9 research outputs found

    Characterizing, optimizing and backtesting metrics of risk

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    Measures of risk and riskmetrics were proposed to quantify the risks people are faced with in financial, statistical, and economic practice. They are widely discussed and studied by literature in the context of financial regulation, insurance, operations research, and statistics. Several major research topics on riskmetrics remain to be important in both academic study and industrial practice. First, characterization, especially axiomatic characterization of riskmetrics, lays essential theoretical foundation of specific classes of riskmetrics about why they are widely adopted in practice and research. It usually involves challenging mathematical approaches and deep practical insights. Second, riskmetrics are used by researchers in optimization as the objective functionals of decision makers. This links riskmetrics to the literature of operations research and decision theory, and leads to wide applications of riskmetrics to portfolio management, robust optimization, and insurance design. Third, relevant statistical models of estimation and hypothesis tests for riskmetrics need to be established to serve for practical risk management and financial regulation. In particular, risk forecasts and backtests of different riskmetrics are always the main concern and challenge for risk managers and financial regulators. In this thesis, we investigate several important questions in characterization, optimization, and backtest for measures of risk with different focuses on establishing theoretical framework and solving practical problems. To offer a comprehensive theoretical toolkit for future study, in Chapter 2, we propose the class of distortion riskmetrics defined through signed Choquet integrals. Distortion riskmetrics include many classic risk measures, deviation measures, and other functionals in the literature of finance and actuarial science. We obtain characterization, finiteness, convexity, and continuity results on general model spaces, extending various results in the existing literature on distortion risk measures and signed Choquet integrals. To explore deeper applications of distortion riskmetrics in optimization problems, in Chapter 3, we study optimization of distortion riskmetrics with distributional uncertainty. One of our central findings is a unifying result that allows us to convert an optimization of a non-convex distortion riskmetric with distributional uncertainty to a convex one, leading to practical tractability. A sufficient condition to the unifying equivalence result is the novel notion of closedness under concentration, a variation of which is also shown to be necessary for the equivalence. Our results include many special cases that are well studied in the optimization literature, including but not limited to optimizing probabilities, Value-at-Risk, Expected Shortfall, Yaari's dual utility, and differences between distortion risk measures, under various forms of distributional uncertainty. We illustrate our theoretical results via applications to portfolio optimization, optimization under moment constraints, and preference robust optimization. In Chapter 4, we study characterization of measures of risk in the context of statistical elicitation. Motivated by recent advances on elicitability of risk measures and practical considerations of risk optimization, we introduce the notions of Bayes pairs and Bayes risk measures. Bayes risk measures are the counterpart of elicitable risk measures, extensively studied in the recent literature. The Expected Shortfall (ES) is the most important coherent risk measure in both industry practice and academic research in finance, insurance, risk management, and engineering. One of our central results is that under a continuity condition, ES is the only class of coherent Bayes risk measures. We further show that entropic risk measures are the only risk measures which are both elicitable and Bayes. Several other theoretical properties and open questions on Bayes risk measures are discussed. In Chapter 5, we further study characterization of measures of risk in insurance design. We study the characterization of risk measures induced by efficient insurance contracts, i.e., those that are Pareto optimal for the insured and the insurer. One of our major results is that we characterize a mixture of the mean and ES as the risk measure of the insured and the insurer, when contracts with deductibles are efficient. Characterization results of other risk measures, including the mean and distortion risk measures, are also presented by linking them to different sets of contracts. In Chapter 6, we focus on a larger class of riskmetrics, cash-subadditive risk measures. We study cash-subadditive risk measures without quasi-convexity. One of our major results is that a general cash-subadditive risk measure can be represented as the lower envelope of a family of quasi-convex and cash-subadditive risk measures. Representation results of cash-subadditive risk measures with some additional properties are also examined. The notion of quasi-star-shapedness, which is a natural analogue of star-shapedness, is introduced and we obtain a corresponding representation result. In Chapter 7, we discuss backtesting riskmetrics. One of the most challenging tasks in risk modeling practice is to backtest ES forecasts provided by financial institutions. To design a model-free backtesting procedure for ES, we make use of the recently developed techniques of e-values and e-processes. Model-free e-statistics are introduced to formulate e-processes for risk measure forecasts, and unique forms of model-free e-statistics for VaR and ES are characterized using recent results on identification functions. For a given model-free e-statistic, optimal ways of constructing the e-processes are studied. The proposed method can be naturally applied to many other risk measures and statistical quantities. We conduct extensive simulation studies and data analysis to illustrate the advantages of the model-free backtesting method, and compare it with the ones in the literature

    Measuring farmers’ risk and uncertainty attitudes: an interval prospect experiment

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    Attitudes to risk have generated a lot of attention over the years due to its vital importance in decision-making processes that are necessary for life and livelihoods. Attitudes towards uncertainty have received less attention even though arguably most important decisions are under uncertainty rather than risk. In addition, many studies modelling attitudes to risk have adopted experiments that place significant cognitive burden on respondents. Crucially, they are also framed in a way that do not reflect everyday problems. Specifically, the most common way of eliciting attitudes is to ask decision makers to choose between discrete monetary lotteries with known probabilities attached to the payoffs. Yet, arguably, the vast majority of choices that people make in their day-to-day lives are with respect to continuous non-monetary outcomes. To address these gaps, this thesis investigates responses to continuous ‘prospects’ across different conditions (risk & uncertainty), contexts (monetary & time) and content domains (gain, loss & mixed). Further, this thesis examines the link between attitudes to risk/uncertainty and mental health related factors and the effect of attitudes to risk and uncertainty on farmers’ decisions both for themselves and for others. This thesis uses both non-parametric methods - relating to the patterns that characterise participants’ choices and their determinants; and parametric models – based upon cumulative prospect theory (CPT) as it extends to continuous prospects. The data were gathered using lab-in-field experiments in which Nigerian farmer’s chose between pairs of prospects with continuous distributions, which were not exclusively monetary in nature. Attitudes towards risk, as opposed to uncertainty were elicited by specifying that all outcomes over the specified interval were ‘equally likely’ (thus a uniform probability density). Uncertainty was specified by indicating to farmers that one outcome within the specified interval would be realised but without the specification of an associated probability density. Key findings are that attitudes differ under different conditions, contexts and content domains. Using continuous prospects, respondents did not treat equally likely outcomes as ‘equally likely’ and appear to demonstrate cumulative probability distribution warping consistent with the CPT. However, there were behaviours that are difficult to reconcile with CPT such as the preferences of many respondents could only be modelled using “extreme curvature” of the value function. This was induced by what we term negligible gain avoidance (i.e. avoiding prospects with zero lower bound in the gain domain) or negligible loss seeking (i.e. preferring prospects with zero upper bound in the loss domain) behaviours. CPT, Salience theory, Heuristics and other theories examined in this study could not alone explain these behaviours. Results from investigating the effect of bipolar disorder tendencies (BD) on risk attitudes show that BD significantly affects the shape of the value and probability weighting functions; and farmers that have BD are more likely to make random choices. Other results show that risk aversion for losses increases participation in off-farm income generating activities; and that farmers’ likelihood to engage in specific types of offfarm activities is determined by their risk and uncertainty attitudes

    Decision behaviour under risk: Experimental studies and new theoretical approaches

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    Diese kumulative Dissertation beschÀftigt sich mit individuellem Entscheidungsverhalten unter Risiko. In Teil A werden neue Experimente dargestellt, die das UnabhÀngigkeitsaxiom und den Erwartungsnutzen testen. Teil B enthÀlt dagegen im Wesentlichen neue Resultate im Rahmen der Prospect Theorie.This thesis is a collection of essays on decision making under risk. Part A presents new experiments which aim at testing the independence axiom and expected utility. Part B mainly contains new theoretical results in the framework of prospect theory

    Axiomatizations of signed discrete Choquet integrals

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    We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lov\'asz extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory

    Theoretical aspects of long-term evaluation in environmental economics

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    The present work is dedicated to theoretical aspects of long-term evaluation with a focus on time and uncertainty structure. Motivated along the lines of global warming, the analysis renders contributions to the fields of environmental economics, decision theory, the economics of sustainability and cost benefit analysis. The thesis is structured in three parts. The first part examines the relation between the concepts of weak and strong sustainability and the weight given to future consumption. The second part introduces a generalized evaluation model and a new concept of risk aversion. The latter concept, termed intertemporal risk aversion, takes up an important concern of the precautionary principle. The third part extends the underlying model and analyzes the interaction with other characteristics of intertemporal decision making. The latter include an implied preference for the timing of uncertainty resolution as well as different stationarity assumptions. The first part of the thesis relates to the sustainability debate and the concepts of weak versus strong sustainability. While the advocates of the weak sustainability concept consider man made goods and capital a fair substitute for environmental goods and capital, the advocates of the strong sustainability concept judge such substitutability as highly limited. I show in a stylized growth model, how social discount rates generally fall for a weak sustainability specification of welfare, while they grow for a strong sustainability specification. It turns out that under the given assumptions a strong sustainability specification of welfare implies a lower weight given to future consumption streams than a weak sustainability specification. The second part of the thesis introduces the concept of intertemporal risk aversion in a didactically simplified two period framework. The concept takes up an important concern of the precautionary principle regarding a higher willingness to undergo preventive measures in order to avoid a threat of harm. I show that the concern is substantiated as well by von Neumann Morgenstern’s widespread axioms for choice under uncertainty when carefully integrated into a temporal setting. In such a generalized framework, the standard model of intertemporally additive expected utility corresponds to intertemporal risk neutrality. In contrast to the classical concept of (atemporal) risk aversion, the concept of intertemporal risk aversion can be applied immediately to the multi-commodity setting. For the one commodity special case, the concept closely relates to the attempts of disentangling atemporal risk aversion from intertemporal substitutability. The third part of the thesis extends the model to an arbitrary finite time horizon with generalized preferences and elaborates the corresponding axiomatic and functional characterizations of intertemporal risk aversion. Moreover, I identify different assumptions that allow to simplify the model structure. On the one hand, these assumptions are concerned with a stationary evaluation of certain and uncertain consumption plans. On the other hand, they relate to a deduced preference for the timing of uncertainty resolution. The resulting simplifications allow to characterize intertemporal risk aversion in a single parameter, as well as to disentangle atemporal risk aversion from intertemporal substitutability in a non-recursive evaluation structure. Finally, I show that a normatively motivated combination of the assumptions implies that a time consistent, intertemporal risk averse decision maker has to choose a zero rate of pure time preference. Instead of devaluing the future for reasons of sheer impatience, such a decision maker is only allowed to give reduced weight to future welfare if uncertainty increases over time. The major implications of the present work can be divided into two fields. The first field relates to the sustainability debate and the evaluation of the long run. In this regard, the analysis in the first part of the thesis shows that the characterization of weak and strong sustainability through the degree of substitutability between environmental and produced goods stands in a surprising and possibly unwanted relation to the sustainability demand in the sense of a stronger commitment to future consumption streams. The analyses carried out in the last part of the thesis implies that a zero rate of pure time preference cannot only be founded on moral considerations, but also on assumptions concerning a time consistent evaluation of uncertainty. The second field of implications concerns the handling of uncertainty. In particular, the concept of intertemporal risk aversion mediates between the advocates and the opponents of the precautionary principle. On the one hand, it takes up the concern regarding a higher willingness to undergo preventive action than implied by the standard model. On the other hand, intertemporal risk aversion formalizes this concern and reconciles it with the standard assumptions underlying economic evaluation. That way, it encounters the critique of the precautionary principle as being vague, arbitrary and, thus, paralyzing
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