MODERN RESERVING TECHNIQUES FOR THE INSURANCE BUSINESS

Katholieke Universiteit Leuven FACULTEIT WETENSCHAPPEN DEPARTEMENT WISKUNDE MODERN RESERVING TECHNIQUES FOR THE INSURANCE BUSINESS door Tom HOEDEMAKERS Promotor: Prof. Dr. J. Beirlant Prof. Dr. J. Dhaene Proefschrift ingediend tot het behalen van de graad van Doctor in de Wetenschappen Leuven 2005

Acknowledgments Four years ago I became part of the stimulating and renowned academic environment at K. U. Leuven, the Department of Applied Economics, and the AFI Leuven Research Center in particular. As a researcher, I had the opportunity to interact, work with and learn from many interesting people. I consider myself extremely fortunate to have had the following people in support for the realization of this thesis. I feel very privileged to have worked with my two supervisors, Jan Beirlant and Jan Dhaene. To each of them I owe a great debt of gratitude for their continuous encouragement, patience, inspiration and friendship. I especially want to thank them for the freedom they allowed me to seek satisfaction in research, for supporting me in my choices and for believing in me. They carefully answered the many (sometimes not well-defined) questions that I had and they always found a way to make themselves available for yet another meeting. Each chapter of this thesis has benefitted from their critical comments, which often inspired me to do further research and to improve the vital points of the argument. It has been a privilege to study under Jan and Jan, and to them goes my highest personal and professional respect. I am also grateful to Marc Goovaerts for giving me the opportunity to start my thesis in one of the world-leading actuarial research centers. Marc Goovaerts and Jan Dhaene have taught me a great deal about the field of actuarial science by sharing with me the joy of discovery and investigation that is the heart of research. They brought me in contact with a lot of interesting people in the actuarial world and gave me the possibility to present my work at different congresses all over the world. I would also like to thank the other members of the doctoral committee Michel Denuit, Rob Kaas, Wim Schoutens and Jef Teugels for their valuable contributions as committee members. Their detailed comments as i

ii Acknowledgments well as their broader reactions definitely helped me to improve the quality of my research and its write-up. Many thanks go also to my (ex-)colleagues Ales, Björn, David, Grzegorz, Katrien, Marc, Piotr, Steven and Yuri for their enthusiasm and stimulating cooperation. A lot of sympathy goes to Emiliano Valdez for the serious discussions, and even more important, for the fun we had during his stay at the K. U. Leuven in the beginning of this year. After the professionals, a word of thanks is addressed to all my friends and fellow students for their friendship and support. Finally, not least, I would like to thank my parents and my sister Leen for their love, guidance and support. They constantly reminded me of their confidence and encouraged me to pursue my scientific vocation, especially in moments of doubt. You have always believed in me and that was a great moral support. Tom Leuven, 2005

Table of Contents Acknowledgments Preface Publications List of abbreviations and symbols i vii xix xxi 1 Risk and comonotonicity in the actuarial world 1 1.1 Fundamental concepts in actuarial risk theory........ 1 1.1.1 Dependent risks.................... 2 1.1.2 Risk measures..................... 4 1.1.3 Actuarial ordering of risks............... 10 1.2 Comonotonicity......................... 15 2 Convex bounds 21 2.1 Introduction........................... 21 2.2 Convex bounds for sums of dependent random variables.. 23 2.2.1 The comonotonic upper bound............ 25 2.2.2 The improved comonotonic upper bound....... 26 2.2.3 The lower bound.................... 28 2.2.4 Moments based approximations............ 29 2.3 Upper bounds for stop-loss premiums............ 30 2.3.1 Upper bounds based on lower bound plus error term 31 2.3.2 Bounds by conditioning through decomposition of the stop-loss premium................. 33 2.3.3 Partially exact/comonotonic upper bound...... 35 2.3.4 The case of a sum of lognormal random variables.. 35 iii

iv Table of Contents 2.4 Application: discounted loss reserves............. 47 2.4.1 Framework and notation................ 48 2.4.2 Calculation of convex lower and upper bounds... 52 2.5 Convex bounds for scalar products of random vectors... 56 2.5.1 Theoretical results................... 58 2.5.2 Stop-loss premiums................... 61 2.5.3 The case of log-normal discount factors....... 62 2.6 Application: the present value of stochastic cash flows... 68 2.6.1 Stochastic returns................... 68 2.6.2 Lognormally distributed payments.......... 72 2.6.3 Elliptically distributed payments........... 77 2.6.4 Independent and identically distributed payments. 84 2.7 Proofs.............................. 89 3 Reserving in life insurance business 93 3.1 Introduction........................... 93 3.2 Modelling stochastic decrements............... 96 3.3 The distribution of life annuities............... 100 3.3.1 A single life annuity.................. 100 3.3.2 A homogeneous portfolio of life annuities...... 113 3.3.3 An average portfolio of life annuities........ 119 3.3.4 A numerical illustration................ 120 3.4 Conclusion........................... 125 4 Reserving in non-life insurance business 127 4.1 Introduction........................... 127 4.2 The claims reserving problem................. 131 4.3 Model set-up: regression models............... 133 4.3.1 Lognormal linear models................ 135 4.3.2 Loglinear location-scale models............ 137 4.3.3 Generalized linear models............... 141 4.3.4 Linear predictors and the discounted IBNR reserve. 146 4.4 Convex bounds for the discounted IBNR reserve...... 148 4.4.1 Asymptotic results in generalized linear models... 148 4.4.2 Lower and upper bounds............... 151 4.5 The bootstrap methodology in claims reserving....... 157 4.5.1 Introduction...................... 157 4.5.2 Central idea....................... 158

Table of Contents v 4.5.3 Bootstrap confidence intervals............. 158 4.5.4 Bootstrap in claims reserving............. 159 4.6 Three applications....................... 163 4.6.1 Lognormal linear models................ 164 4.6.2 Loglinear location-scale models............ 171 4.6.3 Generalized linear models............... 177 4.7 Conclusion........................... 183 5 Other approximation techniques for sums of dependent random variables 185 5.1 Introduction........................... 185 5.2 Moment matching approximations.............. 187 5.2.1 Two well-known moment matching approximations. 187 5.2.2 Application: discounted loss reserves......... 190 5.3 Asymptotic approximations.................. 192 5.3.1 Preliminaries for heavy-tailed distributions..... 192 5.3.2 Asymptotic results................... 194 5.3.3 Application: discounted loss reserves......... 198 5.4 The Bayesian approach.................... 201 5.4.1 Introduction...................... 201 5.4.2 Prior choice....................... 203 5.4.3 Iterative simulation methods............. 205 5.4.4 Bayesian model set-up................. 207 5.5 Applications in claims reserving................ 209 5.5.1 The comonotonicity approach versus the Bayesian approximations..................... 209 5.5.2 The comonotonicity approach versus the asymptotic and moment matching approximations........ 216 5.6 Proofs.............................. 220 Samenvatting in het Nederlands (Summary in Dutch) 227 Bibliography 237

Preface Uncertainty is very much a part of the world in which we live. Indeed, one often hears the well-known cliche that the only certainties in life are death and taxes. However, even these supposed certainties are far from being completely certain, as any actuary or accountant can attest. For although one s eventual death and the requirement that one pay taxes may be facts of life, the timing of one s death and the amount of taxes to pay are far from certain and are generally beyond one s control. Uncertainty can make life interesting. Indeed, the world would likely be a very dull place if everything were perfectly predictable. However, uncertainty can also cause grief and suffering. Actuarial science is the subject whose primary focus is analyzing the financial consequences of future uncertain events. In particular, it is concerned with analyzing the adverse financial consequences of large, unpredictable losses and with designing mechanisms to cushion the harmful financial effects of such losses. Insurance is based on the premise that individuals faced with large and unpredictable losses can reduce the variability of such losses by forming a group and sharing the losses incurred by the group as a whole. This important principle of loss sharing, known as the insurance principle, forms the foundation of actuarial science. It can be justified mathematically using the Central Limit Theorem from probability theory. For the insurance principle to be valid, essentially four conditions should hold (or very nearly hold). The losses should be unpredictable. The risks should be independent in the sense that a loss incurred by one member of the group makes additional losses by other members of the group no more or less likely. The risks should be homogeneous in the sense that a loss incurred by one member of the group is not expected to be any different in size or likelihood from losses incurred by other members of the group. Finally, vii

viii Preface the group should be sufficiently large so that the portion of the total loss that each individual is required to pay becomes relatively certain. In practice, risks are not truly independent or homogeneous. Moreover, there will always be situations where the condition of unpredictability is violated. Actuarial science seeks to address the following three problems associated with any such insurance arrangement: 1. Given the nature of the risk being assumed, what price (i.e. premium) should the insurance company charge? 2. Given the nature of the overall risks being assumed, how much of the aggregate premium income should the insurance company set aside in a reserve to meet contractual obligations (i.e. pay insurance claims) as they arise? 3. Given the importance to society and the general economy of having sound financial institutions able to meet all their obligations, how much capital should an insurance company have above and beyond its reserves to absorb losses that are larger than expected? Given the actual level of an insurance company s capital, what is the probability of the company remaining solvent? These are generally referred to as the problems of pricing, reserving, and solvency. This thesis focuses on the problem of reserving and total balance sheet requirements. A reserving analysis involves the determination of the random present value of an unknown amount of future loss payments. For a property/casualty insurance company this uncertain amount is usually the most important number on its financial statement. The care and expertise with which that number is developed are crucial to the company and to its policyholders. It is important not to let the inherent uncertainties serve as an excuse for providing anything less than a rigorous scientific analysis. Among those who rely on reserve estimates, interests and priorities may vary. To company management the reserve estimate should provide reliable information in order to maximize the company s viability and profitability. To the insurance regulator, concerned with company solvency, reserves should be set conservatively to reduce the probability of failure of the insurance company. To the tax agent charged with ensuring timely reporting

Preface ix of earned income, the reserves should reflect actual payments as nearly as it is possible to ascertain them. The policyholder is most concerned that reserves are adequate to pay insured claims, but does not want to be overcharged. Besides all the techniques, the primary goal of the reserving process can be stated quite simply. As of a given date, an insurer is liable for all claims incurred from that date on. As well as for claims that arise from already occurred events as for claims that arise from risks covered by the insurer but for which the uncertain event has not yet occurred. Costs associated with these claims fall into two categories: those which have been paid and those which have not. The primary goal of the reserving process is to estimate those which have not yet been paid (i.e. unpaid losses). As of a given reserve date, the distribution of possible aggregate unpaid loss amounts may be represented as a probability density function. Much has been written about the statistical distributions that have proven to be most useful in the study of risk and insurance. In practice full information about the underlying distributions is hardly ever available. For this reason one often has to rely on partial information, for example estimations of the first couple of moments. Not only the basic summary measures, but also more sophisticated risk measures (such as measures of skewness or extreme percentiles of the distribution) which require much deeper knowledge about the underlying distributions are of interest. The computation of the first few moments may be seen as just a first attempt to explore the properties of a random distribution. Moreover in general the variance does not appear to be the most suitable risk measure to determine the solvency requirements for an insurance portfolio. As a two-sided risk measure it takes into account both positive and negative discrepancies which leads to underestimation of the reserve in the case of a skewed distribution. Moreover it does not emphasize the tail properties of the distribution. In this case it seems much more appropriate to use the Value-at-Risk (the p-th quantile) or also the Tail Value-at-Risk (which is essentially the same as an average of all quantiles above a predefined level p). Also risk measures based on stoploss premiums (for example the Expected Shortfall) can be used in this context. These trends are also reflected in the recent regulatory changes in banking and insurance (Basel 2 and Solvency 2) which stress the role of the risk-based approach in asset-liability management. This creates a need for new methodological tools which allow to obtain more sophisticated information about the underlying risks, like the upper quantiles, stop-loss

x Preface premiums and others. There is little in the actuarial literature which considers the adequate computation of the distribution of reserve outcomes. Several methods exist which allow to approximate efficiently the distribution functions for sums of independent risks (e.g. Panjer s recursion, convolution,...). Moreover if the number of risks in an insurance portfolio is large enough, the Central Limit Theorem allows to obtain a normal approximation for aggregate claims. Therefore even if the independence assumption is not justified (e.g. when it is rejected by formal statistical tests), it is often used in practice because of its mathematical convenience. In a lot of practical applications the independence assumption may be often violated, which can lead to significant underestimation of the riskiness of the portfolio. This is the case for example when the actuarial technical risk is combined with the financial investment risk. Unlike in finance, in insurance the concept of stochastic interest rates emerged quite recently. Traditionally actuaries rely on deterministic interest rates. Such a simplification allows to treat efficiently summary measures of financial contracts such as the mean, the standard deviation or the upper quantiles. However due to a high uncertainty about future investment results, actuaries are forced to adopt very conservative assumptions in order to calculate insurance premiums or mathematical reserves. As a result the diversification effects between returns in different investment periods cannot be taken into account (i.e. the fact that poor investment results in some periods are usually compensated by very good results in others). This additional cost is transferred either to the insureds who have to pay higher insurance premiums or to the shareholders who have to provide more economic capital. For these reasons the need for introducing models with stochastic interest rates has been well-understood also in the actuarial world. The move toward stochastic modelling of interest rates is additionally enhanced by the latest regulatory changes in banking and insurance (Basel 2, Solvency 2) which promote the risk-based approach to determine economic capital, i.e. they state that traditional actuarial conservatism should be replaced by the fair value reserving, with the regulatory capital determined solely on the basis of unexpected losses which can be estimated e.g. by taking the Value-at-Risk measure at appropriate probability level p. Projecting cash flows with stochastic rates of return is also crucial in pricing applications

Preface xi in insurance, like the embedded value (the present value of cash flows generated only by policies-in-force) and the appraisal value (the present value of cash flows generated both by policies-in-force and by new business, i.e. the policies which will be written in the future). A mathematical description of the discussed problem can be summarized as follows. Let X i denote a random amount to be paid at time t i, i = 1,..., n and let V i denote the discount factor over the period [0, t i ]. We will consider the present value of future payments being a scalar product of the form S = X i V i. (1) The random vector X = (X 1, X 2,..., X n ) may reflect e.g. the insurance or credit risk while the vector V = (V 1, V 2,..., V n ) represents the financial/investment risk. In general we assume that these vectors are mutually independent. In practical applications the independence assumption is often violated, e.g. due to an inflation factor which strongly influences both payments and investment results. One can however tackle this problem by considering sums of the form S = X i Ṽ i, where X i = X i /Z i and Ṽi = V i Z i are the adjusted values expressed in real terms (Z i denotes here an inflation factor over period [0, t i ]). For this reason the assumption of independence between the insurance risk and the financial risk is in most cases realistic and can be efficiently applied to obtain various quantities describing risk within financial institutions, e.g. discounted insurance claims or the embedded/appraisal value of a company. Typically these distribution functions are rather involved, which is mainly due to two important reasons. First of all, the distribution of the sum of random variables with marginal distributions in the same distribution class in general does not belong to the particular distribution class. Secondly, the stochastic dependence between the elements in the sum precludes convolution and complicates matters considerably.

xii Preface Consequently, in order to compute functionals of sums of dependent random variables, approximation methods are generally indispensable. Provided that the whole dependency structure is known, one can use Monte Carlo simulation to obtain empirical distribution functions. However, this is typically a time consuming approach, in particular if we want to approximate tail probabilities, which would require an excessive number of simulations. Therefore, alternative methods need to be explored. In this thesis we discuss the most frequent used approximation techniques for reserving applications. The central idea in this work is the concept of comonotonicity. We suggest to solve the above described problem by calculating upper and lower bounds for the sum of dependent random variables making efficient use of the available information. These bounds are based on a general technique for deriving lower and upper bounds for stop-loss premiums of sums of dependent random variables, as explained in Kaas et al. (2000), Dhaene et al. (2002a,b), among others. The first approximation we will consider for the distribution function of the discounted reserve is derived by approximating the dependence structure between the random variables involved by a comonotonic dependence structure. In this way the multi-dimensional problem is reduced to a twodimensional one which can easily be solved by conditioning and using some numerical techniques. It is argued that this approach is plausible in actuarial applications because it leads to prudent and conservative values of the reserves and solvency margin. If the dependency structure between the summands of S is strong enough, this upper bound in convex order performs reasonably well. The second approximation, which is derived by considering conditional expectations, takes part of the dependence structure into account. This lower bound in convex order turns out to be extremely useful to evaluate the quality of approximation provided by the upper bound. The lower bound can also be applied as an approximation of the underlying distribution. This choice is not actuarially prudent, however the relative error of this approximation significantly outperforms the relative error of the upper bound. For this reason, the lower bound will always be preferable in the applications which require high precision of approximations, like pricing of exotic derivatives (e.g. Decamps et al. (2004), Deelstra et al. (2004) and Vyncke et al. (2004)) or optimal portfolio selection problems (e.g. Dhaene et al. (2005)).

Preface xiii This thesis is set out as follows. The first chapter recalls the basics of actuarial risk theory. We define some frequently used measures of dependence and the most important orderings of risks for actuarial applications. We further introduce several well-known risk measures and the relations that hold between them. We summarize properties of these risk measures that can be used to facilitate decision-taking. Finally, we provide theoretical background for the concept of comonotonicity and we review the most important properties of comonotonic risks. In Chapter 2 we recall how the comonotonic bounds can be derived and illustrate the theoretical results by means of an application in the context of discounted loss reserves. The advantage of working with a sum of comonotonic variables has to be that the calculation of the distribution of such a sum is quite easy. In particular this technique is very useful to find reliable estimations of upper quantiles and stop-loss premiums. In practical applications the comonotonic upper bound seems to be useful only in the case of a very strong dependency between successive summands. Even then the bounds for stop-loss premiums provided by the comonotonic approximation are often not satisfactory. In this chapter we present a number of techniques which allow to determine much more efficient upper bounds for stop-loss premiums. To this end, we use on the one hand the method of conditioning as in Curran (1994) and in Rogers and Shi (1995), and on the other hand the upper and lower bounds for stop-loss premiums of sums of dependent random variables. We show also how to apply the results to the case of sums of lognormally distributed random variables. Such sums are widely encountered in practice, both in actuarial science and in finance. We derive comonotonic approximations for the scalar product of random vectors of the form (1) and explain a general procedure to obtain accurate estimates for quantiles and stop-loss premiums. We study the distribution of the present value function of a series of random payments in a stochastic financial environment described by a lognormal discounting process. Such distributions occur naturally in a wide range of applications within fields of insurance and finance. Accurate approximations are obtained by developing upper and lower bounds in the convex order sense for

xiv Preface such present value functions. Finally, we consider several applications for discounted claim processes under the Black & Scholes setting. In particular we analyze in detail the cases when the random variables X i denote insurance losses modelled by lognormal, normal (more general: elliptical) and gamma or inverse Gaussian (more general: tempered stable) distributions. As we demonstrate by means of a series of numerical illustrations, the methodology provides an excellent framework to get accurate and easily obtainable approximations of distribution functions for random variables of the form (1). Chapters 3 and 4 apply the obtained results to two important reserving problems in insurance business and illustrate them numerically. In Chapter 3 we consider an important application in the life insurance business. We aim to provide some conservative estimates both for high quantiles and stop-loss premiums for a single life annuity and for a whole portfolio. We focus here only on life annuities, however similar techniques may be used to get analogous estimates for more general life contingencies. Our solution enables to solve with a great accuracy personal finance problems such as: How much does one need to invest now to ensure given a periodical (e.g. yearly) consumption pattern that the probability of outliving ones money is very small (e.g. less than 1%)? The case of a portfolio of life annuity policies has been studied extensively in the literature, but only in the limiting case for homogeneous portfolios, when the mortality risk is fully diversified. However the applicability of these results in insurance practice may be questioned: especially in the case of the life annuity business a typical portfolio does not contain enough policies to speak about full diversification. For this reason we propose to approximate the number of active policies in subsequent years using a normal power distribution (by fitting the first three moments of the corresponding binomial distributions) and to model the present value of future benefits as a scalar product of mutually independent random vectors. Chapter 4 focuses on the claims reserving problem. To get the correct picture of its liabilities, a company should set aside the correctly estimated amount to meet claims arising in the future on the written policies. The past data used to construct estimates for the future payments consist of a

Preface xv triangle of incremental claims. The purpose is to complete this run-off triangle to a square, and even to a rectangle if estimates are required pertaining to development years of which no data are recorded in the run-off triangle at hand. To this end, the actuary can make use of a variety of techniques. The inherent uncertainty is described by the distribution of possible outcomes, and one needs to arrive at the best estimate of the reserve. In this chapter we look at the discounted reserve and impose an explicit margin based on a risk measure from the distribution of the total discounted reserve. We will model the claim payments using lognormal linear, loglinear location-scale and generalized linear models, and derive accurate comonotonic approximations for the discounted loss reserve. The bootstrap technique has proved to be a very useful tool in many statistical applications and can be particularly interesting to assess the variability of the claim reserving predictions and to construct upper limits at an adequate confidence level. Its popularity is due to a combination of available computing power and theoretical development. One advantage of the bootstrap is that the technique can be applied to any data set without having to assume an underlying distribution. Moreover, most computer packages can handle very large numbers of repeated samplings, and this should not limit the accuracy of the bootstrap estimates. In the last chapter we derive, review and discuss some other methods to obtain approximations for S. In the first section we recall two wellknown moment matching approximations: the lognormal and the reciprocal gamma approximation. Practitioners often use a moment matching lognormal approximation for the distribution of S. The lognormal and reciprocal gamma approximations are chosen such that their first two moments are equal to the corresponding moments of S. Although the comonotonic bounds in convex order have proven to be good approximations in case the variance of the random sum is sufficiently small, they perform much worse when the variance gets large. In actuarial applications it is often merely the tail of the distribution function that is of interest. Indeed, one may think of Value-at-Risk, Conditional Tail Expectation or Expected Shortfall estimations. Therefore, approximations for functionals of sums of dependent random variables may alternatively be obtained through the use of asymptotic relations. Although asymptotic results are valid at infinity, they may as well serve as approximations near

xvi Preface infinity. We establish some asymptotic results for the tail probability of a sum of heavy tailed dependent random variables. In particular, we derive an asymptotic result for the randomly weighted sum of a sequence of non-negative numbers. Furthermore, we establish under two different sets of conditions, an asymptotic result for the randomly weighted sum of a sequence of independent random variables that consist of a random and a deterministic component. Throughout, the random weights are products of i.i.d. random variables and thus exhibit an explicit dependence structure. Since the early 1990 s, statistics has seen an explosion in applied Bayesian research. This explosion has had little to do with a warming of the statistics and econometrics communities to the theoretical foundation of Bayesianism, or to a sudden awakening to the merits of the Bayesian approach over frequentist methods, but instead can be primarily explained on pragmatic grounds. Bayesian inference is the process of fitting a probability model to a set of data and summarizing the result by a probability distribution on the parameters of the model and on unobserved quantities such as predictions for new observations. Simple simulation methods exist to draw samples from posterior and predictive distributions, automatically incorporating uncertainty in the model parameters. An advantage of the Bayesian approach is that we can compute, using simulation, the posterior predictive distribution for any data summary, so we do not need to put a lot of effort into estimating the sampling distribution of test statistics. The development of powerful computational tools (and the realization that existing statistical tools could prove quite useful for fitting Bayesian models) has drawn a number of researchers to use the Bayesian approach in practice. Indeed, the use of such tools often enables researchers to estimate complicated statistical models that would be quite difficult, if not virtually impossible, using standard frequentist techniques. The purpose of this third section is to sketch, in very broad terms, basic elements of Bayesian computation. Finally, we compare these approximations with the comonotonic approximations of the previous chapter in the context of claims reserving. In case the underlying variance of the statistical and financial part of the discounted IBNR reserve gets large, the comonotonic approximations perform worse. We will illustrate this observation by means of a simple example and propose to solve this problem using the derived asymptotic results for the tail probability of a sum of dependent random variables, in the presence of heavy-tailedness conditions. These approximations are compared with

Preface xvii the lognormal moment matching approximations. We finally consider the distribution of the discounted loss reserve when the data in the run-off triangle is modelled by a generalized linear model and compare the outcomes of the Bayesian approach with the comonotonic approximations.