows for a counter-cyclical variation of risk aversion. Our results show that at four-week horizon we can reject the hypothesis that between October 1996 and March 2000 the risk-neutral densities provide accurate predictions of the distributions of future realisations of the IBEX 35 index at a four-week horizon. When forecasting through risk-adjusted densities the performance of this period is statistically improved and we no longer reject that hypothesis. All risk-adjusted densities generate similar forecasting statistics. Then, at least for a horizon of four-weeks, the actual risk adjustment does not seem to be the issue. By contrast, at the one-week horizon risk-adjusted densities do not improve the forecasting ability of the risk-neutral counterparts.

Markov Switching Risk Premium and the term structure of interest rates. Empirical evidence from US post-war interest rates

This paper considers the basic present value model of interest rates under rational expectations with two additional features. First, following McCallum (1994), the model assumes a policy reaction function where changes in the short-term interest rate are determined by the long-short spread. Second, the short-term interest rate and the risk premium processes are characterized by a Markov regime-switching model. Using US post-war interest rate data, this paper finds evidence that a two-regime switching model fits the data better than the basic model. The estimation results also show the presence of two alternative states displaying quite different features.

Expected Fair Allocation in Farsighted Network Formation

I consider cooperation situations where players have network relations. Networks evolve according to a stationary transition probability matrix and at each moment in time players receive payoffs from a stationary allocation rule. Players discount the future by a common factor. The pair formed by an allocation rule and a transition probability matrix is called expected fair if for every link in the network both participants gain, marginally, and in discounted, expected terms, the same from it; and it is called a pairwise network formation procedure if the probability that a link is created (or eliminated) is positive if the discounted, expected gains to its two participants are positive too. The main result is the existence, for the discount factor small enough, of an expected fair and pairwise network formation procedure where the allocation rule is component balanced, meaning it distributes the total value of any maximal connected subnetwork among its participants. This existence result holds for all discount factors when the pairwise network formation procedure is restricted. I finally provide some comparison with previous models of farsighted network formation.

A colorectal cancer genome-wide association study in a Spanish cohort identifies two variants associated with colorectal cancer risk at 1p33 and 8p12

Background: Colorectal cancer (CRC) is a disease of complex aetiology, with much of the expected inherited risk being due to several common low risk variants. Genome-Wide Association Studies (GWAS) have identified 20 CRC risk variants. Nevertheless, these have only been able to explain part of the missing heritability. Moreover, these signals have only been inspected in populations of Northern European origin. Results: Thus, we followed the same approach in a Spanish cohort of 881 cases and 667 controls. Sixty-four variants at 24 loci were found to be associated with CRC at p-values <10-5. We therefore evaluated the 24 loci in another Spanish replication cohort (1481 cases and 1850 controls). Two of these SNPs, rs12080929 at 1p33 (P-replication=0.042; P-pooled=5.523x10(-03); OR (CI95%)=0.866(0.782-0.959)) and rs11987193 at 8p12 (P-replication=0.039; P-pooled=6.985x10(-5); OR (CI95%)=0.786(0.705-0.878)) were replicated in the second Phase, although they did not reach genome-wide statistical significance. Conclusions: We have performed the first CRC GWAS in a Southern European population and by these means we were able to identify two new susceptibility variants at 1p33 and 8p12 loci. These two SNPs are located near the SLC5A9 and DUSP4 loci, respectively, which could be good functional candidates for the association signals. We therefore believe that these two markers constitute good candidates for CRC susceptibility loci and should be further evaluated in other larger datasets. Moreover, we highlight that were these two SNPs true susceptibility variants, they would constitute a decrease in the CRC missing heritability fraction.

Choice under Risk : Theory and Applications

Es útil para estudiantes de postgrado (Master y Doctorado) en cursos de Economía o de Microeconomía en los que se analicen problemas de Decisión en condiciones de Riesgo o Incertidumbre. El documento comienza explicando la Teoría de la Utilidad Esperada. A continuación se estudian la aversión al riesgo, los coeficientes de aversión absoluta y relativa al riesgo, la relación “más averso que” entre agentes económicos y los efectos riqueza sobre las decisiones en algunas relaciones de preferencia utilizadas frecuentemente en el análisis económico. La sección 4 se centra en la comparación entre alternativas arriesgadas en términos de rendimiento y riesgo, considerando la dominancia estocástica de primer y segundo orden y algunas extensiones posteriores de esas relaciones de orden. El documento concluye con doce ejercicios resueltos en los que se aplican los conceptos y resultados expuestos en las secciones anteriores a problemas de decisión en varios contextos

Elements of Decision under Uncertainty with Applications

This work presents the basic elements for the analysis of decision under uncertainty: Expected Utility Theory and its citicisms and risk aversion and its measurement. The concepts of certainty equivalent, risk premium, absolute risk aversion and relative risk aversion, and the "more risk averse than" relation are discussed. The work is completed with several applications of decision making under uncertainty to different economic problems: investment in risky assets and portfolio selection, risk sharing, investment to reduce risk, insurance, taxes and income underreporting, deposit insurance and the value of information.

Economía de la salud, modalidades de evaluación económica

La salud es un aspecto muy importante en la vida de cualquier persona, de forma que, al ocurrir cualquier contingencia que merma el estado de salud de un individuo o grupo de personas, se debe valorar estrictamente y en detalle las distintas alternativas destinadas a combatir la enfermedad. Esto se debe a que, la calidad de vida de los pacientes variará dependiendo de la alternativa elegida. La calidad de vida relacionada con la salud (CVRS) se entiende como el valor asignado a la duración de la vida, modificado por la oportunidad social, la percepción, el estado funcional y la disminución provocadas por una enfermedad, accidente, tratamiento o política (Sacristán et al, 1995). Para determinar el valor numérico asignado a la CVRS, ante una intervención, debemos beber de la teoría económica aplicada a las evaluaciones sanitarias para nuevas intervenciones. Entre los métodos de evaluación económica sanitaria, el método coste-utilidad emplea como utilidad, los años de vida ajustado por calidad (AVAC), que consiste, por un lado, tener en cuenta la calidad de vida ante una intervención médica, y por otro lado, los años estimados a vivir tras la intervención. Para determinar la calidad de vida, se emplea técnicas como el Juego Estándar, la Equivalencia Temporal y la Escala de Categoría. Estas técnicas nos proporcionan un valor numérico entre 0 y 1, siendo 0 el peor estado y 1 el estado perfecto de salud. Al entrevistar a un paciente a cerca de la utilidad en términos de salud, puede haber riesgo o incertidumbre en la pregunta planteada. En tal caso, se aplica el Juego Estándar con el fin de determinar el valor numérico de la utilidad o calidad de vida del paciente ante un tratamiento dado. Para obtener este valor, al paciente se le plantean dos escenarios: en primer lugar, un estado de salud con probabilidad de morir y de sobrevivir, y en segundo lugar, un estado de certeza. La utilidad se determina modificando la probabilidad de morir hasta llegar a la probabilidad que muestra la indiferencia del individuo entre el estado de riesgo y el estado de certeza. De forma similar, tenemos la equivalencia temporal, cuya aplicación resulta más fácil que el juego estándar ya que valora en un eje de ordenadas y abscisas, el valor de la salud y el tiempo a cumplir en esa situación ante un tratamiento sanitario, de forma que, se llega al valor correspondiente a la calidad de vida variando el tiempo hasta que el individuo se muestre indiferente entre las dos alternativas. En último lugar, si lo que se espera del paciente es una lista de estados de salud preferidos ante un tratamiento, empleamos la Escala de Categoría, que consiste en una línea horizontal de 10 centímetros con puntuaciones desde 0 a 100. La persona entrevistada coloca la lista de estados de salud según el orden de preferencia en la escala que después es normalizado a un intervalo entre 0 y 1. Los años de vida ajustado por calidad se obtienen multiplicando el valor de la calidad de vida por los años de vida estimados que vivirá el paciente. Sin embargo, ninguno de estas metodologías mencionadas consideran el factor edad, siendo necesario la inclusión de esta variable. Además, los pacientes pueden responder de manera subjetiva, situación en la que se requiere la opinión de un experto que determine el nivel de discapacidad del aquejado. De esta forma, se introduce el concepto de años de vida ajustado por discapacidad (AVAD) tal que el parámetro de utilidad de los AVAC será el complementario del parámetro de discapacidad de los AVAD Q^i=1-D^i. A pesar de que este último incorpora parámetros de ponderación de edad que no se contemplan en los AVAC. Además, bajo la suposición Q=1-D, podemos determinar la calidad de vida del individuo antes del tratamiento. Una vez obtenido los AVAC ganados, procedemos a la valoración monetaria de éstos. Para ello, partimos de la suposición de que la intervención sanitaria permite al individuo volver a realizar las labores que venía realizando. De modo que valoramos los salarios probables con una temporalidad igual a los AVAC ganados, teniendo en cuenta la limitación que supone la aplicación de este enfoque. Finalmente, analizamos los beneficios derivados del tratamiento (masa salarial probable) si empleamos la tabla GRF-95 (población femenina) y GRM-95 (población masculina).