This research studies two modelling techniques that help seek optimal strategies in financial risk management. Both are based on the stochastic programming methodology. The first technique is concerned with market risk management in portfolio selection problems; the second technique contributes to operational risk management by optimally allocating workforce from a managerial perspective. The first model involves multiperiod decisions (portfolio rebalancing) for an asset and liability management problem and deals with the usual uncertainty of investment returns and future liabilities. Therefore it is well-suited to a stochastic programming approach. A stochastic dominance concept is applied to control the risk of underfunding. A small numerical example and a backtest are provided to demonstrate advantages of this new model which includes stochastic dominance constraints over the basic model. Adding stochastic dominance constraints comes with a price: it complicates the structure of the underlying stochastic program. Indeed, new constraints create a link between variables associated with different scenarios of the same time stage. This destroys the usual tree-structure of the constraint matrix in the stochastic program and prevents the application of standard stochastic programming approaches such as (nested) Benders decomposition and progressive hedging. A structure-exploiting interior point method is applied to this problem. Computational results on medium scale problems with sizes reaching about one million variables demonstrate the efficiency of the specialised solution technique. The second model deals with operational risk from human origin. Unlike market risk that can be handled in a financial manner (e.g. insurances, savings, derivatives), the treatment of operational risks calls for a “managerial approach”. Consequently, we propose a new way of dealing with operational risk, which relies on the well known Aggregate Planning Model. To illustrate this idea, we have adapted this model to the case of a back office of a bank specialising in the trading of derivative products. Our contribution corresponds to several improvements applied to stochastic programming modelling. First, the basic model is transformed into a multistage stochastic program in order to take into account the randomness associated with the volume of transaction demand and with the capacity of work provided by qualified and non-qualified employees over the planning horizon. Second, as advocated by Basel II, we calculate the probability distribution based on a Bayesian Network to circumvent the difficulty of obtaining data which characterises uncertainty in operations. Third, we go a step further by relaxing the traditional assumption in stochastic programming that imposes a strict independence between the decision variables and the random elements. Comparative results show that in general these improved stochastic programming models tend to allocate more human expertise in order to hedge operational risks. The dual solutions of the stochastic programs are exploited to detect periods and nodes that are at risk in terms of expertise availability.

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It is often desirable to compare risky investments in the context of economic decision theory. Expected utility analyses are means by which stochastic alternatives can be ranked by re-weighting the probability mass using a decision-making agent’s utility function. By maximizing expected utility, an agent seeks to balance expected returns with the inherent risk in each investment alternative. This can be accomplished by ranking prospects based on the certainty equivalent associated with each alternative. In instances where only a small sample of observed data is available to estimate the underlying distributions of the risky options, reliable inferences are difficult to make. In this process of comparing alternatives, when estimating explicit probability forms or nonparametric densities, the variance of the estimate, in this case the certainty equivalent, is often ignored. Resampling methods allow for estimating dispersion for a statistic when no parametric assumptions are made about the underlying distribution. An objective of this dissertation is to utilize these methods to estimate confidence regions for the sample certainty equivalents of the alternatives over a subset of the parameter space of the utility function. A second goal of this research is to formalize a testing procedure when dealing with preference ranking with respect to utility. This is largely based on Meyer’s work (1977b) developing stochastic dominance with respect to a function and more specific testing procedures outlined by Eubank et. al. (1993). Within this objective, the asymptotic distribution of the test statistic associated with the hypothesis of preference of one risky outcome over another given a sub-set of the utility function parameter space is explored.

Title: Multivariate stochastic dominance and its application in portfolio optimization Problems Author: Barbora Petrová Department: Department of Probability and Mathematical Statistics Supervisor: doc. RNDr. Ing. Miloš Kopa, Ph.D., Department of Probability and Mathematical Statistics Abstract: This thesis discusses the concept of multivariate stochastic dominance, which serves as a tool for ordering random vectors, and its possible usage in dynamic portfolio optimization problems. We strictly focus on different types of the first-order multivariate stochastic dominance for which we describe their generators in the sense of von Neumann-Morgenstern utility functions. The first one, called strong multivariate stochastic dominance, is generated by all nondecreasing multivariate utility functions. The second one, called weak multivariate stochastic dominance, is defined by relation between survival functions, and the last one, called the first-order linear multivariate stochastic dominance, applies the first-order univariate stochastic dominance notion to linear combinations of marginals. We focus on the main characteristics of these types of stochastic dominance, their relationships as well as their relation to the cumulative and marginal distribution functions of considered random vectors. Formulated...

Title: Almost stochastic dominance Author: Adam Štefánik Department: Probability and Mathematical Statistics Supervisor: RNDr. Ing. Miloš Kopa, PhD. Department of Probability and Mathematical Statistics, MFF UK Abstract: In the presented work we study the almost stochastic dominance and it's properties. Almost stochastic dominance is a relaxation of stochastic dominance. Almost stochastic dominance also deals with paradox situations occurring in case of stochastic dominance. This is a situation when stochastic dominance determines indifferent relation- ship between two portfolios, but in fact almost all investors can choose the better one. The original almost stochastic dominance presented by Leshno and Levy (2002) is compu- tationally expensive. Lizyayev and Ruszczy'nski (2012) suggested an alternative approach. This work introduces both approaches. The most interesting part of this work is a search for efficient portfolio with respect to the almost stochastic dominance by the simple linear programming. Lizyayev and Ruszczy'nski (2012) approach is applied to Kopa and Chovanec (2008) quantile approach for portfolio efficiency testing with respect to second order stochastic dominance. Keywords: almost stochastic dominance, efficiency, CVaR

[pt] Esta dissertação tem como objetivo comparar os índices de desempenho de média-variância com os critérios de dominância estocástica de primeira, segunda e terceira ordens para fundos de gerenciamento ativo presentes no mercado brasileiro. Foram analisados 84 fundos de ações entre maio de 1999 e abril de 2001. Para o cálculo da dominância estocástica foi criada uma função em Matlab que, a partir dos retornos dos fundos, compara-os entre si e retorna quais os fundos mais dominantes em relação aos outros. O que se concluiu é que os indivíduos que selecionam seus investimentos com base somente nos índices de média-variância podem tomar decisões que contrariam seus critérios de aversão ao risco e de aversão crescente ao risco. Igualmente, o desempenho de fundos de investimento medido apenas através dos critérios de dominância estocástica não significará necessariamente um maior excesso de retorno com relação ao risco corrido. Para se tomar uma decisão de investimento bem estruturada, o investidor deve considerar todos os momentos da distribuição dos retornos e realizar uma análise tanto por média-variância quanto por dominância estocástica. / [en] The scope of this dissertation is the comparison between the meanvariance based performance measurers of active management Brazilian-based stock funds and stochastic dominance of first, second and third orders criteria. 84 funds were considered and the period studied goes from May 1999 to April 2001. For the stochastic dominance calculus a Matlab function was created so that, with the funds returns as inputs, it gives the most dominating funds in relation to the others. The conclusion of this study is that individuals that chose investments taking account solely mean-variance measurers can make decisions that goes against their criteria of risk aversion and absolute decreasing risk aversion. In the same way, investments funds performance measured only by stochastic dominance criteria will not lead necessarily to a highest reward-to- risk ratio. Regarding a well structured investment decision, investors should consider all moments of the distribution of returns and perform not only a mean- variance but also a stochastic dominance analysis.

abstract: Prior research on consumer behavior in health insurance markets has primarily focused on individual decision making while relying on strong parametric assumptions about preferences. The aim of this dissertation is to improve the traditional approach in both dimensions. First, I consider the importance of joint decision-making in individual insurance markets by studying how married couples coordinate their choices in these markets. Second, I investigate the robustness of prior studies by developing a non-parametric method to assess decision-making in health insurance markets. To study how married couples make choices in individual insurance markets I estimate a stochastic choice model of household demand that takes into account spouses' risk aversion, spouses' expenditure risk, risk sharing, and switching costs. I use the model estimates to study how coordination within couples and interaction between couples and singles affects the way that markets adjust to policies designed to nudge consumers toward choosing higher value plans, particularly with respect to adverse selection. Finally, to assess consumer decision-making beyond standard parametric assumptions about preferences, I use second--order stochastic dominance rankings. Moreover, I show how to extend this method to construct bounds on the welfare implications of choosing dominated plans. / Dissertation/Thesis / Doctoral Dissertation Economics 2020

Travel time reliability is deemed as one of the most important factors affecting travelers’ route choice decisions. However, existing practices mostly consider average travel time only. This dissertation establishes a methodology framework to overcome such limitation. Semi-standard deviation is first proposed as the measure of reliability to quantify the risk under uncertain conditions on the network. This measure only accounts for travel times that exceed certain pre-specified benchmark, which offers a better behavioral interpretation and theoretical foundation than some currently used measures such as standard deviation and the probability of on-time arrival. Two path finding models are then developed by integrating both average travel time and semi-standard deviation. The single objective model tries to minimize the weighted sum of average travel time and semi-standard deviation, while the multi-objective model treats them as separate objectives and seeks to minimize them simultaneously. The multi-objective formulation is preferred to the single objective model, because it eliminates the need for prior knowledge of reliability ratios. It offers an additional benefit of providing multiple attractive paths for traveler’s further decision making. The sampling based approach using archived travel time data is applied to derive the path semi-standard deviation. The approach provides a nice workaround to the problem that there is no exact solution to analytically derive the measure. Through this process, the correlation structure can be implicitly accounted for while simultaneously avoiding the complicated link travel time distribution fitting and convolution process. Furthermore, the metaheuristic algorithm and stochastic dominance based approach are adapted to solve the proposed models. Both approaches address the issue where classical shortest path algorithms are not applicable due to non-additive semi-standard deviation. However, the stochastic dominance based approach is preferred because it is more computationally efficient and can always find the true optimal paths. In addition to semi-standard deviation, on-time arrival probability and scheduling delay measures are also investigated. Although these three measures share similar mathematical structures, they exhibit different behaviors in response to large deviations from the pre-specified travel time benchmark. Theoretical connections between these measures and the first three stochastic dominance rules are also established. This enables us to incorporate on-time arrival probability and scheduling delay measures into the methodology framework as well.

At the beginning of this thesis we discuss DEA methods, which measure efficiency of Decision Making Units by comparing weighted inputs and outputs. First we describe basic DEA models without random inputs and outputs then stochastic DEA models which are derived from the deterministic ones. We describe more approaches to stochastic DEA models, for example using scenario approach or chance constrained programming problems. Another approach for measuring efficiency employs stochastic dominance. Stochastic dominance is a relation that allows to compare two random variables. We describe the first and second order stochastic dominance. First we consider pairwise stochastic efficiency, then we discuss the first and second order stochastic dominance portfolio efficiency. We describe different tests to measure this type of efficiency. At the end of this thesis we study efficiency of US stock portfolios using real historical data and we compare results obtained when using stochastic DEA models and stochastic dominance. Powered by TCPDF (www.tcpdf.org)

This thesis focuses on stochastic dominance in portfolio selection problems. The thesis recalls basic knowledge from the area of portfolio optimization with utility functions and first, second, $N$-th and infinite order of stochastic dominance. It sumarizes Post's, Kuosmanen's and Kopa's criteria for portfolio efficiency and necessary and sufficient conditions of stochastic dominance for discrete and continuous probability distributions. The thesis also contains formulations of optimization problems with second order stochastic dominance constraints derived for discrete and continuous probability distributions. A practical application is also a part of the thesis, where the optimization problems for monthly returns of Czech stocks are solved using optimization software GAMS.