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# Modern Actuarial Risk Theory-Using R

(Parte **1** de 6)

Modern Actuarial Risk Theory Modern Actuarial Risk Theory

Jan Dhaene • Michel Denuit

Theory Modern Actuarial Risk

Using R Second Edition

Rob Kaas • Marc Goovaerts

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UvA / KE Roetersstraat 1 1018 WB Amsterdam The Netherlands r.kaas@uva.nl

Professor Marc Goovaerts UvA / KE Roetersstraat 1 1018 WB Amsterdam The Netherlands

Professor Jan Dhaene AFI (Accountancy, Finance, Insurance) Research Group

Naamsestraat 69 3000 Leuven Belgium jan.dhaene@econ.kuleuven.be

Faculteit Economie en Bedrijfswetenschappen Professor Michel Denuit

Belgium michel.denuit@uclouvain.be

ISBN: 978-3-540-70992-3Violations are liable for prosecution under the German Copyright Law. e-ISBN: 978-3-540-70998-5

Institut de Statistique K.U. Leuven

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Professor Rob Kaas marc.goovaerts@econ.kuleuven.be

K.U. Leuven University of Amsterdam

Voie du Roman Pays 20 Université Catholique de Louvain

3000 Leuven

1348 Louvain-la-Neuve

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University of Amsterdam

Study the past if you would deﬁne the future — Confucius, 551 BC - 479 BC

Risk Theory has been identiﬁed and recognized as an important part of actuarial education; this is for example documented by the Syllabus of the Society of Actuaries and by the recommendations of the Groupe Consultatif. Hence it is desirable to have a diversity of textbooks in this area.

This text in risk theory is original in several respects. In the language of ﬁgure skating or gymnastics, the text has two parts, the compulsory part and the freestyle part. The compulsory part includes Chapters 1–4, which are compatible with ofﬁcial material of the Society of Actuaries. This feature makes the text also useful to students who prepare themselves for the actuarial exams. Other chapters are more of a free-style nature, for example Chapter 7 (Ordering of Risks, a speciality of the authors). And I would like to mention Chapters 9 and 1 in particular. To my knowledge, this is the ﬁrst text in risk theory with an introduction to Generalized Linear Models.

Special pedagogical efforts have been made throughout the book. The clear language and the numerous exercises are an example for this. Thus the book can be highly recommended as a textbook.

I congratulate the authors to their text, and I would like to thank them also in the name of students and teachers that they undertook the effort to translate their text into English. I am sure that the text will be successfully used in many classrooms.

Lausanne, 2001 Hans Gerber

Foreword to the First Edition

what is called the Worldwide Web | Now even my cat has its |

When I took ofﬁce, only high energy physicists had ever heard of own page — Bill Clinton, 1996

This book gives a comprehensive survey of non-life insurance mathematics. Originally written for use with the actuarial science programs at the Universities of Amsterdam and Leuven, it is now in use at many other universities, as well as for the non-academic actuarial education program organized by the Dutch Actuarial Society. It provides a link to the further theoretical study of actuarial science. The methods presented can not only be used in non-life insurance, but are also effective in other branches of actuarial science, as well as, of course, in actuarial practice.

Apart from the standard theory, this text contains methods directly relevant for actuarial practice, for example the rating of automobile insurance policies, premium principles and risk measures, and IBNR models. Also, the important actuarial statistical tool of the Generalized Linear Models is studied. These models provide extra possibilities beyond ordinary linear models and regression that are the statistical tools of choice for econometricians. Furthermore, a short introduction is given to credibility theory. Another topic which always has enjoyed the attention of risk theoreticians is the study of ordering of risks. The book reﬂects the state of the art in actuarial risk theory; many results presented were published in the actuarial literature only recently.

In this second edition of the book, we have aimed to make the theory even more directly applicable by using the software R. It provides an implementation of the language S, not unlike S-Plus. It is not just a set of statistical routines but a fullﬂedged object oriented programming language. Other software may provide similar capabilities, but the great advantage of R is that it is open source, hence available to everyone free of charge. This is why we feel justiﬁed in imposing it on the users of this book as a de facto standard. On the internet, a lot of documentation about R can be found. In an Appendix, we give some examples of use of R. After a general introduction, explaining how it works, we study a problem from risk management, trying to forecast the future behavior of stock prices with a simple model, based on stock prices of three recent years. Next, we show how to use R to generate pseudorandom datasets that resemble what might be encountered in actuarial practice.

vii

Preface to the Second Edition viii Preface

Models and paradigms studied The time aspect is essential in many models of life insurance. Between paying premiums and collecting the resulting pension, decades may elapse. This time element is less prominent in non-life insurance. Here, however, the statistical models are generally more involved. The topics in the ﬁrst ﬁve chapters of this textbook are basic for non-life actuarial science. The remaining chapters contain short introductions to other topics traditionally regarded as non-life actuarial science.

1. The expected utility model The very existence of insurers can be explained by the expected utility model. In this model, an insured is a risk averse and rational decision maker, who by virtue of Jensen’s inequality is ready to pay more than the expected value of his claims just to be in a secure ﬁnancial position. The mechanism through which decisions are taken under uncertainty is not by direct comparison of the expected payoffs of decisions, but rather of the expected utilities associated with these payoffs.

2. The individual risk model In the individual risk model, as well as in the collective risk model below, the total claims on a portfolio of insurance contracts is the random variable of interest. We want to compute, for example, the probability that a certain capital will be sufﬁcient to pay these claims, or the value-at-risk at level 9.5% associated with the portfolio, being the 9.5% quantile of its cumulative distribution function (cdf). The total claims is modeled as the sum of all claims on the policies, which are assumed independent. Such claims cannot always be modeled as purely discrete random variables, nor as purely continuous ones, and we use a notation, involving Stieltjes integrals and differentials, encompassing both these as special cases.

The individual model, though the most realistic possible, is not always very convenient, because the available dataset is not in any way condensed. The obvious technique to use in this model is convolution, but it is generally quite awkward. Using transforms like the moment generating function sometimes helps. The Fast Fourier Transform (FFT) technique gives a fast way to compute a distribution from its characteristic function. It can easily be implemented in R.

We also present approximations based on ﬁtting moments of the distribution. The

Central Limit Theorem, ﬁtting two moments, is not sufﬁciently accurate in the important right-hand tail of the distribution. So we also look at some methods using three moments: the translated gamma and the normal power approximation.

3. Collective risk models A model that is often used to approximate the individual model is the collective risk model. In this model, an insurance portfolio is regarded as a process that produces claims over time. The sizes of these claims are taken to be independent, identically distributed random variables, independent also of the number of claims generated. This makes the total claims the sum of a random number of iid individual claim amounts. Usually one assumes additionally that the number of claims is a Poisson variate with the right mean, or allows for some overdispersion by taking a negative

Preface ix binomial claim number. For the cdf of the individual claims, one takes an average ally tractable model. Several techniques, including Panjer’s recursion formula, to compute the cdf of the total claims modeled this way are presented.

For some purposes it is convenient to replace the observed claim severity distribution by a parametric loss distribution. Families that may be considered are for example the gamma and the lognormal distributions. We present a number of such distributions, and also demonstrate how to estimate the parameters from data. Further, we show how to generate pseudo-random samples from these distributions,

4. The ruin model The ruin model describes the stability of an insurer. Starting from capital u at time t = 0, his capital is assumed to increase linearly in time by ﬁxed annual premiums, but it decreases with a jump whenever a claim occurs. Ruin occurs when the capital is negative at some point in time. The probability that this ever happens, under the assumption that the annual premium as well as the claim generating process remain unchanged, is a good indication of whether the insurer’s assets match his liabilities sufﬁciently. If not, one may take out more reinsurance, raise the premiums or increase the initial capital.

Analytical methods to compute ruin probabilities exist only for claims distributions that are mixtures and combinations of exponential distributions. Algorithms exist for discrete distributions with not too many mass points. Also, tight upper and lower bounds can be derived. Instead of looking at the ruin probability ψ(u) with initial capital u, often one just considers an upper bound e−Ru for it (Lundberg), where the number R is the so-called adjustment coefﬁcient and depends on the claim size distribution and the safety loading contained in the premium.

Computing a ruin probability assumes the portfolio to be unchanged eternally.

Moreover, it considers just the insurance risk, not the ﬁnancial risk. Therefore not much weight should be attached to its precise value beyond, say, the ﬁrst relevant decimal. Though some claim that survival probabilities are ‘the goal of risk theory’, many actuarial practitioners are of the opinion that ruin theory, however topical still in academic circles, is of no signiﬁcance to them. Nonetheless, we recommend to study at least the ﬁrst three sections of Chapter 4, which contain the description of the Poisson process as well as some key results. A simple proof is provided for Lundberg’s exponential upper bound, as well as a derivation of the ruin probability in case of exponential claim sizes.

5. Premium principles and risk measures Assuming that the cdf of a risk is known, or at least some characteristics of it like mean and variance, a premium principle assigns to the risk a real number used as a ﬁnancial compensation for the one who takes over this risk. Note that we study only risk premiums, disregarding surcharges for costs incurred by the insurance company. By the law of large numbers, to avoid eventual ruin the total premium should be at least equal to the expected total claims, but additionally, there has to be a loading in of the cdfs of the individual policies. This leads to a close ﬁtting and computationbeyond the standard facilities offered by R.

x Preface the premium to compensate the insurer for making available his risk carrying capacity. From this loading, the insurer has to build a reservoir to draw upon in adverse times, so as to avoid getting in ruin. We present a number of premium principles, together with the most important properties that characterize premium principles. The choice of a premium principle depends heavily on the importance attached to such properties. There is no premium principle that is uniformly best.

Risk measures also attach a real number to some risky situation. Examples are premiums, inﬁnite ruin probabilities, one-year probabilities of insolvency, the required capital to be able to pay all claims with a prescribed probability, the expected value of the shortfall of claims over available capital, and more.

6. Bonus-malus systems With some types of insurance, notably car insurance, charging a premium based exclusively on factors known a priori is insufﬁcient. To incorporate the effect of risk factors of which the use as rating factors is inappropriate, such as race or quite often sex of the policy holder, and also of non-observable factors, such as state of health, reﬂexes and accident proneness, many countries apply an experience rating system. Such systems on the one hand use premiums based on a priori factors such as type of coverage and list-price or weight of a car, on the other hand they adjust these premiums by using a bonus-malus system, where one gets more discount after a claim-free year, but pays more after ﬁling one or more claims. In this way, premiums are charged that reﬂect the exact driving capabilities of the driver better. The situation can be modeled as a Markov chain.

The quality of a bonus-malus system is determined by the degree in which the premium paid is in proportion to the risk. The Loimaranta efﬁciency equals the elasticity of the mean premium against the expected number of claims. Finding it involves computing eigenvectors of the Markov matrix of transition probabilities. R provides tools to do this.

7. Ordering of risks It is the very essence of the actuary’s profession to be able to express preferences between random future gains or losses. Therefore, stochastic ordering is a vital part of his education and of his toolbox. Sometimes it happens that for two losses X and Y, it is known that every sensible decision maker prefers losing X, because Y is in a sense ‘larger’ than X. It may also happen that only the smaller group of all risk averse decision makers agree about which risk to prefer. In this case, risk Y may be larger than X, or merely more ‘spread’, which also makes a risk less attractive. When we interpret ‘more spread’ as having thicker tails of the cumulative distribution function, we get a method of ordering risks that has many appealing properties. For example, the preferred loss also outperforms the other one as regards zero utility premiums, ruin probabilities, and stop-loss premiums for compound distributions with these risks as individual terms. It can be shown that the collective model of Chapter 3 is more spread than the individual model it approximates, hence using the collective model, in most cases, leads to more conservative decisions regarding premiums to be asked, reserves to be held, and values-at-risk. Also, we can prove

(Parte **1** de 6)