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The fundamental theorem of actuarial risk science S´ergio B. Volchan∗

Abstract In this paper we present one of the main results of collective risk theory in non-life insurance mathematics, which says that for small claims the ruin probability decreases exponentially fast. The discussion is made in the context of the classical Cram´er-Lundberg model using the martingale technique. Keywords: risk theory, insurance, martingales

1

Introduction

Like most areas of mathematics, probability theory emerged from the need to deal with real-life problems. Besides its use in games of chance and astronomical data analysis, the field experienced a great impetus in the 19th century from applications in the social sciences: statistics, demography, economics, insurance and even law. It is then not totally surprising, though still remarkable, that the pioneering work on the most sophisticated branch of modern probability theory, that of continuous-time stochastic processes, were also related to social science applications. 1 First and foremost is Louis Bachelier’s 1900 famous thesis titled “Th´eorie de la sp´eculation” in which for the first time a (semi-rigorous) mathematical treatment of Brownian motion is used to describe price fluctuations at the Paris stock exchange. Written under the supervision of none other than Henri Poincar´e, it antedated Einstein’s work (1905) on the physical explanation of Brownian motion and Wiener’s (1923) rigorous mathematical construction, namely of Wiener process (with continuous nowhere differentiable sample paths). It antedated as well the now almost Pontif´ıcia Universidade Cat´ olica do Rio de Janeiro, Departamento de Matem´ atica, Rua Marquˆes de S˜ ao Vicente 225,G´ avea, 22453-900 Rio de Janeiro, Brasil [email protected] 1 Incidentally, the earlier use of probabilistic ideas in kinetic gas theory in the hands of Maxwell and Boltzmann was in part inspired by statistics as applied to social sciences. It seems that the first evolution equation for a probability density ever written was Boltzmann’s integro-differential equation in 1872. ∗

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universally accepted axiomatization of probability theory laid down in 1933 by Andrei Kolmogorov (who was critical of Bachelier’s lack or rigor but acknowledged his influence in his work on continuous-time Markov processes), not to mention Itˆo’s stochastic calculus developed in the 1940’s, also influenced by Bachelier. As is well known, this work was almost forgotten (partly due to its novelty and partly to its lack of rigor) until it was rediscovered in the 1950’s and early 60’s by mathematicians, physicists and economists. It gained renewed significance during the intense academic work on financial mathematics in the wake of the deep structural changes in the global financial markets of the seventies (and the concomitant computer and telecommunications revolution), a work that culminated in the Nobel-winning (1997) Black-Scholes-Merton model. By that time, the theory of martingales, developed in the fifties by Doob (having Wiener process as the quintessential example) and Itˆo’s stochastic calculus were in full bloom. Then in the eighties the connection of martingales to the crucial financial concept of arbitrage was clarified and widely explored. Nowadays, stochastic analysis is a standard tool of the modern theoretical and applied finance specialist and the importance of Bachelier’s work was internationally recognised through the celebration of the First World Congress of the Bachelier Finance Society held in Paris (2000). [14] What is perhaps less well known is that almost at the same time as Bachelier’s thesis, another pioneering work, this turn in the field of actuarial science, was done by the Swede Filip Lundberg. In his 1903 Uppsala thesis he uses yet another important example of stochastic process, to wit, Poisson process (with discontinuous sample paths), in modelling the ruin problem for an insurance company. Extended and rigorized by Harald Cram´er in the thirties, the so-called Cram´er-Lundberg model is still a landmark of insurance mathematics (non-life branch). It wouldn’t be unfair to say that it has a similar role in actuarial science as the Black-Scholes-Merton model in finance. [12] It is fascinating to realize that from such applications in finance and insurance, developed almost simultaneously, the two most important examples of stochastic processes came to life. 2 Interestingly, actuarial/insurance mathematics was also important in the quest for the foundations of probability theory. In fact, David Hilbert, in his famous address at 1900 International Congress of Mathematics held in Paris, included the axiomatization of probability theory as part his 6th problem (on the axiomatization of physics of which probability was thought to be a part). He makes reference to a lecture on the subject by insurance mathematician Georg Bohlmann (published in 1900) in the context of life-insurance problems (who, in turn, cites Poincar´e’s 1896 textbook on probability as a main source!). [7] In this paper we discuss the classical Cr´amer-Lundberg model, in particular the use of martingale methods to derive the Cram´er-Lundberg estimate. The paper is structured as follows. We first discuss the central but subtle concept of risk in finance and insurance and the role of the actuary as a risk manager. We then describe the classical Cram´er-Lundberg model and the related ruin problem for insurance risk. Finally we derive the Cram´er-Lundberg estimate, after which we make some concluding remarks. 2

Which are the basic examples of the class of of L´evy processes [1, 17], i.e., with stationary and independent increments and right-continuous with left limits sample paths.

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2

The elusive concept of risk

Everyone agrees that the concept of risk is central to the disciplines of finance and actuarial science. Ironically, however, there is no consensus on what it precisely means. It generally has the negative connotation of a “loss” 3 usually measured in monetary units. And the fact is that it may have different meanings to different people in different situations. Thus one speaks of various “risk factors” in a given context. These factors are usually interconnected in complex ways and one tries to devise associated “risk measures” in order to obtain some quantitative estimate (and hopefully some control) of one’s vulnerability to such factors. 4 Risk is also commonly associated with uncertainty, a psychological category, which in turn manifests itself due to the unpredictability of certain future events. This is a typical situation in finance and insurance, as various transactions and exchanges are contingent on the occurrence or not of certain future events, generating uncertainty regarding what could happen in-between. Consider for example the risk of default of a company (or government) which is unable to honour its contractual obligations due to unforeseen changes in the political, social-economic and even natural scenarios. Now, the unpredictability of a certain phenomenon or process has many sources, for instance: (a) Ignorance, lack of information or poor/partial knowledge about the laws and mechanisms that rule this phenomenon. This is an unavoidable fact of the human condition. It can in principle be mitigated with improved and continuous research; so in a sense, science can be viewed as a collective effort to curb such human limitations. But for an individual, incomplete information is the rule. (b) It might be the case that even knowing in detail the laws and mechanisms involved, still the process or system dealt with has some intrinsic instability; for example, it could display sensitivity to small perturbations causing an amplification of errors, leading to a disruption of any long-time predictability. This is thought to be the case of so-called “chaotic dynamical systems” (e.g., in meteorology). (c) It could also happen that the system is stochastic by nature. Of course, that doesn’t mean that the system’s behaviour is arbitrary; quite the opposite, it means that its behaviour (or some aspects of it) is governed by the (very stringent) laws of probability. All of the items above (and others) can of course happen simultaneously in a given situation. However, item (c), that we may call “stochastic hypothesis”, is very popular in many models in finance and actuarial science. So much so that in these models “risk” is simply defined as a certain random variable (or some parameter linked to it, like standard deviation) representing an uncertain future payment/liability. In other words, for the sake of mathematical modelling, one avoids discussing the origin of randomness and just identifies it to risk. Thus, in finance models all the complexities of market price determination are reduced to 3

But not always, as it can mean gains from investment or speculation. Incidentally, the word risk in Chinese is the concatenation of “danger” with “opportunity”, which seems to better capture the gist of the concept. 4 By the way, the charge of interest is an old form of protection against the risk of lending money, so in this sense it is a primitive form of risk measure.

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the hypothesis that the prices of risky assets are described by some given stochastic processes, typically related to Wiener’s. Similarly, in actuarial science it is common to suppose that claims (corresponding, say, to car accidents) happen at random according to a given probability distribution, typically linked to the Poisson process (see section 3 below). As uncertainty cannot be completely eliminated in human affairs, individuals and society tried to devise means to at least minimise its impact. In other words, whatever the origins or type of risk, its management or administration is one of the primary tasks of the risk analyst or technologist (a function in great demand at major financial and governmental institutions). Related tasks are planning and forecasting, designing of appropriate management tools (e.g., models, risk measures and estimation), etc. The fundamental goal is to improve the decision-making process at all levels of the organisation, avoiding real and potential losses and increasing its overall efficiency. This is accomplished through an analysis of the vulnerabilities of the firm to all kinds of risks and impacts that could compromise its future revenues. This calls for a careful analysis of such risks based on sound economic fundamentals, reliable data and good mathematical modelling. 5 A decision that comes out of such analysis should concern, for example, the amount of reserve capital that the institution should keep in order to absorb the impact of possible losses due to adverse changes of price of assets in the firm’s portfolio (and the choice of such assets itself), the so-called market risk. Such a decision is made not only in the private sector but also by the regulatory authorities, such as governments and central banks, when they demand safety margins in business transactions. Clearly the importance and responsibility of the risk manager are enormous. A wrong decision regarding asset allocation based on a faulty risk analysis could cripple the competitiveness of the firm (or even jeopardise its very existence). A series of recent management/financial scandals [9] such as Barings Bank, Proctor & Gamble, Metallgeselschaft and Long Term Capital Management 6 , just to mention a few, brought public opinion and governments to demand greater controls and responsible management in a context of global, fast, interdependent and volatile (i.e., unstable) markets. As the level of sophistication and sheer complexity of financial systems (and society as a whole) increased so did the mathematical tools needed to tackle models describing them. This partly explains the demand for people with such technical expertise: statisticians, mathematicians, engineers and physicists. While this trend opened a new market for these professionals, it also meant an increase in the cost of specialised personnel to the firms, which inevitably raises the price of their services. Moreover, it is not enough to have a good technical expertise, one also needs the ability to critically evaluate the strengths and weaknesses of the models, particularly with respect to its presuppositions and unavoidable simplifications. This could help minimise yet another form of risk, namely modelling risk, a type of operational risk arising from errors and/or inadequacies of the model used. So, for example, a common simplifying assumptions common to many finance models is that the firm’s assets are always convertible to cash, i.e., that there is no (or very low) liquidity risk. Illiquidity usually manifests itself as a difficult in changing one’s position without appreciably depressing market prices. This increases the time needed to 5

The lack of reliable mathematical models for premium calculation seems to be the reason why till around the middle of 19th century insurance was a very risky business indeed: around 60% of the companies went bankrupt in 10 years! [4] 6 In whose team, by the way, belonged Nobel prize winners Myron Scholes and Robert Merton.

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complete such transactions which in turn might increase the firm’s exposure to market fluctuations which, in turn might increase transaction costs, spreads, etc. Such situations are bound to happen sometime in the real world and the risk manager has to be prepared for those, maybe rare, but quite real, occasions. The actuary is that kind of risk technologist specialised in insurance, a social mechanism for compensating individuals and institutions for financial losses due to unforeseen events or circumstances. Though traditionally associated with property and life insurance (covering for instance accidents, theft, natural disasters, etc) the field has considerably enlarged its scope in the last 30 years. The renowned actuary Hans B¨ uhlmann (1979) proposed the following classification of insurance kinds, roughly following the historical trend of the field: • Insurance of the first kind: life-insurance, based on mortality tables and interest rate calculations using mainly non-probabilistic techniques; • Insurance of the second kind: characterised by the introduction of probabilistic tools and results such as the Law of Large Numbers and the Central Limit Theorem in non-life insurance; • Insurance of the third kind insurance: characterised by a deep interaction with finance and heavily involving the whole area of stochastic analysis, advanced statistics and other tools. Though insurance of the third kind seems to be a new trend, one easily recognises that many crucial notions in modern finance have a distinctly actuarial flavor. For example, the concept of hedging can be viewed as a kind of protection against unfavourable market behaviour. And although derivative instruments (options, futures, swaps, etc) have a clear speculative aim, they are also mechanisms of risk transfer.

3

The Ruin Problem

Risks can of course be minimised simply by avoiding them altogether or by trying to predict the occurrence of unfavourable events. As these alternatives are not always feasible, a third method, which is at the heart of the insurance industry, is that of risk transfer from a group of people (or institutions) to another. A classical mechanism is pooling 7 : many individuals transfer their risks to a collective body, like an insurance company or a pension fund, which is better equipped to handle them. Human behaviour regarding risk and uncertainty is quite complicated. We feel discomfort when facing uncertainty and studies show that people (even experts) perform poorly when handling probabilistic concepts. Our behaviour is not always “rational” 8 and depends on many variables like level of education, how far in the future a possibly hazardous event is, emotional and affective dispositions, social factors, etc. In any case, a traditional working hypothesis is that “rational” individuals tend to be risk averse: they would rather face a small but certain loss than a large uncertain one. As unfavourable events are (conceivably) rare, the risk from 7

Which is akin to the concept of diversification, well known in finance. Think of smoking habits, drug abuse and the socially accepted though highly dangerous car driving activity. 8

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the many to which the insured-against event did not happen can be transfered to the few for whom it did. This is an insight gained from the Law of Large Numbers: the “collective” or average (or, depending on the context, long-term) effect of many small independent random causes is nonrandom. It justifies the motto “certum ex incertis” adopted by the oldest actuarial organisation, the Institute of Actuaries in London (1848). [5] From the viewpoint of an individual buying an insurance contract he gets, through an affordable price paid regularly, protection against unexpected and potentially costly events (e.g., a car crash, illness). On the other side, the insurance company selling the contract charges a relatively low price (the premium) very many clients in such a way that it will always be able to pay for claims, as these occur randomly but not very frequently. Therefore, running an insurance company requires first and foremost the maintenance of a healthy balance of cash inflows and outflows, which is the basic principle of double-entry bookkeeping going back to Luca Pacioli (1496). [6] Of course, the company has to gauge the cost of its diverse administration services, taxes and other expenses and would invest part of its capital to earn profits. A great innovation of Lundberg’s approach in studying the interaction of insurer and insured was to switch from an individual model (which focuses on an individual’s portfolio) to a collective model, focusing on aggregate quantities. Thus, ignoring for simplicity other kinds of inflow (investment revenues, new capital, interests and dividends received, rental incomes, etc) and outflows (commissions and other administrative and operational costs, taxes, interests and dividends payed, etc), [8] the basic ingredients of his model are: • the premium charged by the company: call Πt be the aggregate (or total) premium collected up to time t > 0. • the claims arriving at some random claim times 0 = T0 < T1 < T2 < ..., where the corresponding amounts to be paid at these times are described by some non-negative random variables X1 , X2 , ..., called the claim sizes. Let Nt = sup{n ≥ 1 : Tn ≤ t} (sup(∅) ≡ 0) be the number of claims arriving in the interval [0, t]. Then the aggregate claim amount process {St }t≥0 is given by the random sum:  Nt X     Xk , N t > 0  k=1 St = (1)     0, Nt = 0.

The capital surplus of the insurance company is the stochastic process {Ut }t≥0 , called the risk process, defined on some probability space (Ω, F , P), with initial capital U0 = u ≥ 0 and given, for t > 0 by Ut = u + Π t − S t . The classical Cram´er-Lundberg model has the following additional hypothesis: (0) the premium income is linear deterministic: Πt = ct, where c > 0 is the premium income rate;

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(1) the claim size process {Xn }n≥1 is a sequence of positive i.i.d. random variables with common distribution function F and finite mean µ = E[X1 ]; (2) the inter-arrival times Tn − Tn−1 , n ≥ 1 are i.i.d. exponentially distributed random variables with parameter λ > 0; (3) the processes {Tn }n≥1 and {Xn }n≥1 are independent. It follows from (2) that the claim number process {Nt }t≥0 is a homogeneous Poisson process with intensity λ, that is (N0 = 0 a.s.): P(Nt = k) = e

−λt (λt)

k!

k

, k = 0, 1, 2, . . . ,

so that {St }t≥0 is a compound Poisson process, which is a L´evy process. [18]

Ut

u PSfrag replacements

0

t

τ

Figure 1: A realisation of the risk process It is clearly of great practical importance to evaluate or estimate the aggregate claim size distribution Gt (x) = P(St ≤ x) =

∞ X

e−λt

k=0

(λt)k n∗ F (x), x, t ≥ 0, k!

where the F n∗ (x) = P(X1 + . . . + Xn ≤ x) is the n-fold convolution of the claim size distribution F (F 0∗ is the Heaviside function). In what follows, however, we’ll focus on another (and in a sense preliminary) issue: the long-term stability of the risk process. 7

As the service offered by an insurance company is of the kind in which one pays first and takes later (hopefully never!), a basic requirement (and there are explicit legal regulations in this respect) is that the company will still exist and be solvent if and when one needs it. More precisely, one needs to evaluate the ruin probability of the risk process: ψ(u) = P(Ut < 0, for some t ≥ 0), which can alternatively be expressed through the associated ruin time: τ = τu = inf{t ≥ 0 : Ut < 0}, (inf{∅} ≡ +∞), i.e., the first time that the company is “in the red”. This is a stopping-time with respect to the filtration F = {Ft }t≥0 , where Ft = σ(Us : s ≤ t) is the σ-algebra describing the history of the risk process up to time t and we have, ψ(u) = P(τ < ∞). A fundamental problem is then to determine the adequate initial capital u > 0 and insurance premium rate c > 0. Using ψ(u) as a measure of solvency a first natural requirement is that that 0 ≤ ψ(u) < 1, otherwise ψ(u) = 1 and ruin is (almost) sure. 9 It is therefore reasonable to impose that E[Ut ] > 0 for all t > 0. A simple calculation shows that E[Ut ] ≥ E[Ut − U0 ] = ct − µE[Nt ] = t(c − λµ), so that one assumes the so-called net profit condition (“the premium has to be greater than the average total claim”): c > λµ, which already gives a basic estimate of the premium rate. Define the parameter c ρ= − 1 > 0, λµ called the safety loading. As the premium income up to time t is ct = (1 + ρ)λµt, ρ can be interpreted as a risk premium rate, that extra the insurance company charges to avoid certain ruin. Notice that by the Law of Large Numbers, under the net profit condition Ut → +∞ almost surely as t ↑ ∞, though the process can attain negative values in some instants. It can be shown [11] that under the net profit condition: ∞

ρ X (1 + ρ)−n FIn∗ (x) 1 − ψ(u) = 1 + ρ n=0 where

Z 1 x ¯ F (y)dy, FI (x) = µ 0 called the “integrated tail distribution”. Though this a very important formula, one cannot hope to obtain closed-form expressions from it apart for some special cases (for example, for the exponential distribution). So it is important to have at least an estimate of ψ(u) guaranteeing that it is “sufficiently” small for all u ≥ 0, say, below a prescribed threshold of 5%. It turns out that if one assumes a certain condition bearing on the tail of the claim size distribution, then one can get such an estimate in terms of the initial capital u. This is discussed next. 9

Also one can check that limu→∞ ψ(u) = 0.

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4

The Cram´ er-Lundberg estimate

We can at last state one of the classical results of actuarial mathematics: Theorem (Cram´ er-Lundberg). Consider the above model under the net-profit condition and suppose there exists R > 0 (called the Lundberg exponent or adjustment coefficient) satisfying the following Lundberg or small-claim condition: Z ∞ c eRx (1 − F (x))dx = . (∗) λ 0 Then for all t > 0 ψ(u) ≤ e−Ru . The original proof was based on complicated computations based on inversion of Laplace-Stieltjes transforms. We will follow the very elegant “exponential martingale technique” that goes back to Gerber [15]. But before embarking on the proof, let us discuss the meaning of condition (*). It presupposes that the left-hand side exists in a neighbourhood of the origin. Consider the function Z ∞   (esx − 1)dF (x) = E esX1 − 1, h(s) = 0

which is essentially the Laplace-Stieltjes transform of F , well defined for all s ≤ 0. If there is an s0 > 0 such that h(s0 ) < ∞, then h(s) < ∞ and infinitely differentiable for s ∈ (−∞, s0 ). Moreover, for s > 0, an application of Fubini’s theorem gives Z ∞ Z ∞ sx h(s) = s e (1 − F (x))dx = s esx F¯ (x)dx, 0

0

where F¯ (x) = 1 − F (x) is the tail of F . We then see that (*) can be rephrased as: there exists a positive root R of the function g(s) = λh(s) − cs. By Markov’s inequality, for all x > 0   F¯ (x) = 1 − F (x) = P(X1 > x) ≤ e−Rx E eRX1 , so that (*) implies that large claims have exponentially small probabilities, justifying the name “small-claims condition”. Now, g(0) = 0, g 0 (s) = λh0 (s) − c = λE[X1 esX1 ] − c, so that g 0 (0) = λE[X1 ] − c = λµ−c < 0 by the net-profit condition. Furthermore, g 00 (s) = λh00 (s) = λE[X12 esX1 ] ≥ 0, so that g is convex. Hence, if g has a non-zero root, it is positive and unique. The existence of a positive root is achieved if h(s) grows fast enough in relation to the linear function cs. A sufficient condition is the following: there exists s∞ > 0 (possibly = ∞) such that h(s) < ∞ for s < s∞ and h(s) ↑ +∞ ass ↑ +∞. The case s∞ finite is immediate. For the case s∞ = ∞, picking xˆ with F (ˆ x) < 1 one gets for s big enough that Z +∞ h(s) ≥ esx dF (x) − 1 ≥ esˆx F¯ (ˆ x) − 1, x ˆ

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g(s)

PSfrag replacements 0

s∞

R

s

Figure 2:

which grows faster than any linear function (see fig. 2). Proof of the theorem: The proof rests on the fact that the process {Mt }t≥0 , where Mt = e−rUt −tg(r) ,   is a continuous-time F-martingale for every value of r for which E erX1 < ∞. First notice that this process is non-negative and obviously F-adapted. Also, ∞ X  −r(Ut −U0 )   r P Nt X   P Nt  −rct −rct i i=0 E e =e E e =e E er i=0 Xi |Nt = k P(Nt = k) = k=0

e−rct

∞ X

(1 + h(r))n

k=0

e−λt (λt)n = e−rct eλth(r) = etg(r) , n!

and similar calculation shows that for 0 ≤ s < t one has E[e−r(Ut −Us ) ] = e(t−s)g(r) . It follows that for all t ≥ 0 E[Mt ] = e−ru < ∞. Furthermore, as the risk process has independent increments, then for 0 ≤ s ≤ t,       E Mt |Fs = E e−rUt −tg(r) |Fs = E e−r(Ut −Us )−(t−s)g(r) e−rUs −sg(r) |Fs =   e−rUs −sg(r) E e−r(Ut −Us )−(t−s)g(r) |Fs = e−rUs −sg(r) = Ms P − a.s.,

which is the basic martingale property.

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Next, let t0 < ∞ and consider the bounded stopping time t0 ∧ τ = min(t0 , τ ). Applying the martingale stopping theorem (see Kallenberg [17], Thm. 6.12, p. 104) we obtain: E[Mt0 ∧τ ] = E[M0 ] = e−ru . But remembering that Uτ < 0, we have that   E e−rUt0 ∧τ −(t0 ∧τ )g(r) =     E e−rUt0 ∧τ −(t0 ∧τ )g(r) |τ ≤ t0 P(τ ≤ t0 ) + E e−rUt0 ∧τ −(t0 ∧τ )g(r) |τ > t0 P(τ > t0 ) ≥     E e−rUτ −τ g(r) |τ ≤ t0 P(τ ≤ t0 ) ≥ E e−τ g(r) |τ ≤ t0 P(τ ≤ t0 ) ≥

inf (e−tg(r) )P(τ ≤ t0 ).

0≤t≤t0

Therefore, P(τ ≤ t0 ) ≤

e−ru = e−ru sup (etg(r) ), inf 0≤t≤t0 (e−tg(r) ) 0≤t≤t0

an estimate that is interesting in itself. Taking now t0 ↑ ∞, we get P(τ < ∞) ≤ e−ru sup(etg(r) ), 0≤t

and one would like to substitute for the greatest possible r, subject to the condition sup(etg(r) ) < ∞. In other words, we look for sup(r : g(r) ≤ 0), which is exactly the 0≤t

0≤t

Lundberg exponent R. Hence, P(τ < ∞) ≤ e−Ru , finishing the proof.

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Conclusions

The Cram´er-Lundberg theorem is a nice example of the use of martingale methods in applied probability. But the discussion presented here is just the tip of the iceberg: there is a huge literature and ongoing research activity on various generalisations of, improvements and related topics on the ruin problem. To begin with, the classical model is not very realistic. A basic criticism is that it ignores possible effects of dependence, seasonality, clustering and other inhomogeneities of claim distributions. An even more serious problem is that many distributions used in practice to fit empirical claim-size data, like Pareto’s, violate Lundberg’s condition. That is, they display heavy tails and the theory has to be reconfigured to deal with such “dangerous” non-Cram´er regime. Within the classical model, there is a host of interesting questions we did not touch here. For example, we have seen that even if ruin happens, under the net profit condition the firm will eventually come out of insolvency. One would then like to know, among other things, for how long the firm stays insolvent, how severe is its deficit during that period, etc. Also, as in the long run the firm’s capital surplus increases without limits, one could impose some upper barrier to model the 11

allocation of capital to pay for costs, dividends, etc. More generally, there is the ever-growing interplay of insurance with finance, witnessing a growing exchange of methods and ideas from the two disciplines. Another “hot” topic is reinsurance, a risk-reducing mechanism in which an insurance company buys insurance (from another company) to cover its own portfolio. A motivation for this move is the occurrence of catastrophic events, like natural disasters (hurricane, earthquakes, tsunamis, etc) or man-made ones (accidents, bankruptcies, etc) whose costs far exceeds the covering capacity of an isolated firm. And in some cases not even reinsurance is enough to cope with such disasters, which explains the importance of the recent research effort on rare but extremal events. Finally, there are some bigger questions: are modern societies, with their increasing complexity and interdependence, more vulnerable to systemic risks? Are insurance technologies making our societies safer or not? What is the role and responsibilities of risk managers in the new global scenario? These are formidable challenges to modern society whose answers will keep firms, governments and risk managers extremely busy for the coming decades. ACKNOWLEGMENTS. Work partially supported by FINEP (Pronex project). The author is thankful to the anonymous referee for many valuable criticisms and suggestions.

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