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.

The customizable area whole farm insurance (CAWFI) is proposed and evaluated as a possible wholearm revenue protection design for crop farms. The evaluation included assessing appropriate weight, optimal scale, and optimal coverage level. The optimal CAWFI was tested against no insurance program, 90% farm level whole farm insurance (90% CFWFI), and CAWFI with scale and coverage level as provisioned in GRP product (restricted CAWFI) in representative farm in Kansas, North Dakota, Illinois, and Mississippi. The study finds the optimal CAWFI outperforms no insurance program and restricted CAWFI asserting that CAWFI is a workable insurance model and relaxing restriction on scale and coverage level can increase expected utility of farmers. The optimal CAWFI results in a risk reduction roughly equal with 90% farm-level wholefarm insurance though the expected indemnities in it are at least three fold.

Ziel dieser Dissertation ist es, den Begriff des Risikos unter den Aspekten seiner Quantifizierung durch robuste Darstellungen zu untersuchen. In einem ersten Teil wird Risiko anhand Kontext-Invarianter Merkmale betrachtet: Diversifizierung und Monotonie. Wir führen die drei Schlüsselkonzepte, Risikoordnung, Risikomaß und Risikoakzeptanzfamilen ein, und studieren deren eins-zu-eins Beziehung. Unser Hauptresultat stellt eine eindeutige duale robuste Darstellung jedes unterhalbstetigen Risikomaßes auf topologischen Vektorräumen her. Wir zeigen auch automatische Stetigkeitsergebnisse und robuste Darstellungen für Risikomaße auf diversen Arten von konvexen Mengen. Diese Herangehensweise lässt bei der Wahl der konvexen Menge viel Spielraum, und erlaubt damit eine Vielfalt von Interpretationen von Risiko: Modellrisiko im Falle von Zufallsvariablen, Verteilungsrisiko im Falle von Lotterien, Abdiskontierungsrisiko im Falle von Konsumströmen... Diverse Beispiele sind dann in diesen verschiedenen Situationen explizit berechnet (Sicherheitsäquivalent, ökonomischer Risikoindex, VaR für Lotterien, "variational preferences"...). Im zweiten Teil, betrachten wir Präferenzordnungen, die möglicherweise zusätzliche Informationen benötigen, um ausgedrückt zu werden. Hierzu führen wir einen axiomatischen Rahmen in Form von bedingten Präferenzordungen ein, die lokal mit der Information kompatibel sind. Dies erlaubt die Konstruktion einer bedingten numerischen Darstellung. Wir erhalten eine bedingte Variante der von Neumann und Morgenstern Darstellung für messbare stochastische Kerne und erweitern dieses Ergebnis zur einer bedingten Version der "variational preferences". Abschließend, klären wir das Zusammenpiel zwischen Modellrisiko und Verteilungsrisiko auf der axiomatischen Ebene. / The goal of this thesis is the conceptual study of risk and its quantification via robust representations. We concentrate in a first part on context invariant features related to this notion: diversification and monotonicity. We introduce and study the general properties of three key concepts, risk order, risk measure and risk acceptance family and their one-to-one relations. Our main result is a uniquely characterized dual robust representation of lower semicontinuous risk orders on topological vector space. We also provide automatic continuity and robust representation results on specific convex sets. This approach allows multiple interpretation of risk depending on the setting: model risk in the case of random variables, distributional risk in the case of lotteries, discounting risk in the case of consumption streams... Various explicit computations in those different settings are then treated (economic index of riskiness, certainty equivalent, VaR on lotteries, variational preferences...). In the second part, we consider preferences which might require additional information in order to be expressed. We provide a mathematical framework for this idea in terms of preorders, called conditional preference orders, which are locally compatible with the available information. This allows us to construct conditional numerical representations of conditional preferences. We obtain a conditional version of the von Neumann and Morgenstern representation for measurable stochastic kernels and extend then to a conditional version of the variational preferences. We finally clarify the interplay between model risk and distributional risk on the axiomatic level.