Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing

Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing.Dependence structure, Extremal values, Copula modeling, Copula review

Copula-based probabilistic assessment of intensity and duration of cold episodes: A case study of Malayer vineyard region

Frost, particularly during the spring, is one of the most damaging weather phenomena for vineyards, causing significant economic losses to vineyards around the world each year. The risk of tardive frost damage in vine-yards due to changing climate is considered as an important threat to the sustainable production of grapes. Therefore, the cold monitoring strategies is one of the criteria with significant impacts on the yields and prosperity of horticulture and raisin factories. Frost events can be characterized by duration and severity. This paper investigates the risk and impacts of frost phenomenon in the vineyards by modeling the joint distribution of duration and severity factors and analyzing the influential parameter’s dependency structure using capabilities of copula functions. A novel mathematical framework is developed within this study to understand the risk and uncertainties associate with frost events and the impacts on yields of vineyards by analyzing the non-linear dependency structure using copula functions as an efficient tool. The developed model was successfully vali-dated for the case study of vineyard in Malayer city of Iran. The copula model developed in this study was shown to be a robust tool for predicting the return period of the frost events

Essays in Quantitative Risk Management for Financial Regulation of Operational Risk Models

An extensive amount of evolving guidance and rules are provided to banks by financial regulators. A particular set of instructions outline requirements to calculate and set aside loss-absorbing regulatory capital to ensure the solvency of a bank. Mathematical models are typically used by banks to quantify sufficient amounts of capital. In this thesis, we explore areas that advance our knowledge in regulatory risk management. In the first essay, we explore an aspect of operational risk loss modeling using scenario analysis. An actuarial modeling method is typically used to quantify a baseline capital value which is then layered with a judgemental component in order to account for and integrate what-if future potential losses into the model. We propose a method from digital signal processing using the convolution operator that views the problem of the blending of two signals. That is, a baseline loss distribution obtained from the modeling of frequency and severity of internal losses is combined with a probability distribution obtained from scenario responses to yield a final output that integrates both sets of information. In the second essay, we revisit scenario analysis and the potential impact of catastrophic events to that of the enterprise level of a bank. We generalize an algorithm to account for multiple level of intensities of events together with unique loss profiles depending on the business units effected. In the third essay, we investigate the problem of allocating aggregate capital across sub-portfolios in a fair manner when there are various forms of interdependencies. Relevant to areas of market, credit and operational risk, the multivariate shortfall allocation problem quantifies the optimal amount of capital needed to ensure that the expected loss under a convex loss penalty function remains bounded by a threshold. We first provide an application of the existing methodology to a subset of high frequency loss cells. Lastly, we provide an extension using copula models which allows for the modeling of joint fat-tailed events or asymmetries in the underlying process

Modeling and pricing cyber insurance: Idiosyncratic, systematic, and systemic risks

The paper provides a comprehensive overview of modeling and pricing cyber insurance and includes clear and easily understandable explanations of the underlying mathematical concepts. We distinguish three main types of cyber risks: idiosyncratic, systematic, and systemic cyber risks. While for idiosyncratic and systematic cyber risks, classical actuarial and financial mathematics appear to be well-suited, systemic cyber risks require more sophisticated approaches that capture both network and strategic interactions. In the context of pricing cyber insurance policies, issues of interdependence arise for both systematic and systemic cyber risks; classical actuarial valuation needs to be extended to include more complex methods, such as concepts of risk-neutral valuation and (set-valued) monetary risk measures

Risk Aggregation in the presence of Discrete Causally Connected Random Variables

Risk aggregation is a popular method used to estimate the sum of a collection of financial assets or events, where each asset or event is modelled as a random variable. Applications include insurance, operational risk, stress testing, and sensitivity analysis. In practice the sum of a set of random variables involves the use of two well-known mathematical operations: n-fold convolution (for a fixed number n) and N-fold convolution, defined as the compound sum of a frequency distribution N and a severity distribution, where the number of constant n-fold convolutions is determined by N. Where the severity and frequency variables are independent, and continuous, currently numerical solutions such as, Panjer’s recursion, Fast Fourier transforms and Monte Carlo simulation produce acceptable results. However, they have not been designed to cope with new modelling challenges that require hybrid models containing discrete explanatory (regime switching) variables or where discrete and continuous variables are inter-dependent and may influence the severity and frequency in complex, non-linear, ways. This paper de-scribes a Bayesian Factorization and Elimination (BFE) algorithm that performs convo

Doctor of Philosophy

dissertationWater resources are limited and disproportionately distributed in time and place. Moreover, complex interactions among different components of the water system, changes in population and urbanization growth rates, and climate change have increased the uncertainty influencing water resource planning. The ultimate question arising for water managers considering the complexity of water systems is how to determine if management strategies are effective and improve the performance of a water system. Generally, decision-makers assess the system’s condition based on a univariate measure of reliability or vulnerability. However, these measures do not deliver sufficient information, and present a limited view about the system’s performance. There is a known need to study water resources in an integrated fashion to effectively manage for the present and the future. In this dissertation, a new comprehensive integrated modeling and performance assessment framework is offered. First, a new approach is designed to assess vulnerability of a water system based on important factors including exposure, sensitivity, severity, potential severity, social vulnerability, and adaptive capacity. Then, instead of an individual metric, the joint probability distribution of reliability and vulnerability based on copula function is developed to estimate a new index, the Water System Performance Index (WSPI), to evaluate the reliability and vulnerability of a water system simultaneously. To test the effectiveness of the framework and demonstrate the advances of the new performance index, a practical application is conducted for the Salt Lake City Department of Public Utilities (SLCDPU) water system. For this purpose, an integrated water resource management (IWRM) model is developed using system dynamics approach for the case study. Management alternatives are incorporated into the model using a decision support tool designed for use by water managers and stakeholders. Results of the study show an inconsistency in the degree of vulnerability between traditionally used and the new vulnerability assessment approaches. The use of the integrated model and new vulnerability approach is also shown to provide more informative guidance for decision makers evaluating alternative management strategies during failure events. Furthermore, results illustrate the effectiveness of the WSPI to identify critical conditions when there is a need for a combined measure of performance. In terms of water management decision making, the final results of this dissertation indicate centralized water storage solutions improve water system performance better than rainwater harvesting for the Salt Lake City case study