Chapter One
Motivation for Heavy-Tailed Models

1.1 Structure of the Book

This book is split into a few core components covering fundamental concepts:

• Chapter 1 motivates the need to consider heavy-tailed loss process models in operational risk (OpRisk) and insurance modeling frameworks. It provides a basic introduction to the concept of separating the modeling of the central loss process and the tails of the loss process through splice models. It also sets out the key statistical questions one must consider studying when performing analysis and modeling of high consequence rare-event loss processes.
• Chapter 2 covers all the fundamental properties one may require in univariate loss process modeling under an extreme value theory (EVT) approach. Of importance is the detailed discussion on the associated statistical assumptions that must be made regarding the properties of any data utilized in model estimation when working with EVT models. This chapter provides a relatively advanced coverage of generalized extreme value (GEV) family of models, block maximum and peaks over threshold frameworks. It provides detailed discussion on statistical estimation that should be utilized in practice for such models and how one may adapt such methods to small sample settings that may arise in OpRisk settings. In the process, the chapter details clearly how to construct several loss distributional approach models based on EVT analysis. It then concludes with results of EVT in the context of compound processes.
• Chapter 3 provides a set of formal mathematical definitions for different notions regarding a heavy-tailed or fat-tailed loss distribution and its properties. It is important that when modeling such loss processes, especially the asymptotic properties of compound process models built with heavy-tailed loss models, aclear understanding of the tail properties of such loss models is understood. In this regard, we discuss the family of sub-exponential loss models, the family of regularly varying and slowly varying models. There are within these large classes of models sub-categorizations that are often of use to understand when thinking about risk measures resulting from such loss models, these are also detailed, for example, long-tailed models, subversively varying models and extended regular variation. In addition, the chapter opens with a basic introduction to key notations and properties of asymptotic notations that are utilized throughout the book.
• Chapter 4 begins with a basic introduction to properties of mathematical representations and characterizations of heavy-tailed loss models through the characteristic function and its representation. It then details the notions of divisibility, self-decomposability and the resulting consequences such distributional properties have on loss distributional approach compound process models. The remainder of the chapter provides a detailed coverage of the family of univariate -stable models, detailing their characterization, the parameterizations, density and distribution representations and parameter estimation. Such a family of models is becoming increasingly interesting for OpRisk modeling and insurance. It is recognized that such a family of models possesses many relevant and useful features that will capture aspects of OpRisk and insurance loss processes accurately and with advantageous features when used in a compound process model under a loss distributional approach structure.
• Chapter 5 provides the representations of flexible severity models based on tempering or exponential tilting of the -stable family of loss models. Under this concept, there are many families of tempered stable models available; this chapter characterizes each and discusses the mathematical properties of each sub-class of models and how they may be used in compound process models for heavy-tailed loss models in OpRisk and insurance. In addition, it discusses the aspects of model estimation and simulation for such models. The chapter then finishes with a detailed discussion on quantile-transformed-based heavy-tailed loss models for OpRisk and insurance, such as the Tukey transforms and the sub-family of the g-and-h distributions that have been popular in OpRisk.
• Chapter 6 discusses compound processes and convolutional semi-group structures. This then leads to developing representations of closed-form compound process loss distributions and densities that admit heavy-tailed loss processes. The chapter characterizes several classes of such models that can be used in practice, which avoid the need for computationally costly Monte Carlo simulation when working with such models.
• Chapter 7 discusses many properties of different classes of heavy-tailed loss processes with regard to asymptotic representations and properties of the tail of both partial sums and compound random sums. It does so under first-, second- and third-order asymptotic expansions for the tail process of such heavy-tailed loss processes. This is achieved under many different assumptions relating to the frequency and severity distribution and the possible dependence structures in such loss processes.
• Chapter 8 extends the results of the asymptotics for the tail of heavy-tailed loss processes partial sums and compound random sums to the asymptotics of risk measures developed from such loss processes. In particular, it discusses closed-form single-loss approximations and first-order, second-order and higher order expansion representations. It covers value-at-risk, expected shortfall and spectral risk measure asymptotics. This chapter also covers some alternative risk measure asymptotic results based on EVT known as penultimate approximations.
• Chapter 9 rounds off the book with a coverage of numerical simulation and estimation procedures for rare-event simulations in heavy-tailed loss processes, primarily for the estimation of properties of risk measures that provides an efficient numerical alternative procedure to utilization of such asymptotic closed-form representations.

1.2 Dominance of the Heaviest Tail Risks

In this book, we develop and discuss models for OpRisk to better understand statistical properties and capital frameworks which incorporate risk classes in which infrequent, though catastrophic or high consequence loss events may occur. This is particularly relevant in OpRisk as can be illustrated by the historical events which demonstrate just how significant the appropriate modeling of OpRisk can be to a financial institution.

Examples of large recent OpRisk losses are

• J.P. Morgan, GBP 3760 million in 2013-US authorities demand money because of mis-sold securities to Fannie Mae and Freddie Mac;
• Madoff and investors, GBP 40,819 million in 2008-B. Madoff's Ponzi scheme;
• Société Générale, GBP 4548 million in 2008-a trader entered futures positions circumventing internal regulations.

Other well-known examples of OpRisk-related events include the 1995 Barings Bank loss of around GBP 1.3 billion; the 2001 Enron loss of around USD 2.2 billion and the 2004 National Australia Bank loss of AUD 360 m.

The impact that such significant losses have had on the financial industry and its perceived stability combined with the Basel II regulatory requirements BCBS (2006) have significantly changed the view that financial institutions have regarding OpRisk. Under the three pillars of the Basel II framework, internationally active banks are required to set aside capital reserves against risk, to implement risk management frameworks and processes for their continual review and to adhere to certain disclosure requirements. There are three broad approaches that a bank may use to calculate its minimal capital reserve, as specified in the first Basel II pillar. They are known as basic indicator approach, standardized approach and advanced measurement approach (AMA) discussed in detail in Cruz et al. (2015). AMA is of interest here because it is the most advanced framework with regards to statistical modeling. A bank adopting the AMA must develop a comprehensive internal risk quantification system. This approach is the most flexible from a quantitative perspective, as banks may use a variety of methods and models, which they believe are most suitable for their operating environment and culture, provided they can convince the local regulator (BCBS 2006, pp. 150-152). The key quantitative criterion is that a bank's models must sufficiently account for potentially high impact rare events. The most widely used approach for AMA is loss distribution approach (LDA) that involves modeling the severity and frequency distributions over a predetermined time horizon so that the overall loss of a risk over this time period (e.g. year) is

where is the frequency modeled by random variable from discrete distribution and the independent severities from continuous distribution . There are many important aspects of LDA such as estimation of frequency and severity distributions using data and expert judgements or modeling dependence between risks considered in detail in Cruz et al. (2015). In this book, we focus on modeling heavy-tailed severities.

Whilst many OpRisk events occur frequently and with low impact (indeed, are 'expected losses'), others are rare and their impact may be as extreme as the total collapse of the bank. The modeling and development of methodology to capture, classify and understand properties of operational...