This chapter implements a coherent statistical framework for validation of economic capital models via copula methods using a unique dataset to aggregate credit, market, operational, and interest rate risks. This framework includes benchmarking with alternative copula models and backtesting with alternative penalty functions, in addition to stability and stress tests of economic capital estimates. The analysis is expanded to include the latest supervisory guidance on model validation (i.e. SR11-7) and Basel Accord changes (i.e., Basel III). Second, proprietary confidential loss data is used from major US banks for market risk and operational risk. Lastly, both analytic and visual goodness-of-fit tests for copula models are included. For the data used in this study, the T copula with 4 degree of freedom provides a good statistical fit, superior backtesting performance, reasonable model stability and sufficient sensitivity to stress. In addition, the results provide some support for regulators’ hesitation to recognize diversification benefits by demonstrating a wide range of diversification benefits across risk types under different dependence models.

Supervisory applications are rife with examples where models are used and consequently require appropriate validation. This chapter compares and contrasts the notions of model risk and model uncertainty as they relate to both model choice and validation strategy. The chapter uses a decision-theoretic architecture (cf. ch. 7 of Optimal Statistical Decisions by M. DeGroot, 1970) and advances a thesis of how risk model validation can be actuated via utility optimization. An empirical exercise where the aforementioned themes conclude the chapter using a Home Mortgage Disclosure Act (HMDA) data set.

The Conway–Maxwell–Poisson distribution has garnered interest in and development of other flexible alternatives to classical distributions. This chapter introduces various distributional extensions and generalities motivated by functions of COM–Poisson random variables, including Conway–Maxwell-inspired generalizations of the Skellam distribution, binomial distribution, negative binomial distribution, the Katz class of distributions, two flexible series system life length distributions, and generalizations of the negative hypergeometric distribution.

This chapter considers various models that focus largely on serially dependent variables and the respective methodologies developed with a COM–Poisson underpinning. This chapter first introduces the reader to the various stochastic processes that have been established, including a homogeneous COM–Poisson process, a copula-based COM–Poisson Markov model, and a COM–Poisson hidden Markov model. Meanwhile, there are two approaches for conducting time series analysis on time-dependent count data. One approach assumes that the time dependence occurs with respect to the intensity vector. Under this framework, the usual time series models that assume a continuous variable can be applied. Alternatively, the time series model can be applied directly to the outcomes themselves. Maintaining the discrete nature of the observations, however, requires a different approach referred to as a thinning-based method. Different thinning-based operators can be considered for such models. The chapter then broadens the discussion of dependence to consider COM–Poisson-based spatio-temporal models, thus allowing both for serial and spatial dependence among variables.

This chapter examines wholesale credit risk models and their validation at US banking institutions. The most common practice in wholesale credit risk modeling for loss estimation among large US banking institutions today is to use expected loss models, typically at the loan level. The chapter discusses the quantification and validation of three key risk parameters in this modeling approach, namely, probability of default (PD), loss given default (LGD), and exposure at default (EAD).

This chapter provides an overview of the validation of models that are used in interest rate risk of the banking book (IRRBB). These includes models used for Funds Transfer Pricing (FTP) as well as asset–liability management (ALM). FTP is a charge (for assets) or a credit (for liabilities) that is charged (credited) by the corporate treasury to the business unit in order to isolate the business unit from market interest rate fluctuations for the life of the asset (liability). ALM involves modeling of principal and interest cash flows – positive cash flows for assets and negative cash flows for liabilities.