Modeling the dependence structures of financial assets through the Copula Quantile-on-Quantile approach

This paper considers a new approach of analyzing asset dependence by estimating how the distributions (in particular, quantiles) of assets are related. Combining the techniques of quantile regression and copula modeling, I propose the Copula Quantile-on-Quantile Regression approach to estimate the correlation that is associated with the quantiles of asset returns, which is able to uncover obscure nonlinear characteristics in asset dependence. The estimation procedure proposed here can also be used for analyzing dependence structures in other settings, such as for studying how macroeconomic covariates are nonlinearly related by looking at the relationship between their quantiles.     

A Spatial Cross-Sectional Credibility Model with Dependence Among Risks

A Bühlmann-Straub type credibility model with dependence structure among risk parameters and conditional spatial cross-sectional dependence is studied. Predictors of future losses for the model under both types of dependence are derived by minimizing the expected quadratic loss function, and nonparametric estimators of structural parameters are considered in the spatial statistics context. Predictions and estimations made for the proposed model are examined and compared to other models in an application with crop insurance data and in a simulation study.     

Cybersecurity Insurance: Modeling and Pricing

Cybersecurity risk has attracted considerable attention in recent decades. However, the modeling of cybersecurity risk is still in its infancy, mainly because of its unique characteristics. In this study, we develop a framework for modeling and pricing cybersecurity risk. The proposed model consists of three components: the epidemic model, loss function, and premium strategy. We study the dynamic upper bounds for the infection probabilities based on both Markov and non-Markov models. A simulation approach is proposed to compute the premium for cybersecurity risk for practical use. The effects of different infection distributions and dependence among infection processes on the losses are also studied.     

The Predictive Power of Anisotropic Spatial Correlation Modeling in Housing Prices

This paper develops a method to capture anisotropic spatial autocorrelation in the context of the simultaneous autoregressive model. Standard isotropic models assume that spatial correlation is a homogeneous function of distance. This assumption, however, is oversimplified if spatial dependence changes with direction. We thus propose a local anisotropic approach based on non-linear scale-space image processing. We illustrate the methodology by using data on single-family house transactions in Lucas County, Ohio. The empirical results suggest that the anisotropic modeling technique can reduce both in-sample and out-of-sample forecast errors. Moreover, it can easily be applied to other spatial econometric functional and kernel forms.     

Multivariate Cox Hidden Markov models with an application to operational risk

ABSTRACT

Modeling multivariate time-series aggregate losses is an important actuarial topic that is very challenging due to the fact that losses can be serially dependent with heterogeneous dependence structures across loss types and business lines. In this paper, we investigate a flexible class of multivariate Cox Hidden Markov Models for the joint arrival process of loss events. Some of the nice properties possessed by this class of models, such as closed-form expressions, thinning properties and model versatility are discussed in details. We provide the expectation-maximization (EM) algorithm for efficient model calibration. Applying the proposed model to an operational risk dataset, we demonstrate that the model offers sufficient flexibility to capture most characteristics of the observed loss frequencies. By modeling the log-transformed loss severities through mixture of Erlang distributions, we can model the aggregate losses. Finally, out-of-sample testing shows that the proposed model is adequate to predict short-term future operational risk losses.     

Estimation of risk contributions with MCMC

Determining risk contributions of unit exposures to portfolio-wide economic capital is an important task in financial risk management. Computing risk contributions involves difficulties caused by rare-event simulations. In this study, we address the problem of estimating risk contributions when the total risk is measured by value-at-risk (VaR). Our proposed estimator of VaR contributions is based on the Metropolis-Hasting (MH) algorithm, which is one of the most prevalent Markov chain Monte Carlo (MCMC) methods. Unlike existing estimators, our MH-based estimator consists of samples from the conditional loss distribution given a rare event of interest. This feature enhances sample efficiency compared with the crude Monte Carlo method. Moreover, our method has consistency and asymptotic normality, and is widely applicable to various risk models having a joint loss density. Our numerical experiments based on simulation and real-world data demonstrate that in various risk models, even those having high-dimensional (≈500) inhomogeneous margins, our MH estimator has smaller bias and mean squared error when compared with existing estimators.     

Flexible dependence modeling of operational risk losses and its impact on total capital requirements

Operational risk data, when available, are usually scarce, heavy-tailed and possibly dependent. In this work, we introduce a model that captures such real-world characteristics and explicitly deals with heterogeneous pairwise and tail dependence of losses. By considering flexible families of copulas, we can easily move beyond modeling bivariate dependence among losses and estimate the total risk capital for the seven- and eight-dimensional distributions of event types and business lines. Using real-world data, we then evaluate the impact of realistic dependence modeling on estimating the total regulatory capital, which turns out to be up to 38% smaller than what the standard Basel approach would prescribe.     

Structure and estimation of Lévy subordinated hierarchical Archimedean copulas (LSHAC): Theory and empirical tests

Lévy subordinated hierarchical Archimedean copulas (LSHAC) are flexible models in high dimensional modeling. However, there is limited literature discussing their applications, largely due to the challenges in estimating their structures and their parameters. In this paper, we propose a three-stage estimation procedure to determine the hierarchical structure and the parameters of a LSHAC. This is the first paper to empirically examine the modeling performances of LSHAC models using exchange traded funds. Simulation study demonstrates the reliability and robustness of the proposed estimation method in determining the optimal structure. Empirical analysis further shows that, compared to elliptical copulas, LSHACs have better fitting abilities as well as more accurate out-of-sample Value-at-Risk estimates with less parameters. In addition, from a financial risk management point of view, the LSHACs have the advantage of being very flexible in modeling the asymmetric tail dependence, providing more conservative estimations of the probabilities of extreme downward co-movements in the financial market.     

Nonlinear term structure dependence: Copula functions,empirics, and risk implications

This paper documents nonlinear cross-sectional dependence in the term structure of US-Treasury yields and points out risk management implications. The analysis is based on a Kalman filter estimation of a two-factor affine model which specifies the yield curve dynamics. We then apply a broad class of copula functions for modeling dependence in factors spanning the yield curve. Our sample of monthly yields in the 1982–2001 period provides evidence of upper tail dependence in yield innovations; i.e., large positive interest rate shocks tend to occur under increased dependence. In contrast, the best-fitting copula model coincides with zero lower tail dependence. This asymmetry has substantial risk management implications. We give an example in estimating bond portfolio loss quantiles and report the biases which result from an application of the normal dependence model.     

A simple approach to standardized-residuals-based higher-moment tests

We propose a new approach to the higher-moment tests for evaluating the standardized error distribution hypothesis of a conditional mean-and-variance model (such as a GARCH-type model). Our key idea is to purge the effect of estimating the conditional mean-and-variance parameters on the estimated higher moments by suitably using the first and second moments of the standardized residuals. The resulting higher-moment tests have a simple invariant form for various conditional mean-and-variance models, and are also applicable to the symmetry or independence hypothesis that does not involve a complete standardized error distribution. Thus, our tests are simple and flexible. Using our approach, we establish a class of skewness–kurtosis tests, characteristic-function-based moment tests, and Value-at-Risk tests for exploring the standardized error distribution and higher-order dependence structures. We also conduct a simulation to show the validity of our approach in purging the estimation effect, and provide an empirical example to show the usefulness of our tests in exploring conditional non-normality.     

Canonical vine copulas in the context of modern portfolio management: Are they worth it?

In the context of managing downside correlations, we examine the use of multi-dimensional elliptical and asymmetric copula models to forecast returns for portfolios with 3–12 constituents. Our analysis assumes that investors have no short-sales constraints and a utility function characterized by the minimization of Conditional Value-at-Risk (CVaR). We examine the efficient frontiers produced by each model and focus on comparing two methods for incorporating scalable asymmetric dependence structures across asset returns using the Archimedean Clayton copula in an out-of-sample, long-run multi-period setting. For portfolios of higher dimensions, we find that modeling asymmetries within the marginals and the dependence structure with the Clayton canonical vine copula (CVC) consistently produces the highest-ranked outcomes across a range of statistical and economic metrics when compared to other models incorporating elliptical or symmetric dependence structures. Accordingly, we conclude that CVC copulas are ‘worth it’ when managing larger portfolios.     

Evaluation of volatility predictions in a VaR framework

The evaluation of volatility forecasts is not straightforward and some issues can arise. A standard approach relies on statistical loss functions. Another approach bases the evaluation of the volatility predictions on utility functions or Value at Risk (VaR) measures. This work aims to combine the two approaches, using the VaR measures within the loss functions. By means of this method, the VaR measures obtained from a set of competing models are plugged into two loss functions, the magnitude loss function and a proposed new one. This latter loss function more heavily penalizes the models with a number of VaR violations greater than the expected one. The loss function values are evaluated against a benchmark obtained from the inclusion of a consistent estimate of the VaR measures in the loss function. In order to investigate the performance of the proposed method and the new loss function, a Monte Carlo experiment and an empirical analysis of a stock listed on the New York Stock Exchange are provided. The proposed strategy helps with the selection of a superior model, in terms of forecast accuracy, when the cited approaches do not clearly and uniquely identify it. Moreover, the new asymmetric loss function allows a greater discrimination with regard to models, helping to find the best volatility model.     

Spatial Distribution of Retail Sales

We examine the distribution of sales for a retail chain in the Houston market using a spatial gravity model. Unlike previous empirical studies, our approach models spatial dependencies among both consumers and retailers. The results show that both forms of spatial dependence exert statistically and economically significant impacts on the estimates of parameters in retail gravity models. Contrary to the suggestions of (Gautschi, D. A. (1981). “Specification of Patronage Models for Retail Center Choice,” Journal of Marketing Research 18, 162–174.) as well as (Eppli, M. J., and J. D. Shilling. (1996). “How Critical Is a Good Location to a Regional Shopping Center?” Journal of Real Estate Research 12, 459–468.), our results show the importance of the distance parameter in retail gravity models may be greatly understated. Thus, ignoring spatial dependence may lead to overestimation of the deterministic extent of trade areas, and underestimate the importance of good locations.     

Multivariate Lévy processes with dependent jump intensity

In this work we propose a new and general approach to build dependence in multivariate Lévy processes. We fully characterize a multivariate Lévy process whose margins are able to approximate any Lévy type. Dependence is generated by one or more common sources of jump intensity separately in jumps of any sign and size and a parsimonious method to determine the intensities of these common factors is proposed. Such a new approach allows the calibration of any smooth transition between independence and a large amount of linear dependence and provides greater flexibility in calibrating nonlinear dependence than in other comparable Lévy models in the literature. The model is analytically tractable and a straightforward multivariate simulation procedure is available. An empirical analysis shows an accurate multivariate fit of stock returns in terms of linear and nonlinear dependence. A numerical illustration of multi-asset option pricing emphasizes the importance of the proposed new approach for modeling dependence.     

A Markov-switching generalized additive model for compound Poisson processes,with applications to operational loss models

This paper is concerned with modelling the behaviour of random sums over time. Such models are particularly useful to describe the dynamics of operational losses, and to correctly estimate tail-related risk indicators. However, time-varying dependence structures make it a difficult task. To tackle these issues, we formulate a new Markov-switching generalized additive compound process combining Poisson and generalized Pareto distributions. This flexible model takes into account two important features: on the one hand, we allow all parameters of the compound loss distribution to depend on economic covariates in a flexible way. On the other hand, we allow this dependence to vary over time, via a hidden state process. A simulation study indicates that, even in the case of a short time series, this model is easily and well estimated with a standard maximum likelihood procedure. Relying on this approach, we analyse a novel data-set of 819 losses resulting from frauds at the Italian bank UniCredit. We show that our model improves the estimation of the total loss distribution over time, compared to standard alternatives. In particular, this model provides estimations of the 99.9% quantile that are never exceeded by the historical total losses, a feature particularly desirable for banking regulators.     

Volatility dependences of stock markets with structural breaks

We develop a Vector Heterogeneous Autoregression model with Continuous Volatility and Jumps (VHARCJ) where residuals follow a flexible dynamic heterogeneous covariance structure. We employ the Bayesian data augmentation approach to match the realised volatility series based on high-frequency data from six stock markets. The structural breaks in the covariance are captured by an exogenous stochastic component that follows a three-state Markov regime-switching process. We find that the stock markets have higher volatility dependence during turmoil periods and that breakdowns in volatility dependence can be attributed to the increase in market volatilities. We also find positive correlations between the Asian stock markets, the European stock market, and the UK stock market. The US stock market has positive correlations with all other markets for most of the sample periods, indicating the leading position of US stock market in the global stock markets. In addition, the proposed three-state VHARCJ model with Dynamic Conditional Correlation (DCC) and break structure under student-t distribution has a superior density forecast performance as compared to the competing models. The forecast models with structural breaks outperform those without structural breaks based on the log predicted likelihood, the log Bayesian factor, and the root mean square loss function.     

Intraday risk management in International stock markets: A conditional EVT approach

The study compares the predictive ability of various models in estimating intraday Value-at-Risk (VaR) and Expected Shortfall (ES) using high frequency share price index data from sixteen different countries across the world for a period of seven and half months from September 20, 2013 to May 07, 2014. The main emphasis of the study has been given to Extreme Value Theory (EVT) and to evaluate how well Conditional EVT model performs in modeling tails of distributions and in estimating and forecasting intraday VaR and ES measures. We have followed McNeil and Frey's (2000) two stage approach called Conditional EVT to estimate dynamic intraday VaR and ES. We have compared the accuracy of Conditional EVT approach to intraday VaR and ES estimation with other competing models. The best performing model is found to be the Conditional EVT in estimating both the quantiles for the entire sample. The study is useful for market participants (such as intraday traders and market makers) involved in frequent intraday trading in such equity markets.     

Modeling Operational Risk With Bayesian Networks

Bayesian networks is an emerging tool for a wide range of risk management applications, one of which is the modeling of operational risk. This comes at a time when changes in the supervision of financial institutions have resulted in increased scrutiny on the risk management of banks and insurance companies, thus giving the industry an impetus to measure and manage operational risk. The more established methods for risk quantification are linear models such as time series models, econometric models, empirical actuarial models, and extreme value theory. Due to data limitations and complex interaction between operational risk variables, various nonlinear methods have been proposed, one of which is the focus of this article: Bayesian networks. Using an idealized example of a fictitious on line business, we construct a Bayesian network that models various risk factors and their combination into an overall loss distribution. Using this model, we show how established Bayesian network methodology can be applied to: (1) form posterior marginal distributions of variables based on evidence, (2) simulate scenarios, (3) update the parameters of the model using data, and (4) quantify in real‐time how well the model predictions compare to actual data. A specific example of Bayesian networks application to operational risk in an insurance setting is then suggested.     

A Bayesian approach for more reliable tail risk forecasts

This paper demonstrates that existing quantile regression models used for jointly forecasting Value-at-Risk (VaR) and expected shortfall (ES) are sensitive to initial conditions. Given the importance of these measures in financial systems, this sensitivity is a critical issue. A new Bayesian quantile regression approach is proposed for estimating joint VaR and ES models. By treating the initial values as unknown parameters, sensitivity issues can be dealt with. Furthermore, new additive-type models are developed for the ES component that are more robust to initial conditions. A novel approach using the open-faced sandwich (OFS) method is proposed which improves uncertainty quantification in risk forecasts. Simulation and empirical results highlight the improvements in risk forecasts ensuing from the proposed methods.     

Credit Spreads in Illiquid Markets: Model and Implementation

This paper presents a methodology for estimating a family of credit spread term structures in a market with few transactions. The authors propose partitioning the market into risk classes and modeling credit spread term structures for each risk class using a multifactor Vasicek model with some common and some risk class-specific factors. The approach uses information on the cross section and time series of corporate bonds in all the risk classes to estimate the term structure of credit spreads in each risk class. The model is jointly estimated using an extended Kalman filter and implemented using Chilean corporate and government bonds.