Introduction

In this chapter we continue our study of the classical risk model. We start with a useful result concerning the probability that ruin occurs without the surplus process first attaining a specified level. This result will be applied in Sections 8.4 and 8.5 and Exercise 9. We then consider the insurer's deficit when ruin occurs and provide a means of finding the distribution of this deficit. We extend this study by considering the insurer's largest deficit before the surplus process recovers to level 0. Following this, we consider the distribution of the insurer's surplus immediately prior to ruin. We then consider the distribution of the time to ruin, and conclude with a discussion of a problem which involves modifying the surplus process through the payment of dividends.

In this chapter we use the same assumptions and notation as in Chapter 7.

A barrier problem

Let us consider the following question: what is the probability that ruin occurs from initial surplus u without the surplus process reaching level b > u prior to ruin? An alternative way of expressing this question is to ask what is the probability that ruin occurs in the presence of an absorbing barrier at b? We denote this probability by ξ(u, b), and let χ(u, b) denote the probability that the surplus process attains the level b from initial surplus u without first falling below zero.

Introduction

In this chapter we consider the aggregate claims arising from a general insurance risk over a short period of time, typically one year. We use the term ‘risk’ to describe a collection of similar policies, although the term could also apply to an individual policy. As indicated in Chapter 1, at the start of a period of insurance cover the insurer does not know how many claims will occur, and, if claims do occur, what the amounts of these claims will be. It is therefore necessary to construct a model that takes account of these two sources of variability. In the following, we consider claims arising over a one-year time interval purely for ease of presentation, but any unit of time can be used.

We start by modelling aggregate claims in Section 4.2 as a random variable, S, and derive expressions for the distribution function and moments of S. We then consider the important special case when the distribution of S is compound Poisson, and we give an important result concerning the sum of independent compound Poisson random variables. In Section 4.4 we consider the effect of reinsurance on aggregate claims, both from the point of view of the insurer and the reinsurer.

The remainder of the chapter is devoted to the important practical question of calculating an aggregate claims distribution. In Section 4.5 we introduce certain classes of counting distribution for the number of claims from a risk.

This book is designed for final-year university students taking a first course in insurance risk theory. Like many textbooks, it has its origins in lectures delivered in university courses, in this case at Heriot-Watt University, Edinburgh, and at the University of Melbourne. My intention in writing this book is to provide an introduction to the classical topics in risk theory, especially aggregate claims distributions and ruin theory.

The prerequisite knowledge for this book is probability theory at a level such as that in Grimmett and Welsh (1986). In particular, readers should be familiar with the basic concepts of distribution theory and be comfortable in the use of tools such as generating functions. Much of Chapter 1 reviews distributions and concepts with which the reader should be familiar. A basic knowledge of stochastic processes is helpful, but not essential, for Chapters 6 to 8. Throughout the text, care has been taken to use straightforward mathematical techniques to derive results.

Since the early 1980s, there has been much research in risk theory in computational methods, and recursive schemes in particular. Throughout the text, recursive methods are described and applied, but a full understanding of such methods can only be obtained by applying them. The reader should therefore by prepared to write some (short) computer programs to tackle some of the examples and exercises.

Many of these examples and exercises are drawn from materials I have used in teaching and examining, so the degree of difficulty is not uniform.

Introduction

Ruin theory is concerned with the level of an insurer's surplus for a portfolio of insurance policies. In Chapter 4 we considered the aggregate amount of claims paid out in a single time period. We now consider the evolution of an insurance fund over time, taking account of the times at which claims occur, as well as their amounts. To make our study mathematically tractable, we simplify a real life insurance operation by assuming that the insurer starts with some non-negative amount of money, collects premiums and pays claims as they occur. Our model of an insurance surplus process is thus deemed to have three components: initial surplus (or surplus at time zero), premiums received and claims paid. For the model discussed in this chapter, if the insurer's surplus falls to zero or below, we say that ruin occurs.

The aim of this chapter is to provide an introduction to the ideas of ruin theory, in particular probabilistic arguments. We use a discrete time model to introduce ideas that we apply in the next two chapters where we consider a continuous time model. Indeed, we will meet analogues of results given in this chapter in these next two chapters. We start in Section 6.2 by describing our model, then in Section 6.3 we derive a general equation for the probability of ruin in an infinite time horizon, and consider situations in which it is possible to obtain an explicit solution for this probability.

Introduction

This book is about risk theory, with particular emphasis on the two major topics in the field, namely risk models and ruin theory. Risk theory provides a mathematical basis for the study of general insurance risks, and so it is appropriate to start with a brief description of the nature of general insurance risks. The term general insurance essentially applies to an insurance risk that is not a life insurance or health insurance risk, and so the term covers familiar forms of personal insurance such as motor vehicle insurance, home and contents insurance, and travel insurance.

Let us focus on how a motor vehicle insurance policy typically operates from an insurer's point of view. Under such a policy, the insured party pays an amount of money (the premium) to the insurer at the start of the period of insurance cover, which we assume to be one year. The insured party will make a claim under the insurance policy each time the insured party has an accident during the year that results in damage to the motor vehicle, and hence requires repair costs. There are two sources of uncertainty for the insurer: how many claims will the insured party make, and, if claims are made, what will be the amounts of those claims? Thus, if the insurer were to build a probabilistic model to represent its claims outgo under the policy, the model would require a component that modelled the number of claims and another that modelled the amounts of those claims.

INTRODUCTION

On February 13, 2002, Japanese Prime Minister Koizumi directed ministries to formulate emergency countermeasures to deflation, stating, in a press release that “while deflation has a varied background, the resolution of the immediate financial problems is indispensable to overcoming deflation” (italics added). Koizumi linked deflation, shown in Figure 9.1, to banking problems and a credit crunch, and called on the Bank of Japan to take action. The same press release goes on to state, “In particular, with regard to the disposal of non-performing loans which presently stands as a top-priority issue, actions shall be taken to achieve concrete progress in line with the ongoing process of special inspections, thereby solidifying the prospects for an early resolution of the so-called ‘non-performing loans problem.’ Meanwhile, the government shall stand ready to take any and all necessary measures to prevent a financial crisis. Overcoming deflation will require the government and the Bank of Japan to act in unison. Therefore, we turn to the Bank of Japan with the request that it act with due audacity in implementing monetary policy” (italics added; Japan Ministry of Foreign Affairs 2002).

The call by the Prime Minister for the government to take emergency action reflects the concern that the deflation in Japan, if allowed to persist, would gradually erode the foundation of the economy and eventually lead to a much deeper and dramatic decline reminiscent of the Great Depression in the 1930s. Initially viewed by policy makers as a “good deflation,” price declines were now seen as a harbinger of economic collapse.

Both of us, independently and jointly, have devoted many years to studying inflation and hyperinflation and analyzing central banking institutions. Indeed, it was our shared interests in these areas that led us to engage in joint research. In 1998, we began to think seriously about whether the incipient deflations in Asia, most notably in China and Japan, amounted to something other than simply the flip side of inflation. It quickly became apparent to us there was, potentially at least, something substantially different about studying an economy in the throes of a deflation. We also decided, early on, that the best way to proceed would be to call upon several interested academics to engage in research that would parallel our own interests, enabling us to provide a much broader perspective on historical and modern experiences with deflation. The end result is the present edited volume. The contributed papers were presented at two major conferences on the topic. One was held in April 2000 at Claremont McKenna College. We also organized a session at the XIIIth International Economic History Congress in Buenos Aires in 2002 that was devoted to deflation.

The rather long gestation period for this project partly reflects the fact that the study of deflation is, in many ways, in its infancy, especially regarding the conduct of monetary policy in such an environment. We were also determined to take the time to hone our ideas through seminar and conference participation, and in other ways to seek out respected colleagues' opinions. We trust that the end product will serve as a useful starting point for academics and policy makers wishing to learn more about the economics of deflation.

INTRODUCTION

The link between monetary policy and asset price movements has been of perennial interest to policy makers. The 1920s stock market boom and 1929 crash and the 1980s Japanese asset bubble are two salient examples in which asset price reversals were followed by protracted recessions and deflation. The key questions that arise from these episodes is whether the monetary authorities could have been more successful in preventing the consequences of an asset market bust or whether it was appropriate for the authorities only to react to these events ex post.

In this chapter we consider the potential cases for proactive versus reactive monetary policy based on the situation in which asset price reversals can have serious effects on real output.

Our analysis is based on a stylized model of the dilemma with which the monetary authorities are faced in asset price booms. On the one hand, letting the boom go unchecked entails the risk that it will be followed by a bust accompanied by a collateral-induced credit crunch. Restricting monetary policy can be thought of as insurance against the risk of a credit crunch. On the other hand, this insurance does not come free: Restricting monetary policy implies immediate costs in terms of lower output and inflation. The optimal monetary policy depends on the relative cost and benefits of the insurance.

Although the model is quite stylized, we find that the optimal monetary policy depends on the economic conditions – including the private sector's beliefs – in a rather complex way.

INTRODUCTION

After the U.S. stock market plunged in October 1929, deflation in asset prices was soon followed by declining goods prices as well. Although this episode has attracted enormous scrutiny over the years, surprisingly little attention has been paid to the fact that one of the very few, if not only, areas of appreciation during the bleak years after 1929 was in the stock prices of gold and silver producers. The largest gold producer at the time, Homestake Mining, enjoyed greater than 500 percent capital appreciation between October 1929 and December 1935, even as the Dow Jones index lost almost two-thirds of its value. These gains occurred in the midst of financial turmoil and massive bank failures, with nearly one-third of all U.S. banks failing between 1930 and 1933. At the same time, the great out-performance of gold stocks over this period proves that gold stocks do not require inflationary conditions in order to prosper. Indeed, the recent strength of gold stocks after 2000 has again occurred under conditions in which the threat of deflation again appears much more imminent than that of inflation – as reflected in the sharp rate cuts eventually taking the federal funds rate target down to 1 percent.

In this chapter, we assess the performance of gold stocks following two more recent stock market crashes (post–October 1987 and post–March 2000) in relation to the experience with Homestake Mining after the 1929 crash.

SEPARATE REGIONS WITH SEPARATE BANKING SYSTEMS

The Great Contraction of 1929 to 1939, as Friedman and Schwartz (1963) designated it, must surely be the most carefully studied episode in American monetary history. Their work, and that of Kindleberger (1973), Temin (1976, 1989), Eichengreen (1992a), and many other scholars focusing on particular aspects of the contraction, has done much to clarify the causes and consequences of the collapse, and to clarify the policy mistakes made by the Federal Reserve System. Nevertheless, the flood of books and papers on monetary problems during the Depression shows no signs of abating, suggesting that we have not yet reached the point where we believe that existing explanations are entirely sufficient. Moreover, as this is written, the renewed threat of falling prices makes it more important than ever to examine how financial system performance may be affected by deflationary pressures.

Important advances have been made in assessing contagion effects, as opposed to the role of fundamentals, in causing the collapse of a banking system. In this chapter, I will argue that the banking crises of the early 1930s were similar to the “twin crises” – banking and balance of payments crises – that have occurred in developing countries in recent years. The rural regions of the United States were like the developing countries today: dependent on the export of a few commodities in which they had a comparative advantage and with independent and relatively weak banking systems. Declining demand and export prices after 1929 weakened the banking systems in rural regions because they raised doubts about the ability of bank borrowers to repay their loans and led to balance of payments deficits.