Banking is a risk and return business; in order to obtain the desired returns, banks are required to take on risks. Following the demise of Lehman Brothers in September 2008, the Basel III Accord proposed considerable increases in capital charges for banks. Whilst this ensures greater economic stability, banks now face an increasing risk of becoming capital inefficient. Furthermore, capital analysts are not only required to estimate capital requirements for individual business lines, but also for the organization as a whole. Copulas are a popular technique to model joint multi-dimensional problems, as they can be applied as a mechanism that models relationships among multivariate distributions. Firstly, a review of the Basel Capital Accord will be provided. Secondly, well known risk measures as proposed under the Basel Accord will be investigated. The penultimate chapter is dedicated to the theory of copulas as well as other measures of dependence. The final chapter presents a practical illustration of how business line losses can be simulated by using the Gaussian, Cauchy, Student t and Clayton copulas in order to determine capital requirements using 95% VaR, 99% VaR, 95% ETL, 99% ETL and StressVaR. The resultant capital estimates will always be a function of the choice of copula, the choice of risk measure and the correlation inputs into the copula calibration algorithm. The choice of copula, the choice of risk measure and the conservativeness of correlation inputs will be determined by the organization’s risk appetite. / Sc (Applied Mathematics), North-West University, Potchefstroom Campus, 2014

Banking is a risk and return business; in order to obtain the desired returns, banks are required to take on risks. Following the demise of Lehman Brothers in September 2008, the Basel III Accord proposed considerable increases in capital charges for banks. Whilst this ensures greater economic stability, banks now face an increasing risk of becoming capital inefficient. Furthermore, capital analysts are not only required to estimate capital requirements for individual business lines, but also for the organization as a whole. Copulas are a popular technique to model joint multi-dimensional problems, as they can be applied as a mechanism that models relationships among multivariate distributions. Firstly, a review of the Basel Capital Accord will be provided. Secondly, well known risk measures as proposed under the Basel Accord will be investigated. The penultimate chapter is dedicated to the theory of copulas as well as other measures of dependence. The final chapter presents a practical illustration of how business line losses can be simulated by using the Gaussian, Cauchy, Student t and Clayton copulas in order to determine capital requirements using 95% VaR, 99% VaR, 95% ETL, 99% ETL and StressVaR. The resultant capital estimates will always be a function of the choice of copula, the choice of risk measure and the correlation inputs into the copula calibration algorithm. The choice of copula, the choice of risk measure and the conservativeness of correlation inputs will be determined by the organization’s risk appetite. / Sc (Applied Mathematics), North-West University, Potchefstroom Campus, 2014

Financial contagion refers to a scenario in which small shocks, which initially affect only a few financial institutions or a particular region of the economy, spread to the rest of the financial sector and other countries whose economies were previously healthy. This resembles the “transmission” of a medical disease. Financial contagion happens both at domestic level and international level. At domestic level, usually the failure of a domestic bank or financial intermediary triggers transmission by defaulting on inter-bank liabilities, selling assets in a fire sale, and undermining confidence in similar banks. An example of this phenomenon is the failure of Lehman Brothers and the subsequent turmoil in the US financial markets. International financial contagion happens in both advanced economies and developing economies, and is the transmission of financial crises across financial markets. Within the current globalise financial system, with large volumes of cash flow and cross-regional operations of large banks and hedge funds, financial contagion usually happens simultaneously among both domestic institutions and across countries. There is no conclusive definition of financial contagion, most research papers study contagion by analyzing the change in the variance-covariance matrix during the period of market turmoil. King and Wadhwani (1990) first test the correlations between the US, UK and Japan, during the US stock market crash of 1987. Boyer (1997) finds significant increases in correlation during financial crises, and reinforces a definition of financial contagion as a correlation changing during the crash period. Forbes and Rigobon (2002) give a definition of financial contagion. In their work, the term interdependence is used as the alternative to contagion. They claim that for the period they study, there is no contagion but only interdependence. Interdependence leads to common price movements during periods both of stability and turmoil. In the past two decades, many studies (e.g. Kaminsky et at., 1998; Kaminsky 1999) have developed early warning systems focused on the origins of financial crises rather than on financial contagion. Further authors (e.g. Forbes and Rigobon, 2002; Caporale et al, 2005), on the other hand, have focused on studying contagion or interdependence. In this thesis, an overall mechanism is proposed that simulates characteristics of propagating crisis through contagion. Within that scope, a new co-evolutionary market model is developed, where some of the technical traders change their behaviour during crisis to transform into herd traders making their decisions based on market sentiment rather than underlying strategies or factors. The thesis focuses on the transformation of market interdependence into contagion and on the contagion effects. The author first build a multi-national platform to allow different type of players to trade implementing their own rules and considering information from the domestic and a foreign market. Traders’ strategies and the performance of the simulated domestic market are trained using historical prices on both markets, and optimizing artificial market’s parameters through immune - particle swarm optimization techniques (I-PSO). The author also introduces a mechanism contributing to the transformation of technical into herd traders. A generalized auto-regressive conditional heteroscedasticity - copula (GARCH-copula) is further applied to calculate the tail dependence between the affected market and the origin of the crisis, and that parameter is used in the fitness function for selecting the best solutions within the evolving population of possible model parameters, and therefore in the optimization criteria for contagion simulation. The overall model is also applied in predictive mode, where the author optimize in the pre-crisis period using data from the domestic market and the crisis-origin foreign market, and predict in the crisis period using data from the foreign market and predicting the affected domestic market.

In many complex water supply systems, the next generation of water resources planning models will require simultaneous probabilistic streamflow inputs at multiple locations on an interconnected network. To make use of the valuable multicentury records provided by tree-ring data, reconstruction models must be able to produce appropriate multisite inputs. Existing streamflow reconstruction models typically focus on one site at a time, not addressing intersite dependencies and potentially misrepresenting uncertainty. To this end, we develop a model for multisite streamflow reconstruction with the ability to capture intersite correlations. The proposed model is a hierarchical Bayesian nonhomogeneous hidden Markov model (NHMM). A NHMM is fit to contemporary streamflow at each location using lognormal component distributions. Leading principal components of tree rings are used as covariates to model nonstationary transition probabilities and the parameters of the lognormal component distributions. Spatial dependence between sites is captured with a Gaussian elliptical copula. Parameters of the model are estimated in a fully Bayesian framework, in that marginal posterior distributions of all the parameters are obtained. The model is applied to reconstruct flows at 20 sites in the Upper Colorado River Basin (UCRB) from 1473 to 1906. Many previous reconstructions are available for this basin, making it ideal for testing this new method. The results show some improvements over regression-based methods in terms of validation statistics. Key advantages of the Bayesian NHMM over traditional approaches are a dynamic representation of uncertainty and the ability to make long multisite simulations that capture at-site statistics and spatial correlations between sites.

Modern portfolio theory (MPT) is an investment theory which was introduced by Harry Markowitz in 1952 and describes how risk averse investors can optimize their portfolios. The objective of MPT is to assemble a portfolio by maximizing the expected return given a level of market risk or minimizing the market risk given an expected return. Although MPT has gained popularity over the years it has also been criticized for several theoretical and empirical shortcomings such as using variance as a measure of risk, measuring the dependence with linear correlation and assuming that returns are normally distributed when in fact empirical data suggests otherwise. When moving away from the assumption that returns are elliptical distributed, for example normally distributed, we can not use linear correlation as a measure of dependence in an accurate way. Copulas are a flexible tool for modeling dependence of random variables and enable us to separate the marginals from any joint distribution in order to extract the dependence structure. The objective of this paper was to examine the applicability of a copula-CVaR framework in portfolio optimization compared to the traditional MPT. Further, we studied how the presence of memory, when calibrating the copulas, affects portfolio optimization. The marginals for the copula based portfolios were constructed using Extreme Value Theory and the market risk was measured by Conditional Value at Risk. We implemented a dynamic investing strategy where the portfolios were optimized on a monthly basis with two different length of rolling calibration windows. The portfolios were backtested during a sample period from 2000-2016 and compared against two benchmarks; Markowitz portfolio based on normally distributed returns and an equally weighted, non optimized portfolio. The results demonstrated that portfolio optimization is often preferred compared to choosing an equally weighted portfolio. However, the results also indicated that the copula based portfolios do not always beat the traditional Markowitz portfolio. Furthermore, the results indicated that the choice of length of calibration window affects the selected portfolios and consequently also the performance. This result was supported both by the performance metrics and the stability of the estimated copula parameters.

Valuation of R&D projects is quite complex due to the substantial uncertainties in a project's life-cycle phases. The sequential nature of R&D projects continuously provides decision-makers with choices regarding whether and when to undertake future potential investment opportunities. This means that when valuing R&D projects decision-makers should take these factors into account. But R&D project usually takes long time to complete processes for commercialization. If the time to complete is longer, it is easier to trigger the crisis for capital shortage. So it seems very important modeling the capital shortage risk to induce the probability of failure in the pricing model. In this thesis we try to apply the analogy of financial securities subject to credit risk of Jarrow & Turnbull (1995) and attempt to value patents with capital shortage risk in an arbitrage free environment using the martingale measure technique. Furthermore, derive closed form formula for patents valuation which makes application easier than that of the theoretic option model. The major findings are: (1) when considering the effect of the failure frequency (capital shortage risk), the patent value will grow rapidly and then converge in the short run, no matter how other parameters incorporated into the robust analysis; (2) when increasing in the volatility of market revenues with synchronized higher volatility of investment cost, the volatility curve will be distorted to be U-shaped. Meanwhile, lower failure frequency could aggravate the decreasing in the option value. Another issue is when the manager exercises the project with multiple underlying assets, where the assets returns are of non-linear correlation particularly in the non-Normal environment. Non-parametric dependence measures may better employed when explaining co-movement. We focus on the value of a (such as resources development) project in general depends on the price of the multiple products; these are usually correlated to some extent. So the project was treated as having a rainbow option, whose underlying asset prices correlate with each other, and also as having uncertainties that decrease according to the project stage. Based on Cherubini and Luciano’s framework (2002), the risk-neutral copula models are derived to figure decision flexibilities out easily. The main framework studies the valuation of a project (call on Max) by determining the joint risk-neutral distribution of the underlying assets (products) using copulas. Monte-Carlo simulations show that the higher default risk and association among the assets and the expected cost to completion contributes the higher risk premium in our model with dependence structure of Archimedean copula family than traditional Black-Scholes environment. / Valuation of R&D projects is quite complex due to the substantial uncertainties in a project's life-cycle phases. The sequential nature of R&D projects continuously provides decision-makers with choices regarding whether and when to undertake future potential investment opportunities. This means that when valuing R&D projects decision-makers should take these factors into account. But R&D project usually takes long time to complete processes for commercialization. If the time to complete is longer, it is easier to trigger the crisis for capital shortage. So it seems very important modeling the capital shortage risk to induce the probability of failure in the pricing model. In this thesis we try to apply the analogy of financial securities subject to credit risk of Jarrow & Turnbull (1995) and attempt to value patents with capital shortage risk in an arbitrage free environment using the martingale measure technique. Furthermore, derive closed form formula for patents valuation which makes application easier than that of the theoretic option model. The major findings are: (1) when considering the effect of the failure frequency (capital shortage risk), the patent value will grow rapidly and then converge in the short run, no matter how other parameters incorporated into the robust analysis; (2) when increasing in the volatility of market revenues with synchronized higher volatility of investment cost, the volatility curve will be distorted to be U-shaped. Meanwhile, lower failure frequency could aggravate the decreasing in the option value. Another issue is when the manager exercises the project with multiple underlying assets, where the assets returns are of non-linear correlation particularly in the non-Normal environment. Non-parametric dependence measures may better employed when explaining co-movement. We focus on the value of a (such as resources development) project in general depends on the price of the multiple products; these are usually correlated to some extent. So the project was treated as having a rainbow option, whose underlying asset prices correlate with each other, and also as having uncertainties that decrease according to the project stage. Based on Cherubini and Luciano’s framework (2002), the risk-neutral copula models are derived to figure decision flexibilities out easily. The main framework studies the valuation of a project (call on Max) by determining the joint risk-neutral distribution of the underlying assets (products) using copulas. Monte-Carlo simulations show that the higher default risk and association among the assets and the expected cost to completion contributes the higher risk premium in our model with dependence structure of Archimedean copula family than traditional Black-Scholes environment.

Large-scale circulation patterns have a significant modulating influence on local hydro-meteorological variables, and consequently on water availability. An understanding of the influence of these patterns on the hydrological cycle, and the ability to timely predict their impacts, is crucial for water resources planning and management. This dissertation focusses on the influence of two major large-scale circulation patterns, the El Niño Southern Oscillation (ENSO) and the Pacific Decadal Oscillation (PDO), on the Rio Grande basin and the state of Texas, US. Both study areas are subject to a varying climate, and are extremely vulnerable to droughts, which can have devastating socio-economic impacts. The strength and spatial correlation structure of the climate indices on gauged precipitation was first established. Precipitation is not linearly related to water availability; therefore a land surface model (LSM), with land use land cover constant, was used to create naturalized flow, as it incorporates all necessary hydro-meteorological factors. As not all ENSO events are created equal, the influence of individual El Niño and La Niña events, classified using four different metrics, on water availability was examined. A general increase (decrease) in runoff during El Niños (La Niñas) was noted, but some individual events actually caused a decrease (increase) in water availability. Long duration El Niños have more influence on water availability than short duration high intensity events. Positive PDO enhances the effect of El Niño, and dampens the negative effect of La Niña, but when it is in its neutral or transition phase, La Niña tends to dominate climatic conditions and reduce water availability. LSM derived runoffs were converted into 3-month Standardized Runoff Indices (SRI 3) from which water deficit durations and severities were extracted. Conditional probability models of duration and severity were developed and compared with that based on observed precipitations. It was found that model derived information can be used in regions having limited ground observation data, or can be used in tandem with observation driven conditional probabilities for more efficient water resources planning and management. Finally a multidimensional model was developed, using copulas, to predict precipitation based on the phase of ENSO and PDO. A bivariate model, with ENSO and precipitation, was compared to a trivariate model, which incorporates PDO, and it was found that information on the state of PDO is important for efficient precipitation predictions.

This thesis considers statistical issues in the analysis of data in the studies of chronic diseases which involve modeling dependencies between life history processes using copula functions. Many disease processes feature recurrent events which represent events arising from an underlying chronic condition; these are often modeled as point processes. In addition, however, there often exists a random variable which is realized upon the occurrence of each event, which is called a mark of the point process. When considered together, such processes are called marked point processes. A novel copula model for the marked point process is described here which uses copula functions to govern the association between marks and event times. Specifically, a copula function is used to link each mark with the next event time following the realization of that mark to reflect the pattern in the data wherein larger marks are often followed by longer time to the next event. The extent of organ damage in an individual can often be characterized by ordered states, and interest frequently lies in modeling the rates at which individuals progress through these states. Risk factors can be studied and the effect of therapeutic interventions can be assessed based on relevant multistate models. When chronic diseases affect multiple organ systems, joint modeling of progression in several organ systems is also important. In contrast to common intensity-based or frailty-based approaches to modelling, this thesis considers a copula-based framework for modeling and analysis. Through decomposition of the density and by use of conditional independence assumptions, an appealing joint model is obtained by assuming that the joint survival function of absorption transition times is governed by a multivariate copula function. Different approaches to estimation and inference are discussed and compared including composite likelihood and two-stage estimation methods. Special attention is paid to the case of interval-censored data arising from intermittent assessment. Attention is also directed to use of copula models for more general scenarios with a focus on semiparametric two-stage estimation procedures. In this approach nonparametric or semiparametric estimates of the marginal survivor functions are obtained in the first stage and estimates of the association parameters are obtained in the second stage. Bivariate failure time models are considered for data under right-censoring and current status observation schemes, and right-censored multistate models. A new expression for the asymptotic variance of the second-stage estimator for the association parameter along with a way of estimating this for finite samples are presented under these models and observation schemes.

This paper studies the dependence structure between the real estate and financial markets in the United States from roughly 1975 to 2010, including the stock, bond and foreign exchange markets. This analysis uses dynamic copulas, including the Gaussian, Gumbel and Clayton copula. The Gumbel and Clayton copulas are used to separately capture the tail dependence of data. The dependence between the property indices (HPI and NCREIF) and the three financial markets is analyzed using the parameters of the copula. The property indices are divided in two different ways: by different regions and by different types of real estate. Although we study the dependence between the real estate and the financial markets in the U.S., the main objective of this paper is to analyze the change in the dependence structure when financial disasters occur. This study indicates that the real estate and the stock markets were positively related during this time period, and this dependence drove extreme movement when financial crises occurred. This dependence differed depending on the type of financial crisis, such as the Internet bubble crisis or the financial crisis in 2008. The dependence between the real estate and bond markets was also positively related, and extreme movement also occurred during financial crises. As for the dependence between the real estate and foreign exchange markets, although the results shows that dependence decreased when financial crises occurred, this is because the value of U.S. dollars are opposite to those of the index, and the left tail dependence exists as previous result. When looking at different regions or types of property, the differences in dependence structure were not obvious, although they were positively related. Both right and left tail dependences existed for most regions and property types, although some regions or types showed either right or left tail dependences alone. Therefore, investors should focus on the relationship between different markets, not on the region or type of real estate.

The asset combination of the multi-target credit derivatives and the pricing model of credit risk, the dependence in credit default in credit derivatives is an important connection factor. Copula functions represent a methodology which has recently become the most significant new tool to handle in a flexible way the comovement between markets, risk factors and other relevant variables studied in finance. Besides, Copula functions have been applied to the solution of the need to reach effective diversification has led to new investment products, bound to exploit the credit risk features of the assets. It is particularly for the evaluation of these new products, such as securitized assets (asset-backed securities, such as CDO and the like) and basked credit derivatives (nth to default options) that the need to account for comovement among non-normally distributed variabes has become an unavoidable task. This article attempts utilizes the credit yield spread between the non-risk bond and the common corporation bond in the market and using Copula functions to make up the relation composition of asset combination. Then, penetrates through the Monte-Carlo Simulation to estimated the default time of asset combination and princing the credit risk spread in the tranche of the Collateralized Debt Obligation (CDO). Besides, this article aims at the asset default recovery rate, the discount rate and the correlation coefficient of asset combination and so on three factors makes the sensitivity analysis, we find that the most effect of the credit default spread in the Collateralized Debt Obligation is asset default recovery rate, next is the correlation coefficient of asset combination, the influence of discount rate is not obvious.