Ever since the work of Swedish actuary Filip Lundberg on collective risk, Ruin Theory, also known as Risk Theory, always stands in the center of the stage in insurance mathematics and actuarial science. In its classical setting, the ruin model describes the cash flow of an insurance company in a simplified way. Here the simplification is mainly in two directions: first, the arrival process of the claims is modelled by a Poisson process, which implies that the time between the arrivals of claims follow an exponential distribution; second, all claims follow a distribution with a tail decaying exponentially fast, or in its simplest version, they follow exactly an exponential distribution. The “double exponential” feature in classical Risk Theory does not reflect stylized features in reality such as heavy-tailed losses. As a consequence, classical Risk Theory does not seem to be directly applicable for insurance business. In particular, with the development of reinsurance, insurers can shift their tail risks to reinsurers and thus mitigate the probability of ruin.
Nevertheless, for reinsurance business itself, the setup of a ruin model is still the best approximation of the reality. The major gap left between classical Risk Theory and applications to reinsurance is how to handle heavy tailed claims and more general models for the arrival process of the claims. The book by Konstantinides provides a rigorous approach on how recent research in Risk Theory handles heavy tails. It therefore bridges theoretical research in Risk Theory with real life applications in (re)insurance. For researchers in this field, this book provides a self-contained treatment moving from the classical setting to the modern frontier.
The 13 chapters in Konstantinides (2017) can be divided into four parts. The first part, Chapters 1–3, reviews the classical Risk Theory with exponentially decaying tail. The second part, Chapters 4–6, deals with heavy tails in Risk Theory. The third part, Chapters 7–10, handles advanced topics in Risk Theory, with a focus on heavy tails. The advanced topics cover the single big jump principle, discounting with interest rate and absolute ruin, etc. Finally, the fourth part, Chapters 11–13, handles various situations in which different independence assumptions in the classical settings are violated.
The first part of the book, as standard in texts on ruin theory, sets a coherent system of notation and lays out the main problem: calculating or approximating the ruin probability. The main problem is discussed both within specific and general contexts. Chapter 1 first handles the specific case where the claim size follows exactly the exponential distribution and the arrival of claims follow a simple Poisson process. Then the chapter moves on to general distributions for the claim size, albeit having an exponentially decaying tail. For general claim size distribution, the chapter provides an instrument to handle the ruin probability, which coincides with the Pollaczek–Khintchine formula in queueing theory. This tool will be useful when handling heavy-tailed distributions in the second part. In Chapter 2, the time between successive claims is allowed to follow a general distribution, which opens the door to Renewal Risk Theory. Under the Cramér condition, which implicitly assumes exponentially decaying tails for the claim size, the ruin probability is still decaying at an exponential speed. Chapter 3 further handles approximations of the ruin probability: if one approximates the ruin probability by an exponentially decaying function, how the parameters in such a function are related to characteristics of the claim distribution.
The second part of the book handles claim sizes following a heavy-tailed distribution. To keep the book self-contained, the author spends Chapters 4 and 5 introducing the concept of heavy tails motivated by Extreme Value Theory. These two chapters can be a self-standing text for readers who are interested in univariate Extreme Value Theory. In particular, the chapters provide a detailed explanation on regular varying functions, including important results such as Potter’s inequality, Karamata theorem, etc. At the end of Chapter 5, the Feller theorem regarding the convolution of two regular varying tails is discussed. This is another key instrument for handling the ruin probability with heavy-tailed claim size.
By reading through Chapter 5, a reader is equipped with both the general risk theory and the theories regarding regularly varying tails (i.e., heavy tails). Starting from Chapter 6, the two theories meet each other: by applying the Pollaczek–Khintchine formula together with proper inequalities for the convolution of subexponential distributions, the ruin probability for subexponentially distributed claim size can be approximated by the tail integral of the claim distribution. This differs substantially from the classical ruin probability, which decays at an exponential speed. Chapter 6 also takes care of how to verify subexponentiality by providing various characterizations of subexponentiality.
The first two parts of this book together form a good introductory text for researchers who intend to get into the field of Risk Theory. The next two parts discuss various advanced topics that are useful for developing new research ideas.
As an example of advanced topics discussed in the third part of the book, Chapter 8 discusses the so-called “single big jump principle.” When claims are heavy-tailed, the large value of a total claim is often driven by one large claim, that is, the maximum of all claims. This principle is a consequence of subexponentiality. It is also the essential reason why the ruin probability is related to the claim distribution when the claim distribution is heavy tailed. Understanding this principle provides a deeper insight on the results in Chapter 6.
Besides theoretical topics, some advanced topics are more close to applications. For example, Chapter 7 deals with random weighted sum of the claims, which is related to discounting the future cash flows. The latter is related to real life applications when interest rate is accounted in the ruin model. Chapter 9 provides a careful treatment when interest rate is constant. Notice that even for constant interest rate, the claims are weighted by random weights related to the arrival times of the claims.
The last part of the book, Chapters 11–13, considers scenarios in which different dependence assumptions are relaxed. Chapter 11 relaxes the independence between arrival time and claim size. Chapter 12 relaxes the independence across claims to assuming asymptotic independence across claims. Chapter 13 relaxes the independence across the arrival times in between claims. Another additional topic handled in all three chapters is investment. Instead of receiving a constant premium, the insurer is now allowed to receive a random return from investment. Involving stochastic returns from investment in Risk Theory is again more close to applications in real life.
Last, but certainly not least, one important highlight of the book is the exercises provided at the end of each chapter. Especially, all exercises are accompanied with extensive hints which can be regarded as solutions. Not only are the exercises useful material for readers to understand and practice corresponding theories in each chapter, but they are also sometimes important results by themselves, which complements the existing theories in the corresponding chapter.
To summarize, the rigorous and detailed treatment of Risk Theory in this book gears up the readers for recent research in this field, in particular when heavy tails are involved. For academic readers, it provides a rather complete view on both the history of Risk Theory and its frontier research. For practitioners in insurance, this book can be used as a standard reference for handling the ruin probability in various real life situations.