MODERN RESERVING TECHNIQUES FOR THE INSURANCE BUSINESS

Transcription

1 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

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3 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

4 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

5 Table of Contents Acknowledgments Preface Publications List of abbreviations and symbols i vii xix xxi 1 Risk and comonotonicity in the actuarial world Fundamental concepts in actuarial risk theory Dependent risks Risk measures Actuarial ordering of risks Comonotonicity Convex bounds Introduction Convex bounds for sums of dependent random variables The comonotonic upper bound The improved comonotonic upper bound The lower bound Moments based approximations Upper bounds for stop-loss premiums Upper bounds based on lower bound plus error term Bounds by conditioning through decomposition of the stop-loss premium Partially exact/comonotonic upper bound The case of a sum of lognormal random variables.. 35 iii

6 iv Table of Contents 2.4 Application: discounted loss reserves Framework and notation Calculation of convex lower and upper bounds Convex bounds for scalar products of random vectors Theoretical results Stop-loss premiums The case of log-normal discount factors Application: the present value of stochastic cash flows Stochastic returns Lognormally distributed payments Elliptically distributed payments Independent and identically distributed payments Proofs Reserving in life insurance business Introduction Modelling stochastic decrements The distribution of life annuities A single life annuity A homogeneous portfolio of life annuities An average portfolio of life annuities A numerical illustration Conclusion Reserving in non-life insurance business Introduction The claims reserving problem Model set-up: regression models Lognormal linear models Loglinear location-scale models Generalized linear models Linear predictors and the discounted IBNR reserve Convex bounds for the discounted IBNR reserve Asymptotic results in generalized linear models Lower and upper bounds The bootstrap methodology in claims reserving Introduction Central idea

7 Table of Contents v Bootstrap confidence intervals Bootstrap in claims reserving Three applications Lognormal linear models Loglinear location-scale models Generalized linear models Conclusion Other approximation techniques for sums of dependent random variables Introduction Moment matching approximations Two well-known moment matching approximations Application: discounted loss reserves Asymptotic approximations Preliminaries for heavy-tailed distributions Asymptotic results Application: discounted loss reserves The Bayesian approach Introduction Prior choice Iterative simulation methods Bayesian model set-up Applications in claims reserving The comonotonicity approach versus the Bayesian approximations The comonotonicity approach versus the asymptotic and moment matching approximations Proofs Samenvatting in het Nederlands (Summary in Dutch) 227 Bibliography 237

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9 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

10 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

11 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

12 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

13 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.

14 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)).

15 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

16 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

17 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

18 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

19 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.

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21 Publications Ahcan A., Darkiewicz G., Hoedemakers T., Dhaene J. and Goovaerts M.J. (2004), Optimal portfolio selection: Applications in insurance business, Proceedings of the 8th International Congress on Insurance: Mathematics & Economics, June 14-16, Rome, pp. 40. Ahcan A., Darkiewicz G., Goovaerts M.J. and Hoedemakers T. (2005), Computation of convex bounds for present value functions of random payments, Journal of Computational and Applied Mathematics, to be published. Antonio K., Goovaerts M.J. and Hoedemakers T. (2004), On the distribution of discounted loss reserves, Medium Econometrische Toepassingen, vol. 12, no. 2, pp Antonio K., Beirlant J. and Hoedemakers T. (2005), Discussion of A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving by Richard Verrall, North American Actuarial Journal, to be published. Antonio K., Beirlant J., Hoedemakers T. and Verlaak R. (2005), On the use of general linear mixed models in loss reserving, North American Actuarial Journal, submitted. Darkiewicz G. and Hoedemakers T. (2005), How the co-integration analysis can help in mortality forecasting, British Actuarial Journal, submitted. Hoedemakers T., Beirlant J., Goovaerts M.J. and Dhaene J. (2003), Confidence bounds for discounted loss reserves, Insurance: Mathematics & Economics, vol. 33, no. 2, pp xix

22 xx Publications Hoedemakers T. and Goovaerts M.J. (2004), Discussion of Risk and discounted loss reserves by Greg Taylor, North American Actuarial Journal, vol. 8, no. 4, pp Hoedemakers T., Beirlant J., Goovaerts M.J. and Dhaene J. (2005), On the distribution of discounted loss reserves using generalized linear models, Scandinavian Actuarial Journal, vol. 2005, no. 1, pp Hoedemakers T., Darkiewicz G. and Goovaerts M.J. (2005), Approximations for life annuity contracts in a stochastic financial environment, Insurance: Mathematics & Economics, to be published. Hoedemakers T., Darkiewicz G., Deelstra G., Dhaene J. and Vanmaele M. (2005), Bounds for stop-loss premiums of stochastic sums (with applications to life contingencies), Scandinavian Actuarial Journal, submitted. Hoedemakers T., Goovaerts M.J. and Dhaene J. (2003), IBNR problematiek in historisch perspectief, De Actuaris, vol. 11, no. 2, pp Hoedemakers T., Goovaerts M.J. and Dhaene J. (2004), De IBNRdiscussie, De Actuaris, vol. 11, no. 4, pp Laeven R.J.A, Goovaerts M.J. and Hoedemakers T. (2005), Some asymptotic results for sums of dependent random variables with actuarial applications, Insurance: Mathematics & Economics, to be published. Vanduffel S., Hoedemakers T. and Dhaene J. (2005), Comparing approximations for risk measures of sums of non-independent lognormal random variables, North American Actuarial Journal, to be published.

23 List of abbreviations and symbols Abbreviation or symbol Explanation ARMA(p, q) AutoRegressive-Moving Average Process of order (p, q) cdf cumulative distribution function c.f. characteristic function CLT Central Limit Theorem Corr(X, Y ) = r(x, Y ) Pearson s correlation coefficient between the r.v. s X and Y Cov[X, Y ] covariance between the r.v. s X and Y D class of dominatedly varying functions d.f. distribution function E exponential r.v. E n ( µ, Σ, φ) n-dimensional elliptical distribution with parameters µ, Σ and φ F d.f. and distribution of a r.v. F tail of the d.f. F : F = 1 F F n n-fold convolution of the d.f. or distribution F Γ(x) gamma function: Γ(x) = 0 t x 1 e t dt, x > 0 Gamma(a, b) gamma distribution with parameters a and b: f(x) = b a (Γ(a)) 1 x a 1 e bx, x 0 I(a, x) incomplete gamma function: Γ a (x) = (Γ(a)) 1 x e t t a 1 dt, GPD Generalized Pareto Distribution x 0 xxi

24 xxii List of abbreviations and symbols I (.) indicator function: I (c) = 1 if the condition c is true and I (c) = 0 if it is not i.i.d. independent, identically distributed L class of long-tailed distributions LLN Law of Large Numbers logn(µ, σ 2 ) lognormal distribution with parameters µ and σ 2 : f(x) = 1 xσ e (log x µ) 2 2π 2σ 2, x > 0 MLE Maximum Likelihood Estimator N(µ, σ 2 ), N(µ, Σ) Gaussian (normal) distribution with mean µ and variance σ 2 or covariance matrix Σ N(0, 1) standard normal distribution o(1) a(x) = o(b(x)) as x x 0 means that lim x x0 a(x)/b(x) = 0 O(1) a(x) = O(b(x)) as x x 0 means that lim x x0 a(x)/b(x) < ϕ X (t) c.f. of the r.v. X: ϕ X (t) = E[e itx ] Φ(.) the cdf of the standard normal r.v. lim sup n (x n ) limit superior of the bounded sequence {x n }: = lim(s n ), where s n = sup k n x k = sup{x n, x n+1,...} lim inf n (x n ) limit inferior of the bounded sequence {x n }: = lim(t n ), where t n = inf k n x k = inf{x n, x n+1,...} p.d.f probability density function Pr[.] probability measure p(..) conditional probability density p(.) marginal distribution R class of the d.f. s with regularly varying right tail R α class of the regularly varying functions with index α R class of the rapidly varying functions r.v. random variable S class of the subexponential distributions σx 2 variance of the r.v. X σ Xi X j Cov[X i, X j ] sign(a) sign of the real number a T S(δ, a, b) tempered stable law with parameters δ, a and b U(a, b) uniform random variable on (a, b) UMVUE Uniformly Minimum Variance Unbiased Estimator Var[X] variance of the r.v. X

25 List of abbreviations and symbols xxiii a(x) b(x) as x x 0 means that lim x x0 a(x)/b(x) = 1 a(x) 0 means a(x) = o(1) a(x) b(x) as x x 0 means that a(x) is approximately (roughly) of the same order as b(x) as x x 0. It is only used in a heuristic sense. a(x) b(x) as x x 0 means that 0 < lim inf x x0 a(x)/b(x) lim sup x x0 a(x)/b(x) < d d convergence in distribution = equal in distribution. floor function: x is the largest integer less than or equal to x. ceiling function: x is the smallest integer greater than or equal to x (x d) + max(x d, 0) =: or := notation

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27 Chapter 1 Risk and comonotonicity in the actuarial world Summary In order to make decisions one has to evaluate the (distribution function of the) multivariate risk (or random variable) one faces. In this chapter we recall 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. 1.1 Fundamental concepts in actuarial risk theory In this section we briefly recall the most important concepts in actuarial risk theory. The study of dependence has become of major concern in actuarial research. We start by defining three important measures of dependence: Pearson s correlation coefficient, Kendall s τ and Spearman s ρ. Once dependence measures are defined, one could use them to compare the strength of dependence between random variables. The determination of capital requirements for an insurance company is a complex and non-trivial task. From their nature, capital requirements are numeric values expressed in monetary units and based on quantifiable 1

28 2 Chapter 1 - Risk and comonotonicity in the actuarial world measures of risks. Formally a risk measure is defined as a mapping from the set of risks at hand to the real numbers. In other words, with any potential loss X one associates a real number ρ[x]. Thus a risk measure summarizes the riskiness of the underlying distribution in one single number. Usually such quantification serves as a risk management tool (e.g. an insurance premium or an economic capital), but it can be also helpful in overall decision making. We review and place the four popular risk measures (Value-at-Risk, Tail Value-at-Risk, Conditional Tail Expectation and Expected Shortfall) in their context. In the actuarial literature, orderings of risks are an important tool for comparing the attractiveness of different risks. The essential tool for the comparison of different concepts of orderings of risks will be the stop-loss transform/premium and its properties. In the actuarial literature it is a common feature to replace a risk by a less favorable risk that has a simpler structure, making it easier to determine the distribution function. We clarify what we mean with a less favorable risk and define the three most important orderings of risks for actuarial applications: stochastic dominance, stop-loss order and convex order. This chapter is essentially based on Dhaene, Denuit, Goovaerts, Kaas & Vyncke (2002a) and Dhaene, Vanduffel, Tang, Goovaerts, Kaas & Vyncke (2004) Dependent risks In risk theory, all the random variables are traditionally assumed to be mutually independent. It is clear that this assumption is made for mathematical convenience. In some situations however, insured risks tend to act similarly. The independence assumption is then violated and is not an adequate way to describe the relations between the different random variables involved. The individual risks of an earthquake or flooding risk portfolio which are located in the same geographic area are correlated, since individual claims are contingent on the occurrence and severity of the same earthquake or flood. On a foggy day all cars of a region have higher probability to be involved in an accident. During dry hot summers, all wooden cottages are more exposed to fire. More generally, one can say that if the density of insured risks in a certain area or organization is high enough, then catastrophes such as storms, explosions, earthquakes, epidemics and

29 1.1. Fundamental concepts in actuarial risk theory 3 so on can cause an accumulation of claims for the insurer. In life insurance, there is ample evidence that the lifetimes of husbands and their wives are positively associated. There may be certain selection mechanisms in the matching of couples ( birds of a feather flock together ): both partners often belong to the same social class and have the same life style. Further, it is known that the mortality rate increases after the passing away of one s spouse (the broken heart syndrome ). These phenomena have implications on the valuation of aggregate claims in life insurance portfolios. Another example in a life insurance context is a pension fund that covers the pensions of persons working for the same company. These persons work at the same location, they take the same flights. It is evident that the mortality of these persons will be dependent, at least to a certain extent. The study of dependence has become of major concern in actuarial research. There are a variety of ways to measure dependence. First Pearson s product moment correlation coefficient, captures the linear dependence between couples of random variables. For a random couple (X 1, X 2 ) having marginals with finite variances, Pearson s product correlation coefficient is defined by Corr(X 1, X 2 ) = Cov[X 1, X 2 ] Var[X1 ]Var[X 2 ]. Pearson s correlation coefficient contains information on both the strength and direction of a linear relationship between two random variables. If one variable is an exact linear function of the other variable, a positive relationship leads to correlation coefficient 1, while a negative relationship leads to correlation coefficient 1. If there is no linear predictability between the two variables, the correlation is 0. Kendall s τ is a nonparametric measure of association based on the probabilities of concordances and discordances in paired observations. Concordance occurs when paired observations vary together, and discordance occurs when paired observations vary differently. Specifically, Kendall s τ for a random couple (X 1, X 2 ) of random variables with continuous cdf s is defined as τ(x 1, X 2 ) = Pr[(X 1 X 1)(X 2 X 2) > 0] Pr[(X 1 X 1)(X 2 X 2) < 0] = 2Pr[(X 1 X 1)(X 2 X 2) > 0] 1,

30 4 Chapter 1 - Risk and comonotonicity in the actuarial world where (X 1, X 2 ) is an independent copy of (X 1, X 2 ). Contrary to Pearson s r, Kendall s τ is invariant under strictly monotone transformations, that is, if φ 1 and φ 2 are strictly increasing (or decreasing) functions on the supports of X 1 and X 2, respectively, then τ ( φ 1 (X 1 ), φ 2 (X 2 ) ) = τ ( ) X 1, X 2 provided the cdf s of X1 and X 2 are continuous. Further, (X 1, X 2 ) are perfectly dependent if and only if, τ(x 1, X 2 ) = 1. Another very useful dependence measure is Spearman s ρ. The idea behind this dependence measure is very simple. Given random variables X 1 and X 2 with continuous cdf s F X1 and F X2, we first create U 1 = F X1 (X 1 ) and U 2 = F X2 (X 2 ), which are uniformly distributed over [0, 1] and then use Pearson s r. Spearman s ρ is thus defined as ρ(x 1, X 2 ) = r(u 1, U 2 ). Dependence measures can be used to compare the strength of dependence between random variables Risk measures Measuring risk and measuring preferences is not the same. When ordering preferences, activities, for example, alternatives A and B with financial consequences X A and X B, are compared in order of preference under conditions of risk. A preference order A B means that A is preferred to B. This order is represented by a preference function Ψ with A B Ψ[X A ] > Ψ[X B ]. In contrast, a risk order A R B means that A is riskier than B and is represented by a function ρ with A R B ρ[x A ] > ρ[x B ]. Every such function ρ is called a risk measure. Models in actuarial science are used both for quantifying risks and for pricing risks. Quantifying risk requires a risk measure to convert a random future gain or loss into a certainty equivalent that can then be used to order different risks and for decision-making purposes. In order to quantify risk, it is necessary to specify the probability distributions of the risks involved and to apply a preference function to these probability distributions. Thus, this process involves both statistical assumptions and economic assumptions. Individuals are assumed to be risk averse and to have a preference to diversify risks. Banks and regulatory agencies use monetary measures of risk to assess the risk taken by financial investors; important examples are given by the so-called Value-at-Risk and Tail Value-at-Risk. Two-sided risk measures measure the magnitude of the distance (in

31 1.1. Fundamental concepts in actuarial risk theory 5 both directions) from X to E[X]. Different functions of distance lead to different risk measures. Looking, for instance, at quadratic deviations, this leads to the risk measure variance or to the risk measure standard deviation. These risk measures have been the traditional measures in economics and finance since the pioneering work of Markowitz. They exhibit a number of nice technical properties. For instance, the variance of a portfolio return is the sum of the variances and covariances of the individual returns. Furthermore, the variance is used as a standard optimization function (quadratic optimization). On the other hand, a two-sided risk measure contradicts the intuitive notion of risk that only negative deviations are dangerous. In addition variance does not account for fat tails of the underlying distribution and for the corresponding tail risk. For this reason, people include higher (normalized) central moments, as for example, skewness and kurtosis, into the analysis to assess risk more properly. Perhaps the most popular risk measure is the Value-at-Risk (VaR). Let L be the potential loss of a financial position. The VaR at confidence level p (0 < p < 1) is then defined by the requirement Pr [ L > VaR p [L] ] = 1 p. (1.1) An intuitive interpretation of the VaR is that of a probable maximum loss or more concrete, a 100 p% maximal loss, because Pr [ L VaR p [L] ] = p, which means that in 100 p% of the cases, the loss is smaller or equal to VaR p [L]. Interpreting the VaR as necessary underlying capital to bear risk, relation (1.1) implies that this capital will, on average, not be exhausted in 100 p% of the cases. Obviously, the VaR is identical to the p-quantile of the loss distribution, that is VaR p [L] = F 1 L (p). It is important to remark that the VaR does not take into account the severity of potential losses in the 100 (1 p)% worst cases. A regulator for instance is not only concerned with the frequency of default, but also about the severity of default. Also shareholders and management should be concerned with the question how bad is bad? when they want to evaluate the risks at hand in a consistent way. Therefore, one often uses another risk measure which is called the Tail Value-at-Risk (TVaR) and defined by TVaR p [L] = 1 1 p 1 p VaR q [L]dq, p (0, 1).

32 6 Chapter 1 - Risk and comonotonicity in the actuarial world It is the arithmetic average of the quantiles of L, from p on. Note that the TVaR is always larger than the corresponding VaR. We will define the other popular risk measures in terms of L for a better comparison to the VaR. The Conditional Tail Expectation (CTE) at confidence level p is defined by CTE p [L] = E [ L L > VaR p [L] ], p (0, 1). On the basis of the interpretation of the VaR as a 100 p%-maximum loss, the CTE can be interpreted as the average maximal loss in the worst 100 (1 p)% cases. Notice that in case of continuous distributions the CTE and TVaR coincide. Measures of shortfall risk are one-sided risk measures and measures the shortfall risk relative to a target variable. This may be the expected value, but in general, it is an arbitrary deterministic target or a stochastic benchmark. The Expected Shortfall (ESF) at confidence level p is defined as ESF p [L] = E[max(L VaR p [L], 0)], p (0, 1). The following relations hold between the four risk measures defined above. Theorem 1 (Relation between VaR, TVaR, CTE and ESF). For p (0, 1), we have that TVaR p [X] = VaR p [X] p ESF p[x], CTE p [X] = VaR p [X] + CTE p [X] = TVaR FX (VaR p[x])[x]. 1 1 F X (VaR p [X]) ESF p[x], Proof. See Dhaene et al. (2004). Researchers always aimed to find a set of properties (axioms) that any risk measure should satisfy. Recently the class of coherent risk measures, introduced in Artzner (1999) and Artzner et al. (1999), has drawn a lot of attention in the actuarial literature. The authors postulated that every coherent risk measure should satisfy the following four properties:

33 1.1. Fundamental concepts in actuarial risk theory 7 1. monotonicity, i.e. X Y ρ[x] ρ[y ]; 2. subadditivity, i.e. ρ[x + Y ] ρ[x] + ρ[y ]; 3. translation invariance, i.e. ρ[x + c] = ρ[x] + c c R; 4. positive homogeneity, i.e. ρ[ax] = aρ[x] a 0. It can be demonstrated that the Value-at-Risk and the Expected Shortfall are in general not subadditive. On the other hand, the TVaR is subadditive. The desirability of the subadditivity property of risk measures has been a major topic for research and discussion. Some researchers believe that the axiom of subadditivity of risk measures used to determine the solvency capital, reflects the risk diversification. However other authors argue that the diversification benefits should be considered rather in terms of subadditivity of the corresponding shortfalls. It is an open question whether the coherent set of axioms is indeed the best one. For a relevant discussion we refer to e.g. Dhaene et al. (2003), Goovaerts et al. (2003, 2004) and Darkiewicz et al. (2005a). It should be noted that in spite of the disagreement in the scientific community about the axioms of coherency, a lot of well-known risk measures satisfy conditions (1)-(4) (e.g. the TVaR). The expressions for the discussed risk measures of normal and lognormal losses are given in the next two examples, which will be used in the remainder of this thesis. For a proof of these examples, we refer to Dhaene et al. (2004). Example 1 (normal losses). Consider a random variable X N(µ, σ 2 ). The VaR, ESF and CTE at confidence level p (p (0, 1)) of X are given by VaR p [X] = µ + σφ 1 (p), (1.2) ESF p [X] = σφ ( Φ 1 (p) ) σφ 1 (p)(1 p), (1.3) CTE p [X] = µ + σ φ( Φ 1 (p) ), (1.4) 1 p where φ(x) = Φ (x) denotes the density function of the standard normal distribution.

34 8 Chapter 1 - Risk and comonotonicity in the actuarial world Example 2 (lognormal losses). Consider a random variable X logn(µ, σ 2 ). The VaR, ESF and CTE at confidence level p (p (0, 1)) of X are given by VaR p [X] = e µ+σφ 1 (p), (1.5) ESF p [X] = e µ+σ2 /2 Φ ( σ Φ 1 (p) ) e µ+σφ 1 (p) (1 p), (1.6) CTE p [X] = e µ+σ2 /2 Φ( σ Φ 1 (p) ). (1.7) 1 p We end this section with a note about inverse distribution functions. Inverse distribution functions The cdf F X (x) = Pr[X x] of a random variable X is a right continuous non-decreasing function with F X ( ) = lim F X(x) = 0, F X (+ ) = lim F X(x) = 1. x x + The classical definition of the inverse of a distribution function is the nondecreasing and left-continuous function F 1 X (p) defined by F 1 X (p) = inf{x R F X(x) p}, p [0, 1] with inf = + by convention. For all x R and p [0, 1], we have F 1 X (p) x p F X(x). (1.8) In this thesis we will use a more sophisticated definition for inverses of distribution functions. For any real p [0, 1], a possible choice for the inverse of F X in p is any point in the closed interval [ inf{x R FX (x) p}, sup{x R F X (x) p} ], where, as before, inf = +, and also sup =. Taking the left hand border of this interval to be the value of the inverse cdf at p, we get F 1 X (p). Similarly, we define F 1+ X (p) as the right hand border of the interval: F 1+ X (p) = sup{x R F x(x) p}, p [0, 1] which is a non-decreasing and right-continuous function. Note that F 1 X (0) =, F 1+ X (1) = + and that all the probability mass of X is contained

35 1.1. Fundamental concepts in actuarial risk theory 9 in the interval [ F 1+ 1 X, (0)FX (1)]. Also note that F 1 1+ X (p) and FX (p) are finite for all p (0, 1). In the sequel we will always use p as a value ranging over the open interval (0, 1), unless stated otherwise. In the following lemma, we state the relation between the inverse distribution functions of the random variables X and g(x) for a monotone function g. Lemma 1 (Inverse distribution function of g(x)). Let X and g(x) be real-valued random variables and 0 < p < 1. (a) If g is non-decreasing and left-continuous, then F 1 g(x) (p) = g( F 1 X (p)). (b) If g is non-decreasing and right-continuous, then F 1+ g(x) (p) = g( F 1+ X (p)). (c) If g is non-increasing and left-continuous, then F 1+ g(x) (p) = g( F 1 X (1 p)). (d) If g is non-increasing and right-continuous, then Proof. See Dhaene et al. (2002a). F 1 g(x) (p) = g( F 1+ X (1 p)). Hereafter, we will reserve the notation U and V for U(0, 1) random variables, i.e. F U (p) = p and F 1 U (p) = p for all 0 < p < 1, and the same for V. One can prove that X = d F 1 X (U) = d F 1+ X (U). (1.9) The first distributional equality is known as the quantile transform theorem and follows immediately from (1.8). It states that a sample of random numbers from a general cumulative distribution function F X can be generated from a sample of uniform random numbers. Note that F X has at most a countable number of horizontal segments, implying that the last two random variables in (1.9) only differ in a null-set of values of U. This means that these random variables are equal with probability one.

36 10 Chapter 1 - Risk and comonotonicity in the actuarial world Actuarial ordering of risks In the actuarial literature, orderings of risks are an important tool for comparing the attractiveness of different risks. Many examples and results can be found in the work of Goovaerts et al. (1990), Van Heerwaarden (1991) and Kaas et al. (1998). The essential tool for the comparison of different concepts of orderings of risks will be the stop-loss transform/premium and its properties. Throughout this section a risk X will be a random variable with finite mean. The distribution function of X is denoted by F X, and F X = 1 F X is the corresponding survival function. In the actuarial literature it is a common feature to replace a risk by a less favorable risk that has a simpler structure, making it easier to determine the distribution function. Of course, we have to clarify what we mean with a less favorable risk. Therefore, we first introduce the notion of stop-loss premium of a distribution function. Definition 1 (Stop-loss premium). The stop-loss premium with retention d of a risk X is defined by π(x, d) := E [ ] (X d) + = F X (x)dx, < d < +, (1.10) with the notation (x d) + = max(x d, 0). d From this formula it is clear that the stop-loss premium with retention d can be considered as the weight of an upper tail of (the distribution function of) X. Indeed, it is the surface between the cdf F X of X and the constant function 1, from d on. For these reasons stop-loss premiums contain a lot of information about riskiness of underlying distributions. The following properties of the stop-loss premium can easily be deduced from the definition. Theorem 2 (Stop-loss properties). The stop-loss premium π(x,.) has the following properties: (i) π(x,.) is decreasing and convex; (ii) The right-hand derivative π + (X,.) exists and 1 π + (X,.) 0; (iii) lim d + π(x, d) = 0.

37 1.1. Fundamental concepts in actuarial risk theory 11 To every function π : R + R, that fulfils (i)-(iii) there is a risk X, such that π is the stop-loss premium of X. The distribution function of X is given by F X (d) = π + (X, d) + 1. There are many concepts for comparing random variables. The most familiar one is the usual stochastic order introduced by Lehmann (1955). In the actuarial and economic literature this ordering is sometimes called stochastic dominance, see e.g. Goovaerts et al. (1990) and Van Heerwaarden (1991). Definition 2 (Stochastic order). We say that risk Y stochastically dominates risk X, written X st Y, if and only if F X (t) F Y (t) for all t R. In other words, X st Y if their corresponding quantiles are ordered. Note that the condition for stochastic dominance is very strong it can be easily seen that X st Y if and only if there exist a bivariate vector (X, Y ) with the same marginal distributions as X and Y and such that X Y almost surely. Several results for this ordering can be found in Shaked and Shanthikumar (1994). In the following theorem, some equivalent characterizations are given for stochastic dominance. Lemma 2 (Characterizations for stochastic dominance). X st Y holds if and only if any of the following equivalent conditions is satisfied: 1. Pr[X t] Pr[Y t], for all t R; 2. Pr[X > t] Pr[Y > t], for all t R; 3. E[φ(X)] E[φ(Y )], for all non-decreasing functions φ(.); 4. E[ψ( X)] E[ψ( Y )], for all non-decreasing functions ψ(.); 5. The function t π(y, t) π(x, t) is non-increasing. A consequence of stochastic order X st Y, i.e. a necessary condition for it, is obviously that E[X] E[Y ], and even E[X] < E[Y ] unless X d = Y. The stochastic dominance has a natural interpretation in terms of utility theory. We have that X st Y holds if and only if E[u( X)] E[u( Y )] for every non-decreasing utility function u. So the pairs of risks X and

38 12 Chapter 1 - Risk and comonotonicity in the actuarial world Y with X st Y are exactly those pairs of losses about which all decision makers with an increasing utility function agree. For actuarial applications the stop-loss order is much more interesting. This ordering was investigated by Bühlmann et al. (1977), Goovaerts et al. (1990) and Van Heerwaarden (1991). It is equivalent to increasing convex order, which is well known in operations research and statistics. Definition 3 (Stop-loss order). If X and Y are two risks, then X precedes Y in stop-loss order, written X sl Y, if and only if π(x, d) π(y, d) for all < d < +. (1.11) In other words two risks are ordered in the stop-loss sense if their corresponding stop-loss premiums are ordered. It is clear that stochastic order induces stop-loss order. Like stochastic order, stop-loss order between two risks X and Y implies a corresponding ordering of their means. To prove this, assume that d < 0. From the expression (1.10) in Definition 1 of stop-loss premiums as upper tails, we immediately find the following equality: d + π(x, d) = and also, letting d, 0 d F X (x)dx + ( ) lim d + π(x, d) = E[X]. d 0 (1 F X (x))dx (1.12) Hence, adding d to both sides of the inequality (1.11) in Definition 3 and taking the limit for d, we get E[X] E[Y ]. A sufficient condition for X sl Y to hold is that E[X] E[Y ], together with the condition that their cumulative distribution functions only cross once. This means that there exists a real number c such that F X (x) F Y (x) for x c, but F X (x) F Y (x) for x < c. Indeed, considering the function f(d) = π(y, d) π(x, d), we have that lim f(d) = E[Y ] E[X] 0, and lim d f(d) = 0. d + Further, f(d) first increases, and then decreases (from c on) but remains non-negative.

39 1.1. Fundamental concepts in actuarial risk theory 13 If two risks X and Y are ordered in the stop-loss sense, X sl Y, this means that X has uniformly smaller upper tails than Y, which in turns means that a risk X is more attractive than a risk Y for an insurance company. Moreover stop-loss order has a natural economic interpretation in terms of expected utility. Indeed, it can be shown that X sl Y if and only if E [ u( X) ] E [ u( Y ) ] holds for all non-decreasing concave real functions u. This means that any risk-averse decision maker will prefer to pay X instead of Y, which implies that acting as if the obligations X are replaced by Y indeed leads to conservative or prudent decisions. This characterization of stop-loss order in terms of utility functions is equivalent to E [ v(x) ] E [ v(y ) ] holding for all non-decreasing convex functions v. For this reason stop-loss order is alternatively called an increasing convex order and denoted by icx. Recall that our original problem was to replace a risk X by a less favorable risk Y, for which the distribution function is easier to obtain. If X sl Y, then also E[X] E[Y ], and it is intuitively clear that the best approximations arise in the borderline case where E[X] = E[Y ]. This leads to the so-called convex order. Definition 4 (Convex order). If X and Y are two risks, then X precedes Y in convex order, written X cx Y, if and only if E[X] = E[Y ] and π(x, d) π(y, d) for all < d < +. (1.13) A sufficient condition for X cx Y to hold is that E[X] = E[Y ], together with the condition that their cumulative distribution functions only cross once. This once-crossing condition can be observed to hold in most natural examples, but it is of course easy to construct examples with X cx Y and distribution functions that cross more than once. It can also be proven that X cx Y if and only E [ v(x) ] E [ v(y ) ] for all convex functions v. This explains the name convex order. Note that when characterizing stop-loss order, the convex functions v are additionally required to be non-decreasing. Hence, stop-loss order is weaker: more pairs of random variables are ordered. In the utility context one will reformulate this condition to E[X] = E[Y ] and E [ u( X) ] E [ u( Y ) ] for all non-decreasing concave functions u. These conditions represent the common preferences of all risk-averse decision makers between risks with equal mean. We summarize the properties of convex order in the following lemma.

40 14 Chapter 1 - Risk and comonotonicity in the actuarial world 1. E[X] = E[Y ] and π(x, d) π(y, d) for all d R; 2. E[X] = E[Y ] and E[(d X) + ] E[(d Y ) + ] for all d R; 3. π(x, d) π(y, d) and E[(d X) + ] E[(d Y ) + ] for all d R; 4. E[X] = E[Y ] and E [ u( X) ] E [ u( Y ) ] for all concave functions u(.); 5. E [ v(x) ] E [ v(y ) ] for all convex functions v(.). In case X cx Y, the upper tails as well as the lower tails of Y eclipse the corresponding tails of X, which means that extreme values are more likely to occur for Y than for X. This observation also implies that X cx Y is equivalent to X cx Y. Hence, the interpretation of risks as payments or as incomes is irrelevant for the convex order. Note that with stop-loss order, we are concerned with large values of a random loss, and call the risk Y less attractive than X if the expected values of all top parts (Y d) + are larger than those of X. Negative values for these random variables are actually gains. With stability in mind, excessive gains might also be unattractive for the decision maker, for instance for tax reasons. In this situation, X could be considered to be more attractive than Y if both the top parts (X d) + and the bottom parts (d X) + have a lower expected value than for Y. Both conditions just define the convex order introduced above. Corollary 1 (Convex order and variance). If X cx Y then Var[X] Var[Y ]. Proof. It suffices to take the convex function v(x) = x 2. Lemma 3 (Characterizations for convex order). X cx Y if and only if any of the following equivalent conditions is satisfied: Notice that the reverse implication does not hold in general. Comparing variances is meaningful when comparing stop-loss premiums of convex ordered risks. The following corollary links variances and stop-loss premiums.

41 1.2. Comonotonicity 15 Corollary 2 (Variance and stop-loss premiums). For any random variable X we can write 1 + ( 2 Var[X] = π(x, t) ( E[X] t ) ) dt. (1.14) + Proof. See e.g. Kaas et al. (1998). From relation (1.14) in Corollary 2 we deduce that if X cx Y, + π(y, t) π(x, t) dt = 2( 1 ) Var[Y ] Var[X]. (1.15) Thus, if X cx Y, their stop-loss distance, i.e. the integrated absolute difference of their respective stop-loss premiums, equals half the variance difference between these two random variables. As the integrand in (1.15) is non-negative, we find that if X cx Y and in addition Var[X] = Var[Y ], then X and Y must have necessarily equal stop-loss premiums and hence the same distribution. We also find that if X cx Y, and X and Y are not equal in distribution, then Var[X] < Var[Y ] must hold. Note that (1.14) and (1.15) have been derived under the additional condition that X and Y have finite second moments, hence both lim x x 2 (1 F X (x)) and lim x x 2 F X (x) are equal to 0 (and similar for Y ). In the following theorem we recall the characterization of stochastic dominance in terms of Value-at-Risk, and a similar result characterizing stoploss order by Tail Value-at-Risk. Theorem 3. For any random pair (X, Y ) we have that 1. X st Y VaR p [X] VaR p [Y ] for all p (0, 1); 2. X sl Y TVaR p [X] TVaR p [Y ] for all p (0, 1). Proof. See Dhaene et al. (2004). 1.2 Comonotonicity In an insurance context, one is often interested in the distribution function of a sum of random variables. Such a sum appears for instance when considering the aggregate claims of an insurance portfolio over a certain reference period. In traditional risk theory, the individual risks of a portfolio are

42 16 Chapter 1 - Risk and comonotonicity in the actuarial world usually assumed to be mutually independent. This is very convenient from a mathematical point of view as the standard techniques for determining the distribution function of aggregate claims, such as Panjer s recursion and convolution, are based on the independence assumption. Moreover, in general the statistics gathered by the insurer only give information about the marginal distributions of the risks, not about their joint distribution, i.e. the way these risks are interrelated. The assumption of mutual independence however does not always comply with reality, which may resolve in an underestimation of the total risk. On the other hand, the mathematics for dependent variables is less tractable, except when the variables are comonotonic. This section provides theoretical background for the concept of comonotonicity. We start by defining a comonotonicity of a set A of n-vectors in R n. We will denote an n-vector (x 1, x 2,..., x n ) by x. For two n-vectors x and y, the notation x y will be used for the componentwise order which is defined by x i y i for all i = 1, 2,..., n. We will denote the (i, j)-projection of a set A in R n by A i,j. It is formally defined by A ij = {(x i, x j ) x A}. Definition 5 (Comonotonic set). The set A R n is said to be comonotonic if for any x and y in A, either x y or y x holds. A set A R n is comonotonic if for any x and y in A, if x i < y i for some i, then x y must hold. Hence, a comonotonic set is simultaneously nondecreasing in each component. Notice that a comonotonic set is a thin set: it cannot contain any subset of dimension larger than 1. Any subset of a comonotonic set is also comonotonic. The proof of the following lemma is straightforward. Lemma 4. A R n is comonotonic if and only if the set A i,j is comonotonic for all i j in {1, 2,..., n}. For a general set A, comonotonicity of the (i, i + 1)-projections A i,i+1, (i = 1, 2,..., n 1), will not necessarily imply that A is comonotonic. As a counter example, consider the set A = {(x 1, 1, x 3 ) 0 < x 1, x 3 < 1}. This set is not comonotonic, although A 1,2 and A 2,3 are comonotonic.

43 1.2. Comonotonicity 17 Next, we define the notion of support of an n-dimensional random vector X = [ (X 1,.. ]., X n ). Any subsect [ A ] R n will be called a support of X if Pr X A = 1 and Pr X / A = 0. In generally we will be interested in supports which are as small as possible. Informally, the smallest support of a random vector X is the subset of R n that is obtained by deleting from R n all points which have a zero-probability neighborhood (with respect to X). This support can be interpreted as the set of all possible outcomes of X. Definition 6 (Comonotonic random vector). A random vector X = (X 1, X 2,..., X n ) is said to be comonotonic if it has a comonotonic support. From Definition 6 we can conclude that comonotonicity is a very strong positive dependency structure. Indeed, if x and y are elements of the comonotonic support of X, i.e. x and y are possible outcomes of X, then they must be ordered component by component. This explains the term comonotonic (common monotonic). Comonotonicity of a random vector X implies that the higher the value of one component X j, the higher the value of any other component X k. This means that comonotonicity entails that no X j is in any way a hedge, for another component X k. In the following theorem, some equivalent characterizations are given for comonotonicity of a random vector. Theorem 4 (Characterizations for comonotonicity). A random vector X = (X 1, X 2,..., X n ) is comonotonic if and only if one of the following equivalent conditions are satisfied: 1. X has a comonotonic support; 2. For all x = (x 1, x 2,..., x n ), we have F X ( x) = min { F X1 (x 1 ), F X2 (x 2 ),..., F Xn (x n ) } ; (1.16) 3. For U U(0, 1), we have X d = ( F 1 X 1 (U), F 1 X 2 (U),..., F 1 X n (U) ) ; (1.17)

44 18 Chapter 1 - Risk and comonotonicity in the actuarial world 4. There exist a random variable Z and non-decreasing functions f i (i = 1, 2,..., n), such that X d = (f 1 (Z), f 2 (Z),..., f n (Z)). Proof. See Dhaene et al. (2002a). From (1.16) we see that, in order to find the probability of all the outcomes of n comonotonic risks X i being less than x i (i = 1,..., n) one simply takes the probability of the least likely of these n events. It is obvious that for any random vector (X 1,..., X n ), not necessarily comonotonic, the following inequality holds: Pr [ X 1 x 1,..., X n x n ] min { FX1 (x 1 ),..., F Xn (x n ) }, (1.18) and it is well-known that the function min { F X1 (x 1 ),..., F Xn (x n } is indeed the multivariate cdf of a random vector ( F 1 X 1 (U),..., F 1 X n (U) ), which has the same marginal distributions as (X 1,..., X n ). Inequality (1.18) states that in the class of all random vectors (X 1,..., X n ) with the same marginal distributions, the probability that all X i simultaneously realize large values is maximized if the vector is comonotonic, suggesting that comonotonicity is indeed a very strong positive dependency structure. In the special case that all marginal distribution functions F Xi are identical, we find from (1.17) that comonotonicity of X is equivalent to saying that X 1 = X 2 = = X n holds almost surely. A standard way of modelling situations where individual random variables X 1,..., X n are subject to the same external mechanism is to use a secondary mixing distribution. The uncertainty about the external mechanism is then described by a structure variable z, which is a realization of a random variable Z and acts as a (random) parameter of the distribution of X. The aggregate claims can then be seen as a two-stage process: first, the external parameter Z = z is drawn from the distribution function F Z of z. The claim amount of each individual risk X i is then obtained as a realization from the conditional distribution function of X i given Z = z. A special type of such a mixing model is the case where given Z = z, the claim amounts X i are degenerate on x i, where the x i = x i (z) are nondecreasing in z. This means that (X 1,..., X n ) d = (f 1 (Z),... f n (Z)) where all functions f i are non-decreasing. Hence, (X 1,..., X n ) is comonotonic. Such a model is in a sense an extreme form of a mixing model, as in this

45 1.2. Comonotonicity 19 case the external parameter Z = z completely determines the aggregate claims. If U U(0, 1), then also 1 U U(0, 1). This implies that comonotonicity of X can also be characterized by X d = ( F 1 X 1 (1 U), F 1 X 2 (1 U),..., F 1 X n (1 U) ). Similarly, one can prove that X is comonotonic if and only if there exist a random variable Z and non-increasing functions f i, (i = 1, 2,..., n), such that X d = (f 1 (Z), f 2 (Z),..., f n (Z)). In the sequel, for any random vector (X 1,..., X n ), the notation (X1 c,..., Xc n) will be used to indicate a comonotonic random vector with the same marginals as (X 1,..., X n ). From (1.17) we find that for any random vector X the outcome of its comonotonic counterpart X c = (X1 c,..., Xc n) lies with probability one in the following set {( F 1 X 1 (p), F 1 X 2 (p),..., F 1 X n (p) ) 0 < p < 1 }. The following theorem states essentially that the comonotonicity of a random vector is equivalent with pairwise comonotonicity. Theorem 5 (Pairwise comonotonicity). A random vector X is comonotonic if and only if the couples (X i, X j ) are comonotonic for all i and j in {1, 2,..., n}. The next theorem characterizes a comonotonic random couple by means of Pearson s correlation coefficient r. Theorem 6 (Comonotonicity and maximum correlation). For any random vector (X 1, X 2 ) the following inequality holds: r(x 1, X 2 ) r ( F 1 X 1 (U), F 1 X 2 (U) ), (1.19) with strict inequalities when (X 1, X 2 ) is not comonotonic. As a special case of (1.19), we find that r ( F 1 X 1 (U), F 1 X 2 (U) ) 0 always holds. In Denuit & Dhaene (2003) it is shown that other dependence measures such as Kendall s τ and Spearman s ρ equal 1 (and thus are also maximal) if and only if the variables are comonotonic. In the following theorem we recall that the Value-at-Risk (VaR p ), the Tail Value-at-Risk (TVaR p ) and the Expected Shortfall (ESF p ) are additive for comonotonic risks.

46 20 Chapter 1 - Risk and comonotonicity in the actuarial world Theorem 7 (Comonotonicity and risk measures). Consider a comonotonic random vector ( X c 1, Xc 2,..., Xc n), and let S c = X c 1 + Xc Xc n. Then for all p (0, 1) one has that VaR p [S c ] = TVaR p [S c ] = ESF p [S c ] = VaR p [X i ]; (1.20) TVaR p [X i ]; (1.21) ESF p [X i ]. (1.22) Proof. See Dhaene et al. (2004). The computation of the most important risk measures is very easy for sums of comonotonic random variables, since it suffices to perform calculations for marginal distributions and add up the resulting values. Throughout the rest of this thesis we will use the property of additivity of a quantile function for comonotonic risks.

47 Chapter 2 Convex bounds Summary In many actuarial and financial problems the distribution of a sum of dependent random variables is of interest. In general, however, this distribution function can not be obtained analytically because of the complex underlying dependency structure. Kaas et al. (2000) and Dhaene et al. (2002a) propose a possible way out by considering upper and lower bounds for (the distribution function of) such a sum that allow explicit calculations of various actuarial quantities. When lower and upper bounds are close to each other, together they can provide reliable information about the original and more complex variable. In particular this technique is very useful to find reliable estimations of upper quantiles and stop-loss premiums. We summarize the main results for deriving lower and upper bounds and we construct sharper upper bounds for stop-loss premiums, based upon the traditional comonotonic bounds. The idea of convex upper and lower bounds is generalized to the case of scalar products of non-negative random variables. We apply the derived results to the case of general discounted cash flows, with stochastic payments. Numerous numerical illustrations are provided, demonstrating that the derived methodology gives very accurate approximations for the underlying distribution functions and the corresponding risk measures, like quantiles and stop-loss premiums. 2.1 Introduction In many financial and actuarial applications where a sum of stochastic terms is involved, the distribution of the quantity under investigation is too difficult to obtain. It is well-known that in general the distribution function 21

48 22 Chapter 2 - Convex bounds of a sum of dependent random variables cannot be determined analytically. Therefore, instead of aiming to calculate the exact distribution, we will look for approximations (bounds), in the convex order sense, with a simpler structure. The first approximation we will consider for the distribution function of a sum of dependent random variables is derived by approximating the dependence structure between the random variables involved by a comonotonic dependence structure. If the dependency structure between the summands of such a sum is strong enough, this upper bound in convex order performs reasonably well. The second approximation, which is derived by considering conditional expectations, partly takes 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, but the relative error of this approximation significantly outperforms the relative error of the upper bound. When lower and upper bounds are close to each other, together they can provide reliable information about the original and more complex variable. We emphasize that the bounds are in convex order, which does not mean that the real value always lies between these two approximations. In particular this technique is very useful to find reliable estimations of upper quantiles and stop-loss premiums. Section 2 recalls these theoretical results of Dhaene et al. (2002a). The lower bound approximates very accurate the real stop-loss premium, but the comonotonic upper bounds perform rather poorly. Therefore, in Section 3 we construct sharper upper bounds based upon the traditional comonotonic bounds. Making use of the ideas of Rogers and Shi (1995), the first upper bound is obtained as the comonotonic lower bound plus an error term. Next, this bound is refined by making the error term dependent on the retention in the stop-loss premium. Further, we study the case that the stop-loss premium can be decomposed into two parts. One part can be evaluated exactly, to another part, comonotonic bounds are applied. The application to the lognormal case is presented at the end of Section 3. In Section 4 we illustrate the accuracy of the comonotonic approximations by means of an application in the context of discounted reserves.

49 2.2. Convex bounds for sums of dependent random variables 23 Section 5 extends the methodology of Dhaene et al. (2002a,b) for deriving lower and upper bounds of a sum of dependent variables to the case of scalar products of independent random vectors. We derive a procedure for calculating the lower and upper bounds in case one of the vectors follows the multivariate lognormal law. In Section 6 we apply these results to the case of general discounted cash flows, with stochastic payments. Numerous numerical illustrations are provided, demonstrating that the derived methodology gives very accurate approximations for the underlying distribution functions and the corresponding risk measures. Section 2 and 3 in this chapter are mainly based on Hoedemakers, Darkiewicz, Deelstra, Dhaene & Vanmaele (2005). The results in Section 4 come from Hoedemakers & Goovaerts (2004). The generalization to the scalar product of two random vectors in Section 5 is based on Hoedemakers, Darkiewicz & Goovaerts (2005) and Section 6 is taken from Ahcan, Darkiewicz, Goovaerts & Hoedemakers (2005). 2.2 Convex bounds for sums of dependent random variables In the actuarial context one encounters quite often random variables of the type S = X 1 + X X n, where the terms X i are not mutually independent, but the multivariate distribution function of the random vector X = (X 1, X 2,..., X n ) is not completely specified and one only knows the marginal distribution functions of the random variables X i. In such cases, to be able to make decisions it may be helpful to find the dependence structure for the random vector (X 1,..., X n ) producing the least favorable aggregate claims S with given marginals. Therefore, given the marginal distributions of the terms in a random variable S = n X i, we shall look for a joint distribution with a smaller resp. larger sum, in the convex order sense. If S consists of a sum of random variables (X 1,..., X n ), replacing the joint distribution of (X 1,..., X n ) by the comonotonic joint distribution yields an upper bound for S in the convex order. On the other hand, applying conditioning to S provides us a lower bound. Finally, if we combine

50 24 Chapter 2 - Convex bounds both ideas, then we end up with an improved upper bound. This is formalized in the following theorem, which is taken from Dhaene et al. (2002a) and Kaas et al. (2000). Theorem 8 (Bounds for a sum of random variables). Consider a sum of random variables S = X 1 + X X n and define the following related random variables: S l = E[X 1 Λ] + E[X 2 Λ] E[X n Λ], (2.1) S c = F 1 X 1 (U) + F 1 X 2 (U) F 1 X n (U), (2.2) S u = F X 1 Λ (U) + FX 2 Λ (U) FX n Λ (U), (2.3) with U a U(0,1) random variable and Λ an arbitrary random variable. Here F 1 X i Λ (U) is the notation for the random variable f i(u, Λ), with the function f i defined by f i (u, λ) = F 1 X i Λ=λ (u). The following relations then hold: S l cx S cx S u cx S c. Proof. See e.g. Dhaene et al. (2002a). The comonotonic upper bound changes the original copula, but keeps the marginal distributions unchanged. The comonotonic lower bound on the other hand, changes both the copula and the marginals involved. Intuitively, one can expect that an appropriate choice of the conditioning variable Λ will lead to much better approximations compared to the upper bound. The upper bound S c is the most dangerous sum of random variables with the same marginal distributions as the original terms X j in S. Indeed, the upper bound S c now consists of a sum of comonotonic variables all depending on the same random variable U. If one can find a conditioning random variable Λ with the property that all random variables E[X j Λ] are non-increasing functions of Λ (or all are non-decreasing functions of Λ), then the lower bound S l = n j=1 E[X j Λ] is also a sum of n comonotonic random variables. We recall from Dhaene et al. (2002a) and the references therein the procedures for obtaining the lower and upper bounds for stop-loss premiums of sums S of dependent random variables by using the notion of comonotonicity.

51 2.2. Convex bounds for sums of dependent random variables The comonotonic upper bound As proven in Dhaene et al. (2002a), the convex-largest sum of the components of a random vector with given marginals is obtained by the comonotonic sum S c = X1 c + Xc Xc n with S c d = F 1 X i (U), (2.4) where U denotes in the following a U(0, 1) random variable. Kaas et al. (2000) have proved that the inverse distribution function of a sum of comonotonic random variables is simply the sum of the inverse distribution functions of the marginal distributions. See also Theorem 7. Therefore, given the inverse functions F 1 X i, the cumulative distribution function of S c = X1 c + Xc Xc n can be determined as follows: F S c(x) = sup {p (0, 1) F S c(x) p} = sup { p (0, 1) F 1 S (p) x } { c } = sup p (0, 1) F 1 X i (p) x. (2.5) Moreover, in case of strictly increasing and continuous marginals, the cdf F S c(x) is uniquely determined by F 1 S (F c S c (x)) = F 1 X i (F S c (x)) = x, F 1+ S (0) < x < F 1 c S (1). c (2.6) Hereafter we restrict ourselves to this case of strictly increasing and continuous marginals. In the following theorem Dhaene et al. (2000) have proved that the stop-loss premiums of a sum of comonotonic random variables can easily be obtained from the stop-loss premiums of the terms. Theorem 9 (Stop-loss premium of comonotonic sum). The stop-loss premium, denoted by π cub (S, d), of the sum S c of the components of the comonotonic random vector (X1 c, Xc 2,..., Xc n) at retention d is given by π cub (S, d) = ( π X i, F 1 ( X FS i c(d) )), ( F 1+ S (0) < d < F 1 c S (1) ). c (2.7)

52 26 Chapter 2 - Convex bounds If the only information available concerning the multivariate distribution function of the random vector (X 1,..., X n ) consists of the marginal distribution functions of the X i, then the distribution function of S c = F 1 X 1 (U) + F 1 X 2 (U) + + F 1 X n (U) is a prudent choice for approximating the unknown distribution function of S = X 1 + X X n. It is a supremum in terms of convex order. It is the best upper bound that can be derived under the given conditions. We end this part about the comonotonic upper bound by summarizing the main advantages of using S c = X1 c + Xc Xc n instead of S = X 1 + X X n : Replacing the distribution function of S by the distribution function of S c is a prudent strategy in the framework of utility theory: the real distribution function is replaced by a less attractive one. The random variables S and S c have the same expected value. As these random variables are ordered in the convex order sense, we have that every moment of order 2k (k = 1, 2,...) of S is smaller than the corresponding moment of S c. Many actuarially relevant quantities reflect convex order, for instance both the ruin probability and the Lundberg upper bound for it increase when the claim size distribution is replaced by a convex larger one. Other examples are zero-utility premiums such as the exponential premium, and of course stop-loss premiums for any retention d. The cdf of S c can easily be obtained; essentially, S c has a onedimensional distribution, depending only on the random variable U. The distribution function of S can only be obtained if the dependency structure is known. Even if this dependency structure is known, it can be hard to determine the distribution function of S from it. The stop-loss premiums of S c follow from stop-loss premiums of the marginal random variables involved. Computing the stop-loss premiums of S can only be carried out when the dependency structure is known, and in general requires n integrations to be performed The improved comonotonic upper bound Let us now assume that we have some additional information available concerning the stochastic nature of (X 1,..., X n ). More precisely, we as-

53 2.2. Convex bounds for sums of dependent random variables 27 sume that there exists some random variable Λ with a given distribution function, such that we know the conditional cumulative distribution functions, given Λ = λ, of the random variables X i, for all possible values of λ. In fact, Kaas et al. (2000) define the improved comonotonic upper bound S u as S u = F X 1 Λ (U) + FX 2 Λ (U) + + FX n Λ (U). (2.8) In order to obtain the distribution function of S u, observe that given the event Λ = λ, the random variable S u is a sum of comonotonic random variables. Hence, F 1 S u Λ=λ (p) = Given Λ = λ, the cdf of S u is defined by { F S u Λ=λ(x) = sup The cdf of S u then follows from F S u(x) = F 1 X i Λ=λ (p), p (0, 1). p (0, 1) + F 1 X i Λ=λ (p) x F S u Λ=λ(x) df Λ (λ). If the marginal cdf s F Xi Λ=λ are strictly increasing and continuous, then F S u Λ=λ(x) is a solution to F 1 ( X i Λ=λ FS u Λ=λ(x) ) ( ) = x, x F 1+ 1 S u Λ=λ (0), FS u Λ=λ (1). ( ) (2.9) In this case, we also find that for any d F 1+ 1 S u Λ=λ (0), FS u Λ=λ (1) : E [ (S u d) + Λ = λ ] = [ ( E X i F 1 ( X i Λ=λ FS u Λ=λ(d) )) ] Λ = λ, + from which the stop-loss premium at retention d of S u, which we will denote by π icub (S, d, Λ), can be determined by weighted integration with respect to λ over the real line. }.

54 28 Chapter 2 - Convex bounds The lower bound Let X = (X 1,..., X n ) be a random vector with given marginal cumulative distribution functions F X1, F X2,..., F Xn. Let us now assume that we have some additional information available concerning the stochastic nature of (X 1,..., X n ). More precisely, we assume that there exists some random variable Λ with a given distribution function, such that we know the conditional distribution, given Λ = λ, of the random variables X i, for all possible values of λ. We recall from Kaas et al. (2000) that a lower bound, in the sense of convex order, for S = X 1 + X X n is S l = E [S Λ]. (2.10) This idea can also be found in Rogers and Shi (1995) for the continuous and lognormal case. Let us further assume that the random variable Λ is such that all E [X i Λ] are non-decreasing and continuous functions of Λ, then S l is a comonotonic sum. The quantiles of the lower bound S l then follow from F 1 S l (p) = F 1 E[X i Λ] (p) = E [ X i Λ = F 1 Λ (p)], p (0, 1), (2.11) and the cdf of S l is according to (2.5) given by { F S l(x) = sup p (0, 1) E [ } X i Λ = F 1 Λ (p)] x. (2.12) Using Theorem 9, the stop-loss premiums with retention d read ( F 1+ (0) S l < d < F 1 (1) ) S l π lb (S, d, Λ) = ( π E[X i Λ], F 1 E[X i Λ]( FS l(d) )). When in addition the cdf s of the random variables E [X i Λ] are strictly increasing and continuous, then the cdf of S l is also strictly increasing and continuous, and we get analogously to (2.6) for all x ( F 1+ (0), F 1 (1) ), S l S l F 1 E[X i Λ]( FS l(x) ) = x [ E X i Λ = F 1 ( Λ FS l(x) )] = x, (2.13)

55 2.2. Convex bounds for sums of dependent random variables 29 which unambiguously determines the cdf of the convex order lower bound S l for S. In order to derive the above equivalence, we used the results of Lemma 1. Invoking Theorem 9, the stop-loss premium π lb (S, d, Λ) of S l can be computed as: π lb (S, d, Λ) = ( π E [ X i Λ ], E [ X i Λ = F 1 ( Λ FS l(d) )]), (2.14) which holds for all retentions d ( F 1+ (0), F 1 (1) ). S l S l So far, we considered the case that all E [X i Λ] are non-decreasing functions of Λ. The case where all E [X i Λ] are non-increasing and continuous functions of Λ also leads to a comonotonic vector ( E [X 1 Λ],..., E [X n Λ] ), and can be treated in a similar way. In case the cumulative distribution functions of the random variables E [X i Λ] are not continuous nor strictly increasing or decreasing functions of Λ, then the stop-loss premiums of S l, which is not comonotonic anymore, can be determined as follows : ( + ) π lb (S, d, Λ) = E [X i Λ = λ] d df Λ (λ) Moments based approximations The lower and upper bounds can be considered as approximations for the distribution of a sum S of random variables. On the other hand, any convex combination of the stop-loss premiums of the lower bound S l and the upper bounds S c or S u also could serve as an approximation for the stop-loss premium of S. Since the bounds S l and S c have the same mean as S, any random variable S m defined by its stop-loss premiums π m (S, d, Λ) = zπ lb (S, d, Λ) + (1 z)π cub (S, d), 0 z 1, will also have the same mean as S. By taking the (right-hand) derivative we find F S m(x) = zf S l(x) + (1 z)f S c(x), 0 z 1, so the distribution function of the approximation can be calculated fairly easily. By choosing the optimal weight z, we want S m to be as close as +

56 30 Chapter 2 - Convex bounds possible to S. In Vyncke et al. (2004) z is chosen as z = Var[Sc ] Var[S] Var[S c ] Var[S l ]. (2.15) This choice does not depend on the retention and it leads to equal variances Var[S m ] = Var[S]. As an alternative one could consider the improved upper bound S u and define a second approximation as follows π m2 (S, d, Λ) = zπ lb (S, d, Λ) + (1 z)π icub (S, d, Λ), now with z = Var[Su ] Var[S] Var[S u ] Var[S l ]. 2.3 Upper bounds for stop-loss premiums One of the most important tasks of actuaries is to assess the degree of dangerousness of a risk X either by finding the (approximate) distribution or at least by summarizing its properties quantitatively by means of risk measures to determine an insurance premium or a sufficient reserve with solvency margin. A stop-loss premium π(x, d) = E[(X d) + ] = E[max(0, X d)] is one of the most important risk measures. The retention d is usually interpreted as an amount retained by an insured (or an insurer) while an amount X d is ceded to an insurer (or a reinsurer). In this case π(x, d) has a clear interpretation as a pure insurance (reinsurance) premium. Another practical application of stop-loss premiums is the following: Suppose that a financial institution faces a risk X to which a capital K is allocated. Then the residual risk R = (X K) + is a quantity of concern to the society and regulators. Indeed, it represents the pessimistic case when the random loss X exceeds the available capital. The value E[R] is often referred to as the expected shortfall as explained in Subsection 1.1.2, with K a VaR at some level. It is not always straightforward to compute stop-loss premiums. In the actuarial literature a lot of attention has been devoted to determine bounds for stop-loss premiums in case only partial information about the

57 2.3. Upper bounds for stop-loss premiums 31 claim size distribution is available (e.g. De Vylder & Goovaerts (1982), Jansen et al. (1986), Hürlimann (1996, 1998), among others). Other types of problems appear in the case of sums of random variables S = X 1 + +X n when full information about marginal distributions is available but the dependency structure is not known. In the previous section it is explained how the upper bound S c of the sum S in so called convex order sense can be calculated by replacing the unknown joint distribution of the random vector (X 1, X 2,..., X n ) by the most dangerous comonotonic joint distribution. One can also obtain a lower bound S l through conditioning. Such an approach allows to determine analytical bounds for stop-loss premiums π lb (S, d, Λ) π(s, d) π cub (S, d). 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 section 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 one the hand the method of conditioning as in Curran (1994) and in Rogers & Shi (1995), and on the other hand the upper and lower bounds for stoploss premiums of sums of dependent random variables as explained in the previous subsection Upper bounds based on lower bound plus error term Following the ideas of Rogers and Shi (1995), we derive an upper bound based on the lower bound S l. Lemma 5. For any random variable X we have the following inequality E[X + ] E[X] Var1/2 (X). (2.16) Proof. Define X + as follows X + := max( X, 0) = ( X) + = min(x, 0).

58 32 Chapter 2 - Convex bounds Using Jensen s inequality twice we have 0 E[X + ] E[X] + = 2{ 1 (E[X+ ) ( ] E[X] + + E[X + ] E[X] ) } + = 1 { } E[X + + X+ 2 ] E[X] + E[X] + = 1 { } E[ X ] E[X] 2 1 E[ X E[X] ] Var1/2 (X) Applying now Proposition 5 for any random variable Y and Z: 0 E [ ] 1 [ ] E [Y + Z] E [Y Z] + 2 E Var[Y Z] (2.17) to the case of Y being S d and Z being our conditioning variable Λ, we obtain an error bound ] 0 E [E [(S d) + Λ] (S l d) + 1 [ ] 2 E Var[S Λ], (2.18) which is only useful if the retention d is strictly positive. Consequently, we find as upper bound for the stop-loss premium of S with π eub (S, d, Λ) given by π(s, d) π eub (S, d, Λ), (2.19) π eub (S, d, Λ) = π lb (S, d, Λ) E [ Var[S Λ] ]. (2.20) The second term on the right hand side takes the form [ ] [ ( E Var[S Λ] = E E [ S 2 Λ ] ( E[S Λ] ) ) ] 2 1/2 (2.21) [ ( = E E [X i X j Λ] ( S l) ) ] 1/2 2, j=1 and once the distributions of X i and Λ are specified and known, it can be written out more explicitly.

59 2.3. Upper bounds for stop-loss premiums Bounds by conditioning through decomposition of the stop-loss premium Decomposition of the stop-loss premium In this part we show how to improve the bounds introduced in Section 2.2 and Subsection By conditioning S on some random variable Λ, the stop-loss premium can be decomposed in two parts, one of which can either be computed exactly or by using numerical integration, depending on the distribution of the underlying random variable. For the remaining part we first derive a lower and an upper bound based on comonotonic risks, and another upper bound equal to that lower bound plus an error term. This idea of decomposition goes back at least to Curran (1994). By the tower property for conditional expectations the stop-loss premium π(s, d) with S = n X i equals E [ E[(S d) + Λ] ], for every conditioning variable Λ, say with cdf F Λ. If in addition there exists a d Λ such that Λ d Λ implies that S d, we can decompose the stop-loss premium of S as follows π(s, d) = dλ E[(S d) + Λ = λ]df Λ (λ) + + d Λ E[S d Λ = λ]df Λ (λ) =: I 1 + I 2. (2.22) Notice that the other case (Λ d Λ implies that S d) can be treated in a similar way with the appropriate integration bounds. In practical applications the existence of such a d Λ depends on the actual form of S and Λ = λ. The second integral can further be simplified to I 2 = + d Λ E [ X i Λ = λ ] ( df Λ (λ) d(1 F Λ dλ ) ), (2.23) and can be written out explicitly if the bivariate distribution of (X i, Λ) is known for all i. Deriving bounds for the first part I 1 in decomposition (2.22) and adding up to the exact part (2.23) gives us the bounds for the stop-loss premium.

60 34 Chapter 2 - Convex bounds Lower bound By means of Jensen s inequality, the first integral I 1 of (2.22) can be bounded below: I 1 dλ ( E[S Λ = λ] d ) + df Λ(λ) = dλ ( ) E[X i Λ = λ] d df Λ(λ). + (2.24) By adding the exact part (2.23) and introducing notation (2.10), we end up with the inequality of Section 2.2.3: π(s, d) π lb (S, d, Λ). When S l is a sum of n comonotonic risks we can apply (2.14) which holds even when we do not know or find a d Λ. When S l is not comonotonic we use the decomposition π lb (S, d, Λ) = dλ ( + + d Λ ) E[X i Λ = λ] d df Λ(λ) + E [ X i Λ = λ ] ( df Λ (λ) d(1 F Λ dλ ) ). Upper bound based on lower bound In this part we improve the bound (2.19) by applying (2.17) to (2.24): 0 E [ E[(S d) + Λ] (S l ] d) + dλ ( = E [ (S d) + Λ = λ ] ( E[S Λ = λ] d ) ) df + Λ (λ) dλ ( Var[S Λ = λ] ) 1 2 df Λ (λ) (2.25) ( E [ Var[S Λ]I (Λ<dΛ )] ) 1 ( 2 E [ ] ) 1 2 I (Λ<dΛ ) =: ε(d Λ ), (2.26) where Hölder s inequality has been applied in the last inequality. We will denote this upper bound by π deub (S, d, Λ). So we have that π deub (S, d, Λ) = π lb (S, d, Λ) + ε(d Λ ). (2.27)

61 2.3. Upper bounds for stop-loss premiums 35 We remark that the error bound (2.18), and hence also the upper bound π eub (S, d, Λ), is independent of d Λ and corresponds to the limiting case of (2.25) where d Λ equals infinity. Obviously, the error bound (2.25) improves the error bound (2.18). In practical applications, the additional error introduced by [ Hölders inequality turns out to be much smaller than Var[S Λ] ] the difference 1 2 E ε(d Λ ) Partially exact/comonotonic upper bound We bound the first term I 1 of (2.22) above by replacing S Λ = λ by its comonotonic upper bound S u (in convex order sense): dλ E [ (S d) + Λ = λ ] df Λ (λ) dλ E [ (S u d) + Λ = λ ] df Λ (λ). (2.28) Adding (2.28) to the exact part (2.23) of the decomposition (2.22) results in the so-called partially exact/comonotonic upper bound for a stop-loss premium. We will use the notation π pecub (S, d, Λ) to indicate this upper bound. It is easily seen that π pecub (S, d, Λ) π icub (S, d, Λ), while for two distinct conditioning variables Λ 1 and Λ 2 it does not necessarily holds that π pecub (S, d, Λ 1 ) π icub (S, d, Λ 2 ) The case of a sum of lognormal random variables We show how to apply our results to the case of sums of lognormal distributed random variables. Such sums are widely encountered in practice, both in actuarial science and in finance. Typical examples are present values of future cash flows with stochastic (Gaussian) returns (see Dhaene et al. (2002b)), Asian options (see e.g. Simon et al. (2000), Vanmaele et al. (2004b) and Albrecher et al. (2005)) and basket options (see Deelstra et al. (2004) and Vanmaele et al. (2004a)).

62 36 Chapter 2 - Convex bounds We assume that X i = α i e Z i with Z i N(E[Z i ], σz 2 i ) and α i R. We develop the expressions for the lower and upper bounds for the following sum S S = X i = α i e Z i. (2.29) In this case the stop-loss premium π(x i, d i ) with some retention d i is wellknown from the following lemma. Lemma 6 (Stop-loss premium of lognormal random variable). Let X be a lognormal random variable of the form αe Z with Z N(E[Z], σ 2 Z ) and α R. Then the stop-loss premium with retention d equals for αd > 0 where σ2 µ+ π(x, d) = sign(α)e 2 Φ ( ) ( ) sign(α)b 1 dφ sign(α)b2, (2.30) The case αd < 0 is trivial. µ = ln α + E[Z] σ = σ Z b 1 = µ + σ2 ln d σ b 2 = b 1 σ. (2.31) We now consider a normally distributed random variable Λ. The following results are analogous to Theorem 1 in Dhaene et al. (2002b). Theorem 10 (Bounds for a sum of lognormal random variables). Let S be given by (2.29) and consider a normally distributed random variable Λ which is such that (Z i, Λ) is bivariate normally distributed for all i. Then the distributions of the lower bound S l, the improved comonotonic upper bound S u and the comonotonic upper bound S c are given by S l = S u = S c = α i e E[Z i]+r i σ Zi Φ 1 (V )+ 2(1 r 1 i 2 )σz 2 i, (2.32) α i e E[Z i]+r i σ Zi Φ 1 (V )+sign(α i ) 1 ri 2 σ Z i Φ 1 (U), (2.33) α i e E[Z i]+sign(α i )σ Zi Φ 1 (U), (2.34)

63 2.3. Upper bounds for stop-loss premiums 37 ( ) Λ E[Λ] where U and V = Φ are mutually independent U(0,1) random σ Λ variables, and r i, i = 1,..., n, are correlations defined by r i = Corr (Z i, Λ) = Cov [Z i, Λ] σ Zi σ Λ. If, for all i sign(α i ) = sign(r i ), or, for all i sign(α i ) = sign(r i ) with r i 0, then S l is comonotonic. Proof. See Dhaene et al. (2002b) Comonotonic upper bound The quantile function of S c results from (1.20) in Theorem 7 and is given by F 1 S (p) = α c i e E[Z i]+sign(α i )σ Zi Φ 1 (p), p (0, 1). (2.35) Since the cdf s F Xi are strictly increasing and continuous, it follows from (2.6) and (2.34) that for x ( F 1+ S (0), F 1 c S (1) ), the cdf of the comonotonic c sum F S c(x) can be found by solving ( ) α i e E[Z i]+sign(α i )σ Zi Φ 1 F S c (x) = x. Combination of Theorem 9 and Lemma 6 yields the following expression for the stop-loss premium of S c at retention d with F 1+ S (0) < d < F 1 c S (1): c π cub (S, d) = α i e E[Z i]+ σ 2 Z i [ 2 Φ sign(α i )σ Zi Φ 1( F S c(d) )] d ( 1 F S c(d) ). Improved comonotonic upper bound We now determine the cdf of S u and the stop-loss premium π icub (S, d, Λ), where we condition on a normally distributed random variable Λ or equivalently on the U(0, 1) random variable introduced in Theorem 10: ( ) Λ E [Λ] V = Φ. The conditional probability F S u V =v(x) also denoted by F S u(x V = v), is the cdf of a sum of n comonotonic random variables and follows for σ Λ

64 38 Chapter 2 - Convex bounds F 1+ S u V =v from: 1 (0) < x < FS u V =v (1), according to (2.9) and (2.33), implicitly ( ) α i e E[Z i]+r i σ Zi Φ 1 (v)+sign(α i ) 1 ri 2 σ Z i Φ 1 F S u (x V =v) = x. (2.36) The cdf of S u is then given by F S u(x) = 1 0 F S u V =v(x)dv. We now look for an expression for the stop-loss premium at retention d with F 1+ 1 S u V =v (0) < d < FS u V =v (1) for Su : π icub (S, d, Λ) = = 1 0 E [ (S u d) + V = v ] dv 1 0 [ (F 1 E X i Λ (U V = v) d ) ] i dv + with d i = F 1 X i Λ( FS u(d V = v) V = v ) and with U a random variable which is uniformly distributed on (0, 1). Since sign(α i )F 1 X i Λ (U V = v) follows a lognormal distribution with mean and standard deviation: µ v (i) = ln α i + E [Z i ] + r i σ Zi Φ 1 (v), σ v (i) = 1 ri 2 σ Z i, one obtains that [ d i = α i exp E[Z i ] + r i σ Zi Φ 1 (v) + sign(α i ) 1 ri 2 σ Z i Φ 1( F S u V =v(d) )]. Formula (2.30) then yields E [ (S u d) + V = v ] = [ sign(α i )e µv(i)+ σ2 v (i) 2 Φ ( ) sign(α i )b i,1 di Φ ( ) ] sign(α i )b i,2, with, according to (2.31), b i,1 = µ v(i) + σv(i) 2 ln d i, b i,2 = b i,1 σ v (i). σ v (i)

65 2.3. Upper bounds for stop-loss premiums 39 Substitution of the corresponding expressions and integration over the interval [0, 1] leads to the following result π icub (S, d, Λ) = α i e E[Z i]+ 1 2 σ2 Z i (1 ri 2 ) 1 e r iσ Zi Φ 1 (v) Φ 0 ( sign(α i ) 1 ri 2 σ Z i Φ 1 ( F S u V =v(d) )) dv d ( 1 F S u(d) ). (2.37) Lower bound In this subsection, we study the case that, for all i, sign(α i ) = sign(r i ) when r i 0. For simplicity we take all α i 0 and assume that the conditioning variable Λ is normally distributed and has the right sign such that the correlation coefficients r i are all positive. These conditions ensure that S l is the sum of n comonotonic random variables. The case that, for all i, sign(α i ) = sign(r i ) when r i 0 can be dealt with in an analogous way. The quantile function of S l results from (1.20) in Theorem 7 and is given by F 1 S l (p) = α i e E[Z i]+r i σ Zi Φ 1 (p)+ 2(1 r 1 i 2 )σz 2 i, p (0, 1). (2.38) Since by our assumptions E[X i Λ] is increasing, we can obtain F S l(x) according to (2.13) and (2.32) from ( ) α i e E[Z i]+r i σ Zi Φ 1 F S l(x) + 2(1 r 1 i 2 )σz 2 i = x. (2.39) Moreover as S l is the sum of n lognormally distributed random variables, the stop-loss premium at retention d (> 0) can be expressed explicitly by invoking Theorem 9 and Lemma 6: π lb (S, d, Λ) = [ α i e E[Z i]+ 1 2 σ2 Z i Φ r i σ Zi Φ 1( F S l(d) )] d ( 1 F S l(d) ). (2.40)

66 40 Chapter 2 - Convex bounds Upper bound based on lower bound From (2.21) we obtain that [ ] E Var[S Λ] = { + j=1 E [ X i X j Λ = λ ] ( E[S Λ = λ] ) } df Λ (λ). (2.41) Now consider the first term in the right hand side of (2.41). Because of the properties of lognormally distributed random variables, the product of lognormals is again lognormal if the underlying vector is multivariate normal distributed, and conditioning a lognormal variate on a normal variate yields a lognormally distributed variable. We can proceed by denoting Z ij = Z i + Z j with E[Z ij ] = E[Z i ] + E[Z j ] and σ 2 Z ij = σ 2 Z i + σ 2 Z j + 2σ Zi Z j, where σ Zi Z j := Cov[Z i, Z j ]. Note that r ij = Cov[Z ij, Λ] σ Zij σ Λ = Cov [Z i, Λ] σ Zij σ Λ + Cov [Z j, Λ] σ Zij σ Λ = σ Z i σ Zij r i + σ Z j σ Zij r j. Conditionally, given Λ = λ, the random variable Z ij is normally distributed with parameters µ(i, j) = E [Z ij ]+r ij σ Λ σ Zij ( ) λ E[Λ] and σ 2 (i, j) = ( ) 1 r 2 ij σ 2 Zij. Hence, conditionally, given Λ = λ, the random variable e Z ij is lognormally distributed with parameters µ(i, j) and σ 2 (i, j). As E [ e Z ij Λ = λ ] = e µ(i,j)+ 1 2 σ2 (i,j), we find E [ e Z ij Λ ] = e E[Z ij]+r ij σ Zij Φ 1 (V )+ 1 2(1 r 2 ij)σ 2 Z ij, ( where the random variable V = Φ Λ E[Λ] the interval (0, 1). σ Λ ) is uniformly distributed on

67 2.3. Upper bounds for stop-loss premiums 41 j=1 Thus, the first term in (2.41) equals E[X i X j Λ] = j=1 ( α i α j exp E[Z ij ] + r ij σ Zij Φ 1 (V ) + 1 ) ( ) 1 r 2 2 ij σ 2 Zij, (2.42) while the second term consists of (2.32). Hence (2.41) can be written out explicitly and by using (2.20), we have that the upper bound (2.19) is given by π eub (S, d, Λ) = α i e E[Z i]+ 1 2 σ2 Z i Φ [r i σ Zi Φ 1( F S l(d) )] d ( 1 F S l(d) ) 1 0 { j=1 ( α i α j e E[Z ij]+r ij σ Zij Φ 1 (v)+ 1 2 (1 r2 ij )σ2 Z ij α i e E[Z i]+r i σ Zi Φ 1 (v)+ 1 2(1 r 2 i )σ 2 Z i ) 2 } 1 2 dv. Bounds by conditioning through decomposition of stop-loss premium In this part we apply the theory of Subsection to the sum of lognormal random variables (2.29). We give here the analytical expressions for the two upper bounds π deub (S, d, Λ) and π pecub (S, d, Λ). For more details concerning the calculation of the bounds the reader is referred to the last section of this chapter. The following auxiliary result is needed in order to write out the bounds explicitly. Lemma 7. For any constant a R and any normally distributed random variable Λ dλ e aφ 1 (v) df Λ (λ) = e a2 2 Φ(d Λ a), (2.43) where d Λ = d Λ E[Λ] σ Λ and Φ 1 (v) = λ E[Λ] σ Λ.

68 42 Chapter 2 - Convex bounds Lower bound Note that the lower bound via the decomposition equals the lower bound without the decomposition. So the lower bound in the lognormal and comonotonic case is given by expression (2.40). Upper bound based on lower bound The upper bound (2.27) can be written out explicitly as follows π deub (S, d, Λ) = α i e E[Z i]+ 1 2 σ2 Z i Φ [ r i σ Zi Φ 1 (F S l(d)) ] d (1 F S l(d)) Φ(d Λ) 1/2 { α i α j e E[Z ij]+ 1 2 (σ2 Z +σ 2 i Z ) j j=1 Φ ( d Λ ( r i σ Zi + r j σ Zj )) ( e σ Zi Z j e σ Z i σ Zj r i r j ) } 1 2.(2.44) Proof. See Section 2.7. Partially exact/comonotonic upper bound The partially exact/comonotonic upper bound of Subsection is given by π pecub (S, d, Λ) = { α i e E[Z i]+ 1 2 σ2 Z i (1 ri 2 r i ) 2 σ2 Z i e 2 Φ(r i σ Zi d Λ) + d ( 1 Φ(d Λ ) Proof. See Section Φ(d Λ ) 0 e r iσ Zi Φ 1 (v) ( Φ sign(α i ) 1 ri 2 σ Z i Φ 1 ( F S u V =v(d) )) dv F S u V =v(d)dv ). (2.45) }

69 2.3. Upper bounds for stop-loss premiums 43 Choice of the conditioning variable If X cx Y, and X and Y are not equal in distribution, then Var[X] < Var[Y ] must hold. An equality in variance would imply that X = d Y. This shows that if we want to replace S by the convex smaller S l, the best approximations will occur when the variance of S l is as close as possible to the variance of S. Hence we should choose Λ such that the goodness-offit expressed by the ratio z = Var[Sl ] is as close as possible to 1. Of course Var[S] one can always use numerical procedures to optimize z but this would outweigh one of the main features of the convex bounds, namely that the different relevant actuarial quantities (quantiles, stop-loss premiums) can be easily obtained. Having a ready-to-use approximation that can be easily implemented and used by all kind of end-users is important from a business point of view. Notice that the expected values of the random variables S, S c and S l are all equal: E[S] = E[S l ] = E[S c ] = while their variances are given by Var[S] = Var[S l ] = j=1 j=1 α i e E[Z i]+ 1 2 σ2 Z i, (2.46) α i α j e E[Z i]+e[z j ]+ 1 2 (σ2 Z +σ 2 )( i Zj e σ Z i Z j 1 ), (2.47) α i α j e E[Z i]+e[z j ]+ 1 2 (σ2 Z +σ 2 i Zj ) ( e r ir j σ Zi σ Zj 1 ) (2.48) and Var[S c ] = α i α j e E[Z i]+e[z j ]+ 1 2 (σ2 Z +σ 2 i Zj ) ( e σ Z i σ Zj 1 ), (2.49) j=1 respectively. We propose here three conditioning random variables. The first two are linear combinations of the random variables Z i : Λ = γ i Z i, (2.50)

70 44 Chapter 2 - Convex bounds for particular choices of the coefficients γ i. Kaas et al. (2000) propose the following choice for the parameters γ i when computing the lower bound S l : γ i = α i e E[Z i], i = 1,..., n. (2.51) This choice makes Λ a linear transformation of a first order approximation to S. This can be seen from the following derivation: S = α i e E[Z i] +(Z i E[Z i ]) = C + α i e E[Zi] (1 + Z i E [Z i ]) α i e E[Zi] Z i, (2.52) where C is constant. Hence S l will be close to S, provided (Z i E[Z i ]) is sufficiently small, or equivalently, σ 2 Z i is sufficiently small. One intuitively expects that for this choice for Λ, E [ Var[S Λ] ] is small and, since Var[S] = E [ Var[S Λ] ] + Var[S l ], this exactly means that one expects the ratio z = Var[Sl ] to be close to one. Var[S] A possible decomposition variable is in this case given by d Λ = d C = d α i e E[Zi] (1 E [Z i ]). Using the property that e x 1+x and (2.52), we have that Λ d Λ implies that S d. A second conditioning variable is proposed by Vanduffel et al. (2004). They propose the following choice for the parameters γ i when computing the lower bound S l : γ i = α i e E[Z i]+ 1 2 σ2 Z i, i = 1,..., n. (2.53) In this case the first order approximation of the variance of S l will be

71 2.3. Upper bounds for stop-loss premiums 45 maximized. Indeed, from (2.48) we find that Var[S l ] α i α j e E[Z i]+e[z j ]+ 1 2 (σ2 Z +σ 2 i Zj ) (r i r j σ Zi σ Zj ) = j=1 ( ) α i α j e E[Z i]+e[z j ]+ 1 2 (σ2 Z +σ 2 i Zj ) Cov[Zi, Λ]Cov[Z j, Λ] Var[Λ] ( [ n ]) Cov α i e E[Z i] σ2 Z i Z i, Λ j=1 = Var[Λ] ( ( 2 [ ] = Corr α i e E[Z i]+ 1 2 σ2 Z i Z i, Λ)) Var α i e E[Z i]+ 1 2 σ2 Z i Z i. Hence, the first order approximation of Var[S l ] is maximized when Λ is given by Λ = α i e E[Z i]+ 1 2 σ2 Z i Z i. (2.54) One can easily prove that the first order approximation for Var[S l ] with Λ given by (2.54) is equal to the first order approximation of Var[S]. This observation gives an additional indication that this particular choice for Λ will provide a good fit. For this maximal variance conditioning variable a possible choice for d Λ is given by ( d Λ = d α i e E[Z i]+ 1 2 σ2 Z i 1 E [Z i ] 1 ) 2 σ2 Z i. (2.55) A third conditioning variable is based on the standardized logarithm of the geometric average G = ( n S)1/n as in Nielsen and Sandmann (2003) ln G E[ln G] n Λ = = (Z i E[Z i ]) Var[ln G] Var[ n Z. i] Using the fact that the geometric average is not greater than the arithmetic average, a possible decomposition variable is here given by so that Λ d Λ implies that S d. d Λ = n ln ( d n) n E[Z i] Var[ n Z, i]

72 46 Chapter 2 - Convex bounds Generalization to sums of lognormals with a stochastic time horizon Suppose that S is a sum of lognormal variables with a stochastic time horizon T T S = α i e Z i, with α i R, T a random variable with life time probability distribution F T (t) and Z i N(E[Z i ], σz 2 i ) independent of T. Using the tower property for conditional expectations, we can calculate the stop-loss premium of S as follows ( T ) π(s, d) = π α i e Z i, d with [( T ) ]] = E T [E α i e Z i d T + ( j ) = Pr[T = j]π α i e Z i, d = j=1 Pr[T = j] π(s j, d), (2.56) j=1 S j := j α i e Z i. Notice that in practical applications the infinite time horizon is often replaced by a finite number. In this part of the thesis, the choice of Λ will be dependent on the time horizon n. To indicate this dependence, we introduce the notation Λ n for the used conditioning variable Λ. It is straightforward to obtain a lower bound, denoted as π lb (S, d, Λ), by looking at the combination π lb (S, d, Λ) = Pr[T = j] π lb (S j, d, Λ j ), j=1 with Λ = Λ 1, Λ 2,... and π lb (S j, d, Λ j ) given by (2.40) for n = j. The same reasoning can be followed for obtaining the comonotonic upper bound

73 2.4. Application: discounted loss reserves 47 π cub (S, d), the improved comonotonic upper bound π icub (S, d, Λ) and the partially exact/comonotonic upper bound π pecub (S, d, Λ). For each term π(s j, d) in the sum (2.56) we can take the minimum of two or more of the above defined upper bounds. We propose two upper bounds based on this simple idea. The first bound takes each time the minimum of the error term (2.18) independent of the retention and the error term (2.26) dependent on the retention. Combining this with the stop-loss premium of the lower bound S l results in the following upper bound π emub (S, d, Λ) = j=1 ( [ ] ) 1 Pr[T = j] min 2 E Var[S j Λ j ], ε(d Λj ) + π lb (S, d, Λ). Calculating for each term the minimum of all the presented upper bounds π min (S, d, Λ) = Pr[T = j] j=1 ( ) min π cub (S j, d), π icub (S j, d, Λ j ), π pecub (S j, d, Λ j ), π emub (S j, d, Λ j ), will of course provide the best possible upper bound. Remark that π emub (S j, d, Λ j ) = π lb (S j, d, Λ j ) + min ( [ ] ) 1 2 E Var[S j Λ j ], ε(d Λj ). 2.4 Application: discounted loss reserves Loss reserving deals with the determination of the random present value of future payments. Since this amount is very important for an insurance company and its policyholders, these inherent uncertainties are no excuse for providing anything less than a rigorous scientific analysis. Since the reserve is a provision for the future payments, the estimated loss reserve should reflect the time value of money. At the same time, it may be necessary or desirable for those reserves to contain a security margin that produces p 100% confidence in their adequacy, where p is a suitably high number.

74 48 Chapter 2 - Convex bounds In many situations knowledge of the d.f. of this discounted reserve is useful, for example dynamic financial analysis, assessing profitability and pricing, identifying risk based capital needs, loss portfolio transfers,etc.. This application is concerned with the evaluation of loss reserves of this type according to financial economics (see Panjer (1998)) Framework and notation Consider an insurance portfolio subject to liability payments L (i) 0 at times i = 1, 2,..., where i = 0 denotes the present. Let L (i) be a random variable and suppose that it is modified by certain forces that influence the liability over time. For example, suppose that L (i) t denotes the amount of liability expressed in money values of time i. Then L (i) t evolves in the sense that L (i) t = L (i) t 1 R Lt, t = 1,..., i, where the R Lt are strictly positive random variables of the form R Lt = 1 + r Lt, with r Lt the inflation of claims costs over interval (t 1, t]. The liability finally paid is L (i) = L (i) s. As an example, L (i) t 1 and R Lt might be independently distributed as follows: L (i) t 1 logn(ν, τ 2 ) and R Lt logn(µ, σ 2 ). It is emphasized that, in this example, r Lt denotes claims inflation. This might include influences other than simple community inflation, such as the particular pressures of the legal and health care environments on claim costs. Similarly, a holding of assets of value A t 1 at time t 1 accumulates at time t to A t = A t 1 R At, with R At = 1 + r At.

75 2.4. Application: discounted loss reserves 49 Assume that R Xt, where X is either A or L, follows the capital Asset Pricing Model (CAPM): r Xt = r F t + β X t + ɛ Xt, (2.57) where r F t is the risk-free rate in period t, β X is the CAPM beta associated with X, ɛ Xt is the idiosyncratic risk associated with X, and t = r Mt r F t, with r M denoting the period increase in value of the economy-wide portfolio of assets. The distribution of t is assumed independent of t. The assumption of CAPM returns is consistent with an assumption that assets and liabilities here are marked to market. Henceforth, it will be assumed that r F t = r F, independent of t. This simplifies the following algebraic development considerably. It should be emphasized, however, that the whole development generalizes to the case in which r F t varies with t. The generalization is theoretically straightforward, but adds considerable notational baggage without yielding any deeper insight. Assume that the ɛ At are i.i.d. and similarly the ɛ Lt. Assume that all variables ɛ At, ɛ Lt and t are stochastically independent, and that E[ɛ Xt ] = 0. Let us further denote the variance of ɛ Xt with ωx 2. It follows that the R At and R Lt are independent and identically distributed. Suppose now the following distribution assumptions: L (i) 0 logn ( ν (i) L0, τ 2(i) ) L0 and RXt logn ( µ X, σx 2 ), (2.58) with stochastic independence between L (i) 0 and R Xt for all i, t, and X = A, L. Denote ρ = Corr(logR At, logr Lt ) and Define the accumulation factor κ (rs) = Corr ( logl (r) 0, ) logl(s) 0. R Xt:u = R X,t+1 R X,t+2... R Xu, for u = t + 1, t + 2,... Note that R Xt:t+1 = R X,t+1.

76 50 Chapter 2 - Convex bounds By relation (2.58) and the independence between distinct time intervals, R Xt:u logn ( (u t)µ X, (u t)σ 2 X). The implicit asset allocation is any that is consistent with relation (2.58). One might assume, for example, a constant allocation by asset sector, with continuous rebalancing and sector-specific returns that are constant over time. As remarked earlier in this section, the last of these assumptions could be weakened. Indeed, if the assumptions of constant returns over time were weakened, no assumption would be required with respect to asset allocation. Define the discounted liability payment V (i) = L (i) i R 1 A0:i = L (i) 0 R L0:iR 1 = L (i) 0 A0:i i (R Lj R 1 j=1 Aj ) logn(α (i), δ 2(i) ), with α (i) = ν (i) L0 + i(µ L µ A ) and δ 2(i) = τ 2(i) L0 + i(σ2 L + σ2 A 2ρσ Lσ A ). The present value S, given by S = V (i) := e Z i, (2.59) with n the number of cash-flow liabilities in the discounted value of the total outstanding losses of the portfolio. by In Taylor (2004), the mean and variance of S are calculated and given E[S] = = E[V (s) ] s=1 [ ( RL E[L (s) 1 + (β 2 0 ] A σm 2 + ω2 A )/ R 2 )] s A R A 1 + β A β L σm 2 / R, A RL

77 2.4. Application: discounted loss reserves 51 Var[S] = = Cov[V (r), V (s) ] r,s=1 r,s=1 ( E[V (r) ]E[V (s) ] exp [ κ (rs) τ (r) L0 τ (s) L0 + min(r, s)[σl 2 + σa 2 2ρσ A σ L ] ] ) 1, with R X = E[R Xt ] and σ 2 M = Var[r Mt]. We will denote the variance of S by σ 2 S. There are now three relevant values of loss reserve: n s=1 E[L(s) 0 ], which is the CAPM-based economic value of the liability. E[S], which is the expected value of the discounted liability cash flows, the discount rate taking into account the insurers asset holdings. A p = F 1 S (p) = E[S]exp(σ SΦ 1 (p) 1 2 σ2 S ), which is the p 100%- confidence loss reserve. It may be convenient to write the last of these conditions in the form A p = [1 + η(ρ, σ S )]E[S], where η(ρ, σ S ) may be regarded as a security loading. Note, however, that the security loading in this formulation is applied to E[S] and not to the economic value of the liability. The first two of the above three possibilities for loss reserve are the ones involved in the current debate over the appropriate rate(s) at which to discount liabilities. The quantity E[S] is obtained using the expectations of discount factors that reflect the insurers expected returns. In broad (though not quite precise) terms, it may be thought of as the amount of assets which, accumulating with expected investment return, will be sufficient to meet liabilities as they are required to be paid. This value depends on the insurer-specific asset holdings, and so cannot be market or fair value of the liabilities. This is given by the first of the above three candidates for loss reserves. Taylor (1996) pointed out for high security margins (Φ 1 (p) > σ S ), the size of the security margin increases with increasing asset beta. However,

78 52 Chapter 2 - Convex bounds for low security margins (Φ 1 (p) < σ S ), the size of the security margin decreases with increasing asset beta. In this latter case the additional yield expected from an increased asset risk outweighs the additional risk. Taylor (2004) defines the security margin for confidence level p as SM p [S] := η(p, σ S ) = (VaR p [S]/E[S]) 1, which is based on the quantile risk measure from the distribution of the discounted reserve S. In general, it is hard or even impossible to determine the quantiles of the discounted reserve analytically, because in any realistic model for the return process the random variable S will be a sum of strongly dependent random variables. Here, S is is a finite sum of correlated lognormal random variables. This implies that its cumulative distribution function cannot be determined exactly and is even too cumbersome to work with. An interesting solution to this difficulty consists of determining the lower bound S l and the upper bound S c as explained earlier in this chapter Calculation of convex lower and upper bounds To calculate the security margin η(p, σ S ) expressions for the quantiles and the expected value of S l and S c are needed. The expressions for the quantile function of the lower and upper bound of a sum of lognormal random variables are given by (2.35) and (2.38) in the case of α i = 1 for all i. The expression for the expected value is given by (2.46). To calculate the lower bound we choose the maximal variance conditioning variable given by (2.50) and (2.53): Λ = e E[Z i]+ 1 2 σ2 Z i Z i. We find that ( RL E[Z i ] = ν (i) L0 + log Var[Z i ] = σ 2 Z i = τ 2(i) L0 + iˆσ2, R A ( 1 + (β 2 A σ 2 M + ω2 A )/ R 2 A 1 + (β 2 L σ2 M + ω2 L )/ R 2 L ) 1/2 ) i, where the variability of the discounting structure ˆσ 2 := σ 2 L + σ2 A 2ρσ Lσ A is given by { [1 + (β 2 log A σm 2 + ω2 A )/ R A 2 ][1 + (β2 L σ2 M + ω2 L )/ R L 2 ] } [1 + β A β L σm 2 / R A RL ] 2.

79 2.4. Application: discounted loss reserves 53 The correlation between Z i and Λ is given by r i = Cov[Z i, Λ] σ Zi σ Λ = n k=1 β k (ˆσ 2 min(i, k) + η (i,k)) n, σ n Zi k=1 l=1 β kβ l (ˆσ 2 min(k, l) + η (k,l) ) with [ η (k,l) = Cov log L (k) 0 ], log L(l) 0 = κ (kl) τ (k) L0 τ (l) L0. Notice that if the liability cash flows are independent η (k,s) = τ 2(k) L0 I (k=s). We will compare the performance of the lower and upper bound approach with the Monte Carlo simulation results, obtained by generating random paths, who serve as a benchmark. Note that the random paths are based on antithetic variables in order to reduce the variance of the Monte Carlo estimate. We use the notation SM p [S l ] and SM p [S c ] to denote the security margin for confidence level p approximated by the lower bound and the upper bound approximation respectively. The different tables display the Monte Carlo simulation result (MC) for the security margin, as well as the procentual deviations of the different approximation methods, relative to the Monte Carlo result. These procentual deviations are defined as follows: LB := SM p[s l ] SM p [S MC ] SM p [S MC ] UB := SM p[s c ] SM p [S MC ] SM p [S MC ] 100%, 100%, where S l and S c correspond to the lower bound approach and the upper bound approach, and S MC denotes the Monte Carlo simulation result. The figures displayed in bold in the tables correspond to the best approximations, this means the ones with the smallest procentual deviation compared to the Monte Carlo results. We set β L equal to zero and choose as financial parameters r F = 6%, E[ ] = 6% and β A = 0.9. The tables list the results for different values of the parameters ω L, ω A, σ M and n. We construct two different cash flow structures. Table 2.1 displays the first structure of the liability cash flows (ex. 1), each of which is assumed lognormally distributed, and all of which are stochastically independent.

80 54 Chapter 2 - Convex bounds Time i E[L (i) ] E[L (i) 0 ] ν(i) 0 τ (i) 0 1 5% 4.7% % 2 15% 13.3% % 3 25% 21.0% % 4 20% 15.8% % 5 15% 11.2% % 6 10% 7.0% % 7 5% 3.3% % 8 5% 3.1% % Total 100% 79.6% Table 2.1: Structure of stochastic liability cash flow (ex. 1). The profile of the cash flows is intended to resemble a medium-term casualty payment pattern. It is assumed that ω L = 5% and as financial parameters σ M = 20% and ω A = 0. It follows from equation (2.57) that R L = Further, we have for this example µ L = and σ L = Table 2.2 summarizes the results for the 70% security margin for different market volatilities σ M. The lower bound turns out to fit the security margins the best for all values of the parameters. Notice that between brackets the standard error of the Monte Carlo estimate is displayed. Table 2.3 compares the approximations for some selected confidence levels p. For this example we have that σ A = 16.1%, σ L = 4.7%, µ A = 9.5% and µ L = 5.7%, with µ X and σx 2 such that R X = exp(µ X σ2 X ). The results are in line with the previous ones. The lower bound approach gives excellent results for high as well as for low values of p. Table 2.4 displays the approximated and simulated 97.5% margins for some selected market volatilities. These parameters are consistent with historical capital market values as reported by Ibbotson Associates (2002). The presented figures again indicate that the lower bound is the most precise method.

81 2.4. Application: discounted loss reserves 55 σ M : LB 0.25% 0.09% 0.12% 0.00% UB % % +5.37% 1.62% MC (s.e ) (1.11) (2.47) (6.15) (8.18) Table 2.2: (ex. 1) Approximations for the security margin SM 0.70 [V ] for different market volatilities and ω L = 0.1 and ω A = p : LB 0.38% 0.21% 0.16% 0.08% 0.00% 0.00% UB % % % % % % MC (s.e ) (2.49) (0.46) (0.26) (0.10) (0.06) (0.04) Table 2.3: (ex. 1) Approximations for some selected confidence levels of SM p [V ]. The market volatility is set equal to 20%. (ω L = 0.05 and ω A = 0) σ M : LB 0.19% 0.15% 0.23% 0.16% 0.11% 0.17% 0.38% UB % % % % % % % MC (s.e ) (0.15) (0.29) (0.41) (0.69) (1.22) (3.78) (4.16) Table 2.4: (ex. 1) Approximations for the security margin SM [V ] for different market volatilities. We include an additional example (ex. 2) with a different stochastic liability cash-flow structure. We fix the number of liabilities at n = 30. Further, we choose ν (i) 0 = 4.46 for i = 1,..., 30 and τ (i) 0 = 5% i 5; 10% 5 < i 15; 15% 15 < i 25; 20% 25 < i 28; 25% 28 < i 30.

82 56 Chapter 2 - Convex bounds p : LB 0.93% 0.04% 0.02% 0.18% 0.03% 0.6% UB % % % % +5.16% 30.40% MC (s.e ) (37.63) (2.99) (7.44) (2.79) (0.78) (0.27) Table 2.5: (ex. 2) Approximations for some selected confidence levels of SM p [V ]. The market volatility is set equal to 25%. This means that the sum of the expected cash flows E[L (i) ] is equal to 100% and E[L (i) 0 ] = 35.51%. In this example we fix the parameters ω L and ω A equal to 10% and 5% respectively. The same conclusions as for ex. 1 can be drawn from the results in Table 2.5. This table reports the discussed approximations for SM p [V ] for different probability levels and a fixed market volatility σ M = Note that for the parameters in Table 2.5 σ A = 20.5%, σ L = 9.4%, µ A = 8.7% and µ L = 5.4%. Overall, the comonotonic lower bound approach provides a very accurate fit under different parameter assumptions. These assumptions are in line with realistic market values. Moreover, the comonotonic approximations have the advantage that they are easy computable for any risk measure that is additive for comonotonic risks, such as Value-at-Risk and the wider class of distortion risk measures (see e.g. Dhaene et al. (2004)). 2.5 Convex bounds for scalar products of random vectors Within the fields of finance and actuarial science one is often confronted with the problem of determining the distribution function of a scalar product of two random vectors of the form S = X i Y ti, (2.60) where the nominal random payments X i are due at fixed and known times t i, i = 1,..., n and Y t denotes the nominal discount factor over the interval [0, t], t 0. This means that the amount one needs to invest at time 0

83 2.5. Convex bounds for scalar products of random vectors 57 to get an amount 1 at time t is the random variable Y t. By nominal we mean that there is no correction for inflation. Notice that here the random vector X = (X 1, X 2,..., X n ) may reflect e.g. the insurance or credit risk while the vector Y = (Y t1, Y t2,..., Y tn ) represents the financial/investment risk. If the payments X i at time t i are independent of inflation, then the vectors X and Y can be assumed to be mutually independent. On the other hand if the payments are adjusted for inflation, the vectors X and Y are not mutually independent anymore. Denoting the inflation factor over the period [0, t] by Z t, the random variable S can be rewritten as S = X i Ỹ ti, where the real payments X i and the real discount factors Ỹt i are given by X i = X i /Z ti and Ỹt i = Y ti Z ti. Hence, in this case S is the scalar product of two mutually independent random vectors ( X 1, X 2,..., X n ) and (Ỹt 1, Ỹt 2,..., Ỹt n ). For this reason the assumption of independence between the insurance risk and the financial risk is in most cases realistic and can be efficiently deployed to obtain various quantities describing risk within financial institutions, e.g. discounted insurance claims or the embedded/appraisal value of a company. Distributions of sums of the form (2.60) are often encountered in practice and need to be analyzed thoroughly by actuaries and other practitioners involved in the risk management process. Not only the basic summary measures (like the first few moments) have to be computed, but also more sophisticated risk measures which require much deeper knowledge about the underlying distributions (e.g. the Value-at-Risk). Unfortunately there are no analytical methods to compute distribution functions for random variables of this form. That is why usually one has to rely on volatile and time consuming Monte Carlo simulations. In spite of the enormous increase in computational power observed within the last few decades, computing time remains a serious drawback of Monte Carlo simulations, especially when one is interested in estimating very high values of quantiles (note that a solvency capital of an insurance company may be determined e.g. as the 99.95%-quantile, which is extremely difficult to estimate within reasonable time by simulation methods). In this section we propose an alternative solution. By extending the methodology of Section 2.2 to the case of scalar products of independent

84 58 Chapter 2 - Convex bounds random vectors, we obtain convex upper and lower bounds for sums of the form (2.60). 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 (2.60). We first give the theoretical foundations for convex lower and upper bounds in the case of scalar products of independent random vectors. Next, we demonstrate how to obtain the bounds for (2.60) in the convex order sense in case when Y follows the lognormal law. Finally, we present several applications for discounted claim processes in a Black & Scholes setting Theoretical results Consider sums of the form: S = X 1 Y 1 + X 2 Y X n Y n, (2.61) where the random vectors X = (X 1, X 2,..., X n ) and Y = (Y 1, Y 2,..., Y n ) are assumed to be mutually independent. Theoretically, the techniques developed in Section 2.2 can be applied also in this case (one can take V j = X j Y j ). Such an approach is however not very practical. First of all, it is not always easy to find the marginal distributions of V j. Secondly, it is usually very difficult to find a suitable conditioning random variable Λ, which will be a good approximation to the whole scalar product, taking into account the riskiness of the random vector X and Y simultaneously. The following theorem provides a more suitable approach to deal with scalar products. Before we prove the theorem we recall a helpful lemma. Lemma 8 (Scalar products and convex order). Assume that X = (X 1,..., X n ), Y = (Y 1,..., Y n ) and Z = (Z 1,..., Z n ) are non-negative random vectors and that X is mutually independent of the vectors Y and Z. If for all possible outcomes x 1,..., x n of X x i Y i cx x i Z i, then the corresponding scalar products are ordered in the convex order sense, i.e. X i Y i cx X i Z i.

85 2.5. Convex bounds for scalar products of random vectors 59 Proof. Let φ be a convex function. By conditioning on X and taking the assumptions into account, we find that [ ( )] E φ X i Y i holds for any convex function φ. [ = E X [E φ [ E X [E φ ( ( [ ( )] = E φ X i Z i X i Y i ) X ]] X i Z i ) X ]] Theorem 11 (Bounds for scalar products of random vectors). Consider the following sum of random variables S = X i Y i. (2.62) Assume that the vectors X = (X 1, X 2,..., X n ) and Y = (Y 1, Y 2,..., Y n ) are mutually independent. Define the following quantities: S c = S l = F 1 X i (U)F 1 Y i (V ), (2.63) E[X i Γ]E[Y i Λ], (2.64) where U and V are independent standard uniform random variables, Γ is a random variable independent of Y and Λ, and the second conditioning random variable Λ is independent of X and Γ. Then, the following relation holds: S l cx S cx S c. Proof. The proof is based on a multiple application of Lemma First, we prove that n X iy i cx n F 1 X i (U)F 1 Y i (V ). From Theorem 8 it follows that for all possible outcomes (x 1,..., x n ) of X the following inequality holds: x i Y i cx F 1 x i Y i (V ) = x i F 1 Y i (V ).

86 60 Chapter 2 - Convex bounds Thus from Lemma 8 it follows immediately that n X iy i cx n X if 1 Y i (V ). The same reasoning can be applied to show that X i F 1 Y i (V ) cx F 1 X i (U)F 1 Y i (V ). 2. In a similar way, one can show that E[X i Γ]E[Y i Λ] cx X i E[Y i Λ] cx X i Y i. Remark 1. Notice that n F 1 X i (U)F 1 Y i (V ) n cx F 1 X i Y i (U). Thus the upper bound (2.63) is improved compared to the comonotonic upper bound. It takes the information into account that the vectors X and Y are independent. Remark 2. One can also calculate the improved upper bound S u = F 1 1 X i Γ (U)FY i Λ (V ), but since the improved upper bound S u is very close to the comonotonic upper bound S c and it requires much more computational time, we concentrate in this thesis only on the lower bound S l and the comonotonic upper bound S c as approximations for S. Remark 3. Having obtained the convex upper and lower bounds one can get also the moments based approximation S m as described in Subsection 2.2.4, i.e. by determining the distribution function as follows: F S m(t) = zf S l(t) + (1 z)f S c(t), (2.65) where z = Var[Sc ] Var[S] Var[S c ] Var[S l ]. (2.66)

87 2.5. Convex bounds for scalar products of random vectors Stop-loss premiums The stop-loss premiums of S c and S l provide natural bounds for the stoploss premiums of the underlying scalar product of random vectors. More precisely, one has the following relationship: π lb (S, d, Γ, Λ) π(s, d) π cub (S, d). The values π cub (S, d) and π lb (S, d, Γ, Λ) can be easily computed. Below we give the computational procedure in detail. First, consider a sum of the form (S c U = u) = F 1 X i (u)f 1 Y i (V ). It can be easily seen that it is a sum of the components of a comonotonic vector, and hence the conditional stop-loss premiums of S c (given U = u) can be found in the case the distribution functions of Y i are continuous and strictly increasing, by applying Theorem 9. Then, the overall stoploss premium of S c can be computed by conditioning π cub (S, d) = E [E [ (S c d) + U ]] = 1 0 ( F 1 X i (u)π Y i, F 1 ( Y FS c i U=u(d) )) du. (2.67) In general it is more difficult to calculate stop-loss premiums for the lower bound. However it can be done similarly as in the case of the upper bound if one additionally assumes that the conditioning variables Γ and Λ can be chosen in such a way that for any fixed γ supp(γ) all components E [ X i Γ = γ ] E [ Y i Λ = λ ] are non-decreasing (or equivalently nonincreasing) in λ. Then the vector ( ) E[X 1 Γ = γ]e[y 1 Λ], E[X 2 Γ = γ]e[y 2 Λ],..., E[X n Γ = γ]e[y n Λ]

88 62 Chapter 2 - Convex bounds is comonotonic and Theorem 9 can be applied. Thus, one gets π lb (S, d, Γ, Λ) = E [E [ (S l d) + Γ ]] 1 ( = E [ X i Γ = F 1 Γ (u)] 0 ( π E[Y i Λ], F 1 E[Y i Λ]( FS l Γ=F 1 Γ (u)(d)))) du. (2.68) Hence if one can only compute stop-loss premiums of Y i and E[Y i Λ], one can also compute stop-loss premiums of S c and S l. Note that stop-loss premiums of the moments based approximation S m can be easily calculated as π m (S, d, Γ, Λ) = zπ lb (S, d, Γ, Λ) + (1 z)π cub (S, d) The case of log-normal discount factors In the sequel we develop a framework for computing convex bounds for random variables of the form: S = α i X i e Z i, (2.69) where the vectors X and Z satisfy the usual conditions (see Section 2.5.1). We assume α i > 0 and Z i N(E[Z i ], σ 2 Z i ). In this section we consider the problem in general, without imposing any conditions on the random variables X i. In particular we don t discuss the choice of the conditioning variable Γ. The upper bound From Theorem 11 it follows that S c = F 1 X i (U)F 1 α i e (V ) Z i = F 1 X i (U)α i e E[Z i]+sign(α i )σ Zi Φ 1 (V ), (2.70) where U and V are independent standard uniform random variables. The cumulative distribution function of S c can be calculated in three steps:

89 2.5. Convex bounds for scalar products of random vectors Suppose that U = u is fixed. Then from (2.70) it follows that conditional quantiles can be computed as F 1 S c U=u (p) = F 1 X i (u)α i e E[Z i]+sign(α i )σ Zi Φ 1 (p) ; (2.71) 2. Obviously for any u the function given by (2.71) is continuous and strictly increasing. Thus for any y 0 one can compute the value of the conditional distribution function using one of the well-known numerical methods (e.g. Newton-Raphson) as a solution of F 1 X i (u)α i e E[Z i]+sign(α i )σ Zi Φ 1 (F S c U=u (y)) = y; (2.72) 3. The cumulative distribution function of S c can be now derived as F S c(y) = 1 0 F S c U=u(y)du. The stop-loss premiums of the upper bound can be computed as follows. For simplicity of notation let us denote d u,i = F 1 ( α i e Z i FS c U=u(d) ) = α i e E[Z i]+sign(α i )σ Zi Φ 1 (F S c U=u (d)). (2.73) Then one has π ( α i e Z ) i, d u,i = αi e E[Z i]+ σ 2 Z i ( 2 Φ where, using Lemma 6, sign(α i )b (1) u,i ) ( d u,i Φ b (1) u,i = E[Z i] + σ 2 Z i ln(d u,i ) σ Zi, b (2) u,i = b(1) u,i σ Z i. sign(α i )b (2) u,i ), (2.74) Then the stop-loss premium of S c with retention d can be computed by plugging (2.74) into (2.67) and is given by

90 64 Chapter 2 - Convex bounds π cub (S, d) = = 1 0 F 1 X i (u)π ( α i e Z i, d u,i ) du α i e E[Z i]+ 1 2 σ2 Z i 1 0 ( F 1 X i (u)φ sign(α i )σ Zi Φ 1 ( F S c U=u(d) ) ) du d (1 F S c(d)). (2.75) The lower bound The computations for the lower bound are performed similarly, however the quality of the bound heavily depends on the choice of the conditioning random variables. Recall that from Theorem 11 it follows that S l = E [ X i Γ ] E [ α i e Z i Λ ], (2.76) where the first conditioning variable Γ is independent of Λ and Y and where the second conditioning variable Λ is independent of Γ and X. In this section the choice of Γ will not be discussed and the random variable Λ will be assumed to be of the maximal variance form (2.54) Λ = β i Z i = α i E[X i ]e E[Z i]+ 1 2 σ2 Z i Z i. (2.77) Under these assumptions the vectors of the form ( Z i, Λ ) have a bivariate normal distribution. Thus, Z i Λ = λ will be normally distributed with mean µ i,λ and variance σi,λ 2 given by µ i,λ = E[Z i ] + Cov[ Z i, Λ ] ( ) λ E[Λ] Var[Λ] and σi,λ 2 = σ2 Z i Cov[ Z i, Λ ] 2. Var[Λ]

91 2.5. Convex bounds for scalar products of random vectors 65 The lower bound (2.76) can be written out as S l = = = E [ X i Γ ] E [ α i e Z i Λ ] E [ X i Γ ] α i e µ i,λ+ σ 2 i,λ 2 E [ X i Γ ] α i e E[Z i]+ 1 2 σ2 Z (1 r 2 i i )+σ Z i r i Φ 1 (U), (2.78) with U a standard uniform random variable and correlations given by r i = Corr (Z i, Λ) = Cov[ Z i, Λ ] σ Zi σ Λ = n j=1 E[X i]e E[Z j]+ 1 2 σ2 Z j σ Zi Z j σ Zi 1 k,l n E[X k]e[x l ]e E[Z k]+e[z l ]+ 1 2 (σ2 Z k +σ 2 Z l ) σzk Z l. (2.79) Note that the r i s are non-negative and the random variable S l is (given a value Γ = γ) the sum of the components of a comonotonic vector. Thus the cumulative distribution function of the lower bound S l can be computed, similar to the case of the upper bound S c, in three steps: 1. From (2.78) it follows that the conditional quantiles (given Γ = γ) can be computed as F 1 S l Γ=γ (p) = E [ X i Γ = γ ] α i e E[Z i]+ 1 2 σ2 Z (1 r 2 i i )+σ Z i r i Φ 1 (p) ; (2.80) 2. The conditional distribution function is computed as the solution of E [ X i Γ = γ ] α i e E[Z i]+ 1 2 σ2 Z (1 r 2 i i )+σ Z i r i Φ 1 (F S l Γ=γ (y)) = y; (2.81) 3. Finally, the cumulative distribution function of S l can be derived as F S l(y) = 1 0 F S l Γ=F 1 (u)(y)du. Γ

92 66 Chapter 2 - Convex bounds The stop-loss premiums are computed as follows. Let us denote d γ,i = F 1 [ E Then one has α i e Z i Λ ] ( F S l Γ=γ(d) ) = α i e E[Z i]+ 1 2 σ2 Z i (1 r 2 i )+σ Z i r i Φ 1 (F S l Γ=γ (d)). π (E [ α i e Z i Λ ] ) (, d γ,i = α i e E[Z i]+ 1 2 σ2 Z i Φ with sign(α i )b (1) γ,i ) ( d γ,i Φ sign(α i )b (2) γ,i b (1) γ,i = E[Z i] σ2 Z i (1 r 2 i ) + σ2 Z i r 2 i ln(d γ,i) σ Zi r i, b (2) γ,i = b(1) γ,i σ Z i r i. ), (2.82) Then the stop loss-premium of S l with retention d can be computed by plugging (2.82) into (2.68) and is given by 1 π lb (S, d, Γ, Λ) = E [ X i Γ = F 1 Γ (u)] π (E [ α i e Z i Λ ] ), d γ,i du = 0 α i e E[Z i]+ 1 2 σ2 Z i 1 Moments based approximations 0 E [ ( X i Γ = F 1 Γ (u)] Φ r i σ Zi Φ 1( F S l Γ=γ(d) )) du d ( 1 F S l(d) ). (2.83) For computing the moments based approximation as defined in (2.65), one has to calculate the variance of S, S l and S c. In general the problem is easy solvable for the upper and the lower bound. For the exact distribution it is more difficult to find a universal solution and the problem needs to be considered individually. In the general case one would face the problem of computing multiple integrals, what requires usually too much computational time. Note that the upper and the lower bound of S, as described in Subsections and 2.5.3, can be seen as a special case of the following random variable X with general form given by X = α i f i (U)g i (V ), (2.84)

93 2.5. Convex bounds for scalar products of random vectors 67 where (α 1, α 2,..., α n ) is a vector of non-negative numbers, f i (.) and g i (.) are non-negative functions and U and V two independent standard uniform random variables. Indeed, in the case of the upper bound one takes f i (U) = F 1 X i (U) and g i (V ) = F 1 e Z i (V ) and in the case of the lower bound f i (U) = E [ X i Γ ] and g i (V ) = E [ e Z i Λ ]. The variance of X in expression (2.84) can be computed as follows Var[X] = E [ Var[X U] ] + Var [ E[X U] ] = 1 0 [ ] Var α i f i (u)g i (V ) du ( 1 [ ] 2 E α i f i (u)g i (V ) du). 0 ( E [ ] ) 2 α i f i (u)g i (V ) du Thus the problem of computing the variance of X is always solvable if one is able to compute the expectation and the variance of random variables X of the form X = α i g i (V ), for any vector of non-negative numbers ( α 1, α 2,..., α n ) (here α i = α i f i (u)). For the comonotonic upper bound (2.70), i.e. g i (V ) = e E[Z i]+σ Zi Φ 1 (V ), the variance of X is given by Var [ X] = j=1 σ α i α j e E[Z Z 2 +σ 2 i]+e[z j ]+ i Z j ( σ 2 e Zi σ Zj 1 ) and for the lower bound (2.76), i.e. g i (V ) = e E[Z i]+ 1 2 σ2 Z (1 r 2 i i ) σ Z i r i Φ 1 (V ), by Var [ σ X] = α i α j e E[Z Z 2 +σ 2 i]+e[z j ]+ i Z j ( r 2 e i r j σ Zi σ Zj 1 ). j=1

94 68 Chapter 2 - Convex bounds 2.6 Application: the present value of stochastic cash flows In this section we derive convex upper and lower bounds for general discounted cash flows of the form S = X i e Y (i), (2.85) where the random variables X i denote future (non-negative) payments due at time i and Y (t) is a stochastic process describing returns on investment in the period (0, t). We give explicit results for convex upper and lower bounds in three specific cases: (i) The vector ln X = ( ln(x 1 ), ln(x 2 ),..., ln(x n ) ) has a multivariate normal distribution and hence the losses are log-normally distributed. (ii) The vector X = ( X 1, X 2,..., X n ) has a multivariate elliptical distribution. Formally the described methodology is valid only in the case when X i > 0. (iii) The yearly payments X i are independent and identically distributed Stochastic returns We start with a general definition of a Gaussian process. Definition 7 (Gaussian process). A stochastic process { Y (t) t 0 } is called Gaussian if for any 0 < t 1 < t 2 <... < t n the vector ( Y (t 1 ), Y (t 2 ),..., Y (t n ) ) has a multivariate normal distribution. Gaussian processes have a lot of desirable properties. They are very easy to handle since they are completely determined by their mean and covariance functions m(t) = E[Y (t)] and c(s, t) = Cov[Y (s), Y (t)]. (2.86) For an introduction to Gaussian processes, see e.g. Karatzas & Shreve (1991). The normality assumption for modelling returns on investment

95 2.6. Application: the present value of stochastic cash flows 69 has been questioned in the financial literature for the short term setting (e.g. daily returns see Schoutens (2003)). In the long term however Gaussian models provide a satisfactory approximation since the Central Limit Theorem is applicable under the reasonable assumptions of independent returns with finite variance (some empirical evidence is provided e.g. in Cesari & Cremonini (2003)). Therefore in the framework of this thesis we restrict ourselves to two simple Gaussian models for future returns Y (t). More precisely, we will focus on modelling returns by means of a Brownian motion with drift (the Black & Scholes model) and an Ornstein-Uhlenbeck process. This limitation is very convenient because it leads to closed-form formulas for convex upper and lower bounds of future cash flows. The Black & Scholes setting (B-SM) We assume that a process X(t) satisfies the following stochastic differential equation: dx(t) = X(t) ( µ σ2) dt + X(t)σdW 1 (t), (2.87) where W 1 (t) denotes a standard Brownian motion. It is well-known that (2.87) has a unique solution of the form X(t) = X(0)e µt+σw 1(t), and thus the return on investment process Y (t) = log with mean and covariance functions given by m(t) = µt and c(s, t) = min(s, t)σ 2. ( ) X(t) X(0) is Gaussian One of the most important features of the return process Y (t) is the property of independent increments. Indeed, it is straightforward to verify that for every 0 < s < t < u one has that Cov [ Y (u) Y (t), Y (t) Y (s) ] = 0. For this reason we often consider yearly rates of return Y i = Y (i) Y (i 1) for i = 1, 2,... (2.88) which are independent and normally distributed with mean equal to µ and variance equal to σ 2.

96 70 Chapter 2 - Convex bounds The Ornstein-Uhlenbeck model (O-UM) In the Ornstein-Uhlenbeck model the return process is described as Y (t) = µt + Z(t), where Z(t) is the solution of the following stochastic differential equation: dz(t) = az(t)dt + σdw 1 (t), with a and σ being positive constants. Then Y (t) is again Gaussian with mean and covariance functions given by m(t) = µt and c(s, t) = σ2 ( e a t s e a(t+s)) (2.89) 2a We refer to e.g. Arnold (1974) for more details about the derivation. Note that for a = 0 the Ornstein-Uhlenbeck process degenerates to an ordinary Brownian motion with drift and is equivalent to the Black & Scholes setting. When a > 0, process Y (t) has no independent increments any more. Moreover, it becomes mean reverting. Intuitively the property of mean reversion means that process Y (t) cannot deviate too far from its mean function m(t). In fact the parameter a measures how strongly paths of Y (t) are attracted by the mean function. The value a = 0 corresponds to the case when there is no attraction and as a consequence the increments become independent. On Figure 2.1 we illustrate typical sample paths of the Ornstein-Uhlenbeck model for different values of parameter a. In particular we will concentrate on the case when Y (i) is defined by one of these models. Then the sum S in (2.85) has a clear interpretation: it is the discounted value of future benefits X i with returns described by one of the well-known Gaussian models. The input variables of the two discussed return models are displayed in Table 2.6.

97 2.6. Application: the present value of stochastic cash flows 71 a) The Ornstein-Uhlenbeck process: a=0 b) The Ornstein-Uhlenbeck process: a=0.02 Y(t) Y(t) t t c) The Ornstein-Uhlenbeck process: a=0.1 d) The Ornstein-Uhlenbeck process: a=0.5 Y(t) Y(t) t t Figure 2.1: Typical paths for the Ornstein-Uhlenbeck process with mean µ = 0.05, volatility σ = 0.07 and different values of parameter a. Model Variable Formula B-SM E[Y (i)] iµ Var[Y (i)] iσ 2 n Var[Λ] j=1 jβ2 j σ2 + 1 j<k n 2jβ jβ k σ 2 n Cov[Y (i), Λ] j=1 min(i, j)β jσ 2 O-UM E[Y (i)] iµ σ Var[Y (i)] 2 2α (1 ( e 2iα ) σ Var[Λ] 2 n 2α j=1 β2 j (1 e 2jα )+ + ) 1 j<k n 2β jβ k (e (k j)α e (j+k)α ) Cov[Y (i), Λ] σ 2 2α n j=1 β j(e i j α e (i+j)α ) Table 2.6: Input variables for returns. We take Λ = n β iy (i).

98 72 Chapter 2 - Convex bounds Lognormally distributed payments Consider a sum of the form S LN = e N i e Y (i), (2.90) where N = ( N 1, N 2,..., N n ) = ( ln(x1 ), ln(x 2 ),..., ln(x n ) ) is a normally distributed random vector with mean µ N = ( µ N1, µ N2,..., µ Nn ) and covariance matrix Σ N = [ σ N ij. The corresponding variances are denoted by σn 2 i ]1 i,j n := σn ii. There are two different approaches to derive convex upper and lower bounds for S LN as defined in (2.90). In the first approach independent parts of the scalar product are treated separately (this approach is consistent with the methodology described in Subsections and 2.5.3). In the second approach we treat S LN unidimensionally, by noticing that it can be rewritten as S LN = ˆX i = e ˆN i, (2.91) where ˆN = ( ˆN1, ˆN 2,..., ˆN n ) = ( N1 Y (1), N 2 Y (2),..., N n Y (n) ) has a multivariate normal distribution with parameters µ ˆN = ( µ ˆN1, µ ˆN2,..., µ ˆNn ) and Σ ˆN = [ σ ˆNij ]1 i,j n, (2.92) with µ ˆNi = µ Ni m(i) and σ ˆNij = σ N ij + c(i, j), where m(.) and c(.,.) denote mean and covariance functions of the process Y (.), as defined in (2.86). We further use the following notations σ 2ˆNi := σ N ii, µ i := m(i) and σ 2 i := c(i, i). Thus one can derive convex upper and lower bounds of (2.91) just by adapting the methodology described in Section Below we work out both approaches explicitly. The main advantage of the first method is a better recognition of the dependency structure and this results in more precise estimates (especially the upper bound). On the other hand the second method is much less time-consuming because the problem is reduced to only one dimension.

99 2.6. Application: the present value of stochastic cash flows 73 The upper bound The upper bound can be written as S c LN = e µ N i +µ i +σ Ni Φ 1 (U)+σ i Φ 1 (V ) and its distribution function can be computed as described in Subsection The lower bound To compute the lower bound we propose to define a conditioning random variable Γ symmetrically to the conditioning variable Λ, i.e. Γ = E [ e Y (i)] ( ) e µ N i σ2 N i N i = e µ N i +µ i + 1 σ 2 N 2 +σ 2 i i N i. The conditioning variable Λ is chosen as in (2.77), which gives after the obvious substitution ( ) σn Λ = 2 +σ 2 i i Y (i). (2.93) e µ N i +µ i Now the corresponding lower bound can be written as S l1 LN = e µ N i +µ i σ2 N (1 r 2 i N )+ 1 i 2 σ2 i (1 r2 i )+σ N i r Ni Φ 1 (U)+σ i r i Φ 1 (V ), where correlations r i = r( Y (i), Λ) are defined as in (2.79) and r Ni = r(n i, Γ) = n j=1 eµ N j +µ j σ Ni n k,l=1 eµ N k +µ Nl +µ k +µ l ( ) σn 2 +σ 2 j j σn ij ( σn 2 +σ 2 k N +σ 2 l k +σ2 l )σ Nkl. Its distribution function can be computed by conditioning on U, as described in Section From Remark 1 it follows that SLN c cx F 1 (U), e ˆN i

100 74 Chapter 2 - Convex bounds and thus we don t consider the comonotonic upper bound for (2.91). To compute the lower bound we apply directly the results of Section Therefore, we take as conditioning random variable Λ = e µ ˆNi (µ)+σ 2ˆNi ˆNi. (2.94) Then the lower bound is given explicitly as where S l2 LN = e µ ˆNi σ2ˆni (1 r 2ˆNi )+σ ˆNi r ˆNi Φ 1 (U), r ˆNi = r( ˆN i, Λ) = n j=1 eµ ˆNj σ2ˆnj σ ˆNij ) σ n ˆNi k,l=1 eµ ˆNk +µ ˆNl + (σ 1 2 2ˆNk +σ 2ˆNl σ ˆNkl Note that in order to obtain a comonotonic lower bound one has to assure additionally that r ˆNi > 0 for all i. Suppose that this lower bound is comonotonic. Then its quantiles are given by a closed-form expression: F 1 (p) = SLN l2 e µ ˆNi σ2ˆni (1 r 2ˆNi )+σ ˆNi r ˆNi Φ 1 (p), from which one can easily find values of the corresponding distribution function e.g. by means of the Newton-Raphson method. The moments based approximation It is also possible to derive the moments based approximations S m1 and S m2 as described in (2.65) since there are explicit solutions for the vari-

101 2.6. Application: the present value of stochastic cash flows 75 ances: Var[S LN ] = Var[S c LN ] = Var[S l1 LN ] = Var[S l2 LN ] = j=1 j=1 j=1 j=1 e µ ˆNi +µ ˆNj e µ ˆNi +µ ˆNj e µ ˆNi +µ ˆNj e µ ˆNi +µ ˆNj ( σ 2ˆNi +σ 2ˆNj ) (e σ ˆNij 1 ), ( σ 2ˆNi +σ 2ˆNj ) (e σ Ni σ Nj +σ i σ j 1 ), ( σ 2ˆNi +σ 2ˆNj ) (e r Ni r Nj σ Ni σ Nj +r i r j σ i σ j 1 ), ( σ 2ˆNi +σ 2ˆNj ) (e r ˆNi r ˆNj σ ˆNi σ ˆNj 1 ). After obvious substitutions in formulas (2.75) and (2.83) one gets the following expressions for stop-loss premiums in the first approach: π cub (S LN, d) = e µ i+ 1 2 σ2 i π lb1 (S LN, d, Γ, Λ) = 1 0 ( ( e µ N i +σ Ni Φ 1 (u) Φ σ i Φ 1( F S u LN U=u(d) )) du d ( 1 F S u LN (d) ), 1 e µ i+ 1 2 σ2 i e µ N i (1 r2 N )σ 2 i N +r Ni σ Ni Φ 1 (u) i 0 ( Φ r i σ i Φ 1( F S l1 d ( 1 F S l1 LN (d) ). LN Γ=F 1 Γ (u)(d))) du In the second approach the expression for stop-loss premiums of the lower bound follows straightforward from (2.40): π lb2 (S LN, d, Λ) = e µ ( ˆNi σ2ˆni Φ r ˆNi σ ˆNi Φ 1( F S l2 (d) )) LN d ( 1 F S l2 LN (d) ). Finally, the corresponding stop-loss premiums for the moments based approximations are given by π m1 (S LN, d) = z 1 π lb1 (S LN, d) + (1 z 1 )π cub (S LN, d), π m2 (S LN, d) = z 2 π lb2 (S LN, d) + (1 z 2 )π cub (S LN, d),

102 76 Chapter 2 - Convex bounds p S l1 LN S l2 LN S m1 LN S m2 LN S c LN MC (s.e ) (0.71) (1.06) (1.45) (2.08) (4.59) Table 2.7: Approximations for some selected quantiles with probability level p of S LN. where z 1 = Var[Sc LN ] Var[S LN ] Var[SLN c ] Var[Sl1 LN ] and z 2 = Var[Sc LN ] Var[S LN ] Var[SLN c ] Var[Sl2 A numerical illustration LN ]. We examine the accuracy and efficiency of the derived approximations for the present values of a cash flow with lognormally distributed payments. For the purpose of this numerical illustration we choose parameters µ Ni = ln(1.01) 2 and σn 2 i = ln(1.01) (Note that this value correspond to E[X] = 1 and Var[X] = 0.01). Moreover, we allow for some dependencies between the payments by imposing correlations between the normal exponents: 1 if i = j 0.5 if i j = 1 r(n i, N j ) = 0.2 if i j = 2, 0 if i j > 2. We restrict ourselves to the case of a Black & Scholes setting with drift µ = 0.05 and volatility σ = 0.1. We compare the distribution functions of the upper bound SLN c and the lower bounds Sl1 LN (obtained by taking two conditioning random variables) and SLN l2 (with 1 conditioning variable) with the original distribution function of S LN obtained by means of a Monte Carlo (MC) simulation based on generating sample paths. Table 2.7 illustrates the performance of the different approximations. One can see that the upper bound SLN c gives a poor approximation. The main reason for that is a relatively weak dependence between payments,

103 2.6. Application: the present value of stochastic cash flows 77 d S l1 LN S l2 LN S m1 LN S m2 LN S c LN MC (s.e ) (4.37) (4.37) (4.11) (2.14) (0.72) (0.25) (0.08) Table 2.8: Approximations for some selected stop-loss premiums with retention d of S LN. for which the comonotonic approximation significantly overestimates the tails. On the other hand, both lower bounds SLN l1 and Sl2 LN give excellent approximations. One may be surprised especially with the performance of the second lower bound it turns out that the results are not less accurate for one conditioning random variable than in the case of two conditioning random variables. In the table we include also two moments based approximations SLN m1 and Sm2 LN, which perform excellent as well. Finally, the stop-loss premiums for the different approximations are compared in Table 2.8. This study confirms the high accuracy of the lower bounds and moments based approximations, which are very close to the Monte Carlo estimates. The overestimation of the stop-loss premiums provided by the convex upper bound is considerable Elliptically distributed payments The class of elliptical distributions is a natural extension of the normal law. We say that a random vector X = ( ) X 1, X 2,..., X n has an n- dimensional elliptical distribution with parameters µ = ( ) µ 1, µ 2,..., µ n, Σ = [ σ ij (symmetric and positive definite matrix) and characteristic generator φ( ), if the characteristic function of X is given ]1 i,j n by ϕ X ( t ) = e i t µ φ ( t Σ t ). We write X E n ( µ, Σ, φ). Obviously the normal distribution satisfies this definition, with φ(y) = e 1 2 y. Elliptical distributions are very useful for several reasons. First of all they are very easy to manipulate because they

104 78 Chapter 2 - Convex bounds inherit surprisingly many properties from the normal law. On the other hand the normal distribution is not very flexible in modelling tails (in practice we often encounter much heavier tails than the Gaussian ones). The class of elliptical laws offers a full variety of random distributions, from very heavy-tailed ones (like Cauchy or stable distributions), distributions with tails of the polynomial-type (t-student), through the exponentially-tailed Laplace and logistic distributions to the light-tailed Gaussian distribution. Below we give a brief overview of the properties of elliptical distributions. For more information about elliptical distributions we refer to Fang et al. (1990). The generalization of some of the results on comonotonic bounds for n X i to the multivariate elliptical case can be found in Valdez & Dhaene (2004). 1. E[X i ] = µ i, Var[X i ] = 2φ (0)σ ii and Cov[X i, X j ] = 2φ (0)σ ii if only the corresponding moments exist. Here, φ ( ) is the first derivative of the characteristic generator φ( ). 2. Let Y = A X + b, where A denote an m n-matrix and b is a vector in R n. Then Y E m ( A µ + b, AΣA, φ ) ; 3. If the density function f X ( ) exists, it is given by the formula f X ( x) = c ( ) det [ Σ ]g ( x µ) Σ 1 ( x µ) for any non-negative function g satisfying 0 < 0 z n 2 1 g(z)dz < and c being a normalizing constant. The function g( ) is called the density generator of the distribution E m ( µ, Σ, φ ). A detailed proof of these results, using spherical transformations of rectangular coordinates, can be found in Landsman & Valdez (2002). 4. Let X = ( X1, X ) 2 denote an En+m ( µ, Σ, φ)-random vector, where µ = ( ) µ 1, µ 2 and ( ) Σ11 Σ Σ = 12. Σ 21 Σ 22

105 2.6. Application: the present value of stochastic cash flows 79 Then, given conditionally that X 2 = x 2, the vector X 1 has the E n ( µ 1 2, Σ 11 2, φ x2 )-distribution with parameters given by µ 1 2 = µ 1 + Σ 12 Σ 1 ( ) 22 x2 µ 2 Σ 11 2 = Σ 11 Σ 12 Σ 1 22 Σ 21. and Notice that in general (unlike in the normal case) the characteristic generator of the conditional distribution is not known explicitly and depends on the value of x 2. Consider now sums of the form S el = X i e Y (i), where the return process Y (t) is, like in the previous example, described by the Black & Scholes model and X = ( X 1, X 2,..., X n ) is elliptically distributed with parameters µ X = ( µ X1, µ X2,..., µ Xn ), Σ X = [ σ X ij ]1 i,j n and characteristic generator φ( ). Here we note only that for φ(u) = e u 2 one gets a multivariate normal distribution with mean parameter µ X and covariance matrix Σ X. Note that elliptical random variables take both positive and negative values and therefore one cannot apply immediately Theorem 11. We propose to consider pragmatically only the cases where the probability Pr[X i < 0] is very small. This can be achieved by choosing the parameters in such a way that µ X i σ Xi is much larger then 0, where we use the conventional notation σ 2 X i := σ X ii. The upper bound The computation of the upper bound is straightforward if the inverse distribution function for the specific elliptical distribution is available in the software package. In other words, the comonotonic upper bound is given by S c el = F 1 E n µ Xi,σ 2 X i,φ ( U ) e µ i +σ i Φ 1 (V ), (2.95)

106 d ( 1 F S u el (d) ) (2.96) 80 Chapter 2 - Convex bounds where by convention µ i = m(i) and σi 2 = c(i, i) for m( ) and c(, ) denoting the mean and covariance functions of the process Y (i) described previously in this subsection. Note that for the most interesting case of a multivariate normal distribution one gets S c N = ( ) µ Xi + σ Xi Φ 1 (U) e µ i+σ i Φ 1 (V ). The corresponding expressions for stop-loss premiums are given by π cub (S el, d) = e µ i+ 1 2 σ2 i 1 0 { ( F 1 (u)φ σ E n µ Xi,σX 2 i,φ i Φ 1( F S u el U=u(d) ))} du and π cub (S N, d) = e µ i+ 1 2 σ2 i 1 0 { ( ( µ + σ Xi Xi (u)) Φ 1 Φ σ i Φ 1( F S u N U=u(d) ))} du d ( 1 F S u N (d) ). The lower bound To compute the lower bound, we define the conditioning random variable Γ as follows Γ = E [ e Y (j)] X j = e µ j+ 1 2 σ2 j X j. j=1 Then a random vector ( X j, Γ ) has a bivariate elliptical distribution, with parameters µ Γ,i = ( ) µ Xi, µ Γ and Σ Γ,i = [ σ Γ,i ] kl 1 k,l 2, where µ Γ = j=1 e µ j+ 1 2 σ2 j µ Xj, j=1

107 2.6. Application: the present value of stochastic cash flows 81 σx 2 i := σ Γ,i 11, σγ,i 12 = σγ,i 21 = e µ j+ 1 2 σ2 j X σ ij σ 2 Γ := σ Γ,i 22 = j=1 k=1 j=1 e µ j+µ k ( σ 2 j +σ2 k ) σ X jk. and From property (4) of the elliptical distributions, it follows that given Γ = γ the r.v. X i is elliptically distributed with parameters µ Xi,Γ = µ Xi + σγ,i ( ) 12 Γ σγ 2 µγ, σ 2 Xi,Γ = σx 2 i ( σ Γ,i 12 σ 2 Γ ) 2 (2.97) (Γ µ Γ ) 2 σ 2 Γ and the unknown characteristic generator φ a ( ) depending on a equals (recall that for the multivariate normal case the conditional distribution remains normal). Note that in our application it does not really matter that the characteristic generator φ a ( ) is not known it suffices to notice that E [ X i Γ] = µ Xi,Γ = µ Xi + σγ,i ( ) 12 Γ σγ 2 µγ. The second conditioning random variable is chosen analogously as in (2.93): Λ = E[X i ]e µ i+ 1 2 σ2 i Y (i) = µ Xi e µ i+ 1 2 σ2 i Y (i). From Section it follows that the lower bound is given by the following expression: S l el = (µ Xi + σγ,i ( 12 F 1 σγ 2 Γ (U) µ Γ) ) e µ i+ 1 2 σ2 i (1 r2 i )+r iσ i Φ 1 (V ), (2.98) where correlations r i = r( Y (i), Λ) are defined as in (2.79) (with E[X i ] substituted by µ Xi ). Note that expression (2.98) simplifies in the normal case to S l N = ( µxi + r Xi σ Xi Φ 1 (U) ) e µ i+ 1 2 σ2 i (1 r2 i )+r iσ i Φ 1 (V )

108 82 Chapter 2 - Convex bounds with r Xi = r(x i, Γ) = n j=1 µ X j e µ j+ 1 2 σ2 j σx ij (. n σ Xi k,l=1 µ X k µ Xl e µ k+µ l + 1 σ 2 k 2+σ2 l )σxkl Finally, the corresponding stop-loss premiums are computed according to the following expressions: π lb (S el, d, Γ, Λ) = π lb (S N, d, Γ, Λ) = e µ i+ 1 2 σ2 i ( Φ 1 0 { ( µ + σγ,i ( 12 Xi F 1 σγ 2 Γ (u) µ Γ) ) r i σ i Φ 1( F S l el Γ=F 1 (u)(d)))} Γ d ( 1 F S l (d) ), el 1 { (µxi + r σ Xi Xi Φ 1 (u) ) e µ i+ 1 2 σ2 i 0 du ( Φ r i σ i Φ 1( F S l N Γ=F 1 (u)(d)))} du Γ d ( 1 F S l N (d) ). The moments based approximation It is also possible to find the moments based approximation SN m from formula (2.65), since one can compute the variance of S N as Var[S N ] = E X [ Var [ S N X ]] + Var X [ E [ S N X ]] = E X [ = j=1 ( ) X i X j e µ i+µ j + 1 σ (e 2 i 2+σ2 j σ ij 1 )] [ ] + Var X X i e µ i+ 1 2 σ2 i j=1 ( σ Xij ) ( ) + µ Xi µ Xj e µ i +µ j + 1 σ 2 i 2+σ2 j +σ ij j=1 µ Xi µ Xj e µ i+µ j ( σ 2 i +σ2 j ).

109 2.6. Application: the present value of stochastic cash flows 83 Here, the variances of the upper and the lower bound are computed as explained in Section We remark that for X having a multivariate elliptical distribution the computations are almost identical, with the only difference in the formula for covariances Cov[X i, X j ] = 2φ (0) X ij. Then the stop-loss premium of the moments based approximation is obtained as a convex combination π m (S el, d, Γ, Λ) = zπ lb (S el, d, Γ, Λ) + (1 z)π cub (S el, d), where z is defined as in (2.66). A numerical illustration We study the case of normally distributed payments with mean µ Xi = 1 and variance σ 2 X i = Note that the mean and the variance are the same as in the lognormal case. Moreover we assume the following correlation pattern for the payments: 1 if i = j 0.5 if i j = 1 r(x i, X j ) = 0.2 if i j = 2, 0 if i j > 2. As in the previous example, we work in the Black & Scholes setting with drift parameter µ = 0.05 and volatility σ = 0.1. We compare the performances of the lower bound SN l, the upper bound Sc N and the moments based approximation SN m with the real distribution of S N of the present value function, obtained by a Monte Carlo simulation (MC) based on simulated paths. The performance of the approximations is illustrated by the numerical values of some upper quantiles displayed in Table 2.9. The same conclusions can be drawn as in the log-normal case the upper bound SN c gives a quite poor approximation, while the lower bound SN l and the moments based approximation perform excellent. The study of stop-loss premiums in Table 2.10 confirms this observation..

110 84 Chapter 2 - Convex bounds p S l N S m N S c N MC (s.e ) (0.70) (1.02) (1.46) (2.11) (4.61) Table 2.9: Approximations for some selected quantiles with probability level p of S N. d S l N S m N S c N MC (s.e ) (4.50) (4.50) (4.16) (2.11) (0.74) (0.25) (0.08) Table 2.10: Approximations for some selected stop-loss premiums with retention d of S N Independent and identically distributed payments Finally, we consider the case where the payments X i are independent and identically distributed. The independence assumption accounts for more flexibility in modelling the underlying marginal distributions, however unlike in the lognormal and elliptical cases it imposes a rigid condition on the dependence structure. We start with defining the class of tempered stable distributions for which the methodology works particularly efficient. Tempered stable distributions The Tempered Stable law T S(δ, a, b) for a, b > 0 and 0 < δ < 1 is a one-dimensional distribution given by the characteristic function: ( ) ϕ T S (t; δ, a, b) = e ab a b 1 δ δ 2it. (2.99)

111 2.6. Application: the present value of stochastic cash flows 85 For more details we refer to e.g. Schoutens (2003). This class of distributions has the special property that the sum of independent and identically distributed tempered stable random variables is again tempered stable. This is formalized in the following lemma: Lemma 9 (Sum of tempered stable random variables). If X i are i.i.d. random variables T S(κ, a, b)-distributed for i = 1, 2,..., n, then their sum X 1 + X X n is T S(κ, na, b)-distributed. Proof. Consider the corresponding characteristic functions. We get ϕ X1 +X 2 + +X n (t) = ( ϕ T S (t; κ, a, b) ) n = e (na)b (na)(b 1 κ 2it) κ = ϕ T S (t; κ, na, b). The first two moments of a random variable X T S(δ, a, b) are given by E[X] = 2aδb δ 1 δ and Var[X] = 4aδ(1 δ)b δ 2 δ. In the sequel we provide more details about two well-known special cases: the gamma distribution and the inverse Gaussian distribution. The gamma distribution Gamma(a, b) corresponds to the limiting case when δ 0. The characteristic function of the gamma distribution is given by ( ϕ(t; a, b) = 1 it ) a. b Notice that for X Gamma(a, b) one has E[X] = a b and Var[X] = a. b 2 The inverse Gaussian distribution is a member of the class of Tempered Stable distributions with δ = 1 2. Thus, the characteristic function is given by ( 2it+b ) ϕ(t; a, b) = e a 2 b. Moreover the mean and variance of X IG(a, b) are given by E[X] = a b and Var[X] = a b 3. We consider now sums of the form S ind = X i e Y (i), (2.100) where the process Y (i) is defined like in the previous examples and the payments X i are independent and follow the law defined by the cdf F X ( ).

112 86 Chapter 2 - Convex bounds The upper bound The computation of the upper bound is straightforward (as described in Section 2.5.3): Sind c = F 1 X (U) e µ i+σ i Φ 1 (V ). (2.101) The stop-loss premiums for the upper bound are given by an expression analogous to (2.75), with S c replaced by S c ind. The lower bound To compute the lower bound, we start with defining the conditioning random variables Γ and Λ. Let Γ = X 1 + X X n. If we know the distributions of X i, the distribution of the sum Γ is also known. In particular, for X i gamma distributed the sum Γ remains gamma distributed and the same for X i inverse Gaussian distributed. Like in the previous examples, the conditional random variable Λ is chosen as Λ = E[X i ]e µ i+ 1 2 σ2 i Y (i). (2.102) Now, the lower bound can be written as S l ind = 1 n F 1 Γ (U) e µ i+ 1 2 (1 r2 i )σ2 i +r iσ i Φ 1 (V ), where the correlations r i = r ( Y (i), Λ ) are defined as in (2.79). Note that the computation of stop-loss premiums of the lower bound is straightforward, by applying (2.83) and replacing S l by S l ind. Cumulative distribution functions In this case there is a more efficient method to compute the distribution functions than this described in Section

113 2.6. Application: the present value of stochastic cash flows 87 Remark 4. The cumulative distribution function of the product W of two non-negative independent variables X and Y can be written as ( z ) ( ) 1 z F W (z) = F Y df X (x) = F Y x 0 F 1 X (u) du. (2.103) Using this result one can compute the cumulative distribution functions of the upper and the lower bound as 1 ( ) y F S c ind (y) = F X 0 F S 1 (v) dv, c 1 ( ) y F S l (y) = F 1 ind n 0 Γ F S 1 dv, (v) l where S c = F 1 S c (v) = e µ i+σ i Φ 1 (V ), Sl = e µ i+σ i Φ 1 (v), e µ i+ 1 2 (1 r2 i )σ2 i +r iσ i Φ 1 (V ), F 1 S l (v) = The moments based approximation e µ i+ 1 2 (1 r2 i )σ2 i +r iσ i Φ 1 (v). The moments based approximation of S ind can be found in a similar way to the moments based approximation for elliptical distributions. The key step is to compute the variance of S ind : [ Var[S ind ] = E X Var [ S ind X ]] [ + Var X E [ S ind X ]] = E X [ = j=1 ( ) X i X j e µ i+µ j + 1 σ (e 2 i 2+σ2 j σ ij 1 )] [ ] + Var X X i e µ i+ 1 2 σ2 i j=1 + ( E[Xi ]E[X j ] ) ( ) e µ i+µ j + 1 σ (e 2 i 2+σ2 j σ ij 1 ) Var[X i ]e 2µ i+σi 2. (2.104)

114 88 Chapter 2 - Convex bounds p S l ind S m ind S c ind MC (s.e ) (0.70) (1.02) (1.46) (2.11) (4.61) Table 2.11: Approximations for some selected quantiles with probability level p of S ind for gamma i.i.d. liabilities. The variances of the upper and the lower bound are computed as explained in Subsection Consequently, the stop-loss premium of the moments based approximation is obtained as a convex combination π m (S ind, d, Γ, Λ) = zπ lb (S ind, d, Γ, Λ) + (1 z)π cub (S ind, d), where z is defined as in (2.66). A numerical illustration We consider in this application independent Gamma(100, 100) distributed future payments. Note that this choice of parameters implies that E[X] = 1 and Var[X] = 0.01 i.e. we take the same mean and variance of liabilities as in the lognormal and normal cases. As before we work in a Black & Scholes setting with drift µ = 0.05 and volatility σ = 0.1. We compare the performances of the lower bound Sind l, the upper bound Sc ind and the moments based approximation Sind m with the real value S ind obtained by a Monte Carlo simulation (MC) based on simulated paths. The results are very similar to the normal and lognormal case. It is worth noticing that the variance of S ind ( ) is a bit lower than in the lognormal case ( ) and in the normal case ( ). This is due to independence of gamma-payments while we imposed a slight positive dependence in the previous cases. The quality of the approximations is illustrated by some upper quantiles displayed in Table The lower bound Sind l and the moments based approximation Sind m perform well, but not as good as in the lognormal and normal cases (probably because the conditioning random variable

115 2.7. Proofs 89 d S l ind S m ind S c ind MC (s.e ) (4.44) (4.44) (4.06) (2.08) (0.77) (0.27) (0.09) Table 2.12: Approximations for some selected stop-loss premiums with retention d of S ind for gamma i.i.d. liabilities. Γ does not take discounting factors into account). The study of stop-loss premiums in Table 2.12 goes in line with these findings. 2.7 Proofs Upper bound based on lower bound (2.44) In the following we shall derive an easily computable expression for (2.26). The second expectation term in the product (2.26) equals, when denoting by F Λ ( ) the normal cumulative distribution function of Λ, E[I (Λ<dΛ )] = 0 Pr[Λ d Λ ] + 1 Pr[Λ < d Λ ] = F Λ (d Λ ) = Φ(d Λ). (2.105) The first expectation term in the product (2.26) can be expressed as E [ Var [S Λ] I (Λ<dΛ )] = E [ E[S 2 Λ]I (Λ<dΛ )] E [ (E[S Λ]) 2 I (Λ<dΛ )]. (2.106) Now consider the second term of the right-hand side of (2.106) E [ (E[S Λ]) 2 ] dλ I (Λ<dΛ ) = (E[S Λ = λ]) 2 df Λ (λ). (2.107) According to (2.32) and using the notation Z ij introduced before, we can

116 90 Chapter 2 - Convex bounds express (2.107) as E [ (E[S Λ]) 2 ] I (Λ<dΛ ) = = = = dλ dλ ( 2 E[X i Λ = λ]) df Λ (λ) ( ) 2 α i e E[Z i]+r i σ Zi Φ 1 (v)+ 1 2(1 ri 2 )σz 2 i df Λ (λ) dλ j=1 j=1 α i α j e E[Z ij]+(r i σ Zi +r j σ Zj )Φ 1 (v) e 2 1 (1 ri 2 )σz 2 +(1 r 2 i j)σz j 2 df Λ (λ) (1 ri 2 )σz 2 +(1 r 2 i j)σz j 2 α i α j e E[Z ij]+ 1 2 dλ e (r iσ Zi +r j σ Zj )Φ 1 (v) df Λ (λ). (2.108) Next, applying Lemma 7 to (2.108) with a = r i σ Zi + r j σ Zj yields E [ (E[S Λ]) 2 I (Λ<dΛ )] = α i α j e E[Z ij]+ 1 2 (σ2 Z +σ 2 i Z +2r i r j σ Zi σ Zj ) ( j Φ d Λ ( )) r i σ Zi + r j σ Zj.(2.109) j=1 Now consider the first term of the right-hand side of expression (2.106), E [ E[S 2 Λ]I (Λ<dΛ )]. The term E[S 2 Λ] is given by (2.42). By applying (2.43) with a = r ij σ Zij = r i σ Zi + r j σ Zj, and simplifying, we obtain

117 2.7. Proofs 91 E [ E[S 2 ] I (Λ<dΛ ) = = = = j=1 j=1 j=1 dλ α i α j e E[Z ij]+r ij σ Zij Φ 1 (v)+ 1 2(1 r 2 ij)σ 2 Z ij df Λ (λ) α i α j e E[Z ij]+ 1 2(1 rij)σ 2 Z 2 dλ ij e r ijσ Zij Φ 1 (v) df Λ (λ) j=1 α i α j e E[Z ij]+ 1 2(1 rij)σ 2 Z 2 r ij 2 σ2 Z + ij ij 2 Φ(d Λ r ij σ Zij ) σ α i α j e E[Z Z 2 ij ij]+ 2 Φ(d Λ (r i σ Yi + r j σ Yj )). (2.110) Combining (2.110) and (2.109) into (2.106), and then substituting (2.105) and (2.106) into (2.26) we get the following expression for the error bound ε(d Λ ) (2.26): ε(d Λ ) = 1 2 (Φ(d Λ)) 1 2 = 1 2 (Φ(d Λ)) 1 2 = 1 2 (Φ(d Λ)) 1 2 { j=1 [ α i α j σ e E[Z Z 2 ij ij]+ 2 Φ ( d Λ ( r i σ Zi + r j σ Zj )) e E[Z ij]+ 1 2 (σ2 Z i +σ 2 Z j +2r i r j σ Zi σ Zj ) Φ ( d Λ ( r i σ Zi + r j σ Zj )) ] } 1 2 { j=1 α i α j e E[Zij] Φ ( d Λ ( )) r i σ Zi + r j σ Zj ( ) } 1 e 1 2 (σ2 Z +σ 2 1 i Z +2σ Zi Z j j ) e 2 (σ2 Z +σ 2 2 i Z +2r i r j σ Zi σ Zj ) j { j=1 α i α j e E[Z ij]+ 1 2 (σ2 Z +σ 2 i Z ) ( j Φ d Λ ( )) r i σ Zi + r j σ Zj ( e σ Z i Z j e σ Z i σ Zj r i r j ) } 1 2.

118 92 Chapter 2 - Convex bounds Partially exact/comonotonic upper bound (2.45) Applying Lemma 7 with a = r i σ Zi, and using (2.32), we can express the second term I 2 in (2.22) in closed-form: + d Λ + = = = E[S d Λ = λ]df Λ (λ) E[S Λ = λ]df Λ (λ) d(1 F Λ (d Λ )) d Λ + α i e E[Z i]+ 2(1 r 1 i 2 )σz 2 i e r iσ Zi Φ 1 (v) df Λ (λ) d(1 Φ(d Λ)) d Λ α i e E[Z i]+ σ 2 Z i 2 Φ(ri σ Zi d Λ) dφ( d Λ). (2.111) Substituting (2.33) in (2.28) we end up with the following upper bound of I 1 similar to (2.37) but now with an integral from zero to Φ(d Λ ): dλ dλ = = E[(S d) + Λ = λ]df Λ (λ) Φ(d Λ ) 0 E[(S u d) + Λ = λ]df Λ (λ) E[(S u d) + V = v] dv α i e E[Z i]+ 1 2 σ2 Z i (1 ri 2 ) Φ(d Λ ) 0 ( d Φ(d Λ) ( e r iσ Zi Φ 1 (v) Φ sign(α i ) Φ(d Λ ) 0 F S u V =v(d)dv 1 ri 2 σ Z i Φ 1 ( F S u V =v(d) )) dv ), (2.112) where we recall that d Λ is defined as in (2.43), and the cumulative distribution F S u(d) is, according to (2.36), determined by α i e E[Z i]+r i σ Zi Φ 1 (v)+sign(α i ) 1 ri 2 σ Z i Φ 1 (F S u(d V =v)) = d. Finally, adding (2.112) to the exact part (2.111) of the decomposition (2.22) results in the partially exact/comonotonic upper bound.

119 Chapter 3 Reserving in life insurance business Summary In the traditional approach to life contingencies only decrements are assumed to be stochastic. In this contribution we consider the distribution of a life annuity (and a portfolio of life annuities) when also the stochastic nature of interest rates is taken into account. Although the literature concerning this topic is already quite rich, the authors usually restrict themselves to the computation of the first two or three moments. However, if one wants to determine e.g. capital requirements using more sofisticated risk measures like Value-at-Risk or Tail Value-at-Risk, more detailed knowledge about underlying distributions is required. For this purpose, we propose to use the theory of comonotonic risks introduced in Chapter 2. This methodology allows to obtain reliable approximations of the underlying distribution functions, in particular very accurate estimates of upper quantiles and stop-loss premiums. Several numerical illustrations confirm the very high accuracy of the methodology. 3.1 Introduction Unlike in finance, in insurance the concept of stochastic interest rates emerged quite recently. In the traditional approach to life contingencies only decrements are assumed to be stochastic see e.g. Bowers et al. (1986), Wolthuis & Van Hoek (1986). Such a simplification allows to treat effectively summary measures of financial contracts such as the mean, the 93

120 94 Chapter 3 - Reserving in life insurance business standard deviation or the upper quantiles. For a more detailed discussion about the distributions in life insurance under deterministic interest rates, see e.g. Dhaene (1990). In non-life insurance the use of deterministic interest rates may be justified by short terms of insurance commitments. In the case of the life insurance and the life annuity business, durations of contracts are typically very long (often 30 or even more years). Then uncertainty about future rates of return becomes very high. Moreover the financial and investment risk unlike the mortality risk cannot be diversified with an increase in the number of policies. Therefore in order to calculate insurance premiums or mathematical reserves, actuaries are forced to adopt very conservative assumptions. As a result the diversification effects between interest rates in different investment periods may not be taken into account (i.e. that poor investment results in some periods are usually compensated by very good ones in others) and the life insurance business becomes too expensive, both for the insureds who have to pay higher insurance premiums and for the shareholders who have to provide more capital than necessary. Profitsharing can partially solve this problem. For these reasons the necessity to introduce models with stochastic interest rates have been well-understood in the actuarial world. In the actuarial literature numerous papers have treated the random interest rates. In Boyle (1976) autoregressive models of order one are introduced to model interest rates. Bellhouse & Panjer (1980, 1981) use similar models to compute moments of insurance and annuity functions. In Wilkie (1976) the force of interest is assumed to follow a Gaussian random walk. Waters (1978) computes the moments of actuarial functions when the interest rates are independent and identically Gaussian distributed. He computes also moments of portfolios of policies and approximates the limiting distribution by Pearson s curves. In Dhaene (1989) the force of interest is modelled as an ARMA(p, d, q) process. He uses this model to compute the moments of present value functions. Norberg (1990) provides an axiomatic approach to stochastic interest rates and the valuation of payment streams. Parker (1994d) compares two approaches to the randomness of interest rates: by modelling only the accumulated interest and by modelling the force of interest. Both methodologies are illustrated by calculating the mean, the standard deviation and the skewness of the annuity-immediate. An overview of stochastic life contingencies with solvency valuation is presented in Frees (1990). In the papers of Beekman & Fuelling (1990,

121 3.1. Introduction ) the mean and the standard deviation of continuous-time life annuities are calculated with the force of mortality modelled as an Ornstein- Uhlenbeck and a Wiener process respectively. In Beekman & Fuelling (1993) expressions are given for the mean and the standard deviation of the future life insurance payments. Norberg (1993) derives the first two moments of the present value of stochastic payment streams. The first three moments of homogeneous portfolios of life insurance and endowment policies are calculated in Parker (1994a,b) and the results are generalized to heterogeneous portfolios in Parker (1997). The same author (1994c, 1996) provides a recursive formula to calculate an approximate distribution function of the limiting homogeneous portfolio of term life insurance and endowment policies. In Dȩbicka (2003) the mean and the variance are calculated for the present value of discrete-time payment streams in life insurance. Although the literature on stochastic interest rates in life insurance is already quite rich, for most of the problems no satisfactory solutions have been found as yet. In almost all papers the authors restrict themselves to calculating the first two or three moments of the present value function (except Waters (1978), Parker (1994d, 1996)). 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. It does not emphasize the tail properties of the distribution and does not give any reliable estimates of the Value-at-Risk or other tail-related risk measures, for which simulation methods have to be deployed. The same applies to risk measures based on stop-loss premiums, like Expected Shortfall. In this chapter we aim to provide some conservative estimates both for high quantiles and stop-loss premiums for an individual policy 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. Using the results of Chapter 2 we will approximate the quantiles of the present value of a life annuity and a portfolio of life annuities. We perform our analysis separately for a single life annuity and a whole portfolio of policies. Our solution enables to solve with a great accuracy

122 96 Chapter 3 - Reserving in life insurance business 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%)? Similar problems were studied by Dufresne (2004) and Milevsky & Wang (2004). 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 and to model the present value of future benefits as a scalar product of mutually independent random vectors. This chapter is mainly based on Hoedemakers, Darkiewicz & Goovaerts (2005) and is organized as follows. In Section 2 we give a summary of the model assumptions and properties for the mortality process that are needed to reach our goal. In the first part of Section 3 we apply the results of Chapter 2 to the present value of a single life annuity policy. In the second part of this section we present the convex bounds for a homogeneous portfolio of policies. A numerical illustration is provided at the end of each part. We also illustrate the obtained results graphically. 3.2 Modelling stochastic decrements A life annuity may be defined as a series of periodic payments where each payment will actually be made only if a designated life is alive at the time the payment is due. Let us consider a person aged x years, also called a life aged x and denoted by (x). We denote his or her future lifetime by T x. Thus x + T x will be the age of death of the person. The future lifetime T x is a random variable with a probability distribution function G x (t) = Pr[T x t] = t q x, t 0. The function G x represents the probability that the person will die within t years, for any fixed t. We assume that G x is known. We define K x = T x, the number of completed future years lived by (x), or the curtate future

123 3.2. Modelling stochastic decrements 97 lifetime of (x). The probability distribution of the integer valued random variable K x is given by Pr[K x = k] = Pr[k T x < k + 1] = k+1 q x k q x = k q x, k = 0, 1,.... Let us denote the lifetime from birth by the random variable T. We assume Pr[T x t] = Pr[T x + t T x]. With this notation, T = d T 0. Further, the ultimate age of the life table is denoted by ω, this means that ω x is the first remaining lifetime of (x) for which ω x q x = 1, or equivalently, G 1 x (1) = ω x. In the remainder of this chapter we will always use the standard actuarial notation: Pr[T x > t] = t p x, Pr[T x > 1] = p x, Pr[T x t] = t q x, Pr[T x 1] = q x. In this chapter we consider three types of annuities. The present value of a single life annuity for a person aged x paying periodically (e.g. yearly) a fixed amount of α i (i = 1,..., ω x ) can be expressed as K x S sp,x = α i e Y (i) = ω x I (Tx>i)α i e Y (i). (3.1) We consider also the present value of a homogeneous portfolio of life annuities this random variable is particularly interesting for an insurer who has to determine a sufficient level of the reserve and the solvency margin. Assuming that every beneficiary gets a fixed amount of α i (i = 1,..., ω x ) per year, the present value can be expressed as follows S pp,x = ω x α i N i e Y (i), (3.2) where N i denotes the remaining number of policies-in-force in year i. Finally, consider a portfolio of N 0 homogeneous life annuity contracts for which the future lifetimes of the insureds T x (1), T x (2),..., T (N 0) x are assumed to be independent. Then the insurer faces two risks: mortality risk and investment risk. Note that from the Law of Large Numbers the

124 98 Chapter 3 - Reserving in life insurance business mortality risk decreases with the number of policies N 0 while the investment risk remains the same (each of the policies is exposed to the same investment risk). Thus, for sufficiently large N 0 we have that ω x ω x ω x α i N i e Y (i) = N 0 N i Y (i) α i e N 0 Y (i) α i i p x e. N 0 Hence in the case of large portfolios of life annuities it suffices to compute risk measures of an average portfolio S app,x given by S app,x = ω x α i i p x e Y (i) = E [ S sp,x Y (1),, Y ( ω x ) ]. (3.3) Remark 5. For the random variables S app,x and S sp,x one has that S app,x cx S sp,x and consequently Var[S app,x ] Var[S sp,x ]. Indeed, let Γ denote a random variable independent of T x. Then, it follows immediately from Theorem 8 that S sp,x = cx = ω x ω x ω x = S app,x. Y (i) I (Tx>i)α i e Y (i) E[I (Tx>i) Γ]α i e Y (i) ip x α i e Obviously S sp,x, S pp,x and S app,x depend on the distribution of the total lifetime T. We assume that T follows the Gompertz-Makeham law, i.e. the force of mortality at age ξ is given by the formula µ ξ = α + βc ξ, where α > 0 is a constant component, interpreted as capturing accident hazard, and βc ξ is a variable component capturing the hazard of aging with β > 0 and c > 1. This leads to the survival probability tp x = Pr[T x > t] = e x+t x µ ξ dξ = s t g cx+t c x,

125 3.2. Modelling stochastic decrements 99 where β s = e α and g = e log c. (3.4) In numerical illustrations we use the Belgian analytic life tables MR and FR for life annuity valuation, with corresponding constants for males: s = , g = and c = and for females: s = , g = and c = Denote by T and T x the corresponding random variables from the Gompertz family the subclass of the Makeham-Gompertz family with the force of mortality given by It is straightforward to show that µ ξ = βcξ. T x d = min(t x, E/α), (3.5) where E denotes a random variable from the standard exponential distribution, independent of T. Indeed, one has that Pr[min(T x, E/α) > t] = Pr[T x > t] Pr[E > αt] x+t = e x µ ξ dξ e αt x+t = e x µ ξ dξ = Pr[T x > t]. The cumulative distribution function for the Gompertz law, unlike for the Makeham-Gompertz law in general, has an analytical expression for the inverse function and therefore (3.5) can be used for simulations. For generating one random variate from Makeham s law, we use the composition method (Devroye, 1986) and perform the following steps 1. Generate G from the Gompertz s law by the well-known inversion method 2. Generate E from the exponential(1) distribution 3. Retain T = min(g, E/α), where α = log s, see (3.4).

126 100 Chapter 3 - Reserving in life insurance business 3.3 The distribution of life annuities This section is organized into 2 subsections. In the first subsection we derive upper and lower bounds in convex order for the distribution of the present value of a single life annuity given a mortality law T and a model for the returns. This distribution is very important in the context of socalled personal finance problems. Suppose that (x) disposes of a lump sum L. What is the amount that (x) can yearly consume to be sure with a sufficiently high probability (e.g. p = 99%) that the money will not be run out before death? Obviously, to answer this question one has to compute the Value-at-Risk measure of the distribution at an appropriate chosen level. In the second part of this section we will consider the distribution of a homogeneous and average portfolio of life annuities. An insurer has to derive this distribution to determine its future liabilities and solvency margin. Notice that the presented methodology is appropriate not only in the case of large portfolios when the limiting distribution can be used on the basis of the law of large numbers but also for portfolios of average size (e.g ) which are typical for the life annuity business. The vector Y = ( Y (1), Y (2),..., Y (n) ) is assumed to have a n-dimensional normal distribution with given mean vector and covariance matrix µ = (µ 1,..., µ n ) = ( E[Y (1)], E[Y (2)],..., E[Y (n)] ) Σ = [σ ij ] 1 i,j n = [ Cov ( Y (i), Y (j) )] 1 i,j n. In the above notation we will denote σ ii by σ 2 i A single life annuity In this subsection we consider a whole life annuity of α i (> 0) payable at the end of each year i while (x) survives, described by the formula K x S sp,x = α i e Y (i) = ω x I (Tx>i)α i e Y (i).

127 3.3. The distribution of life annuities 101 The upper bound The random variable X i = I (Tx>i) is Bernoulli( i p x ) distributed and thus the inverse distribution function is given by { F 1 1 for p > X i (p) = i q x 0 for p i q x. This leads to the following formula for the upper bound S c sp,x = = ω x F 1 Tx (U) F 1 X i (U)F 1 α i e Y (i) (V ) F 1 α i e Y (i) (V ), where U and V are independent standard uniformly distributed random variables. Thus the conditional quantiles are given by t F 1 Ssp,x c Tx=t(p) = F 1 (V ) α i e Y (i) and the conditional distribution function can be computed numerically from the identity t α i e µ i+sign(α i )σ i Φ 1 (F S c sp,x Tx=t(y)) = k α i e µ i+sign(α i )σ i Φ 1 (F S c sp,x Kx=k(y)) = y. Define S k as follows: S k = k α i e Y (i), (3.6)

128 102 Chapter 3 - Reserving in life insurance business then S d k = Ssp,x K x = k. Hence, the distribution function of Ssp,x c can be computed as F S c sp,x (y) = = = ω x k=1 ω x k=1 ω x k=1 Pr[K x = k]f S c sp,x K x=k(y) k q x F Sc k (y) k q x Pr [ k ] α i e µ i+sign(α i )σ i Φ 1 (U) y, with S c k = k F 1 α i e Y (i) (U) and U a standard uniform random variable. The computation of the corresponding stop-loss premiums is also straightforward: π cub (S sp,x, d) = E Kx [ E [ (S c sp,x d) + K x ] ] = = ω x k=1 ω x k=1 k q x π cub ( S k, d) k q x ( k π ( α i e Y (i), d c k,i) ), where d c k,i is defined analogously to (2.73) as d c k,i = α ie µ i+sign(α i )σ i Φ 1 (F Sc k (d)) and the values of π(α i e Y (i), d k,i ) are computed as in (2.74). The stop-loss premium of S c sp,x at retention d can be written out explicitly as follows π cub (S sp,x, d) = ω x k=1 k q x { k α i e µ i+ σ 2 ( i 2 Φ sign(α i )σ i Φ 1( F Sc (d) )) k d ( 1 F Sc k (d) )}.

129 3.3. The distribution of life annuities 103 The lower bound For the lower bound one faces the problem of choosing appropriate conditioning random variables Γ and Λ. The random variables X i are in fact comonotonic and depend only on the future lifetime T x, thus Γ = T x is the most natural choice. As a result one simply gets E [ I (Tx>i) T x ] = I(Tx>i). The choice of the second conditioning random variable Λ is less obvious. We propose two different approaches: 1. Λ (a) = ω x ip x α i e µ i+ 1 2 σ2 i Y (i). Intuitively it means that the conditioning random variable is chosen as a first order approximation to the present value of the limiting portfolio S app,x in (3.3). 2. Consider the maximal variance conditioning random variables of the form Λ j = j α ie µ ( ) i+ 1 2 σ2 i Y (i) j = 1,..., ω x and the corresponding lower bounds S l,j sp,x = K x E [ α i e Y (i) ] Λ j, j = 1,..., ω x from which one chooses the lower bound with the largest variance. The corresponding conditioning random variable will be denoted as Λ (m). This choice can be motivated as follows. For two random variables X and Y with X cx Y one has that Var[X] Var[Y ]. As discussed in Chapter 2 we should choose Λ such that the goodnessof-fit expressed by the ratio z = Var[Sl sp,x] is as close as possible to Var[S sp,x ] 1. Hence one can expect that a lower bound with a larger variance will provide a better fit to the original random variable. Having chosen the conditioning random variable Λ one proceeds as in the case of the upper bound: the first step requires the computation of the conditional distribution of the lower bound from the formula k α i e µ i+ 1 2 σ2 i (1 r2 i )+σ ir i Φ 1 (F S l sp,x Kx=k (y)) = y.

130 104 Chapter 3 - Reserving in life insurance business The cumulative distribution function of S l sp,x can then be computed as F S l sp,x (y) = = = ω x k=1 ω x k=1 ω x k=1 k q x F S l sp,x K x=k(y) k q x F Sl k (y) k q x Pr [ k ] α i e µ i r i σ i Φ 1 (U)+ 1 2 (1 r2 i )σ2 i y, with S l k = E[ S k Λ] and U a standard uniform random variable. The computation of the corresponding stop-loss premium is similar to the one of the upper bound and as a result one gets the following explicit solution π lb (S sp,x, d, Γ, Λ) = E Kx [ E [ (S l sp,x d) + K x ] ] = = ω x k=1 ω x k=1 k q x π lb ( S k, d, Λ) ( k ( k q x π E [ α i e Y (i) Λ ], dk,i) ) l, with d l k,i given by d l k,i = α ie µ i+ 1 2 σ2 i (1 r2 i )+σ ir i Φ 1 (F S l sp,x Kx=k (d)). ( Note that the values of π E [ α i e Y (i) Λ ] ), d l k,i can be computed as in (2.82). The stop-loss premium of Ssp,x l at retention d can be written out explicitly as follows π lb (S sp,x, d, Γ, Λ) = ω x k=1 k q x { k α i e µ i+ σ 2 ( i 2 Φ r i σ i Φ 1( F Sl (d) )) k d ( 1 F Sl k (d) )}.

131 3.3. The distribution of life annuities 105 The lower bound based on a lifetime dependent conditioning random variable In this subsection we show how it is possible to improve the lower bound of a scalar product if one of the vectors is comonotonic. We state this result in the following lemma. Lemma 10. Consider a scalar product of random variables S = n X iy i, where the random vectors X and Y are ( independent and X is additionally assumed to be comonotonic, i.e. X = F 1 X 1 (U), F 1 X 2 (U),..., F 1 X n (U) ). Let Λ(u) be a random variable which is defined for each u (0, 1) separately. Define S cl (u) as follows: S cl (u) = then S cl (u) = d (S cl U = u). distribution function F 1 X i (u) E [ Y i Λ(u) ], Define the random variable S cl through its F S cl(y) = 1 0 F S cl U=u(y)du. Then S cl cx S. Remark 6. Obviously the conditioning random variable U can be replaced by any other random variable which determines the comonotonic vector X by a functional relationship. We consider here the case when X i = I (Tx>i) = I (Kx i) and therefore it is convenient to condition on the future lifetime K x. Proof. Let S(u) denote a random variable distributed as S given that U = u. From Definition 1b of convex order, it follows immediately that S cl (u) cx S(u). Indeed, let v(.) be an arbitrary convex function. Then we get E [ v(s cl ) ] = 1 which completes the proof. 0 E [ v(s cl (u)) ] du 1 0 E [ v(s(u)) ] du = E [ v(s) ],

132 106 Chapter 3 - Reserving in life insurance business Because of Lemma 10, one can determine a lower bound of a single life annuity using the following conditioning random variable: K x Λ Kx = α i e µ i+ 1 2 σ2 i Y (i). Intuitively it is clear that the lower bound defined by the random variable Λ Kx should approximate the underlying distribution better than those defined by the conditioning random variables Λ (a) and Λ (m). As before, one starts with computing the conditional distributions for the lower bound S cl sp,x numerically by considering the equation k with correlations r i,k given by α i e µ i+ 1 2 (1 r2 i,k )σ2 i +r i,kσ i Φ 1 ( F S cl sp,x Kx=k(y) ) = y, r i,k = Cov [ Y (i), Λ k ] Var[Y (i)] Var[Λk ] Consequently, the distribution function of S cl sp,x can be obtained as with F S cl ω x (y) = sp,x k=1 ω x Pr[K x = k]f S cl sp,x K x=k(y) = k=1 k q x F (y), Scl k S cl k = E[ Sk Λ k ]. (3.7) The stop-loss premiums of Ssp,x cl can be computed as follows [ π clb (S sp,x, d, Γ, Λ) = E Kx E [ (Ssp,x cl ] ] d) + K x = = ω x k=1 ω x k=1 k q x π lb ( S k, d, Λ k ) k q x ( k π ( E [ α i e Y (i) Λ k ], d cl k,i) ),

133 3.3. The distribution of life annuities 107 with d cl k,i given by d cl k,i = α ie µ i+ 1 2 σ2 i (1 r2 i,k )+σ ir i,k Φ 1 (F Scl k (d)). The stop-loss premium of S cl sp,x at retention d can be written out explicitly as follows π clb (S sp,x, d, Γ, Λ) = ω x k=1 { k k q x α i e µ i+ σ 2 i 2 Φ (r i,k σ i Φ 1( F (d) )) Scl k d ( 1 F (d) )}. Scl k The moments based approximation Having computed the upper bound Ssp,x c and the lower bounds Ssp,x l and Ssp,x, cl one can compute two moments based approximations as described in Subsection To find the coefficient z given by (2.15) one needs to calculate the variances of S c sp,x, S l sp,x, S cl sp,x and S sp,x. The variance of S c sp,x and S l sp,x can be computed as explained in Subsection The variance of S sp,x and S cl sp,x can be treated very similarly. Indeed, after some simple calculations one gets Var [ Ssp,x cl ] Var [ S sp,x ] [ = E Kx E [ (Ssp,x) cl 2 ] ] ( K x E [ Ssp,x cl ] ) 2 = ω x k=1 k q x E [ ( Scl k ) ] ( 2 E [ Ssp,x] ) cl 2, = E Kx [E [ (S sp,x ) 2 ] ] K x (E [ ] ) 2 S sp,x = ω x k=1 [ ( ) ] ( 2 k q x E Sk E [ ] ) 2 S sp,x,

134 108 Chapter 3 - Reserving in life insurance business S cl k where and S k are defined as in (3.7) and (3.6) respectively. Thus it suffices to plug in E [ ] [ ] Scl k = E Sk [ ( ) ] E Scl 2 k [ ( ) ] 2 E Sk = = = k k j=1 k j=1 α i e µ i+ σ 2 i 2, k α i α j e µ i µ j (σ2 i +σ2 j )+r i,kr j,k σ i σ j, k α i α j e µ i µ j (σ2 i +σ2 j )+σ ij, and E [ S sp,x ] = E [ S cl sp,x ω x ] = k=1 k q x E [ ω x ] Sk = k=1 k q x E [ ] Scl k. Now one can compute distributions of the moment based approximations from the formulas F S m sp,x (y) = z 1 F S l sp,x (y) + (1 z 1 )F S c sp,x (y), F S cm sp,x (y) = z 2 F S cl sp,x (y) + (1 z 2)F S c sp,x (y) and their corresponding stop-loss premiums as π m (S sp,x, d, Γ, Λ) = z 1 π lb (S sp,x, d, Γ, Λ) + (1 z 1 )π cub (S sp,x, d), π cm (S sp,x, d, Γ, Λ) = z 2 π clb (S sp,x, d, Γ, Λ) + (1 z 2 )π cub (S sp,x, d), where z 1 = Var[Sc sp,x] Var[S sp,x ] Var[Ssp,x] c Var[Ssp,x] l and z 2 = Var[Sc sp,x] Var[S sp,x ] Var[Ssp,x] c Var[S cl A numerical illustration sp,x]. We examine the accuracy and efficiency of the derived approximations for a single life annuity of a 65-years old male person with yearly unit payments. We restrict ourselves to the case of a Black & Scholes setting (model BS) with drift µ = 0.05 and volatility σ = 0.1. We assume further that the future lifetime T 65 follows the Makeham-Gompertz law with the corresponding coefficients of the Belgian analytic life table MR (see Section

135 3.3. The distribution of life annuities ). We compare the distribution functions of the upper bound Ssp,65 c and the lower bounds Ssp,65 l and Scl sp,65, as described in the previous sections, with the original distribution function of S sp,65 based on extensive Monte Carlo (MC) simulation. We generated paths and for each estimate we computed the standard error (s.e.). As is well-known, the (asymptotic) 95% confidence interval is given by the estimate plus or minus 1.96 times the standard error. Note also that the random paths are based on antithetic variables in order to reduce the variance. Notice that to compute the lower bound we use as conditioning random variable Λ (m) = Λ 24 (the value j = 24 was found to be the one that maximizes the variance as described in Section 3.3.1). Figure 3.1 shows the cumulative distribution functions of the approximations, compared to the empirical distribution. One can see that the lower bound Ssp,65 cl is almost indistinguishable from the original distribution. In order to have a better view on the behavior of the approximations in the tail, we consider a QQ-plots where the quantiles of Ssp,65 l, Scl sp,65 and Ssp,65 c are plotted against the quantiles of S sp,65 obtained by simulation. The different bounds will be good approximations if the plotted points (F 1 S sp,65 (p), F 1 (p)), (F 1 Ssp,65 l S sp,65 (p), F 1 (p)) and (F 1 Ssp,65 cl S sp,65 (p), F 1 Ssp,65 c (p)) for all values of p in (0, 1) do not deviate too much from the line y = x. From the QQ-plot in Figure 3.2, we can conclude that the comonotonic upper bound slightly overestimates the tails of S sp,65, whereas the accuracy of the lower bounds Ssp,65 l and Scl sp,65 is extremely high; the corresponding QQ-plot is indistinguishable from a perfect straight line. These visual observations are confirmed by the numerical values of some upper quantiles displayed in Table 3.1, which also reports the moments based approximations Ssp,65 m and Scm sp,65. Stop-loss premiums for the different approximations are compared in Figure 3.3 and Table 3.2. This study confirms the high accuracy of the derived bounds. Note that for very high values of d the differences become larger, however these cases don t represent any practical importance. All Monte Carlo estimates are very close to π clb (S sp,65, d, Γ, Λ) and some of them even turn out to be smaller than this lower bound for. This not only demonstrates the difficulty of estimating stop-loss premiums by simulation, but it also indicates the accuracy of the lower bound π clb (S sp,65, d, Γ, Λ). Indeed, since the Monte Carlo estimate is based on random paths, it can be smaller than π clb (S sp,65, d, Γ, Λ) and this is very likely to happen if the

136 110 Chapter 3 - Reserving in life insurance business p Ssp,65 l Ssp,65 cl Ssp,65 m Ssp,65 cm Ssp,65 c MC (s.e ) (0.978) (1.420) (1.896) (2.816) (6.324) Table 3.1: Approximations for some selected quantiles with probability level p of S sp,65. d Ssp,65 l Ssp,65 cl Ssp,65 m Ssp,65 cm Ssp,65 c MC (s.e ) (9.43) (8.67) (5.89) (0.34) (0.21) (0.10) (0.02) (0.004) Table 3.2: Approximations for some selected stop-loss premiums with retention d of S sp,65. lower bound is close to the real stop-loss premium. Table 3.3 compares the stop-loss premium of the comonotonic upper bound with the partially exact/comonotonic upper bound π pecub (S sp,65, d, Λ, Γ) (PECUB) and the two combination bounds π eub (S sp,65, d, Λ, Γ) (EMUB) (upper bounds based on the lower bound Ssp,65 l ) and πmin (S sp,65, d, Λ, Γ) (MIN). For the partial exact/comonotonic upper bound we use the same conditioning variable as for the lower bound Ssp,65 cl. Remark that the decomposition variable is of the form (2.55) with Λ Λ n. For the important retentions d = 5, 10, 15 and 20 the upper bound π min (S sp,65, d, Λ, Γ) really improves the comonotonic upper bound. Notice that for the extreme cases the values are more or less the same.

137 3.3. The distribution of life annuities 111 cdf outcome Figure 3.1: The cdf s of S sp,65 (MC) (solid grey line), Ssp,65 l ( -line), Ssp,65 cl ( -line) and Sc sp,65 (dashed line) Figure 3.2: QQ-plot of the quantiles of Ssp,65 l ( ) versus those of S sp,65 (MC). ( ) / Scl sp,65 ( ) and Sc sp,65

138 112 Chapter 3 - Reserving in life insurance business Stop-loss premium outcome Figure 3.3: Stop-loss premiums for S sp,65 (MC) (solid grey line), Ssp,65 l ( -line), Ssp,65 cl ( -line) and Sc sp,65 (dashed line). d MIN EMUB PECUB CUB MC (s.e ) (9.43) (8.67) (5.89) (0.34) (0.21) (0.10) (0.02) (0.004) Table 3.3: Upper bounds for some selected stop-loss premiums with retention d of S sp,65.

139 3.3. The distribution of life annuities A homogeneous portfolio of life annuities We consider now the distribution of the present value of a homogeneous portfolio of N 0 life annuities paying a fixed amount of α i (> 0) at the end of each year i. This present value can be expressed by the formula S pp,x = ω x N i α i e Y (i), where N i denotes the number of survivals in year i and can be written as N i = I T (1) x >i + I T (2) x >i I T (N 0 ) x >i, where T x (j) denotes the future lifetime of the j-th insured. We assume that these random variables are mutually independent. So the random variables N i are binomially distributed with parameters n = N 0 and success parameter i p x. Note that N 0 S pp,x = S sp,x, (j) (3.8) j=1 with S (j) sp,x given by ω x S sp,x (j) = I α T x >i (j) i e Y (i). The computation of the convex upper and lower bound for the case of a portfolio of life annuities is more complicated than in the case of a single life annuity. The binomial distributed random variables N i are not very useful in practical computations, because there exist no closed-form expressions for the cumulative and the inverse distribution functions. This problem can be dealt with by replacing the random variables N i by more handy continuous approximations Ñi. We propose to approximate the distribution of N i by the Normal Power Approximation (NPA). This allows to incorporate the sknewness in contrast with a Normal approximation, because the binomial distribution is very skewed (unless either the parameter

140 114 Chapter 3 - Reserving in life insurance business n is very high or the success parameter p is close to 1 2 ). The distribution function of the NPA Ñi is given by the formula where FÑi (x) = Φ ( 3 γ Ni + µ Ni = E [N i ] = N 0 i p x, σ 2 N i = Var [N i ] = N 0 i p x i q x, 9 γn 2 + 6(x µ N i ) + 1 i γ Ni σ Ni γ Ni = E [ (N i µ Ni ) 3] σ 3 N i = 1 2 ip x N0 i p x i q x. Then the p-th quantile of Ñ i is given by F 1 Ñ i (p) = µ Ni + σ Ni Φ 1 (p) + γ N i σ Ni 6 The upper bound ) ( ) (Φ 1 (p)) 2 1. (3.9) The upper bound Spp,x c is computed as described in Section The only difference is that in the formulas (2.71), (2.72) and (2.75) F 1 X i (u) has to be replaced by the approximation given in (3.9). The lower bound To compute the lower bound one has to choose two conditioning variables: Γ and Λ. For the first conditioning random variable Γ we propose to take N i0 the number of policies-in-force in the year i 0. Note that E [ N i N i0 = n 0 ] = i i0 p x+i0 n 0 for i i 0. For i < i 0, Pr[N i = n N i0 = n i ] can be computed from Bayes theorem. As a result one gets the following formula for the conditional expectation: E [ N i N i0 = n 0 ] = = = N 0 k Pr[N = n i0 0 N i = k]pr[n i = k] Pr[N i0 = n 0 ] k=n 0 N 0 ( k )( N0 ) n k 0 k i 0 ip n 0 x+i i 0 iq k n 0 x+i ip k x iq N 0 k x ( N0 ) k=n 0 n 0 i 0 p n 0 x i 0 q N 0 n 0 x N 0 k k=n 0 ( ) N0 n 0 k n 0 ip k n 0 x i 0 iq k n 0 x+i, i 0 q N 0 n 0 x iq N 0 k x.

141 3.3. The distribution of life annuities 115 For mathematical convenience we rewrite this formula for non-integer values of N i0 as follows E [ N i N i0 = y ] = N 0 k= y ( ) N0 y k k y ip k y x i 0 iq k y x+i i 0 q N 0 y x iq N 0 k x. (3.10) We propose to take Λ (a), as defined in Section 3.3.1, for the second conditioning random variable Λ. Now one can perform step by step the computations described in Subsection with the only exception that E [ X i Γ = γ ] has to replaced in the formulas (2.80) and (2.81) by E [ N i N i0 = y ] in (3.10). Also the stop-loss premiums are calculated according to the methodology presented in Section 5.3 and with the only difference the replacement of E [ X i Γ = F 1 Γ (u)] in formula (2.83) by the approximation given in (3.10). The moments based approximation As in the case of a single life annuity, the only problem in the computation of the weight z given by (2.66) is to find expressions for the variances of S c pp,x, S l pp,x and S pp,x. For the upper and the lower bound we have deployed a procedure, described in Section 2.5.3, with f i (u) replaced by f i (u) = F 1 Ñ i (u) for the upper bound and f i (u) = E [ N i N i0 = F 1 Ñ i0 (u) ] for the lower bound. The variance of S pp,x can be computed from (3.8) and by noticing that, given the returns Y ( ) = Y (1),..., Y ( ω x ), the random variables S (1) S sp,x, (2)..., S (N 0) sp,x are conditionally independent. Hence, we have that Var [ S pp,x ] = E Y [ Var [ S pp,x Y ]] + Var Y [ E [ S pp,x Y ]] = N 0 E Y [ Var [ S sp,x Y ]] + N 2 0 Var Y [ E [ S sp,x Y ]] = N 0 Var [ S sp,x ] + (N 2 0 N 0 )Var Y [E [ S sp,x Y ]], sp,x,

142 116 Chapter 3 - Reserving in life insurance business p S l pp,65 S m pp,65 S c pp,65 MC (s.e.) (3.90) (5.08) (8.15) (8.80) (22.09) Table 3.4: Approximations for some selected quantiles with probability level p of S pp,65. where Var [ S sp,x ] is calculated in Subsection and Var Y [E [ S sp,x Y ]] = ω x A numerical illustration ω x j=1 ip x j p x α i α j e µ i µ j + σ 2 i +σ2 j ( 2 e σ ij 1 ). To test the quality of the derived approximations we present a numerical illustration similar to this from Subsection As before we work in a Black & Scholes setting with drift µ = 0.05 and volatility σ = 0.1 and we use the Makeham-Gompertz law to describe the mortality process of 65-year old male persons. We compare the performance of the lower bound Spp,65 l, the upper bound Sc pp,65 and the moments based approximation Spp,65 m with the real value S pp,65, obtained by extensive simulation, for a portfolio of policies. The number of policies-in-force after the first year N 1 is taken as the conditioning random variable Γ for the lower bound. This choice seems to us to be reasonable other choices can improve the performance of the lower bound only a bit but with a significant increase in computational time as cost. The Monte Carlo (MC) study of S pp,65 is based on simulated paths. Antithetic variables are used in order to reduce the variance of the Monte Carlo estimates. The quality of the approximations is illustrated in Figure 3.4 and 3.5. One can see that the lower bound Spp,65 l indeed performs very well. The fit of the upper bound is a bit poorer but still reasonable. The moments based approximation Spp,65 m performs extremely well. These visual observations are confirmed by the numerical values of some upper quantiles displayed in Table 3.4 and by the study of stop-loss premiums in Figure 3.6 and in Table 3.5.

143 3.3. The distribution of life annuities 117 cdf outcome Figure 3.4: The cdf s of S pp,65 (MC) (solid grey line), Spp,65 l ( -line), Spp,65 m ( -line) and Sc pp,65 (dashed line) Figure 3.5: QQ-plot of the quantiles of S l pp,65 ( )/Sm pp,65 ( ) and Sc pp,65 ( ) versus those of S pp,65 (MC).

144 118 Chapter 3 - Reserving in life insurance business Stop-loss premium outcome Figure 3.6: Stop-loss premiums for S pp,65 (MC) (solid grey line), Spp,65 l ( -line), Spp,65 m ( -line) and Sc pp,65 (dashed line). d S l pp,65 S m pp,65 S c pp,65 MC (s.e.) (2.11) (2.10) (1.95) (1.78) (1.26) (0.09) (0.02) Table 3.5: Approximations for some selected stop-loss premiums with retention d of S pp,65.

145 3.3. The distribution of life annuities An average portfolio of life annuities As explained in Section 3.2 in the case of large portfolios of life annuities it suffices to compute risk measures of an average portfolio given by S app,x = ω x ip x α i e Y (i), where we assume that the payments α i are positive and due at times i = 1,..., ω x (payable at the end of each year). Notice that S app,x is of the form (2.29) and that S app,x = E[S sp,x Y (1),, Y ( ω x )]. Comonotonic approximations for this type of sums has been studied extensively by Kaas et al. (2000), Dhaene et al. (2002a,b), Vyncke (2003), Darkiewicz (2005b) and Vanduffel (2005), among others. It turns out that for this application the conditioning variable of the maximal variance form gives very accurate results. This means that we define Λ here as ω x ip x α i e µ i+ 1 2 σ2 i Y (i). Notice that this conditioning variable could also be used in order to derive the lower bound for a single life annuity. To compute the comonotonic approximations for the quantiles and stop-loss premiums, notice that the correlations r i are given by r i = Corr(Y (i), Λ) = Cov[Y (i), Λ] σ i σ Λ. Because all correlation coefficients r i are positive, we have seen that the lower bound is a comonotonic sum (all the terms in the sum are nondecreasing functions of the same standard uniform random variable U). This implies that the quantiles related to the lower and upper bound can be computed by summing the corresponding quantiles for the marginals involved. We find the following expressions for the quantiles and stop-loss premiums of S l app,x and S c app,x:

146 120 Chapter 3 - Reserving in life insurance business F 1 (p) = Sapp,x l F 1 ω x ip x α i e µ i+r i σ i Φ 1 (p)+ 1 2(1 r 2 i )σ 2 i, ω x S (p) = ip app,x c x α i e µ i+sign( i p xα i )σ i Φ 1 (p), π lb (S app,x, d, Λ) = π cub (S app,x, d) = ω x ip x α i e µ i+ σ 2 i 2 Φ [r i σ i Φ 1( )] F S l app,x (d) ( ) d 1 F S l app,x (d), ω x ip x α i e µ i+ σ 2 i 2 Φ [sign( i p x α i )σ i Φ 1( )] F S c app,x (d) ( ) d 1 F S c app,x (d). To calculate the moments based approximation we need the expressions for the variances of S app,x, S l app,x and S c app,x. These are given by Var[S app,x ] = Var[S l app,x] = Var[S c app,x] = ω x ω x ω x ω x j=1 ω x j=1 ω x j=1 ip x j p x α i α j e µ i µ j + σ 2 i +σ2 j 2 (e σ ij 1), ip x j p x α i α j e µ i µ j + σ 2 i +σ2 j 2 (e r ir j σ i σ j 1), ip x j p x α i α j e µ i µ j + σ 2 i +σ2 j 2 (e σ iσ j 1) A numerical illustration In this subsection we illustrate our findings numerically and graphically. We use the same parameters for the financial and mortality process as in the two previous illustrations, namely a Black & Scholes model for the returns with µ = 0.05, σ = 0.1 and the Makeham-Gompertz law with corresponding coefficients of the Belgian analytic life table MR. We will compare the different approximations for quantiles and stop-loss premiums with the values obtained by Monte Carlo simulation (MC). The simulation

147 3.3. The distribution of life annuities 121 results are based on generating random paths. The estimates obtained from this time-consuming simulation will serve as benchmark. The random paths are based on antithetic variables in order to reduce the variance of the Monte Carlo estimates. Figure 3.7 shows the distribution functions of the lower bound Sapp,65 l, the upper bound Sapp,65 c, the moment based approximation Sm app,65 and the simulated one S app,65. Again the lower bound and the moments based approximation prove to be very good approximations for the real cumulative distribution function of S app,65. To assess the accuracy of the bounds in the tails, we plot their quantiles against those of S app,65 in Figure 3.8. The largest quantile (p = 0.995) of Sapp,65 m in this QQ-plot underestimates the exact quantile by only 0.06%. Table 3.6 shows the numerical values for some high quantiles. The stop-loss premiums for different choices of d are shown in Figure 3.9 and in Table 3.7. The lower bound and the moments based approximation give very accurate results compared to the real value of the stop-loss premium. The comonotonic upper bound performs rather badly for some retentions. But, using the results of Chapter 2 we can construct sharper upper bounds than the traditional comonotonic upper bounds. In Table 3.8 we compare the stop-loss premium of the comonotonic upper bound with the partially exact/comonotonic upper bound π pecub (S app,65, d, Λ) (PECUB) and the two upper bounds based on the lower bound Sapp,65 l plus an error term dependent of the retention π deub (S app,65, d, Λ) (DEUB) and independent of the retention π eub (S app,65, d, Λ) (EUB). For the partial exact/comonotonic upper bound we use the same conditioning variable as for the lower bound Sapp,65 l. The decomposition variable used in this illustration is given by ω x d Λ = d 2 i ip x α i e µ i+ σ 2 ( 1 + µ i 1 ) 2 σ2 i. The results for the different upper bounds are in line with the previous ones for a single life annuity. Note that for very high values of d the differences become larger, but these cases don t represent any practical importance. We can conclude that in both cases the upper bound based on the lower bound plus an error term dependent on the retention π deub (., d, Λ) performs very well.

148 122 Chapter 3 - Reserving in life insurance business p S l app,65 S m app,65 S c app,65 MC (s.e ) (0.03) (0.07) (0.14) (0.24) (1.58) Table 3.6: Approximations for some selected quantiles with probability level p of S app,65. d S l app,65 S m app,65 S c app,65 MC (s.e ) (8.22) (7.67) (4.45) (1.01) (0.31) (0.01) Table 3.7: Approximations for some selected stop-loss premiums with retention d of S app,65. Notice that for the retention d = 0 all values (except the value for DEUB because there the error term is independent of the retention) in both tables are identical and equal to This follows from the fact that in this case the expected value of S sp,65 equals the expected value of S app,65. Note also that the values in Tables 3.2 and 3.3 are typically larger than the corresponding values in Tables 3.7 and 3.8. This is not surprising. From Remark 5 it immediately follows that S app,65 cx S sp,65 and hence for any retention d > 0 one has π(s app,65, d) π(s sp,65, d).

149 3.3. The distribution of life annuities 123 cdf outcome Figure 3.7: The cdf s of S app,65 (MC) (solid grey line), Sapp,65 l ( -line), Sapp,65 m ( -line) and Sc app,65 (dashed line) Figure 3.8: QQ-plot of the quantiles of S l app,65 ( )/Sm app,65 ( ) and Sc app,65 ( ) versus those of S app,65 (MC).

150 124 Chapter 3 - Reserving in life insurance business Stop-loss premium outcome Figure 3.9: Stop-loss premiums for S app,65 (MC) (solid grey line), Sapp,65 l ( -line), Sm app,65 ( -line) and Sc app,65 (dashed line). d EUB DEUB PECUB CUB MC (s.e ) (8.22) (7.67) (4.45) (1.01) (0.31) (0.01) Table 3.8: Upper bounds for some selected stop-loss premiums with retention d of S app,65.

151 3.4. Conclusion Conclusion In this chapter we studied the case of life annuities. The aggregate distribution function of such stochastic sums of dependent random variables is very difficult to calculate. Usually it is only possible to get formulae for the first couple of moments. To compute more cumbersome risk measures, like stop-loss premiums or upper quantiles, one has to rely on time consuming simulations. We derived comonotonicity based approximations both for the case of a single life annuity and a homogeneous portfolio of life annuities. The numerical illustrations confirm the very high accuracy of the bounds (especially the lower bound). These observations are confirmed by the results of the stop-loss premiums. One maybe gets an impression that the upper bound which performs poorer than the lower bound in all cases is not worth being studied. In actuarial applications, however, the upper bound should draw a lot of attention because one is usually interested in conservative estimates of quantities of interest. Indeed, when an actuary calculates reserves he has to take into account some additional sources of uncertainty, such as the choice of the interest rates model, the estimation of parameters, the assumptions about mortality, the longevity risk and many others. For these the estimates provided by the upper bound in convex order can be in many cases more appropriate than the more accurate approximations obtained from the lower bound in convex order.

152

153 Chapter 4 Reserving in non-life insurance business Summary In this chapter we present some methods to set up confidence bounds for the discounted IBNR reserve. We first model the claim payments by means of a lognormal and a loglinear location-scale regression model. We further extend this to the class of generalized linear models. The knowledge of the distribution function of the discounted IBNR reserve will help us to determine the initial reserve, e.g. through the quantile risk measure. The results are based on the comonotonic approximations explained in Chapter Introduction To get the correct picture of its liabilities, a company should set aside the correctly estimated amount of money 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 triangle of incremental claims Y ij, as depicted in Figure 4.1. This is the simplest shape of data that can be obtained and it avoids having to introduce complicated notation to cope with all possible situations. We use the standard notation, with the random variables Y ij for i = 1, 2,..., t; j = 1, 2,..., s denoting the claim figures for year of origin (or accident year) i and development year j, meaning that the claim amounts were paid in calendar year i+j 1. Year of origin, year of development and calendar year act as possible explanatory variables for the observation Y ij. 127

154 128 Chapter 4 - Reserving in non-life insurance business Year of Development year origin 1 2 j t 1 t 1 Y 11 Y 12 Y 1j Y 1,t 1 Y 1t 2 Y 21 Y 22 Y 2j Y 2,t 1. i Y i1 Y ij. t Y t1 Figure 4.1: Random variables in a run-off triangle Most claims reserving methods assume that t = s. For (i, j) combinations with i + j t + 1, Y ij has already been observed, otherwise it is a future observation. Next to claims actually paid, these figures can also be used to denote quantities such as loss ratios. To a large extent, it is irrelevant whether incremental or cumulative data are used when considering claims reserving in a stochastic context. We consider annual development (the methods can be extended easily to semi-annual, quarterly or monthly development) and we assume that the time it takes for the claims to be completely paid is fixed and known. The triangle is augmented each year by the addition of a new diagonal. The purpose is to complete this run-off triangle to a square, or 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. The choice of an appropriate statistical model is an important matter. Furthermore within a stochastic framework, there is considerable flexibility in the choice of predictor structures. In England & Verrall (2002) the reader finds an excellent review of possible stochastic models. An appropriate model will enable the calculation of the distribution of the reserve that reflects the process variability producing the future payments, and accounts for the estimation error and statistical uncertainty (in the sense given in Taylor & Ashe (1983)). It is necessary to be able to estimate the variability of claims reserves, and ideally to be able to estimate the full dis-

155 4.1. Introduction 129 tribution of possible outcomes so that percentiles (or other risk measures of this distribution) can be obtained. Next, recognizing the estimation error involved with the parameter estimates, confidence intervals for these measures constitute another desirable part of the output. Here, putting the emphasis on the discounting aspect of the reserve, we first consider simple lognormal linear models. Doray (1996) studied the loglinear models extensively, taking into account the estimation error on the parameters and the statistical prediction error in the model. Such models have some significant disadvantages. Predictions from this model can yield unusable results, and we need to impose that each incremental value should be greater than zero. So, it is not possible to model negative or zero claims. From the nature of the claims reserving problem, it is expected that a higher proportion of zeros would be observed in the later stages of the incremental loss data triangle. In reinsurance, zero claims are also frequently observed in incremental loss data triangles for excess layers. Negative incremental values will be the result of salvage recoveries, payments from third parties, total or partial cancellation of outstanding claims, due to initial overestimation of the loss or to possible favorable jury decision in favor of the insurer, rejection by the insurer, or just plain errors. In Goovaerts & Redant (1999) a lognormal linear regression model is used to model the random fluctuations in the direction of the calendar years, taking into account the apparatus of financial mathematics. The results are based on supermodularity order, such that, in the framework of stop-loss ordering one obtains the distribution of the IBNR reserve corresponding to an extremal element in this ordering, when some marginals are fixed. The lognormal linear model is a member of the broader class of loglinear location-scale regression models. In Doray (1994) the reader can find an overview with a lot a characteristics of the different distributions in this class. The logarithm of the error is assumed to follow certain known distributions (normal, extreme value, generalized loggamma, logistic and log inverse Gaussian). Doray studied these models extensively. He has derived certain theoretical properties of these distributions and proved that the MLE s of the regression and scale parameters exist and are unique, when the error has a log-concave density. Claim sizes can often be described by distributions with a subexponential right tail. Furthermore, the phenomena to be modelled are rarely additive in the collateral data. A multiplicative model is much more plausible. These problems cannot be solved by working with ordinary linear

156 130 Chapter 4 - Reserving in non-life insurance business models, but with generalized linear models. The generalization is twofold. First, it is allowed that the random deviations from the mean follow another distribution than the normal. In fact, one can take any distribution from the exponential dispersion family, including for instance the Poisson, the binomial, the gamma and the inverse Gaussian distributions. Second, it is no longer necessary that the mean of the random variable is a linear function of the explanatory variables, but it only has to be linear on a certain scale. If this scale for instance is logarithmic, we have in fact a multiplicative model instead of an additive model. Loss reserving deals with the determination of the (characteristics of the) d.f. of the random present value of an unknown amount of future payments. Since this d.f. is very important for an insurance company and its policyholders, these inherent uncertainties are no excuse for providing anything less than a rigorous scientific analysis. In order for the reserve estimate truly to represent the actuary s best estimate of the needed reserve, both the determination of the expected value of unpaid losses and the appropriate discount should reflect the actuary s best estimates (i.e. should not be dictated by others or by regulatory requirements). Since the reserve is a provision for the future payment of unpaid losses, we believe the estimated loss reserve should reflect the time value of money. In many situations this discounted reserve is useful, for example dynamic financial analysis, assessing profitability and pricing, identifying risk based capital needs, loss portfolio transfers, profit testing, and so on. Ideally the discounted loss reserve would also be acceptable for regulatory reporting. However, many current regulations do not permit it. It could be argued that reserves set on an undiscounted basis include an implicit margin for prudence, although, in the current climate of low interest rates, that margin is very much reduced. If reserves are set on a discounted basis, there is a strong case for including an explicit prudential margin. As such, a risk margin based on a risk measure from a predictive distribution of claims reserves is a strong contender. One of the sub-problems in this respect consists of the discounting of the future estimates in the run-off triangle, where returns (and inflation) are not known for certain. We will model the stochastic discount factor using a Brownian motion with drift. When determining the discounted loss reserve, we impose an explicit margin based on a risk measure (for example Value-at-Risk) from the total distribution of the discounted reserve. Considering the discounted IBNR reserve, we have to incorporate a

157 4.2. The claims reserving problem 131 certain dependence structure. In general, it is hard or even impossible to determine the quantiles of the discounted loss reserve analytically, because in any realistic model for the return process this random variable will be a sum of strongly dependent random variables. The true multivariate distribution function of the lower triangle cannot be determined analytically in most cases, because the mutual dependencies are not known, or are difficult to cope with. We suggest to solve this problem by calculating upper and lower bounds making efficient use of the available information. This chapter is set out as follows. Section 2 places the claims reserving problem in a broader context. Section 3 gives a brief review of loglinear and generalized linear models and their applications to claims reserving. To be able to use the results of Chapter 2 we need some asymptotic results for model parameter estimates in generalized linear models. Section 4 describes how convex lower and upper bounds can be obtained for discounted IBNR evaluations. Some numerical illustrations for a simulated data set are provided in Section 5, together with a discussion of the estimation error using a bootstrap approach. We also graphically illustrate the obtained bounds. The results of this chapter come from Hoedemakers, Beirlant, Goovaerts & Dhaene (2003, 2005). 4.2 The claims reserving problem As a rule not all claims on a general insurance portfolio will have been paid by the end of the calender year of an insurance company. There can be several reasons for the delay in payment, e.g. delays in reporting the claim, long legal procedures, difficulties in determining the size of the claim, and so on. It is also possible that the claim still has to occur, but that the cause of the claim occurs in the past (e.g. exposed to asbestos). This of course depends on what is insured in the policy. The delay in payment can vary from a couple of days up to some years depending on the complexity and the severity of the damage. To be able to pay these claims the insurer has to keep reserves which should enable him to pay all future outstanding claims. Claims reserving is a vital area of insurance company management,

158 132 Chapter 4 - Reserving in non-life insurance business which is receiving close attention from shareholders, auditors, tax authorities and regulators. For insurance companies, the claims reserve is a very substantial balance sheet item, which can be large in relation to shareholders funds. Actuaries are now well-established in the area of claims reserving for non-life insurance business. In many countries there is already a statutory requirement for actuarial certification of reserves. Even in jurisdictions where there is no such requirement, the substantial contribution actuaries can make to estimating future liabilities has been recognized across the market. Failure to reserve accurately for outstanding and IBNR claims will adversely affect a company s future financial development. Any current reserve inadequacy will give rise to losses in subsequent years. Conversely, premium calculations based on a too pessimistic evaluation of current liabilities will damage the company s competitive position. The reserves held by a general insurance company can be divided into the following categories: Claims reserves representing the estimated outstanding claims payments that are to be covered by premiums already earned by the company. These reserves are sometimes called IBNS reserves (Incurred But Not Settled). These can in turn be divided into 1. IBNYR reserves, representing the estimated claims payments for claims which have already Incurred, But which are Not Yet Reported to the company. 2. RBNS reserves, being the reserves required in respect of claims which have been Reported to the company, But are Not yet fully Settled. A special case of RBNS reserves are case reserves, which are the individual reserves set by the claim handlers in the claims handling process. Unearned premium reserves (UPR). Because the insurance premiums are paid up-front, the company will, at any given accounting date, need to hold a reserve representing the liability that a part of the paid premium should be paid back to the policyholder in the event that insurance policies were to be cancelled at that date. Unearned premium reserves are pure accounting reserves, calculated on a pro rata basis.

159 4.3. Model set-up: regression models 133 Unexpired risk reserves (URR). While the policyholder only in special cases has the option to cancel a policy before the agreed insurance term has expired, he certainly always has the option to continue the policy for the rest of the term. The insurance company, therefore, runs the risk that the unearned premium will prove insufficient to cover the corresponding unexpired risk, and hence the unexpired risk reserve is set up to cover the probable losses resulting from insufficient written but yet unearned premiums. CBNI reserves. Essentially the same as unearned premium reserves, but to take into account possible seasonal variations in the risk pattern, they are not necessarily calculated pro rata, so that they also incorporate the function of the unexpired risk reserves. Their purpose is to provide for Covered But Not Incurred (CBNI) claims. The sum of the CBNI and IBNS reserves is sometimes called the Covered But Not Settled (CBNS) reserve. Fluctuation reserves (equalization reserves) do not represent a future obligation, but are used as a buffer capital to safeguard against random fluctuations in future business results. The use of fluctuation reserves varies from country to country. The loss reserves considered here only refer to the claims that result from already occurred events; the so-called IBNS reserves. Notice that often the terminology is not used uniformly: the abbreviation IBNR is used when speaking of loss reserving problems as a whole. 4.3 Model set-up: regression models The problem of estimating IBNR claims consists in predicting, for each accident year, the ultimate amount of claims incurred. The amount paid by the insurance company for those claims, when it comes due, is then subtracted, leaving the reserve the insurer should hold for future payments. To calculate the reserve, all methods or models usually assume that the pattern of cumulative or incremental claims incurred or paid is stable across the development years, for each accident year. Since for the last accident year, only one amount will be available, the reserve will be highly sensitive to this amount. Moreover, because of growth experienced by the company,

160 134 Chapter 4 - Reserving in non-life insurance business it will be larger than any other amount in the data set, hence the importance of verifying that the development pattern of the claims has not changed over the years. One of the earliest methods, and now the most commonly used in the actuarial profession, is the chain-ladder method. Assuming that for each accident year, the development pattern remains stable, development factors are calculated by dividing cumulative paid or incurred claims after j periods by the cumulative amount after j 1 periods. The year-to-year development factors are then applied to the most recent amount for each accident year, i.e. the amounts on the right-most diagonal. Many variations have been presented for the basic chain-ladder method just introduced; a linear trend or an exponential growth may be assumed to be present among the development factors. Instead of taking their weighted average, they could be extrapolated into the future. The chain-ladder method can also be adjusted for inflation. However, the chainladder method suffers from the following deficiencies: 1. It explicitly assumes too many parameters (one for each column). 2. It does not give any idea of the variability of the reserve estimate, or a confidence interval for the reserve. 3. It is negatively biased, which could lead to serious underreserving, a threat to the insurer s solvency. Therefore stochastic models have been developed which enable to calculate an amount such that there is a high probability that the reserve will be sufficient to cover the liabilities generated by the current block of business. In claims reserving, we are interested in the aggregated value t t i=2 j=t+2 i In this section we given an overview of the different regression models used in claims reserving. We use the following notation throughout this section: Y = (Y 11,..., Y t1, Y 21,..., Y t1 ) is the vector of claims, β = (β 1,..., β p ) are model parameters, U is the regression matrix corresponding to the upper triangle of dimension [ t(t+1) 2 ] p and R is the regression matrix corresponding to the complete square of dimension t 2 p. Y ij.

161 4.3. Model set-up: regression models Lognormal linear models We consider the following loglinear regression model in matrix notation Z = ln Y = R β + ɛ, ɛ N(0, σ 2 I), (4.1) where ɛ is the vector of independent normal random errors with mean 0 and variance σ 2. So, the normal responses Z ij are assumed to decompose (additively) into a deterministic non-random component with mean (R β) ij and a homoscedastic normally distributed random error component with zero mean. The parameters are estimated by the maximum likelihood method, which in the case of the normal error structure is equivalent to minimizing the residual sum of squares. The unknown variance σ 2 is estimated by the residual sum of squares divided by the degrees of freedom (the number of observations minus the numbers of regression parameters estimated): ˇσ 2 = 1 n p ( Z U ˆ β) ( Z U ˆ β). (4.2) This is an unbiased estimator of σ 2. The maximum likelihood estimator of σ 2 is given by ˆσ 2 = 1 n ( Z U ˆ β) ( Z U ˆ β), (4.3) while the maximum likelihood estimator of β is ˆ β = (U U) 1 U Z. (4.4) Now we can forecast the total IBNR reserve with t t IBNR reserve = e (R ˆ β)ij +ɛ ij. (4.5) i=2 j=t+2 i This definition of the IBNR reserve can, among others, be found in Doray (1996). Here (R ˆ β)ij and ɛ ij are independent. We have that ɛ ij i.i.d N(0, σ 2 ), (4.6) (R ˆ β)ij N ( (Rβ) ij, σ 2 ( R(U U) 1 R ) ) ij. (4.7)

162 136 Chapter 4 - Reserving in non-life insurance business Starting from model (4.1), we summarize now some properties of the IBNR reserve (4.5), which can be found in Doray (1996). 1. The mean of the IBNR reserve equals W = t t e (R β) ij σ2 (1+(R(U U) 1 R ) ij). (4.8) i=2 j=t+2 i 2. The unique UMVUE of the mean of the IBNR reserve is given by Ŵ U = 0 F 1 ( n p 2 ; SS ) t z 4 t i=2 j=t+2 i where 0 F 1 (α; z) denotes the hypergeometric function. 3. The MLE of the mean of the IBNR reserve: e (R ˆ β)ij, (4.9) Ŵ M = t t e (R ˆ β)ij ˆσ2 (1+(R(U U) 1 R ) ij). (4.10) i=2 j=t+2 i Verrall (1991) has considered an estimator similar to ŴM, but with ˆσ 2 replaced with ˇσ 2 : Ŵ V = t t e (R ˆ β)ij ˇσ2 (1+(R(U U) 1 R ) ij). (4.11) i=2 j=t+2 i The simple estimator Ŵ D = t t e (R ˆ β)ij ˇσ2, (4.12) i=2 j=t+2 i was considered in Doray (1996). Also, we have the order relation Ŵ U < ŴD < ŴV, (4.13) which implies that W = E[ŴU] < E[ŴD] < E[ŴV ]. (4.14)

163 4.3. Model set-up: regression models 137 Hence both the estimators ŴD and ŴV exhibit a positive bias. This Lognormal Linear (LL) model with normal random error is a special case of the class of loglinear location-scale models. Other choices possible for the distribution of the random error are the extreme value distribution, leading to the Weibull-extreme value regression model, the generalized loggamma, the logistic, and the log inverse Gaussian distribution. In what follows we shortly recall this class of regression models Loglinear location-scale models For a general introduction to survival analysis we refer to Kalbfleish & Prentice (1980), Lawless (1982), Cohen & Whitten (1985), among others. In this section we recall the structure of this model and the main characteristics of the distributions for the error component. A location-scale model has a cumulative distribution function of the form ( ) x µ F X (x) = G, (4.15) σ where µ is the location parameter, σ is the scale parameter, and G is the standardized form (µ = 0, σ = 1) of the cumulative distribution function. The parameter vector is θ = (µ, σ). We consider the following Loglinear Location-Scale (LLS) regression model in matrix notation Z = ln Y = R β + σ ɛ, (4.16) where (Rβ) ij is the linear predictor or location parameter for Z ij, σ is the scale parameter and ɛ is a random error with known density f ɛ ( ). It should also be noticed that in general the scale parameter estimator is not independent of the location parameter estimator, as is the case in normal regression. It is clear that the random variable Z ij has the following density ( 1 σ f z ij (R ) β) ij ɛ, σ with < z ij <. This model can only be applied if all data points are non-negative. The parameters are estimated by maximum likelihood.

164 138 Chapter 4 - Reserving in non-life insurance business Doray (1994) showed that the maximum likelihood estimators of the regression and scale parameters exist and are unique when the error ɛ in the loglinear location-scale regression model has a log-concave density. This is the case for the five distributions we consider in Table 4.1. Note that the exponential distribution is a special case of the Weibull distribution when the shape parameter is equal to 1. The generalized gamma distribution is a flexible family of distributions containing as special cases the exponential, the Weibull and the gamma distribution. The IBNR reserve under this class of regression models is given by IBNR reserve = t t e (R β) ij + σɛ ij. i=2 j=t+2 i Table 4.2 displays the mean, cumulative distribution function and inverse distribution function of X ij = e (R β) ij + σɛ ij for the different regression models in the LLS family. Notice that the definition of the IBNR reserve here differs from definition (4.3.2) under the lognormal linear model. We use here e (R β) ij + σɛ ij instead of e (R ˆ β)ij +ˆ σɛ ij, where ˆ β and ˆ σ represent the MLE s of β and σ respectively. Also this definition of the IBNR reserve can, among others, be found in Doray (1996). This approach partly uses the information contained in the upper triangle (through ˆ β), and acknowledges the underlying stochastic structure (through ɛ ij ).

165 4.3. Model set-up: regression models 139 Regression model Density Lognormal linear ɛ ij i.i.d N(0, 1) 1 2π e 1 2 x2 ( < x < ) Weibull-extreme value Logistic Generalized loggamma Log inverse Gaussian ɛ ij Gumbel e x ex ( < x < ) ɛ ij standard logistic e x (1+e x ) 2 ( < x < ) k k 2 1 x Γ(k) e kx ke k ( < x < ) (0 < k < + ) ɛ ij loggamma ɛ ij log inverse Gaussian (2πλ) 1 2 e x 2 e 1 λ e 1 λ cosh(x) ( < x < ) (λ > 0) Table 4.1: Characteristics of the random error ɛ ij in the regression models of the LLS family.

166 Regression model E[X ij ] F Xij (x ij ) F 1 X ij (p) ( ) Lognormal linear e (R β) ij+ σ2 ln(xij) (Rβ) 2 Φ ij σ [ ( ) 1 ] Weibull-extreme value e (R β) ij xij σ Γ(1 + σ) 1 exp e (R β) ij Logistic Generalized loggamma e (R β) ij+ σφ 1 (p) e (R β) ij ( ln(1 p)) σ ) 2 1 ( ) 1 ) 1 ( (e (R σ β) ij (1 2 σ)πcosec(2π σ) (x ij e (R σ ) σ β) ij e (R β) ij p 1 p (1 < σ < 2) e (R β) ij Γ(1 + σ)γ(1 σ) ( σ < 1) k σ k e (R β) ij Γ(k) Γ(k + σ ( ( k) I k, Log inverse Gaussian e (R β) ij Φ [ e 2 λ Φ ) 1 x ij σ k k σ k e (R β) ij e (R β) ij λx ij + x ij [ λe (R β) ij ) ] e (R β) ij λx ij x ij λe (R β) ij / / ] Table 4.2: Characteristics of X ij = e (R β) ij + σɛ ij in the regression models of the LLS family. 140 Chapter 4 - Reserving in non-life insurance business

167 4.3. Model set-up: regression models Generalized linear models For a general introduction to Generalized Linear Models (GLIMs) we refer to McCullagh & Nelder (1992). This family encompasses normal error linear regression models and the nonlinear exponential, logistic and Poisson regression models, as well as many other models, such as loglinear models for categorical data. In this subsection we recall the structure of GLIMs in the framework of claims reserving. The first component of a GLIM, the random component, assumes that the response variables Y ij are independent and that the density function of Y ij belongs to the exponential family with densities of the form f(y ij ; θ ij, φ) = exp {[y ij θ ij b(θ ij )] /a(φ) + c(y ij, φ)}, (4.17) where a( ), b( ) en c(, ) are given functions. The function a(φ) often has the form a(φ) = φ, where φ is called the dispersion parameter. When φ is a known constant, (4.17) simplifies to the natural exponential family f(y ij ; θ ij ) = ã(θ ij ) b(y ij )exp {y ij Q(θ ij )}. (4.18) We identify Q(θ) with θ/a(φ), ã(θ) with exp{ b(θ)/a(φ)}, and b(y) with exp{c(y, φ)}. The more general formula (4.17) is useful for two-parameter families, such as the normal or gamma, in which φ is a nuisance parameter. Denoting the mean of Y ij by µ ij, it is known that µ ij = E[Y ij ] = b (θ ij ) and Var[Y ij ] = b (θ ij )a(φ), (4.19) where the primes denote derivatives with respect to θ. The variance can be expressed as a function of the mean by Var[Y ij ] = a(φ)v (µ ij ) = φv (µ ij ), where V ( ) is called the variance function. The variance function V captures the relationship, if any, between the mean and variance of Y ij. The possible distributions to work with in claims reserving include for instance the normal, Poisson, gamma and inverse Gaussian distributions. Table 4.3 shows some of their characteristics. For a given distribution, link functions other than the canonical link function can also be used. For example, the log-link is often used with the gamma distribution. The systematic component of a GLIM is based on a linear predictor η ij = (R β) ij = β 1 R ij,1 + + β p R ij,p, i, j = 1,..., t. (4.20)

168 142 Chapter 4 - Reserving in non-life insurance business Distribution Density φ Canonical µ(θ) = V (µ) = link θ(µ) b (θ) b (θ) ( ) N(µ, σ 2 1 ) σ exp (y µ)2 2π 2σ σ 2 µ θ 1 2 µ µy Poisson(µ) e y! 1 log(µ) e θ µ Gamma(µ, ν) 1 Γ(ν) IG(µ, σ 2 ) ( ) ν ( νy µ exp νy µ ) ( ) y 3/2 exp (y µ) 2 2πσ 2 2yσ 2 µ 2 1 y 1 ν 1/µ 1/θ µ 2 σ 2 1/µ 2 ( 2θ) 1/2 µ 3 Table 4.3: Characteristics of some frequently used distributions in loss reserving. Various choices are possible for this linear predictor. In Subsection we give a short overview of frequently used parametric structures in claims reserving applications. The link function, the third component of a GLIM, connects the expectation µ ij of Y ij to the linear predictor by η ij = g(µ ij ), (4.21) where g is a monotone, differentiable function. Thus, a GLIM links the expected value of the response to the explanatory variables through the equation g(µ ij ) = (R β) ij i, j = 1,..., t. (4.22) For the canonical link g for which g(µ ij ) = θ ij in (4.17), there is the direct relationship between the natural parameter and the linear predictor. Since µ ij = b (θ ij ), the canonical link is the inverse function of b. Generalized linear models may have nonconstant variances σij 2 for the responses Y ij. Then the variance σij 2 can be taken as a function of the predictor variables through the mean response µ ij, or the variance can be modelled using a parameterized structure (see Renshaw (1994)). Any regression model that belongs to the family of generalized linear models can be analyzed in a unified fashion. The maximum likelihood estimates of the regression parameters can be obtained by iteratively reweighted least

169 4.3. Model set-up: regression models 143 squares (naturally extending ordinary least squares for normal error linear regression models). Supposing that the claim amounts follow a lognormal distribution, taking the logarithm of all Y ij s implies that they have a normal distribution. So, the link function is given by η ij = µ ij and the scale parameter is the variance of the normal distribution, i.e. φ = σ 2. We remark that each incremental claim must be greater than zero, and predictions from this model can yield unusable results. The predicted value under a generalized linear model will be given by IBNR reserve = t t i=2 j=t+2 i with ˆ µ ij = g 1 ( (R ˆ β)ij ) for a given link function g. ˆµ ij, (4.23) We end this section with some extra comments concerning GLIMs. The need for more general GLIM models for modelling claims reserves becomes clear in the column of variance functions in Table 4.3. If the variance of the claims is proportional to the square of the mean, the gamma family of distributions can accommodate this characteristic. The Poisson and inverse Gaussian provide alternative variance functions. However, it may be that the relationship between the mean and the variance falls somewhere between the inverse Gaussian and the gamma models. Quasi-likelihood is designed to handle this broader class of mean-variance relationships. This is a very simple and robust alternative, introduced in Wedderburn (1974), which uses only the most elementary information about the response variable, namely the mean-variance relationship. This information alone is often sufficient to stay close to the full efficiency of maximum likelihood estimators. Suppose that we know that the response is always positive, the data are invariably skew to the right, and the variance increases with the mean. This does not enable us to specify a particular distribution (for example it does not discriminate between Poisson or negative binomial errors), hence one cannot use techniques like maximum likelihood or likelihood ratio tests. However, quasi-likelihood estimation allows one to model the response variable in a regression context without specifying its distribution. We need only to specify the link and variance functions to

170 144 Chapter 4 - Reserving in non-life insurance business estimate regression coefficients. Although the link and variance functions determine a theoretical likelihood, the likelihood itself is not specified so fewer assumptions are required for estimation and inference. This is analogous to the connection between normal-theory regression models and leastsquares estimates. Least-squares estimation provides identical parameter estimates to those obtained from normal-theory models, but least-squares estimation assumes far less. Only second moment assumptions are made by least-squares compared to full distribution assumptions of normal-theory models. For quasi-likelihood, specification of a variance function determines a corresponding quasi-likelihood element for each observation: Q(µ ij ; y ij ) = µij y ij y ij t dt, (4.24) φv (t) where Q(µ ij ; y ij ) satisfies a number of properties in common with the loglikelihood. Specifically, if K = k(µ ij ; Y ij ) = (Y ij µ ij )/(φv (µ ij )), then E(K) = 0 Var(K) = ( ) K E µ ij = 1 φv (µ ij ) 1 φv (µ ij ). (4.25) According to McCullagh & Nelder (1992), since most first-order asymptotic theory regarding likelihood functions is based on the three properties (4.25), we can expect Q(µ ij ; y ij ) to behave like a log-likelihood under certain broad conditions. Summing (4.24) over all y ij -values yields the quasi-likelihood for the complete data. The quasi-deviance D(y ij ; µ ij ) is similarly defined to be the sum over all y ij -values of 2φQ(µ ij ; y ij ) = 2 yij y ij t µ ij V (t) dt. (4.26) Parameter estimation proceeds by maximizing the quasi-likelihood. Since the quasi-likelihood behaves like an ordinary likelihood, it inherits all the large sample properties of likelihoods: approximate unbiasedness and normality of the parameter estimates. For example, through the use of the quasi-likelihood Q(µ ij ; y ij ) = µij y ij Y ij t 1 dt = φt2.5 φµ ij 2.5 ( µ ij y ij ( 1.5) µ2 ij ( 0.5) ) (4.27)

171 4.3. Model set-up: regression models 145 we could model a variance function between those of the gamma and inverse Gaussian families: V (µ ij ) = µ ij 2.5. When using the canonical link function, the quasi-likelihood equations are given by t+1 i j=1 t+1 j µ ij = µ ij = t+1 i j=1 t+1 j Y ij 1 i t; Y ij 1 j t. (4.28) As can easily be seen from these equations in case of the Poisson model with logarithmic link function, it is necessary to impose the constraint that the sum of the incremental claims in every row and column has to be non-negative. For example, this assumption makes the model unsuitable for incurred triangles, which may contain many negatives in the later development periods due to overestimates of case reserves in the earlier development periods. We recall that the only distributional assumptions used in GLIMs are the functional relationship between variance and mean and the fact that the distribution belongs to the exponential family. When we consider the Poisson case, this relationship can be expressed as Var[Y ij ] = E[Y ij ]. (4.29) One can allow for more or less dispersion in the data by generalizing (4.29) to Var[Y ij ]=φe[y ij ] (φ (0, )) without any change in the form and solution of the likelihood equations. For example, it is well known that an over-dispersed Poisson model with the chain-ladder type linear predictor gives the same predictions as those obtained by the deterministic chainladder method (see Renshaw & Verrall, 1994). Modelling the incremental claim amounts as independent gamma response variables, with a logarithmic link function and the chain-ladder type linear predictor produces exactly the same results as obtained by Mack (1991). The relationship between this generalized linear model and the model proposed by Mack was first pointed out by Renshaw & Verrall (1994). The mean-variance relationship for the gamma model is given by Var[Y ij ] = φ (E[Y ij ]) 2. (4.30)

172 146 Chapter 4 - Reserving in non-life insurance business Using this model gives predictions close to those from the deterministic chain-ladder technique, but not exactly the same. Notice that we need to impose that each incremental value should be positive (non-negative) if we work with gamma (Poisson) models. This restriction can be overcome using a quasi-likelihood approach. As in normal regression, the search for a suitable model may encompass a wide range of possibilities. The Bayesian information criterion (BIC) and the Akaike Information Criterion (AIC) are model selection devices that emphasize parsimony by penalizing models for having large numbers of parameters. Tests for model development to determine whether some predictor variables may be dropped from the model can be conducted using partial deviances. Two measures for the goodness-of-fit of a given generalized linear model are the scaled deviance and Pearson s chi-square statistic. In cases where the dispersion parameter is not known, an estimate can be used to obtain an approximation to the scaled deviance and Pearson s chi-square statistic. One strategy is to fit a model that contains a sufficient number of parameters so that all systematic variation is removed, estimate φ from this model, and then use this estimate in computing the scaled deviance of sub-models. The deviance or Pearson s chi-square divided by its degrees of freedom is sometimes used as an estimate of the dispersion parameter φ Linear predictors and the discounted IBNR reserve Various choices are possible for the linear predictor in claims reserving applications. We give here a short overview of frequently used parametric structures. A well-known and widely used predictor is the chain-ladder type η ij = α i + β j, (4.31) (α i is the parameter for each year of origin i and β j for each development year j). It should be noted that this representation implies the same development pattern for all years of origin, where that pattern is defined by the parameters β j. Notice that a parameter, for example β 1, must be set equal to zero, in order to have a non-singular regression matrix. Another natural and frequently used restriction on the parameters is to impose that

173 4.3. Model set-up: regression models 147 β β t = 1, since this allows the β j to be interpreted as the fraction of claims settled in development year j. The separation predictor takes into account the calendar years and replaces in (4.31) α i with γ k (k = i + j 1). It combines the effects of monetary inflation and changing jurisprudence. For a general model with parameters in the three directions, we refer to De Vylder & Goovaerts (1979). We give here some frequently used special cases: The probabilistic trend family (PTF) of models as suggested in Barnett & Zehnwirth (1998) j 1 i+j 2 η ij = α i + β k + γ t, (4.32) k=1 where γ denotes the calendar year effect; it combines the effects of monetary inflation and changing jurisprudence. The Hoerl curve as in Zehnwirth (1985) t=1 η ij = α i + β i log(j) + γ i j (j > 0). (4.33) This model has the advantage that one can predict payments by extrapolation for j > t, because development year j is considered as a continuous covariate. This is useful in estimating tail factors. Wright (1990) extends this Hoerl curve further to model possible claim inflation. A mixture of models (4.31) and (4.33) as in England & Verrall (2001) { αi + β η ij = j if j q; α i + β i log(j) + γ i j if j > q (4.34) for some integer q specified by the modeller. In the case that the type of business allows for discounting we add a discounting process. Of course, the level of the required reserve will strongly

174 148 Chapter 4 - Reserving in non-life insurance business depend on how we will invest this reserve. We define the discounted IBNR reserve S under one of the discussed regression models as follows lognormal linear model: S LL = t t i=2 j=t+2 i loglinear location-scale model: S LLS = generalized linear model: S GLIM = t t e (R ˆ β)ij +ɛ ij Y (i+j t 1), t i=2 j=t+2 i t i=2 j=t+2 i e (R β) ij + σɛ ij Y (i+j t 1), g 1( (R ˆ β)ij ) e Y (i+j t 1), where the returns are modelled by means of a Brownian motion described by the following equation Y (i) = (δ + ς2 )i + ςb(i), (4.35) 2 where B(i) is the standard Brownian motion, ς is the volatility and δ is a constant force of interest. 4.4 Convex bounds for the discounted IBNR reserve Before we can apply the results of Chapter 2 in order to derive the comonotonic approximations for S, we have to specify further the distribution of ˆ µ = g 1( (R ˆ β)ij ). This is done in what follows Asymptotic results in generalized linear models Let ˆφ, ˆ β, ˆ η = R ˆ β and ˆ µ = g 1 (ˆ η) be the maximum likelihood estimates of φ, β, η and µ respectively. The estimation equation for ˆ β is then given by U ŴU ˆ β = U Ŵˆ y, (4.36) where W = diag{w 11,, w t1 }, with w ij = Var[Y ij ] 1 (dµ ij /dη ij ) 2, y = (y 11,, y t1 ), and denoting y ij = η ij + (y ij µ ij )dη ij /dµ ij where y ij denote the sample values. Note that Ŵ is W evaluated at ˆ β. It is well-known that for asymptotically normal statistics, many functions of such statistics

175 4.4. Convex bounds for the discounted IBNR reserve 149 are also asymptotically normal. Because R ˆ ( β = (R ˆ β)11,, (R ˆ ) β)tt is asymptotically multivariate normal with mean Rβ = ((Rβ) 11,, (Rβ) ) tt and variance-covariance matrix Σ(R ˆ β) = Σ a = {σ a ij } = R(U WU) 1 R and g 1 (η 11,, η tt ) has a nonzero differential ψ = (ψ 11,, ψ tt ) at (R β), where ψ ij = dµ ij /dη ij, it follows from the delta method that ] ( ) d [ˆ µ µ N 0, Σ(ˆ µ), (4.37) where Σ(ˆ µ) = ψ Σ a ψ. Hence, for large samples the distribution of ˆ µ = g 1 (R ˆ β) can be approximated by a normal distribution with mean µ and variance-covariance matrix Σ(ˆ µ). Maximum likelihood estimates may be biased when the sample size or the total Fisher information is small. The bias is usually ignored in practice, because it is negligible compared with the standard errors. In small or moderate-sized samples, however, a bias correction can be necessary, and it is helpful to have a rough estimate of its size. In deriving the convex bounds, one need the expected values. Since there is no exact expression for the expectation of ˆ µ, we approximate it using a general formula for the first-order bias of the estimate of µ. Cordeiro & McCullagh (1991) derived the first order bias of ˆ β. In matrix notation this bias reduces to the simple form B( ˆ β) = 1 2 Σb U Σ c d F d 1, (4.38) with Σ b = Σ( ˆ β) = {σ b ij } = (U WU) 1, Σ c = Σ(U ˆ β) = {σ c ij } = UΣ b U, Σ a d = diag{σa 11,, σa tt}, Σ c d = diag{σc 11,, σc t1 }, 1 is a t(t+1) 2 1 vector of ( ) ( ) ones, and F d = diag{f 11,, f t1 } with f ij = Var[Y ij ] 1 dµij d 2 µ ij. dη ij dηij 2 It follows that the n 1 bias of ˆ η also has a simple expression: B(ˆ η) = 1 2 RΣb U Σ c d F d 1. (4.39) To evaluate the n 1 biases of ˆ β and ˆ η we need only the variance and the link functions with their first and second derivatives. In the right-hand sides of equations (4.38) and (4.39), which are of order n 1, consistent estimates of the parameters µ can be inserted to define the corrected maximum

176 150 Chapter 4 - Reserving in non-life insurance business likelihood estimates ˆ η c = ˆ η ˆB(ˆ η) and ˆ βc = ˆ β ˆB( ˆ β), which should have smaller biases than the corresponding ˆ η and ˆ β. From now on ˆB( ) means the value of B( ) at the point ˆ µ. Expressions (4.38) and (4.39) are applicable even if the link is not the same for each observation. For the linear model with any distribution in the exponential family B( ˆ β) and B(ˆ η) are zero. This is to be expected for the normal linear model or for the inverse Gaussian non-intercept linear regression model. However it is not obvious that this happens for any distribution in the exponential family (4.17) with identity link since ˆ β is obtained, apart from these cases, from the non-linear equation (4.36) with and because of the dependence of ˆ β on Ŵ and ˆ y. We now give the n 1 bias of ˆ µ. Because µ ij = g 1 (η ij ) = g 1 ((R β) ij ) and the link function is monotone and twice differentiable, we can apply a Taylor series expansion of ˆµ ij around η ij : ˆµ ij = µij + dµ ij dη ij (ˆη ij η ij ) ˆµ ij µ ij = dµ ij dη ij (ˆη ij η i ) E[ˆµ ij µ ij ] = dµ ij dη ij E[(ˆη ij η ij )] d 2 µ ij dηij 2 (ˆη ij η ij ) 2, d 2 µ ij dηij 2 (ˆη ij η ij ) 2, d 2 µ ij dηij 2 Var[ˆη ij ]. In matrix notation E[ˆ µ µ] = G1 E[(ˆ η η)] G 2[Var(ˆ η)] = 1 2 RΣb U Σ c d F d G 2Σ a d 1 = 1 { } G 2 Σ a 2 d 1 G 1 RΣ b U Σ c d F d 1. So, the first order bias of ˆ µ in matrix notation is given by the following equation: B(ˆ µ) = 1 { } G 2 Σ a 2 d 1 G 1 RΣ b U Σ c d F d 1, (4.40) where 1 is a t 2 1 vector of ones and G 1 = diag{ψ 11,, ψ tt }, G 2 = diag{ϕ 11,, ϕ tt } where ψ ij = dµ ij dη ij and ϕ ij = d2 µ ij. dηij 2

177 4.4. Convex bounds for the discounted IBNR reserve 151 So, we can define adjusted values as ˆ µ c = ˆ µ ˆB(ˆ µ), which should have smaller biases than the corresponding ˆ µ. Note that ˆB( ) means here the value of B( ) taken at ( ˆφ, ˆ µ) Lower and upper bounds In this subsection we will derive the upper and lower bounds in convex order, as described in Chapter 2, for the discounted IBNR reserve S LL, S LLS and S GLIM under the different regression models. Using the results of Chapter 2, we derive a convex lower and upper bound for S = i j X ijz ij given by E[X ij ]E[Z ij Λ] cx X ij Z ij cx F 1 X ij (U)F 1 Z ij (V ), i j } {{ } S l i j } {{ } S i j } {{ } S c with e ɛ ij (S LL ); X ij = e (R β) ij + σɛ ij (S LLS ); ˆµ ij (S GLIM ). e (R ˆ β)ij Y (i+j t 1) (S LL ); Z ij = e Y (i+j t 1) (S LLS ); e Y (i+j t 1) (S GLIM ). We introduce the random variables W ij and W ij defined by with W ij = (R ˆ β)ij Y (i + j t 1) and W ij = Y (i + j t 1), (4.41) E[W ij ] = (Rβ) ij (δ ς2 )(i + j t 1), E[ W ij ] = (δ ς2 )(i + j t 1), Var[W ij ] = σw 2 ij = σ 2 ( R(U U) 1 R ) ij + (i + j t 1)ς2, Var[ W ij ] = σ 2 Wij = (i + j t 1)ς 2.

178 152 Chapter 4 - Reserving in non-life insurance business The lower bound To compute the lower bound we consider the following conditioning normal random variable of the form (2.53) with Λ = t t i=2 j=t+2 i ν ij Y (i + j t 1), (4.42) e (R β) ij e (i+j t 1)δ ] (SLL l ); ν ij = E [e (R β) ij + σɛ ij e (i+j t 1)δ (SLLS l ); (4.43) ) (µ ij + B(ˆ µ) ij e (i+j t 1)δ (SGLIM l ). Notice that (W ij, Λ) has a bivariate normal distribution. Conditionally given Λ = λ, W ij has a univariate normal distribution with mean and variance given by and E[W ij Λ = λ] = E[W ij ] + ρ ij σ Wij σ Λ (λ E[Λ]) (4.44) Var[W ij Λ = λ] = σ 2 W ij ( 1 ρ 2 ij ), (4.45) where ρ ij denotes the correlation between Λ and W ij. The same is true for ( W ij, Λ), where we denote the correlation between Λ and W ij by ρ ij. The lower bound can be written as SLL l = SLLS l = SGLIM l = with E[X ij ] = t t i=2 j=t+2 i t t i=2 j=t+2 i t t i=2 j=t+2 i E[X ij ]e E[W ij]+ρ ij σ Wij Φ 1 (V )+ 1 2 (1 ρ2 ij )σ2 W ij, E[X ij ]e E[ W ij ]+ ρ ij σ Wij Φ 1 (V )+ 1 2 (1 ρ2 ij )σ2 Wij, E[X ij ]e E[ W ij ]+ ρ ij σ Wij Φ 1 (V )+ 1 2 (1 ρ2 ij )σ2 Wij, E [e ɛ ij ] = e 1 2 σ2 ] (S LL ); E [e (R β) ij + σɛ ij = See Table 4.2 (S LLS ); E [g ((R 1 ˆ )] β)ij = µ ij + B(ˆ µ) ij (S GLIM ).

179 4.4. Convex bounds for the discounted IBNR reserve 153 The correlations ρ ij and ρ ij are given by with ρ ij = Cov[Λ, W ij] σ Λ σ Wij, ρ ij = Cov[Λ, W ij ] σ Λ σ Wij, and Cov[Λ, W ij ] = Cov[Λ, W ij ] t t = ς 2 ν kl min(i + j t 1, k + l t 1) k=2 l=t+2 k Var[Λ] = σ 2 Λ = ς 2 t t t t r=2 s=t+2 r v=2 w=t+2 v ν rs ν vw min(r+s t 1, v+w t 1). By conditioning on one of the standard uniform random variables one can compute the distribution function of the lower bound. See Subsection for more details. For the lognormal linear and loglinear location-scale models there exist a closed-form expression for the quantile function of S l. Taking into account that Λ = t t i=2 j=t+2 i ν ijy (i + j t 1) is normally distributed, we find that and hence F 1 Λ (1 p) = E[Λ] σ ΛΦ 1 (p), F 1 S l (p) = F 1 = = t t i=2 j=t+2 i E[X ij]e[z ij Λ] t t i=2 j=t+2 i t t i=2 j=t+2 i F 1 E[X ij ]E[Z ij Λ] (p) (p), p (0, 1) E[X ij ]E[Z ij Λ = F 1 Λ (1 p)], In order to derive the above result, we used the fact that for a nonincreasing continuous function g, we have F 1 g(x) 1 (p) = g(fx (1 p)), p (0, 1). (4.46)

180 154 Chapter 4 - Reserving in non-life insurance business Here, g = E[Z ij Λ] is a non-increasing function of Λ since ρ ij ( ρ ij ) is always negative. So, we have that F 1 S l (p) = t t i=2 j=t+2 i t t i=2 j=t+2 i E[X ij ]e E[W ij] ρ ij σ Wij Φ 1 (p)+ 1 2 (1 ρ2 ij )σ2 W ij, (LL) E[X ij ]e E[ W ij ] ρ ij σ Wij Φ 1 (p)+ 1 2 (1 ρ2 ij )σ2 Wij. (LLS) and F S l(x) can be obtained from solving the equation t t i=2 j=t+2 i t t i=2 j=t+2 i The upper bound E[X ij ]e E[W ij] ρ ij σ Wij Φ 1 (F S l LL (x))+ 1 2 (1 ρ2 ij )σ2 W ij E[X ij ]e E[W ij] ρ ij σ Wij Φ 1 (F S l LLS (x))+ 1 2 (1 ρ2 ij )σ2 W ij The upper bound can be written as = x, (LL) = x. (LLS) S c LL = S c LLS = S c GLIM = t t i=2 j=t+2 i t t i=2 j=t+2 i t t i=2 j=t+2 i F 1 X ij (U)e E[W ij]+σ Wij Φ 1 (V ), F 1 X ij (U)e E[ W ij ]+σ Wij Φ 1 (V ), F 1 X ij (U)e E[ W ij ]+σ Wij Φ 1 (V ), with F 1 X ij (U) = F 1 e ɛ ij (U) = e σφ 1 (U) (S LL ); F 1 (U) = See Table 4.2 (S e (R β) ij +σɛ LLS ); F 1 g 1 (R β)ij ˆ (U) = µ ij + B(ˆ µ) ij + Σ(ˆ µ) ij Φ 1 (p) (S GLIM ).

181 4.4. Convex bounds for the discounted IBNR reserve 155 The cdf of the upper bound can be computed as described in Subsection Using Remark 4 one can calculate the distribution function of SLL c and SLLS c more efficiently. We start with the cdf of Sc LL. From previous results F S c LL (y) = 1 with F N (x) the cdf of N(0, σ 2 ) and and S c LL = = t t 0 i=2 j=t+2 i t t i=2 j=t+2 i ( F N ln(y) ln ( F 1 (u) )) du, SLL c ( ) exp F 1 (R ˆ (U) β)ij Y (i+j t 1) e (R β) ij (δ+ 1 2 ς2 )(i+j t 1) e σ 2 (R(U U) 1 R ) ij +ς 2 (i+j t 1)Φ 1 (p). F 1 (u) = SLL c t t i=2 j=t+2 i e (R β) ij (δ+ 1 2 ς2 )(i+j t 1) e σ 2 (R(U U) 1 R ) ij +ς 2 (i+j t 1)Φ 1 (u). We can write the upper bound of S LLS as with S c LLS = G t t i=2 j=t+2 i e E[ W ij ]+σ Wij Φ 1 (V ) e (R β) ij, e σφ 1 (U) (Lognormal linear); G = ( log(1 U)) σ (Weibull-extreme value); ( ) σ U 1 U (Logistic). The distribution function of G is given by Φ ( ) lnx σ (Lognormal linear); F G (x) 1 e x 1 σ (Weibull-extreme value); ( ) x 1 σ 1 (Logistic).

182 156 Chapter 4 - Reserving in non-life insurance business Using Remark 4 we can write the cdf of SLLS c for the lognormal linear, the weibull-extreme value and the logistic regression model as follows 1 F 1 SLLS(y) c = F G y du. (u) with 0 F 1 S c LLS S c LLS = = t t i=2 j=t+2 i t t i=2 j=t+2 i ( ) exp F 1 (Rβ) (U) ij Y (i+j t 1) e (R β) ij (δ+ 1 2 ς2 )(i+j t 1)+ς i+j t 1Φ 1 (U). and F 1 (u) = SLLS c t t i=2 j=t+2 i e (R β) ij (δ+ 1 2 ς2 )(i+j t 1)+ς i+j t 1Φ 1 (u). Remark 7. Since we have no equality of the first moments in the GLIM framework, the convex order relationship between the two approximations and S is not valid. This does not impose any restrictions on the use of the approximations. In fact, we can say that the convex order only holds asymptotically in this case. Remark 8. The estimator ŴD (4.12), for the mean of the IBNR reserve, constitutes a close upper bound for the UMVUE of the mean of the IBNR reserve if t(t+1) 2 p is large and the residual sum of squares is small. It should be noted that e ((R ˆ β)ij +ˇσ 2 /2) is the estimator of the mean of a lognormal distribution logn((r β) ij, σ 2 ) obtained by replacing the parameters β and σ 2 by their unbiased estimates. Adding now a discount process to Ŵ D gives Ŵ DD = t t i=2 j=t+2 i e (R ˆ β)ij Y (i+j t 1)+ 1 2 ˇσ2. (4.47) Now, we can apply the same methodology as explained before. The results for the lognormal linear model are still applicable. The only difference is

183 4.5. The bootstrap methodology in claims reserving 157 that ɛ ij is changed by 1 2 ˇσ2, with ( 1 n p 2 ˇσ2 Gamma 2, σ 2 ). (4.48) n p 4.5 The bootstrap methodology in claims reserving Introduction The bootstrap technique as an inferential statistical computer intensive device was introduced by Efron (1979) as a quite intuitive and simple way of making approximations to distributions which are very hard or even impossible to compute analytically. This 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 technique is that it 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. Our goal is to obtain quantiles of the loss reserve for which the predictive distribution is not known. If we do not know the distribution, then our best guess at the distribution is provided by the data. The main idea in bootstrapping is that we (a) pretend that the data constitute the population and (b) take samples from this pretended population (which we call resamples ). Substituting the sample for the population means that we are interested in the frequency with which the observed values occurred. This is done by sampling with replacement. From the re-sample, we calculate the statistic we are interested in. This is called a bootstrap statistic. After storing this value, one repeats the above steps collecting a large number (B) of bootstrap statistics. The general idea is that the relationship of the bootstrap statistics to the observed statistic is the same as the relationship of the observed statistic to the true value. Under mild regularity conditions, the bootstrap yields an approximation to the distribution of an estimator or test statistic that is at least as accurate as the approximation obtained from first-order asymptotic theory. For an introduction explaining the bootstrap technique, see Efron & Tibshirani (1993).

184 158 Chapter 4 - Reserving in non-life insurance business Central idea The concept of bootstrap relies on the consideration of the discrete empirical distribution generated by a random sample of size n from an unknown distribution F. This empirical distribution assigns equal probability to each sample item. In the discussion which follows, we will write ˆF n for that distribution. By generating an independent, identically distributed random sequence (resample) from the distribution ˆF n or its appropriately smoothed version, we can arrive at new estimates of various parameters and nonparametric characteristics of the original distribution F. As we have already mentioned, the central idea of bootstrap lies in sampling the empirical cdf ˆF n. This idea is closely related to the following, well-known statistical principle, henceforth referred to as the plug-in principle. Given a parameter of interest θ(f ) depending upon an unknown population cdf F, we estimate this parameter by ˆθ = θ( ˆF n ). That is, we simply replace F in the formula for θ by its empirical counterpart ˆF n obtained from the observed data. The plug-in principle will not provide good results if ˆF n poorly approximates F, or if there is information about F other than that provided by the sample. For instance, in some cases we might know (or be willing to assume) that F belongs to some parametric family of distributions. However, the plug-in principle and the bootstrap may be adapted to this latter situation as well. To illustrate the idea, let us consider a parametric family of cdf s {F µ } indexed by a parameter µ (possibly a vector), and for some given µ 0, let ˆµ 0 denote its estimate calculated from the sample. The plug-in principle in this case states that we should estimate θ(f µ0 ) by θ(fˆµ0 ). In this case, bootstrap is often called parametric, since a resample is now collected from Fˆµ0. Here, we refer to any replica of ˆθ calculated from a resample as a bootstrap estimate of θ(f ) and denote it by ˆθ Bootstrap confidence intervals Let us now turn to the problem of using the bootstrap methodology to construct confidence intervals. This area has been a major focus of theoretical work on the bootstrap, and several different methods of approaching the problem have been suggested. The naive procedure described below is not the most efficient one and can be significantly improved in both rate of convergence and accuracy. It is, however, intuitively obvious and

185 4.5. The bootstrap methodology in claims reserving 159 easy to justify, and seems to be working well enough for the cases considered here. For a complete review of available approaches to bootstrap confidence intervals, see Efron & Tibisharani (1993). Let us consider ˆθ, a bootstrap estimate of θ based on a resample of size n from the original sample X 1,..., X n, and let G be its distribution function given the observed sample values G = Pr[ˆθ x X 1 = x 1,..., X n = x n ]. The bootstrap percentiles method gives G 1 (α) and G 1 (1 α) as, respectively, lower and upper bounds for the (1 2α) confidence interval for ˆθ. Let us note that for most statistics ˆθ, the distribution function of the bootstrap estimator ˆθ is not available. In practice, G 1 (α) and G 1 (1 α) are approximated by taking multiple resamples and then calculating the empirical percentiles. In most cases B 1000 is recommended Bootstrap in claims reserving As already mentioned above, with bootstrapping, we treat the obtained data as if they are an accurate reflection of the parent population, and then draw many bootstrapped samples by sampling, with replacement, from a pseudo-population consisting of the obtained data. Technically, this is called non-parametric bootstrapping, because we are sampling from the actual data and we have made no assumptions about the distribution of the parent population, other than that the raw data adequately reflect the population s shape. If we were willing to make more assumptions, such as an assumption that the parent population follows a normal distribution, then we could do our sampling, with replacement, from a normal distribution. This is called parametric bootstrapping. For a description of the bootstrap methodology in claims reserving we refer to England & Verrall (1999) and Pinheiro et al. (2003). In these papers the bootstrap technique is used to obtain prediction errors for different claims reserving methods, namely methods based on the chain-ladder technique and on generalized linear models. Applications of the bootstrap technique to claims reserving can also be found in Lowe (1994), in Taylor (2000) and in England & Verrall (2002). Starting from the original run-off triangle one can create a large number of bootstrap run-off triangles by repeatedly resampling, with replacement,

186 160 Chapter 4 - Reserving in non-life insurance business from the appropriate residuals. For each bootstrap sample the regression model is refitted and the bootstrap statistic is calculated. In England & Verrall (1999) the bootstrap technique is used to compute the bootstrap root mean squared error of prediction (RMSEP bs ), also known as the bootstrap standard error of prediction. This is equal to, what they call, the square root of the sum of the squares of parameter variability and data variability. For the parameter variability one suggests a correction on the bootstrap standard error to enable a comparison between the analytic standard error and the bootstrap one by taking account of the number of parameters used in fitting the model. The bootstrap standard error is the standard deviation of the bootstrap reserve estimates. So, parameter variability is defined as the bootstrap standard error multiplied by the square root of n divided by n p (n: sample size, p: number of parameters). Data variability is the square root of the uniformly minimum variance unbiased estimator of the variance of the IBNR reserve. This estimator was already calculated by Doray (1996). Note that if the full predictive distribution can be found, the RMSEP can be obtained directly by calculating its standard deviation. Using a normal approximation, a 100(1 α)% bootstrap prediction interval for the total reserve is calculated as [R ± Φ 1 (1 α/2) RMSEP bs (R)], with R the initial forecast of the IBNR reserve. The second approach is more robust against deviations from the hypothesis of the model. For a detailed presentation of this method see Davidson & Hinkley (1997). A new bootstrap statistic is defined here as a function of the bootstrap estimate and a bootstrap simulation of the future reality. This statistic is called the prediction error. (This is very confusing because in the literature the term prediction error is also used for the RMSEP or the standard error of prediction.) For each bootstrap loop the prediction error is then kept in a vector and the percentile method is used to obtain the desired percentile of this prediction error (PPE). In a last stage an upper limit of the prediction interval for the total reserve is calculated as [R + PPE]. The reader can find a complete list of the required steps for those two procedures in the paper of Pinheiro et al. (2003). These authors have also compared and discussed the two bootstrap procedures and the main conclusion is that the differences amongst the results obtained with the two procedures, RMSEP and PPE, are not very important. The PPE procedure generates generally smaller values. Further one suggest to eliminate

187 4.5. The bootstrap methodology in claims reserving 161 the residuals with value 0 and to work with standardized residuals since only the former could be considered as identically distributed. The third approach is explained in England & Verrall (2002). Like in the previous methods, first of all a stochastic model is fitted to the bootstrap sample and a run-off triangle is bootstrapped. For this pseudo triangle the parameters are estimated in order to calculate future incremental claim payments Ŷ ij. The second stage of the procedure replicates the process variance. This is achieved by simulating an observed claim payment for each future cell in the run-off triangle, using the bootstrap value Ŷ ij as the mean, and using the process distribution assumed in the underlying model. For each iteration the reserves are calculated by adding up the simulated forecast payments. The set of reserves obtained in this way forms the predictive distribution. The percentile method is then used to obtain the required prediction interval. In a practical case study one can bootstrap a high percentile of the distribution of the lower bound in order to describe the estimation error involved. Taylor & Ashe (1983) used the terminology estimation error for Var[(R ˆ β)ij ] and statistical or random error for Var[ɛ ij ]. The estimation error arises from the estimation of the vector ˆ β from the data, and the statistical error stems from the stochastic nature of the regression model. We bootstrap an upper triangle using the non-parametric procedure. This involves resampling, with replacement, from the original residuals and then creating a new triangle of past claim payments using the resampled residuals together with the fitted values. With regression type problems the resampling procedure is applied to the residuals of the model. Residuals are approximately independent and identically distributed. In a statistical analysis they are commonly used in order to explore the adequacy of the fit of the model, with respect to the choice of the variance function, link function and terms in the linear predictor. Residuals may also indicate the presence of anomalous values requiring further investigation. For generalized linear models an extended definition of residuals is required, applicable to all the distributions that may replace the normal distribution. It is convenient if these residuals can be used for the same purposes as standard normal residuals. Three well-known forms of general-

188 162 Chapter 4 - Reserving in non-life insurance business ized residuals are the Pearson, Anscombe and deviance residuals. Pearson residuals are easy to interpret: it are just the raw residuals scaled by the estimated standard deviation of the response variable. A disadvantage of the Pearson residual is that the distribution of this residual form for nonnormal distributions is often markedly skewed, and so it may fail to have properties similar to those of a normal theory residual. Anscombe and deviance residuals are more appropriate to check the approximate normality. In general the lower bound S l turns out to perform very well. A final method to obtain a confidence bound for the predictive distribution is a combination of the power of this lower bound and bootstrapping. We will bootstrap a high percentile of the distribution of the lower bound. This is done as follows: 1. The preliminaries: Estimate the model parameters β σ 2 (LL) and σ 2 (LLS) φ (GLIM) e (R ˆ β)ij (LL) Calculate the fitted values: ˆµ ij = See Table 4.2 (LLS) g 1 (R ˆ β)ij (GLIM) (i = 1,..., t; j = 1,..., t + 1 i). z ij ln ˆµ ij (LL) Calculate the residuals: r ij = z ij ln ˆµ ij (LLS) y ij ˆµ ij (GLIM) (i = 1,..., t; j = 1,..., t + 1 i). ˆφV (ˆµij ) 2. Bootstrap loop (to be repeated B times): Generate a set of residuals rij by sampling with replacement from the original residuals (r ij ) (i = 1,..., t; j = 1,..., t + 1 i). Create a new upper triangle y ij : non-parametric bootstrap (NPB) e ln(ˆµ ij)+rij (LL) yij = e ln(ˆµ ij)+rij (LLS) ˆφV (ˆµij )rij + ˆµ ij (GLIM)

189 4.6. Three applications 163 (i = 1,..., t; j = 1,..., t + 1 i). parametric bootstrap (PB) e (R ˆ β)ij +ˇσN(0,1) yij = See Table 4.2 ˆµ ij + B(ˆ µ) ij + Σ(ˆ µ) ij N(0, 1) (i = 1,..., t; j = 1,..., t + 1 i). Now we have bootstrapped a run-off triangle. (LL) (LLS) (GLIM) Calculate for this bootstrapped triangle the parameters ˆ β and (ˇσ 2 ) (LL) (ˇ σ 2 ) (LLS) ˆφ (GLIM) Calculate the percentile k of the distribution of S l, S(k) l, using these parameters. Return to the beginning of step 2 until the B repetitions are completed. 3. Analysis of the bootstrap data: Apply the percentile method to the bootstrap observations to obtain the required prediction interval. 4.6 Three applications In this section we illustrate the effectiveness of the bounds derived for the discounted IBNR reserve S, under the model studied. We investigate the accuracy of the proposed bounds, by comparing their cumulative distribution function to the empirical distribution obtained with Monte Carlo simulation (MC), which serves as a close approximation to the exact distribution of S. The simulation results are based on generating random paths. The estimates obtained from this time-consuming simulation will serve as benchmark. The random paths are based on antithetic variables in order to reduce the variance of the Monte Carlo estimates. In order to illustrate the power of the bounds, namely inspecting the deviation of the cdf of the convex bounds S l and S c from the true distribution of the total IBNR reserve S, we simulate a triangle from a particular model. We created a non-cumulative run-off triangle based on the chainladder predictor (4.31) with parameters given in Table 4.4. So, the run-off

190 164 Chapter 4 - Reserving in non-life insurance business α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8 α 9 α 10 α β 1 β 2 β 3 β 4 β 5 β 6 β 7 β 8 β 9 β 10 β Table 4.4: Model parameters. triangle has only trends in the two main directions, namely in the year of origin and in the development year. The parameter β 1 is set equal to zero in order to have a non-singular regression matrix. We also specify the multivariate distribution function of the random vector (Y 1, Y 2,..., Y t 1 ). In particular, we will assume that the random variables Y i are i.i.d. and N(δ ς2, ς 2 ) distributed with δ = 0.08 and ς = This enables now to simulate the cdf s while there is no way to compute them analytically Lognormal linear models The simulated run-off triangle for this model is displayed in Table 4.5. Fitting the lognormal linear model with a chain-ladder type predictor gives the parameter estimates and standard errors shown in Table 4.6.

191 , , , , , , ,373 95,703 71,742 53,788 35, , , , , , , , ,272 78,515 58, ,154 1,096, , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,636 1,342, , , , ,641 1,219, , , , , , , , ,084,253 Table 4.5: Simulated run-off triangle with non-cumulative claim figures for the lognormal linear regression model Three applications 165

192 166 Chapter 4 - Reserving in non-life insurance business Parameter Value Estimate Standard error α α α α α α α α α α α β β β β β β β β β β σ Table 4.6: Model specification, maximum likelihood estimates and standard errors for the run-off triangle in Table 4.5. Figure 4.2 shows the cdf s of the upper and lower bounds, compared to the empirical distribution based on randomly generated, normally distributed vectors (Y 1, Y 2,..., Y t 1 ) and ɛ. Since SLL l cx S LL cx SLL c, the same ordering holds for the tails of their respective distribution functions which can be observed to cross only once. We see that the cdf of SLL l is very close to the distribution of S LL. The real standard deviation equals 1,617,912 whereas the standard deviation of the lower bound equals 1,590,233. A lower bound for the 95th percentile is given by 13,638,620. The comonotonic upper bound SLL c performs badly in this case. This comes from the fact that in order to determine SLL l, we make use of the (estimated values of the) correlations between the cells of the lower triangle, whereas in the case of SLL c, the distribution is an upper bound (in the sense of convex order) for any possible dependence structure between the components of the vector V. The standard deviation of the upper bound is given by 1,890,298. The 95th percentile of the upper bound now equals

193 4.6. Three applications ,207,619, which is of course much higher than the 95th percentile of SLL l. Table 4.7 summarizes the numerical values of the 95th percentiles of the two bounds SLL l and Sc LL, together with their means and standard deviations. This is also provided for the row totals S LL,i = t j=t+2 i e (R ˆ β)ij Y (i+j t 1)+ɛ ij, i = 2,, t. (4.49) We can conclude that the lower bound approximates the real discounted reserve very well. In order to have a better view on the behavior of the upper bound SLL c and of the lower bound Sl LL in the tails, we consider a QQ-plot where the quantiles of SLL c and of the lower bound Sl LL are plotted against the quantiles of S LL. The upper bound SLL c and the lower bound Sl LL will be a good approximation for S LL if the plotted points (F 1 S LL (p), F 1 SLL(p)), c respectively (F 1 S LL (p), F 1 (p)), for all values of p in (0, 1) do not devi- SLL l ate too much from the line y = x. From the QQ-plot in Figure 4.3, we can conclude that the upper bound (slightly) overestimates the tails of S, whereas the accuracy of the lower bond is extremely high for the chosen set of parameter values. Table 4.8 confirms these observations. We remark that the improved upper bound SLL u is very close to the comonotonic upper bound SLL c. This could be expected because ρ ij is close to ρ kl for any pair (ij, kl) ( with ij and kl sufficient close. This implies that ) for any such pair (ij, kl) F 1 e (R ˆ β)ij (U), F 1 Y (i+j t 1) Λ e (R ˆ β)kl (U) Y (k+l t 1) Λ ( ) is close to F 1 1 e (R ˆ β)ij (U), F (U) Y (i+j t 1) e (R ˆ β)kl. Since the improved upper bound requires more computational time, the results for the Y (k+l t 1) improved upper bound are not displayed in this thesis.

194 168 Chapter 4 - Reserving in non-life insurance business cum. distr *10^6 8*10^6 10^7 1.2*10^7 1.4*10^7 1.6*10^7 1.8*10^7 discounted IBNR reserve Figure 4.2: The cdf s of S LL (MC) (solid line), SLL l (dashed line) for the run-off triangle in Table 4.5. S c LL (dotted line) and 8*10^6 10^7 1.2*10^7 1.6*10^7 8*10^6 10^7 1.2*10^7 1.4*10^7 1.6*10^7 Figure 4.3: QQ-plot of the quantiles of SLL l ( ) and Sc LL of S LL (MC). ( ) versus those

195 SLL l S LL SLL c year 95% mean st. dev. 95% mean st. dev. 95% mean st. dev. 2 41,913 36,694 3,043 43,742 36,690 4,072 43,796 36,694 4, , ,522 18, , ,510 21, , ,522 22, , ,596 33, , ,570 36, , ,596 39, , ,861 51, , ,817 53, , ,861 60, , ,311 66, , ,252 68, , ,311 77, ,515,794 1,206, ,414 1,526,990 1,206, ,891 1,570,251 1,206, , ,976,955 1,574, ,804 1,986,766 1,574, ,635 2,053,898 1,574, , ,392,268 1,095, ,894 1,403,295 1,095, ,890 1,449,017 1,095, , ,641,355 1,287, ,051 1,657,107 1,286, ,005 1,713,161 1,287, , ,423,367 4,267, ,616 5,473,462 4,266, ,975 5,674,518 4,267, ,003 total 13,638,620 10,815,543 1,590,233 13,718,215 10,814,002 1,617,912 14,207,619 10,815,543 1,890,298 Table 4.7: 95th percentiles, means and standard deviations of the distributions of S l LL and Sc LL vs. S LL (MC) Three applications 169

196 170 Chapter 4 - Reserving in non-life insurance business p SLL l S LL SLL c ,638,620 13,718,215 14,207, ,303,311 14,411,869 15,035, ,122,153 15,166,753 16,066, ,709,687 15,710,588 16,813, ,003,250 17,003,255 18,479,550 Table 4.8: Approximations for some selected quantiles with probability level p of S LL. Distribution of bootstrapped Simulated distribution 95th percentiles of SLL l of F 1 S LL (0.95) 1 st percentile 13,587,825 13,578, th percentile 13,589,852 13,579,131 5 th percentile 13,597,445 13,585, th percentile 13,616,522 13,598, th percentile 13,627,692 13,619, th percentile 13,637,841 13,634, th percentile 13,647,654 13,651, th percentile 13,661,140 13,669, th percentile 13,671,003 13,678, th percentile 13,678,085 13,685, th percentile 13,680,785 13,688,379 Table 4.9: Percentiles of the bootstrapped 95th percentile of the distribution of the lower bound Sl(95) B vs. the simulation. Finally, for each bootstrap sample, we calculate the desired percentile of the distribution of SLL l. This two-step procedure is repeated a large number of times. The first column of Table 4.9 shows the results, concerning the 95th percentile, for 5000 bootstrap samples applied to the run-off triangle in Table 4.5. When compared with the simulated distribution of F 1 S LL (0.95) (obtained through 5000 simulated triangles), we can conclude that the bootstrap distribution yields appropriate confidence bounds.

197 4.6. Three applications 171 Parameter Value Estimate Standard error α α α α α α α α α α α β β β β β β β β β β σ Table 4.10: Model specification, maximum likelihood estimates and standard errors for the run-off triangle in Table Loglinear location-scale models Table 4.11 displays the simulated run-off triangle for the logistic regression model with given parameters displayed in Table 4.4. Fitting the logistic regression model with a chain-ladder type predictor gives the parameter estimates and standard errors shown in Table 4.10.

198 , , , , , , ,722 98,983 70,587 50,118 36, , , , , , , , ,641 77,890 58, ,562 1,109, , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,881 1,341, , , , ,565 1,230, , , , , , , , ,093,549 Table 4.11: Simulated run-off triangle with non-cumulative claim figures for the logistic regression model. 172 Chapter 4 - Reserving in non-life insurance business

199 4.6. Three applications 173 We will compare the derived bounds with a time consuming Monte Carlo simulation based on randomly generated, normally distributed vectors (Y 1, Y 2,..., Y t 1 ) and e σ ɛ. Using the following properties, the simulation of these last terms can be done in any statistical software package. If ɛ ij is Gumbel distributed, then we have that e σɛ ij is Weibull distributed with location parameter 1/ σ and scale parameter equal to 1. If ɛ ij is generalized loggamma distributed with parameter k, then we have that e σɛ ij is generalized gamma distributed with parameters γ = 1/( σ k) and α = k σ k. One can generate a random number from a generalized gamma distribution as follows: 1. Generate G k from the gamma distribution with location parameter k and scale parameter 1 2. Retain α(g k ) 1 λ. If ɛ ij is log inverse Gaussian distributed, then we have that e σɛ ij is inverse Gaussian distributed with location parameter and scale parameter equal to 1/ σ. Michael et al. (1976) describe an algorithm to generate a random number from an inverse Gaussian distribution with parameters α and β as follows: 1. Generate C from the χ 2 (1) distribution 2. Calculate x 1 = α β + C 2β 1 ( ) β α x 1 2β 3. Generate U Uniform(0, 1) 4. Retain x 2 if U p 1, else x 1. 4αC + C 2, x 2 = α2 β 2 x 1 and p 1 = On Figures 4.4 and 4.5 we compare the approximations (the convex upper and lower bounds) for the distribution of the discounted loss reserve S LLS to the empirical distribution function obtained by a Monte Carlo (MC) simulation study. One can see that the upper bound SLLS c gives a poor approximation. We observe that this upper bound has heavier tails than the original distribution the deviation for upper quantiles reaches 25%. The main reason for that is a relatively weak dependence between claims, for which the comonotonic approximation significantly overestimates the

200 174 Chapter 4 - Reserving in non-life insurance business p SLLS l S LLS SLLS c ,517,204 13,524,010 14,125, ,175,492 14,165,083 14,950, ,988,558 15,009,978 15,979, ,573,369 15,483,938 16,724, ,865,068 16,623,928 18,386,959 Table 4.12: Approximations for some selected quantiles with probability level p of S LLS. tails, which is very clear both from the plot of cdf s and from the QQ-plot. On the other hand the lower bound gives a much better fit to the original distribution. These findings are confirmed in Table 4.12 for some chosen quantiles. Similar conclusions can be drawn from the study of the reserves for the row totals given by S LLS,i = t j=t+2 i e (R β) ij + σɛ ij Y (i+j t 1), i = 2,, t. (4.50) Table 4.13 summarizes the numerical values of the 95th percentiles of the two bounds SLLS l and Sc LLS, together with their means and standard deviations. We end this illustration with a bootstrap study in order to incorporate the estimation error involved. Starting from the run-off triangle in Table 4.11 we bootstrap 5000 pseudo run-off triangles and calculate for each bootstrap sample the 95% percentile of the distribution of SLLS l. Table 4.14 displays the results of this study. One can observe that, compared to the simulated distribution of F 1 S LLS (0.95), the bootstrap distributions performs very well.

201 4.6. Three applications 175 cum. distr *10^6 8*10^6 10^7 1.2*10^7 1.4*10^7 1.6*10^7 1.8*10^7 discounted IBNR reserve Figure 4.4: The cdf s of S LLS (MC) (solid line), SLLS l (dotted line) and (dashed line) for the run-off triangle in Table S c LLS 8*10^6 10^7 1.2*10^7 1.6*10^7 8*10^6 10^7 1.2*10^7 1.4*10^7 Figure 4.5: QQ-plot of the quantiles of SLLS l ( ) and Sc LLS those of S LLS (MC). ( ) versus

202 SLLS l S LLS SLLS c year 95% mean st. dev. 95% mean st. dev. 95% mean st. dev. 2 41,609 36,990 2,705 44,057 36,990 4,124 44,146 36,990 4, , ,309 16, , ,309 20, , ,309 21, , ,778 30, , ,778 35, , ,778 38, , ,969 48, , ,969 53, , ,969 60, , ,188 63, , ,188 67, , ,188 77, ,473,927 1,178, ,351 1,484,295 1,178, ,499 1,535,494 1,178, , ,971,886 1,576, ,631 1,979,501 1,576, ,440 2,057,661 1,576, , ,384,144 1,090, ,670 1,386,684 1,090, ,568 1,443,768 1,090, , ,593,022 1,248, ,941 1,589,451 1,248, ,174 1,663,714 1,248, , ,438,603 4,278, ,272 5,443,520 4,278, ,437 5,693,117 4,278, ,183 total 13,517,204 10,743,220 1,559,369 13,524,010 10,743,220 1,583,892 14,125,203 10,743,220 1,884,508 Table 4.13: 95th percentiles, means and standard deviations of the distributions of S l LLS and Sc LLS vs. S LLS (MC). 176 Chapter 4 - Reserving in non-life insurance business

203 4.6. Three applications 177 Distribution of bootstrapped Simulated distribution 95th percentiles of SLLS l of F 1 S LLS (0.95) 1 st percentile 13,123,442 13,111, th percentile 13,227,201 13,216,739 5 th percentile 13,314,139 13,301, th percentile 13,363,615 13,340, th percentile 13,434,055 13,421, th percentile 13,510,262 13,501, th percentile 13,583,175 13,585, th percentile 13,646,792 13,654, th percentile 13,691,476 13,698, th percentile 13,716,004 13,731, th percentile 13,730,976 13,740,991 Table 4.14: Percentiles of the bootstrapped 95th percentile of the distribution of the lower bound Sl(95) B vs. the simulation Generalized linear models In this last illustration we model the incremental claims Y ij with a logarithmic link function to obtain a multiplicative parametric structure and we link the expected value of the response to the chain-ladder type linear predictor. Formally, this means that E[Y ij ] = µ ij, Var[Y ij ] = φµ κ ij, log(µ ij ) = η ij, η ij = α i + β j. (4.51) The choice of the error distribution is determined by κ. More specific we consider model (4.51) with the Poisson error distribution (κ=1 and φ = 1). The simulated triangle for this model is depicted in Table Parameter estimates and standard errors for this fit are shown in Table Since this model is a generalized linear model, standard statistical software can be used to obtain maximum (quasi) likelihood parameter estimates, fitted and predicted values. Standard statistical theory also suggests goodness-of-fit measures and appropriate residual definitions for diagnostic checks of the fitted model.

204 , , , , , , ,293 95,961 70,812 53,395 35, , , , , , , , ,760 78,736 58, ,843 1,100, , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,897 1,341, , , , ,268 1,217, , , , , , , , ,087,672 Table 4.15: Simulated run-off triangle with non-cumulative claim figures for the Poisson regression model. 178 Chapter 4 - Reserving in non-life insurance business

205 4.6. Three applications 179 Parameter Value Estimate Standard error α α α α α α α α α α α β β β β β β β β β β φ Table 4.16: Model specification, maximum likelihood estimates and standard errors for the run-off triangle in Table Figure 4.6 shows the distribution functions of the different bounds compared to the empirical distribution obtained by Monte Carlo simulation (MC). The distribution functions are remarkably close to each other and enclose the simulated cdf nicely. This is confirmed by the QQ-plot in Figure 4.7 where we also see that the comonotonic upper bound has somewhat heavier tails. Numerical values of some high quantiles of S GLIM, SGLIM l and SGLIM c are given in Table Table 4.17 summarizes the numerical values of the 95th percentiles of the two bounds SGLIM l and Sc GLIM vs. S GLIM, together with their means and standard deviations. This is also provided for the row totals S GLIM,i = t j=t+2 i ˆµ ij e Y (i+j t 1), i = 2,..., t. (4.52)

206 180 Chapter 4 - Reserving in non-life insurance business cum. distr ^7 1.5*10^7 2*10^7 discounted IBNR reserve Figure 4.6: The cdf s of S GLIM (MC) (solid line), SGLIM l (dotted line) and SGLIM c (dashed line) for the run-off triangle in Table *10^6 10^7 1.2*10^7 1.6*10^7 8*10^6 10^7 1.2*10^7 1.4*10^7 1.6*10^7 Figure 4.7: QQ-plot of the quantiles of SGLIM l ( ) and Sc GLIM those of S GLIM (MC). ( ) versus

207 SGLIM l S GLIM SGLIM c year 95% mean st. dev. 95% mean st. dev. 95% mean st. dev. 2 43,622 36,623 4,041 43,624 36,623 4,042 43,631 36,623 4, , ,600 21, , ,600 21, , ,600 22, , ,318 35, , ,318 35, , ,318 39, , ,089 52, , ,089 53, , ,089 60, , ,289 67, , ,289 67, , ,289 78, ,514,480 1,205, ,099 1,516,799 1,205, ,658 1,567,945 1,205, , ,977,737 1,575, ,703 1,980,868 1,575, ,343 2,054,475 1,575, , ,390,601 1,093, ,320 1,392,957 1,093, ,862 1,444,660 1,093, , ,632,675 1,278, ,110 1,634,653 1,278, ,693 1,702,375 1,278, , ,439,986 4,276, ,280 5,446,107 4,276, ,472 5,685,932 4,276, ,741 total 13,631,905 10,810,476 1,594,152 13,648,695 10,810,476 1,597,507 14,200,226 10,810,476 1,896,219 Table 4.17: 95th percentiles, means and standard deviations of the distributions of S l GLIM and Sc GLIM vs. GLIM (MC) Three applications 181

208 182 Chapter 4 - Reserving in non-life insurance business p SGLIM l S GLIM SGLIM c ,631,905 13,648,695 14,200, ,296,448 14,305,657 15,027, ,115,189 15,122,840 16,057, ,702,702 15,709,497 16,804, ,996,374 17,018,860 18,469,110 Table 4.18: Approximations for some selected quantiles with probability level p of S GLIM. Distribution of bootstrapped Simulated distribution 95th percentiles of SGLIM l of F 1 S GLIM (0.95) 1 st percentile 13,614,404 13,604, th percentile 13,617,028 13,609,425 5 th percentile 13,619,474 13,613, th percentile 13,622,664 13,618, th percentile 13,626,759 13,624, th percentile 13,631,651 13,631, th percentile 13,636,506 13,638, th percentile 13,641,168 13,645, th percentile 13,643,882 13,649, th percentile 13,646,720 13,652, th percentile 13,648,833 13,656,178 Table 4.19: Percentiles of the bootstrapped 95th percentile of the distribution of the lower bound Sl(95) B vs. the simulation. The bootstrap results in Table 4.19 are in line with the results of the previous applications. We can conclude that in the discussed applications the lower bound approximates the real discounted reserve very well. The precision of the bounds only depends on the underlying variance of the statistical and financial part. As long as the yearly volatility does not exceed ς = 35%, the financial part of the comonotonic approximation provides a very accurate fit. These parameters are consistent with historical capital market values as reported by Ibbotson Associates (2002). The underlying variance of the statistical part depends on the estimated dispersion parameter and error distribution or mean-variance relationship. For example, in case of the gamma distribution one obtains excellent results as long

209 4.7. Conclusion 183 as the dispersion parameter is smaller than 1. This is again in line with the volatility structure in practical IBNR data sets. Since the parameters in the paper for the statistical part of the bounds, obtained through the quasi-likelihood approach, have small standard errors, it follows that results would be similar when simulating from a GLIM with the same linear predictor, but for instance with another distribution type. In that sense our findings are robust. 4.7 Conclusion In this chapter, we considered the problem of deriving the distribution function of the present value of a triangle of claim payments that are discounted using some given stochastic return process. We started to model the claim payments by means of a lognormal linear model which is also included in the larger class of loglinear location-scale models. The use of generalized linear models offers a great gain in modelling flexibility over the simple lognormal model. The incremental claim amounts can for instance be modelled as independent normal, Poisson, gamma or inverse Gaussian response variables together with a logarithmic link function and a specified linear predictor. Because an explicit expression for the distribution function is hard to obtain, we presented some approximations for this distribution function, in the sense that these approximations are larger or smaller in convex order sense than the exact distribution. When lower and upper bounds are close to each other, together they can provide reliable information about the original and more complex variable. An essential point in the derivation of the presented convex lower bound approximations is the choice of the conditioning random variable Λ. When dealing with very large variances in the statistical and financial part of our model, an adaptation of the random variable Λ will be necessary or one can use other approximation techniques. This will be the topic of the next chapter.

210

211 Chapter 5 Other approximation techniques for sums of dependent random variables Summary In this chapter we derive some asymptotic results for the tail distribution of sums of heavy tailed dependent random variables. We show how to apply the obtained results to approximate certain functionals of (the d.f. of) sums of dependent random variables. Our numerical results demonstrate that the asymptotic approximations are typically close to the Monte Carlo value. We will further briefly recall the mathematical techniques behind the moment matching approximations and the Bayesian approach. Finally, we compare these approximations with the comonotonic approximations of the previous chapter in the context of claims reserving. 5.1 Introduction Many quantities of relevance in actuarial science concern functionals of (the d.f. of) sums of dependent random variables. For example, one can think of the Value-at-Risk of a stochastically discounted life annuity, or the stop-loss premium for the aggregate claim amount of a number of interrelated policies. Therefore, distribution functions of sums of dependent random variables are of particular interest. Typically these distribution functions are of a complex form. Consequently, in order to compute functionals of sums of dependent random variables, approximation methods 185

212 186 Chapter 5 - Approximation techniques for sums of r.v. s are generally indispensable. Obviously, in many cases we could 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. Practitioners often use moment matching techniques to approximate (the d.f. of) a sum of dependent lognormal random variables. In Section 2 we recall the lognormal and reciprocal gamma moment matching approach. Both approximations are chosen such that their first two moments are equal to the corresponding moments of the random variable of interest. In Chapter 2 we discussed the concept of comonotonicity to obtain bounds in convex order for sums of dependent random variables. Although these 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. Section 3 establishes some asymptotic results for the tail probability of a sum of dependent random variables, in the presence of heavy-tailedness conditions. Section 4 sketches, in very broad terms, basic elements of Bayesian computation. We discuss two major obstacles to its popularity. The first is how to specify prior distributions, and the second is how to evaluate the integrals required for inference, given that for most models, these are analytically intractable. In the last section we compare the discussed 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 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. This chapter is based on Laeven, Goovaerts & Hoedemakers (2005), Vanduffel, Hoedemakers & Dhaene (2004) and Antonio, Beirlant & Hoedemakers (2005).

213 5.2. Moment matching approximations Moment matching approximations Consider a sum S given by S = α i e Z i. (5.1) Here, the α i are non-negative real numbers and (Z 1, Z 2,..., Z n ) is a multivariate normal distributed random vector. The accumulated value at time n of a series of future deterministic saving amounts α i can be written in the form (5.1), where Z i denotes the random accumulation factor over the period [i, n]. Also the present value of a series of future deterministic payments α i can be written in the form (5.1), where now Z i denotes the random discount factor over the period [0, i]. The valuation of Asian or basket options in a Black & Scholes model and the setting of provisions and required capitals in an insurance context boils down to the evaluation of risk measures related to the distribution function of a random variable S as defined in (5.1). The r.v. S defined in (5.1) will in general be a sum of non-independent lognormal r.v. s. Its distribution function cannot be determined analytically and is too cumbersome to work with. In the literature, a variety of approximation techniques for this distribution function has been proposed. Practitioners often use a moment matching lognormal approximation for the distribution of S. The lognormal approximation is chosen such that its first two moments are equal to the corresponding moments of S. The present value of a continuous perpetuity with lognormal return process has a reciprocal gamma distribution, see for instance Milevsky (1997) and Dufresne (1990). This present value can be considered as the limiting case of a random variable S as defined above. Motivated by this observation, Milevsky & Posner (1998) and Milevsky & Robinson (2000) propose a moment matching reciprocal gamma approximation for the d.f. of S such that the first two moments match. They use this technique for deriving closed form approximations for the price of Asian and basket options Two well-known moment matching approximations It belongs to the toolkit of any actuary to approximate the distribution function of an unknown r.v. by a known distribution function in such a

214 188 Chapter 5 - Approximation techniques for sums of r.v. s way that the first moments are preserved. In this section we will briefly describe the reciprocal gamma and the lognormal moment matching approximations. These two methods are frequently used to approximate the distribution function of the r.v. S defined by (5.1). The reciprocal gamma approximation A r.v. X is said to be gamma distributed when its probability density function is given by f X (x; α, β) = βα Γ(α) xα 1 e βx, x > 0, (5.2) where α > 0, β > 0 and Γ(.) denotes the gamma function. Consider now the r.v. Y = 1/X. This r.v. is said to be reciprocal gamma distributed. Its p.d.f. is given by f Y (y; α, β) = f X (1/y; α, β)/y 2, y > 0. (5.3) It is straightforward to prove that the quantiles of Y are given by F 1 Y (p) = 1 F 1 p (0, 1), (5.4) (1 p; α, β), X where F X (.; α, β) is the cdf of the gamma distribution with parameters α and β. Since the inverse of the gamma distribution function is readily available in many statistical software packages, quantiles can easily be determined. The first two moments of the reciprocal gamma distributed r.v. Y are given by 1 E[Y ] = β(α 1), α > 1 (5.5) and E[Y 2 ] = 1 β 2, α > 2. (5.6) (α 1)(α 2) Expressing the parameters α and β in terms of E[Y ] and E[Y 2 ] gives and α = 2E[Y 2 ] E[Y ] 2 E[Y 2 ] E[Y ] 2 (5.7) β = E[Y 2 ] E[Y ] 2 E[Y ]E[Y 2. (5.8) ]

215 5.2. Moment matching approximations 189 The d.f. of the r.v. defined in (5.1) is now approximated by a reciprocal gamma distribution with first two moments (2.46) and (2.47). The coefficients α and β of the reciprocal gamma approximation follow from (5.7) and (5.8). The reciprocal gamma approximation for the quantile function is then given by (5.4). The reciprocal gamma moment matching method appears naturally in case one wants to approximate the d.f. of stochastic present values. Indeed, for the limiting case of the constant continuous perpetuity : ] S = exp [ (µ σ2 )τ σb(τ) dτ, (5.9) 2 0 where B(τ) represents a standard Brownian motion and µ > σ2 2, the risk measures can be calculated very easily since Dufresne (1990) proved that S 1 is gamma distributed with parameters 2µ 1 and σ2 σ 2 2. An elegant proof for this result can be found in Milevsky (1997). Expression (5.9) can be seen as a continous counterpart of a discounted sum such as in (5.1). One expects that the present value of a finite discrete annuity with a normal logreturn process with independent periodic returns, can be approximated by a reciprocal gamma distribution, provided the time period involved is long enough. This idea was set forward and explored in Milevsky & Posner (1998), Milevsky & Robinson (2000) and Huang et al. (2004). The lognormal approximation A r.v. X is said to be lognormally distributed if its p.d.f. is given by f X (x; µ, σ 2 1 ) = xσ 2π e (log x µ) 2 2σ 2, x > 0, (5.10) where σ > 0. The quantiles of X are given by F 1 X (p) = eµ+σφ 1 (p), p (0, 1). (5.11) The first two moments of X are given by E[X] = e µ+ 1 2 σ2 (5.12) and E[X 2 ] = e 2µ+2σ2. (5.13)

216 190 Chapter 5 - Approximation techniques for sums of r.v. s Expressing the parameters µ and σ 2 of the lognormal distribution in terms of E[X] and E[X 2 ] leads to ( ) E[X] 2 µ = log (5.14) E[X 2 ] and σ 2 = log ( E[X 2 ) ] E[X] 2. (5.15) The same procedure as the one explained in the previous subsection can be followed in order to obtain a lognormal approximation for S, with the first two moments matched. Dufresne (2002) obtains a lognormal limit distribution for S as volatility σ tends to zero and this provides a theoretical justification for the use of the lognormal approximation Application: discounted loss reserves We calculate the lognormal moment matching approximations for the application considered in Section 2.4 and compare the results with the convex lower bound. The results are given below. We use the notation SM p [V l ] and SM p [V LN ] to denote the security margin for confidence level p approximated by the lower bound and by the lognormal moment matching technique respectively. The different tables display the Monte Carlo simulation result (MC) for the security margin, as well as the procentual deviations of the different approximation methods, relative to the Monte Carlo result. These procentual deviations are defined as follows: LB := SM p[v l ] SM p [V MC ] SM p [V MC ] LN := SM p[v LN ] SM p [V MC ] SM p [V MC ] 100%, 100%, where V l and V LN correspond to the lower bound approach and the lognormal moment matching approach, and V MC denotes the Monte Carlo simulation result. The figures displayed in bold in the tables correspond to the best approximations, this means the ones with the smallest procentual deviation compared to the Monte Carlo results. Overall the comonotonic lower bound approach provides a very accurate fit under different parameter assumptions. These assumptions are

217 5.2. Moment matching approximations 191 σ M : LB 0.25% 0.09% 0.12% 0.00% LN 1.66% +1.28% +4.09% +7.52% MC (s.e ) (1.11) (2.47) (6.15) (8.18) Table 5.1: (ex. 1) Approximations for the security margin SM 0.70 [V ] for different market volatilities and ω L = 0.1 and ω A = p : LB 0.38% 0.21% 0.16% 0.08% 0.00% 0.00% LN 4.30% 2.96% 2.29% 1.43% 0.11% +1.74% MC (s.e ) (2.49) (0.46) (0.26) (0.10) (0.06) (0.04) Table 5.2: (ex. 1) Approximations for some selected confidence levels of SM p [V ]. The market volatility is set equal to 20%. (ω L = 0.05 and ω A = 0) σ M : LB 0.19% 0.15% 0.23% 0.16% 0.11% 0.17% 0.38% LN 4.94% 3.92% 3.17% 2.49% 1.95% 1.56% 1.30% MC s.e.( 10 5 ) (0.15) (0.29) (0.41) (0.69) (1.22) (3.78) (4.16) Table 5.3: (ex. 1) Approximations for the security margin SM [V ] for different market volatilities. p : LB 0.93% 0.04% 0.02% 0.18% 0.03% 0.6% LN 3.94% +3.78% +7.22% % % % MC s.e.( 10 5 ) (37.63) (2.99) (7.44) (2.79) (0.78) (0.27) Table 5.4: (ex. 2) Approximations for some selected confidence levels of SM p [V ]. The market volatility is set equal to 25%. in line with the realistic market values. Moreover the comonotonic approximations have the advantage that they are easy computable for any risk measure that is additive for comonotonic risks, such as Value-at-Risk

218 192 Chapter 5 - Approximation techniques for sums of r.v. s and Tail Value-at-Risk. We believe the comonotonic approach is preferred to any moment matching approximation, because it is more stable and accurate across all levels of volatility. 5.3 Asymptotic approximations 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 (the d.f. 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 infinity. This section establishes some asymptotic results for the tail probabilities related with a sum of heavy tailed dependent random variables. In particular, we establish 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. Next, we present an application that demonstrates how the derived asymptotic results can be employed to approximate certain functionals of sums of (the d.f. of) dependent random variables. To explore the quality of the asymptotic approximations, we also provide a numerical illustration that compares the asymptotic approximation values to Monte Carlo simulated values Preliminaries for heavy-tailed distributions First we introduce some notational conventions. For a random variable X with a distribution function F, we denote its tail probability by F (x) = 1 F (x) = Pr[X > x]. For two independent r.v. s X and Y with d.f. s F and G supported on (, + ), we write by F G(x) = + F (x t)g(dt), < x < +, the convolution of F and G. We denote by F n = F F the n-fold convolution of F, and we write by F G the d.f. of XY.

219 5.3. Asymptotic approximations 193 Throughout, unless otherwise stated, all limit relations are for x +. Let a(x) 0 and b(x) > 0 be two functions satisfying l a(x) a(x) lim inf lim sup x + b(x) x + b(x) l+. We write a(x) = O (b(x)) if l + < +, a(x) = o (b(x)) if l + = 0 and a(x) b(x) if both l + < + and l > 0. We write a(x) b(x) if l + = 1, a(x) b(x) if l = 1 and a(x) b(x) if both l + = 1 and l = 1. We say that a(x) and b(x) are weakly equivalent if a(x) b(x), and say that a(x) and b(x) are (strongly) equivalent if a(x) b(x). A r.v. X or its d.f. F is said to be heavy-tailed if E[e γx ] = + for any γ > 0. Below we introduce some important classes of heavy-tailed distributions. A d.f. F supported on (0, + ) belongs to the subexponential class S if lim x + F n (x)/f (x) = n (5.16) for any (or equivalently, for some) n 2. More generally, a d.f. F supported on (, + ) belongs to the class S if F (x) = F (x)i (x>0) does. A d.f. F supported on (, + ) belongs to the long-tailed class L if for any real number y (or equivalently, for y = 1) we have that lim F (x + y) /F (x) = 1. (5.17) x + A class of heavy-tailed distributions that is closely related to the classes S and L, is the class D of d.f. s with dominatedly varying tails. A d.f. F supported on (, + ) belongs to the class D if its tail F is of dominated variation in the sense that lim sup x + F (xy) F (x) < + (5.18) for any 0 < y < 1 (or equivalently for some 0 < y < 1). It is well-known that D L S L. See e.g. Embrechts et al. (1997). We remark that the intersection D L contains many useful heavy-tailed distributions. In particular, the intersection D L covers the class R, which consists of all d.f. s with regularly

220 194 Chapter 5 - Approximation techniques for sums of r.v. s varying tails. A d.f. F supported on (, + ) has a regularly varying tail if there is some α > 0 such that the relation F (xy) lim x + F (x) = y α holds true for any y > 0. We denote F R α. In addition to the classes of heavy-tailed distributions introduced above, we introduce the class R of d.f. s with rapidly varying tails, containing both heavy-tailed and light-tailed distributions. For a d.f. F supported on (, + ) satisfying F (x) > 0 for any x > 0, F belongs to the class R if { F (xy) 0, for any y > 1; lim x + F (x) = (5.19) +, for any 0 < y < 1. We remark that the intersection S R contains e.g. lognormal distributions and certain Weibull distributions, which are prominent distributions in actuarial applications. For an elaboration on the classes of heavy-tailed distributions and the class of rapidly varying tailed distributions, and their applications in insurance and finance, the interested reader is referred to Bingham et al. (1987), Embrechts et al. (1997) and Beirlant et al. (2004). In Table 5.5 we list some well-known distributions and their corresponding distribution class Asymptotic results In this subsection, we derive some asymptotic results for the tail probability of sums of dependent r.v. s, in the presence of heavy-tailedness. In the following, we let {X n, n = 1, 2,...} and {Y n, n = 1, 2,...} denote two sequences of i.i.d. r.v. s that are mutually independent. We write by F X the d.f. of a r.v. X of which X n, n = 1, 2,..., are considered to be independent replicates, and assume it is supported on (, + ). Similarly, we write by F Y the d.f. of a r.v. Y of which Y n, n = 1, 2,..., are considered to be independent replicates, and assume it is supported on (0, + ). For notational convenience, we will use the device of independent replicates throughout.

221 Name d.f. or density f Parameters Class Lognormal f(x) = 1 2πσx e 1 log x µ ( ) 2 σ 2, (µ R, σ > 0) R S Weibull F (x) = 1 e cxβ (c > 0, 0 < β < 1) R S Benktander-I F (x) = 1 cx α 1 e β(log x)2 (α + 2β log x) (c, α, β > 0) R S Benktander-II F (x) = 1 cαx (1 β) exp{ (α/β)x β } (c, α > 0, 0 < β < 1) R S Pareto F (x) = 1 ( x β ) α (α, β > 0) R Burr F (x) = 1 (1 + xτ β ) α (α, β, τ > 0) R Loggamma f(x) = βα Γ(α)x (log x)α 1 x β (α, β > 0) R Transformed β f(x) = a B(p,q) xap 1 (1 + x a ) (p+q) (a R, p, q > 0) R Truncated F (x) = Pr[ X x], X α-stable (1 < α < 2) R α-stable Table 5.5: Some well-known distributions and their distribution class Asymptotic approximations 195

222 196 Chapter 5 - Approximation techniques for sums of r.v. s We state the following theorem: Theorem 12. Let Z i = Y 1 Y 2 Y i and 0 < a i < +, i = 1, 2,.... If F Y S R, then it holds for each n = 1, 2,... and x + that Proof. See section 5.6. [ ] Pr a i Z i > x Pr [a i Z i > x]. (5.20) In an actuarial context the sequence {a i, i = 1, 2,...} can be regarded as a sequence of deterministic payments. The following theorem applies to the case in which the payments consist of both a deterministic and a random component, and the deterministic component is either an additive or a multiplicative constant. The theorem is an extension of Theorems 5.1 and 5.2 of Tang & Tsitsiashvili (2003): Theorem 13. Let Z i = Y 1 Y 2 Y i and 0 < a i < +, i = 1, 2,.... conditions are valid: If the following 1. F X D L, 2. F Y R, then it holds for each n = 1, 2,... and x + that [ ] Pr (a i + X i )Z i > x Pr [(a i + X)Z i > x]. (5.21) Furthermore, it holds for each n = 1, 2,... and x + that [ ] Pr (a i X i )Z i > x Pr [(a i X)Z i > x]. (5.22) Proof. See section 5.6.

223 5.3. Asymptotic approximations 197 Corollary 3. Under the conditions stated in Theorem 13, we have for each n = 1, 2,... and x + that [ ] [ n 1 ] Pr (a i + X i )Z i > x Pr (a i + X i )Z i > x Pr [(a n + X)Z n > x]. (5.23) Furthermore, it holds for each n = 1, 2,... and x + that [ ] [ n 1 ] Pr (a i X i )Z i > x Pr (a i X i )Z i > x Pr [a n XZ n > x]. (5.24) Proof. See section 5.6. Corollary 4. If condition 1. stated in Theorem 13 is replaced by F X R α, while the other conditions remain the same, then it holds for each n = 1, 2,... and x + that [ ] Pr (a i + X i )Z i > x F X (x a i ) (E[Y α ]) i. (5.25) and [ ] Pr (a i X i )Z i > x F X (x) Proof. See section 5.6. a α i (E[Y α ]) i. (5.26) We remark that the particular case of lognormally distributed payments is not covered by Theorem 13, since the lognormal distribution does not belong to the intersection D L. The lognormal distribution has a moderately heavy tail and and has been a popular model for loss severity distributions. Hence, we state the following theorem: Theorem 14. Relations (5.21), (5.22), (5.23) and (5.24) remain valid if conditions 1. and 2. stated in Theorem 13 are replaced by 1 X logn(µ X, σ 2 X ), < µ X < + and σ X > 0, 2 Y logn(µ Y, σ 2 Y ), < µ Y < + and σ Y > 0,

224 198 Chapter 5 - Approximation techniques for sums of r.v. s 3 σ X > σ Y. Proof. See section Application: discounted loss reserves In this subsection, we consider the problem of determining stop-loss premiums and quantiles for discounted loss reserves. We denote by the r.v. X i from the i.i.d. sequence {X i, i = 1,..., n}, the net loss in year i. Furthermore, the positive r.v. Y i from the i.i.d. sequence {Y i, i = 1,..., n} represents the present value discounting factor from year i to year i 1. The two sequences {X i, i = 1,..., n} and {Y i, i = 1,..., n} are considered to be mutually independent. Then the discounted loss reserve S is given by i S = Y j. (5.27) X i j=1 Henceforth, we impose that E[ SI ( S>0)] < +, which is implied by the condition that E[XI (X>0) ] < + and E[Y ] < +. Approximate values for the stop-loss premium and quantiles of the discounted loss reserve S may be obtained by using the previously obtained asymptotic results. In particular, if X and Y satisfy the corresponding conditions under which Theorem 13 or Theorem 14 holds, then for sufficiently large values of the retention d, the stop-loss premium can be approximated by π( S, d) + d F X i j=1 Y j (s)ds = ( π X i j=1 ) Y j, d. (5.28) Since the d.f. of X i j=1 Y j will generally not be analytically tractable, Monte Carlo simulation may still be required. However, the number of simulations has been reduced considerably. In case F X R α, 0 < α < +, and F Y R, the asymptotic approximations for the stop-loss premium of S reduce to π( S, d) + d (E[Y α ]) i F X (s)ds = (E[Y α ]) i π(x, d). (5.29)

225 5.3. Asymptotic approximations 199 Furthermore, in this case we have for sufficiently large values of p, that the asymptotic approximation for the p-quantile is given by { } F 1 (p) inf s : (E[Y α ]) i F X (s) 1 p. (5.30) S Under the conditions of Theorem 14, we have for sufficiently large values of p, that the asymptotic approximation for the p-quantile is given by { } F 1 (p) inf s : F S X i j=1 Y (s) 1 p. (5.31) j We emphasize that the approximation (5.31) is not in general valid under the conditions of Theorem 13; it requires the additional condition that F X R α, 0 < α <. As an example, we consider X i GPD(α, β) and Y i logn(µ, σ 2 ), i = 1,..., n, in which GPD(α, β) denotes the generalized Pareto distribution with d.f. F X (x) = 1 (1 + x β ) α, x > 0, where α > 0 and β > 0. Then, clearly we have that F X R α and F Y R. Hence, the asymptotic approximations (5.29) and (5.30) are valid. Notice that for the example considered, the asymptotic approximations can even be computed analytically. We performed Monte Carlo (MC) simulations for quantiles and stop-loss premiums to assess the quality of the asymptotic approximations (5.29) and (5.30), under various specifications of the parameter n. We fix the parameter values: α = 1.5, β = 1, µ = 0.04 and σ = The results are presented in Table 5.6. Ndiff. refers to the normalized difference defined as MC Appr. MC 100%. Our numerical results demonstrate that the asymptotic approximations are typically close to the Monte Carlo value.

226 200 Chapter 5 - Approximation techniques for sums of r.v. s n=3 d MC Appr. Ndiff. p MC Appr. Ndiff % % % % % % % % % % % % % % % % % n = 5 d MC Appr. Ndiff. p MC Appr. Ndiff % % % % % % % % % % % % % % % n = 10 d MC Appr. Ndiff. p MC Appr. Ndiff % % % % % % % % % % % % % Table 5.6: Approximations for stop-loss premiums and quantiles of S for Pareto claim sizes and lognormal present value discounting factors.

227 5.4. The Bayesian approach The Bayesian approach Some comments on notation are needed at this point. First p(..) denotes a conditional probability density with the arguments determined by the context, and similarly for p( ), which denotes a marginal distribution. The same notation is used for continuous density functions and discrete probability mass functions Introduction Bayesian theory is a powerful branch of statistics not yet fully explored by practitioner actuaries. One of its main benefits, which is the core of its philosophy, is the ability of including subjective information in a formal framework. Apart from this, the wide range of models presented by this branch of statistics is also one of the main reasons why it has been so much studied recently. Since the early 1990s, statistics (and to a lesser extent econometrics) has seen an explosion in applied Bayesian research. This explosion has had little to do with a renewed interest 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. The recent developments are mainly due to, firstly, the recent computer developments that have made it easier to perform calculation by simulations and, secondly, to the failure of classical statistical methods to give solutions to many problems. 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. But, although so many developments have been occurring in Bayesian statistics, very few actuaries are aware of them and even fewer make use of them. The purpose of this section is to sketch, in very broad terms, basic elements of Bayesian computation. Classical statistics provides methods to analyze data, from simple descriptive measures to complex and sophisticated models. The available data are processed and then conclusions about a hypothetical population, of which the data available is supposed to be a representative sample, are drawn. It is not hard to imagine situations, however, in which data are not the only available source of information about the population. Bayesian

228 202 Chapter 5 - Approximation techniques for sums of r.v. s methods provide a principled way to incorporate this external information into the data analysis process. To do so, however, Bayesian methods have to change entirely the vision of the data analysis process with respect to the classical approach. In a Bayesian approach, the data analysis process starts already with a given probability distribution. As this distribution is given before any data is considered, it is called prior distribution. Bayesian methods allow us to assign prior distributions to the parameters in the model which capture known qualitative and quantitative features, and then to update these priors in the light of the data, yielding a posterior distribution via Bayes theorem Posterior Likelihood Prior, where denotes that two quantities are proportional to each other. Hence the posterior distribution is found by combining the prior distribution for the parameters with the probability of observing the data given the parameters (the likelihood). The ability to include prior information in the model is not only an attractive pragmatic feature of the Bayesian approach, it is theoretically vital for guaranteeing coherent inferences. More formally Bayes theorem is defined as follows. Consider a process in which observations ( Y is the vector of observations) are to be taken from a distribution for which the probability density function is p( Y θ), where θ is a set of unknown parameters. Before any observation is made, the analyst would include all his previous information and judgements of θ in a prior distribution p( θ), that would be combined with the observations to give a posterior distribution p( θ Y ) in the following way: p( θ Y ) p( Y θ)p( θ) Bayesian modelling involves integrals over the parameters, whereas non- Bayesian methods often rely on optimization of the parameters. The main difference between these methods is that optimization fails to take into account the inherent uncertainty in the parameters. There is no true value for each of the parameters which can be found by optimization. Instead, there is a range of plausible values, each with some associated density. The mechanisms of the Bayesian approach to model fitting to make inferences consists of three basic steps: 1. Assign priors to all the unknown parameters;

229 5.4. The Bayesian approach Write down the likelihood of the data given the parameters; 3. Determine the posterior distribution of the parameters given the data using Bayes theorem. Bayesian inference is quite simple to describe probabilistically, but there have been two major obstacles to its popularity. The first is how to specify prior distributions, and the second is how to evaluate the integrals required for inference, given that for most models, these are analytically intractable. This will be discussed in short in the next two subsections Prior choice The prior distribution can arise from data previously observed, or it can be the subjective assessment of some domain expert and, as such, it represents the information we have about the problem at hand, that is not conveyed by the sample data. Several methods for eliciting prior densities from experts exist. See, e.g. O Hagan (1994) for a comprehensive review. A common approach is to choose a prior distribution with density function similar to the likelihood function. In doing so, the posterior distribution of θ will be in the same class and the prior is said to be conjugate to the likelihood. The conjugate family is mathematically convenient in that the posterior distribution follows a known parametric form. Of course, if information is available that contradicts the conjugate parametric family, it may be necessary to use a more realistic, if inconvenient, prior distribution. The basic justification for the use of conjugate prior distributions is similar to that for using standard models for the likelihood: it is easy to understand the corresponding results, which can often be put in analytic form. Next, they are often a good approximation, and they simplify computations. Although they can make interpretations of posterior inferences less transparent and computation more difficult, non-conjugate prior distributions do not pose any new conceptual problem. In practice, for complicated models, conjugate prior distributions may not even be possible. In general, the exponential families are the only classes of distributions that have natural conjugate distributions, since, apart from certain irregular cases, the only distributions having a fixed number of sufficient statistics are of the exponential type.

230 204 Chapter 5 - Approximation techniques for sums of r.v. s Kass and Wasserman (1996) survey formal rules that have been suggested for choosing a prior. Many of these rules reflect the desire to let the data speak for themselves, so that inferences are unaffected by information external to the current data. This has led to variety of priors with names like conventional, default, diffuse, flat, formal, generic, indifference, neutral, non-informative, objective, reference, and vague priors. Prior distributions playing a minimal role in the posterior distribution are called reference prior distributions. One interpretation of letting the data speak for themselves is to use classical techniques. Maximum likelihood estimates are rationalizable in a Bayesian framework by appropriate choice of prior distribution, specifically a uniform prior. There are many ways of defining a non-informative prior. The main objective is to give as little subjective information as possible. So, usually a prior distribution with a large value for the variance is used. Another way of including the minimal prior information is to find estimates of the parameters of the prior distribution, using the data. This last approach is called the empirical Bayes method, but often there is a relationship between those two approaches non-informative and empirical Bayes. A commonly used reference prior in Bayesian analysis is Jeffreys prior (See Jeffreys (1946)). This choice is based on considering one-to-one transformations of the parameter h( θ). Jeffreys general principle is that any rule for determining the prior density p( θ) should yield an equivalent result if applied to the transformed parameter. This non-informative prior is obtained by applying Jeffreys rule, which is to take the prior density to be proportional to the square root of the determinant of the Fisher information matrix. This prior exhibits many nice features that make it an attractive reference prior. One such property is parametrization invariance. Although Jeffreys rule has many desirable properties, it should be used with caution. In most cases, Jeffreys prior is technically not a probability distribution, since the density function does not have a finite integral over the parameter space. It is then termed an improper prior. It is often the case that Bayesian inference based on improper priors returns proper posterior distributions which then turn out to be numerically equivalent to the results of classical inference. Problems related to the use of improper prior distributions can be overcome by assigning prior distributions that are as uniform as possible but still remain probability distributions. The use of uniform prior distributions to represent uncertainty clearly assumes that

231 5.4. The Bayesian approach 205 equally probable is an adequate representation of lack of information. Theoretically, a prior distribution could be included for all the parameters that are unknown in a model, so that any model could be represented in a Bayesian way. However, this often leads to intractable problems (mainly integrals without solution). So the main limitation of Bayesian theory is the difficulty, and in many cases the impossibility, of solving the required equations analytically. In the last decade many simulation techniques have been developed in order to solve this problem and to obtain estimates of the posterior distribution. These techniques were turning points for the Bayesian theory, making it possible to apply many of its models. On one hand, the use of a final and closed formula for a solution is, generally speaking, more satisfactory than the use of an approximation through simulation. On the other hand, simulation gives a larger range of models for which solutions (or at least good approximations) can be obtained Iterative simulation methods In order to illustrate the simulation philosophy, suppose that the posterior of a specific parameter θ is needed. If an analytical solution was available, a formula would be derived, where the observed data and known parameters would be included, defining a final result. But, depending on the model, this solution will not be possible. In such cases an approximation for the posterior distribution of θ is needed. One way of finding this approximation is by simulation, that substitutes the posterior distribution by a large sample of θ based on the characteristics of the model. With this large sample of θ many summary statistics could be calculated, like the mean, variance or histogram, extracting all the information needed from this sample of the posterior distribution. There are a number of ways of simulating and in all of them some checking should be carried out to guarantee that the simulation set is really representative for the required distribution. For instance, it must be checked whether the simulation is mixing well or, in other words, if the simulation procedure is visiting all the possible values for θ. It should be also considered how large the sample should be, and whether the initial point where the simulation starts does not play a big role. Among many other issues, the moment when convergence to the true distribution of θ is achieved should also be monitored.

232 206 Chapter 5 - Approximation techniques for sums of r.v. s The most popular type of simulation in Bayesian theory are the Markov Chain Monte Carlo (MCMC) methods. This class of simulation models has been used in a large number and wide range of applications, and has been found to be very powerful. The essence of the MCMC method is that by sampling from specific simple distributions (derived from the combination of the likelihood and prior distributions), a sample from the posterior distribution will be obtained in an asymptotic way. Iterative simulation methods, particularly the Gibbs sampler and the Metropolis Hastings algorithm are powerful statistical tools that facilitate computation in a variety of complex models. Though these two algorithms are commonly presented as useful yet distinct instruments for simulating joint posteriors, this distinction is rather artificial - indeed, one can regard the Gibbs sampler as a special case of the Metropolis-Hastings algorithm where jumps along the complete conditional distributions are accepted with probability one. In conditionally conjugate models, the Gibbs sampler is typically the algorithm of choice (since the complete posterior conditionals are easily sampled). The general strategy with iterative methods is to follow the steps of the algorithms to generate a series of draws (sometimes called a parameter chain), say θ 0, θ 1, θ 2,... that converge in distribution to some target density - in our case, the posterior f(θ Y ). The algorithms are constructed so that the posterior f(θ Y ) is the unique stationary distribution of the parameter chain. Once convergence to the target density is achieved we can use these draws in the same way as with direct Monte Carlo integration to calculate posterior means, posterior standard deviations, and so on. In practice, we take care to diagnose that the parameter chain has approached convergence to the target density, to discard the initial set of the preconvergence draws (often called the burn-in period), and then to use the post-convergence sample to calculate the desired quantities. Unlike the non-iterative methods discussed previously, the post-convergence draws we obtain using these iterative methods will prove to be correlated, as the distribution of, say, θ t depends on the last parameter sampled in the chain, θ t 1. If the correlation among the draws is severe, it may prove to be difficult to traverse the entire parameter space, and the numerical standard errors associated with the point estimates can be quite large. When the simulations are highly correlated, and our chain makes only small local movements from iteration to iteration, we refer to this as slow mixing of

233 5.4. The Bayesian approach 207 the parameter chain. One can find an excellent overview and a detailed discussion of examples of MCMC algorithms in, for example, Gilks et al. (1996). Here we will describe Gibbs Sampling (GS), a special case of Metropolis-Hastings algorithms, which is becoming increasingly popular in the statistical community. GS is an iterative method that produces a Markov Chain, that is a sequence of values { θ (0), θ (1), θ (2),...} such that θ (i+1) is sampled from a distribution that depends on the current state i of the chain. The algorithm works as follows. Let θ (0) = {θ (0) 1,..., θ(0) k } be a vector of initial values of θ and suppose that the conditional distributions of θ i (θ 1,..., θ i 1, θ i+1,..., θ k, Y ) are known for each i. The first value in the chain is simulated as follows: θ (1) 1 is sampled from the conditional distribution of θ 1 (θ (1) 2,..., θ(1) k, Y ); θ (1) 2 is sampled from the conditional distribution of θ 2 (θ (1) 1, θ(1) 3,..., θ(1) k, Y ); θ (1) k is sampled from the conditional distribution of θ k (θ (1) 1, θ(1) 2,..., θ(1) k 1, Y ); Then θ (0) is replaced by θ (1) and the simulation is repeated to generate θ (2), and so forth. In general, the i-th value in the chain is generated by simulating from the distribution of θ conditional on the value previously generated θ (i 1). After an initial long chain, called burn-in, of say b iterations, the values { θ (b+1), θ (b+2), θ (b+3),...} will be approximately a sample from the posterior distribution of θ, from which empirical estimates of the posterior means and any other function of the parameters can be computed. Critical issues for this method are the choice of the starting value θ (0), the length of the burn-in and the selection of a stopping rule. The program WinBugs provides an implementation of GS suitable for problems in which the likelihood function satisfies certain factorization properties Bayesian model set-up In this subsection we explain how to set up the relevant Bayesian models and draw samples from posterior distributions for parameters θ and future observables Ỹ. We show how simple simulation methods can be used to draw samples from posterior and predictive distributions, automatically incorporating

234 208 Chapter 5 - Approximation techniques for sums of r.v. s uncertainty in the model parameters, and draw samples for posterior predictive checks. The simplest and most widely used version of this model is the normal linear model, in which the distribution of the response variable Y given the regression matrix X is normal with mean a linear function of X: E[Y i β, X] = β 1 x i1 + + β k x ik, for i = 1,..., n. We further restrict to the case of ordinary linear regression, in which the conditional variances are equal, Var[Y i θ, X] = σ 2 for all i, and the observations are conditionally independent given θ, X. The parameter vector is then θ = (β 1,..., β k, σ 2 ). Under a standard non-informative prior distribution, the Bayesian estimates and standard errors coincide with the classical results. In the simplest case, called ordinary linear regression, the observation errors are independent and have equal variance. In vector notation given by Y β, σ 2, X N n (X β, σ 2 I), where I is the n n identity matrix. In the normal regression model, a convenient non-informative prior distribution is uniform on ( β, log σ) or, equivalently, p( β, σ 2 X) σ 2 When there are many data points and only a few parameters, the noninformative prior distribution is useful it gives acceptable results and takes less effort than specifying prior knowledge in probabilistic form. For a small sample size or a large number of parameters, the likelihood is less sharply peaked, and so prior distributions are more important. We determine first the posterior distribution for β, conditional on σ 2, and then the marginal posterior distribution for σ 2. That is, we factor the joint posterior distribution for β and σ 2 as p( β, σ 2 Y ) = p( β σ 2, Y )p(σ 2 Y ). 1. Conditional posterior distribution of β given σ 2 β σ 2, Y N( ˆ β, V β σ 2 ), with and ˆ β = (X X) 1 X Y V β = (X X) 1

235 5.5. Applications in claims reserving Marginal posterior distribution of σ 2 σ 2 Y Inv χ 2 (n k, s 2 ), where s 2 = 1 n k ( Y X ˆ β) ( Y X ˆ β). The marginal posterior distribution of β y, averaging over σ 2, is multivariate t with n k degrees of freedom, but we rarely use this fact in practice when drawing inferences by simulation, since to characterize the joint posterior distribution we can draw simulations of σ 2 and then β σ 2. The standard non-bayesian estimates of β and σ 2 are ˆ β and s 2, respectively, as just defined. The classical standard error estimate for β is obtained by setting σ 2 = s 2. It is easy to draw samples from the posterior distribution: Compute first ˆ β, V β and s 2 and draw then σ 2 from the scaled inverse-χ 2 distribution and β from the multivariate normal distribution. The posterior predictive distribution of unobserved data, p( Ỹ Y ), has two components of uncertainty: 1. The fundamental variability of the model, represented by the variance σ 2 in Y, and 2. The posterior uncertainty in β and σ 2 due to the finite sample size of Y. As the sample size n, the variance due to posterior uncertainty in ( β, σ 2 ) decreases to zero, but the predictive uncertainty remains. 5.5 Applications in claims reserving The comonotonicity approach versus the Bayesian approximations In this subsection we apply a Bayesian model in the context of discounted loss reserves. The outcomes of this approach are compared with the comonotonic approximations for the distribution of the discounted loss reserve when the run-off triangle is modelled by a generalized linear model.

236 210 Chapter 5 - Approximation techniques for sums of r.v. s We realize that the Bayesian posterior predictive distribution is a very general workhorse, which takes into account all sources of uncertainty in the model formulation and is applicable to different statistical domains, whereas the comonotonic approximations originate from a specific actuarial context. We want to illustrate however that the predictive distribution based on the comonotonic bounds provides results that are close to the results obtained via MCMC. The main advantage of the bounds is that several risk measures such as percentiles (VaRs), expected shortfalls (stoploss premiums) and TailVaRs can be calculated easily from it. As illustrated by Verrall (2004) (for GLIMs) and in earlier work by (for instance) de Alba (2002) (for lognormal models) Bayesian techniques are useful in this area as they provide the posterior predictive distribution of the reserve. Bayesian methods for the analysis of GLIMs We consider Bayesian methods for the analysis of generalized linear models, which provide a general framework for cases in which normality and linearity are not viable assumptions. These cases point out the major computational bottleneck of Bayesian methods: when the assumptions of normality and/or linearity are removed, usually the posterior distribution cannot be computed in closed form. We will discuss some computational methods to approximate this distribution. Generalized linear models provide a unified framework to encompass several situations which are not adequately described by the assumptions of normality of the data and linearity in the parameters. As described in Chapter 4 (Section 4.3.3), the features of a GLIM are the fact that the distribution of Y θ ( θ is used to denote the parameter vector) belongs to the exponential family, and that a transformation of the expectation of the data, g( µ), is a linear function of the linear predictor R β. The parameter vector is made up of β and of the dispersion parameter φ. Classical analyses of generalized linear models allow for the possibility of variation beyond that of the assumed sampling distribution, called overdispersion. A prior distribution can be placed on the dispersion parameter, and any prior information about p( β, φ) can be described conditional on the dispersion parameter; that is, p( β, φ) = p(φ)p( β φ). The classical analysis of generalized linear models is obtained if a noninformative or flat prior distribution is assumed for β. The posterior mode

237 5.5. Applications in claims reserving 211 corresponding to a noninformative uniform prior density is the maximum likelihood estimate for the parameter β, which can be obtained using iterative weighted linear regression. The problem with a Bayesian analysis of GLIMs is that, in general, the posterior distribution of β cannot be calculated exactly, since the marginal density of the data p( Y ) = p( Y θ)p( θ)d θ (5.32) cannot be evaluated in closed form. Numerical integration techniques can be exploited to approximate (5.32), from which a numerical approximation of the posterior density of β can be found. When numerical integration techniques become infeasible, we are left with two main ways to perform approximate posterior analysis: (i) to provide an asymptotic approximation of the posterior distribution or (ii) to use stochastic methods to generate a sample from the posterior distribution. When the sample size is large enough, posterior analysis can be based on an asymptotic approximation of the posterior distribution by using a normal distribution with some mean and variance. This idea generalizes the asymptotic normal distribution of the maximum likelihood estimates when their exact sampling distribution cannot be derived or it is too difficult to be used. Asymptotic normality of the posterior distribution provides notable computational advantages, since marginal and conditional distributions are still normal, and hence inference on parameters of interest can be easily carried out. However, for relatively small samples, the assumption of asymptotic normality can be inaccurate. For relatively small samples, stochastic methods (or Monte Carlo methods) provide an approximate posterior analysis based on a sample of values generated from the posterior distribution of the parameters. The task reduces to generating a sample from the posterior distribution of the parameters. A numerical illustration Consider now the run-off triangle in Table 5.7, taken from Taylor & Ashe (1983) and used in various other publications on claims reserving. These data are modelled using a gamma GLIM (see expression (4.51) with κ = 2) with logarithmic link function.

238 , , , , , , , , ,299 67, , , ,894 1,183, , , , , , ,507 1,001, ,219 1,016, , , , , ,608 1,108, ,189 1,562, , , , , , , , , , , , , , , , ,631 1,131,398 1,063, ,480 1,061,648 1,443, , , ,014 Table 5.7: Run-off triangle with non-cumulative claim figures. 212 Chapter 5 - Approximation techniques for sums of r.v. s

239 5.5. Applications in claims reserving year of origin development year calendar year Figure 5.1: Weighted residuals for linear predictor in (5.33), together with average lines, which represent the average of the weighted standardized residuals in each period of interest. Note that the average is zero when no observations occur. Although the linear predictor in the Probabilistic Trend Family of models (4.32) is over-parameterized, it provides a flexible modelling structure. For example, one might begin with three parameters (α, β, γ), with one accident period level parameter, one development period trend parameter and one calendar period trend parameter, which equates to a linear predictor with the following form, η ij = α + (j 1)β + (i + j 2)γ. (5.33) Adding more accident, development and calendar period parameters where necessary, allows the structure to be extremely flexible. The weighted residuals for this model (Figure 5.1) indicate that there are major trends in the development period that are not being captured. There also appears to be a level change between accident periods one, two-

240 214 Chapter 5 - Approximation techniques for sums of r.v. s year of origin development year calendar year Figure 5.2: Weighted residuals for linear predictor in (5.34), together with average lines, which represent the average of the weighted standardized residuals in each period of interest. Note that the average is zero when no observations occur. three, four and five. To capture these trends extra development period trend parameters and extra accident period level parameters are required. The new form of the linear predictor is given by η ij = α 1 I () + α 2 I (i=2,3) + α 3 I (i=4) + α 4 I (i>4) + β 1 I (j>1) + β 2 I (j>4) + (j 5)β 3 I (5<j<9) + 3β 3 I (j>8). (5.34) The weighted residuals for this updated model (Figure 5.2) indicate that (5.34) appears to capture the significant levels and trends in the data. We recall from the previous chapter the definition of the discounted IBNR reserve under a generalized linear model and normal logreturn process. S GLIM = t t i=2 j=t+2 i g 1( (R ˆ β)ij ) e Y (i+j t 1),

241 5.5. Applications in claims reserving 215 year S l GLIM S c GLIM Bayesian 2 360, , , , , , , , , ,441,016 1,513,186 1,448, ,913,383 1,977,934 1,919, ,519,292 2,614,564 2,558, ,557,014 3,702,302 3,641, ,573,767 4,770,944 4,727, ,577,925 5,821,804 5,638,301 total 20,949,190 21,988,048 20,360,196 Table 5.8: 95th percentile of the predictive distribution of S GLIM where the returns are modelled by means of a Brownian motion described by the following equation Y (i) = (δ + ς2 )i + ςb(i), 2 where B(i) is the standard Brownian motion, ς is the volatility and δ is a constant force of interest. The discounting process (with δ = 0.08 and ς = 0.11) is incorporated in the WinBugs code for the gamma GLIM. To enable comparisons with the results from the comonotonic bounds, flat priors were used both for the row and column parameters of the linear predictor and for the scale parameter in the gamma model. Table 5.8 contains the results obtained via MCMC simulations with the WinBugs program. A burn-in of iterations was allowed, after which another iterations were performed. The bounds for the discounted loss reserve use the maximum likelihood estimates of the parameters in the linear predictor. To incorporate the error arising from the estimation of these parameters we apply the bootstrap algorithm as explained in Section 4.5. We bootstrapped 1000 times, computed each time (analytically) the 95th percentile of upper and lower bound. Table 5.8 compares the Bayesian 95th percentile and the bootstrapped 95th percentile of the lower and upper bound for the different reserves. The results for the upper and lower bounds in convex order are given in the same table. One can see that the results from the comonotonic bounds are close to the results obtained via MCMC simulation. Thus, at least for

242 216 Chapter 5 - Approximation techniques for sums of r.v. s this example, these bounds provide actuaries with accurate information concerning the predictive distribution of discounted loss reserves The comonotonicity approach versus the asymptotic and moment matching approximations 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 by means of a simple example in the context of loss reserving and propose to solve this problem using the asymptotic approximations introduced in Section 5.3. In the following, we assume that the r.v. s Y ij, i, j = 1,..., t can be expressed as products of a deterministic component and an i.i.d. random component. In particular, we consider the following model Y ij = a ij Y ij, i, j = 1,..., t, (5.35) in which Y ij, i, j = 1,..., t are i.i.d. r.v. s and a ij > 0, i, j = 1,..., t are positive numbers. We will consider in this part the simple lognormal linear model (4.1) ln Y = R β + ɛ, ɛ N(0, σ 2 I), with Y as before the vector of historical claim figures. The accumulated IBNR reserve is given by IBNR reserve = t t a ij Y ij. (5.36) i=2 j=t+2 i We will again incorporate stochastic discounting factors. We let the positive r.v. V k from the i.i.d. sequence {V k, k = 1,..., t 1} denote the present value discounting factor from year k to year k 1 and consider the two sequences {Y ij, i = 2,..., t; j = t + 2 i,..., t} and {V k, k = 1,..., t 1} to be mutually independent. Furthermore, for notational convenience, we introduce the positive r.v. Z k = V 1 V 2 V k, k = 1,..., t 1. Then the discounted IBNR reserve S is given by S = t t a ij Y ij Z i+j t 1. (5.37) i=2 j=t+2 i

243 5.5. Applications in claims reserving 217 Henceforth, we impose that E[S] < +. Approximate values for stoploss premiums and quantiles for S may be obtained by using asymptotic results. In particular, if {Y ij, i = 2,..., t; j = t+2 i,..., t} and {V k, k = 1,..., t 1} satisfy the corresponding conditions under which Theorem 13 or Theorem 14 is valid, then for sufficiently large values of d, we have that π(s, d) t t i=2 j=t+2 i a ij π ( Y Z i+j t 1, d/a ij ). (5.38) Furthermore, if either F Y R α for some 0 < α < +, and F V R, or the conditions of Theorem 14 apply, then for sufficiently large values of p, we have that F 1 S (p) inf s : t t i=2 j=t+2 i F Y Zi+j t 1 (s/a ij ) 1 p. (5.39) As an example, we consider a lognormal linear regression model with chainladder linear predictor to describe the random claims and we use a geometric Brownian motion with drift to represent the stochastic discount factors. We remark that for this specification Theorem 14 applies. Furthermore, for this specification the products Y ij Z i+j t 1, i = 2,..., t; j = t + 2 i,..., t are lognormal and therefore the present value of the IBNR reserve becomes a linear combination of dependent lognormal r.v. s, given by S = t t i=2 j=t+2 i a ij Y ij Z i+j t 1 = t t i=2 j=t+2 i e η ij e ε ij e Y (i+j t 1). (5.40) Notice that this definition is the same as (4.35) for the special case of the lognormal linear model (with σ = σ) and chain-ladder type linear predictor η ij = (R β) ij = α i + β j. In this illustration, we start with a given set of parameters and define the reserve as expressed in (5.40). In a real reserving exercise, one has to build an appropriate statistical model based on the incremental claims in the run-off triangle and to estimate the parameters from this model. Using the same notation as in the previous chapter we have for W ij := Y (i + j t 1) that E[ W ij ] = (δ ς2 )(i + j t 1), Var[ W ij ] = σ 2 Wij = (i + j t 1)ς 2.

244 218 Chapter 5 - Approximation techniques for sums of r.v. s The asymptotic approximations (5.38) and (5.39) become π(s, d) t F 1 S (p) inf t i=2 j=t+2 i Φ s : e η ij+e[ W ij ]+ 1 2 (σ2 Wij +σ 2 ) ( (ηij + E[ W ij] + σ 2 W ij + σ 2 log(d) ) / ( (ηij dφ + E[ W ij ] log(d) ) ) / σ 2 Wij + σ 2 t t i=2 j=t+2 i F LN (s) 1 p, p (0, 1), in which F LN is the cdf of logn (η ij + E[ W ) ij ], σ 2 Wij + σ 2. ) σ 2 Wij + σ 2, d R +, To compute the lognormal moment matching approximations as described in Section 5.2 we need expressions for the mean and variance of S. These are given by E[S] = Var[S] = where σ 2 = t t i=2 j=t+2 i t t e η ij+e[ W ij ]+ 1 2 t t i=2 j=t+2 i k=2 l=t+2 k { σ 2 if i, j = k, l; 0 if i, j k, l. (σ 2 Wij +σ 2 ), ( ) ) e σ2 + η ij +η kl +E[ W ij ]+E[ W kl ] + (σ Wij +σ 2 Wkl ( e ς2 min(i+j t 1,k+l t 1)+σ 2 1 We arbitrarily set σ = 3, δ = 0.07, ς = 0.2 and t = 5 and use the following chain-ladder parameters: α 1 α 2 α 3 α 4 α 5 = , β 1 β 2 β 3 β 4 β 5 = ),

245 5.5. Applications in claims reserving 219 d MC Appr. 1 Appr. 2 Appr. 3 Ndiff. 1 Ndiff. 2 Ndiff % -36.0% -21.9% % -41.0% -24.2% % -48.8% -27.4% % -54.7% -29.5% % -59.5% -31.1% % -63.7% -32.3% % -70.5% -34.1% % -76.1% -35.3% % -86.8% -37.2% % -91.5% -38.2% % % -39.1% % % -39.2% % % -39.1% % % -38.9% % % -38.1% % % -37.2% p MC Appr. 1 Appr. 2 Appr. 3 Ndiff. 1 Ndiff. 2 Ndiff % 44.3% 12.7% % 26.8% -1.7% % 3.8% -16.3% % -12.3% -23.3% % -45.5% -31.2% Table 5.9: Monte Carlo (MC) versus approximate values of stop-loss premiums and quantiles for chain-ladder claim sizes and lognormal present value discounting factors. In Table 5.9 we numerically compare the asymptotic approximations with a Monte Carlo (MC) study based on simulations. Numerical results of the comonotonic and moment matching approximations have also been included. Appr. 1 refers to the asymptotic approximation, Appr. 2 to the convex upper bound and Appr. 3 to the lognormal moment matching approach. Ndiff. refers to the normalized difference defined as MC Appr. MC 100%. The numerical results demonstrate that the asymptotic approximation values generally outperform the comonotonic upper bound and the lognormal moment matching technique. Because the comonotonic lower bound performed remarkably bad, its numerical values were left out of the table.

246 220 Chapter 5 - Approximation techniques for sums of r.v. s 5.6 Proofs Theorem 12 In order to prove the theorem, we first establish the following result from Tang & Tsitsiashvili (2004): Lemma 11. Let F 1, F 2 and G be three d.f. s. Suppose that F i (x) > 0 for any real number x, F i (0)G(0) = 0, i = 1, 2, and G R. If F 1 (x) F 2 (x), then F 1 G(x) F 2 G(x). (5.41) Proof. From the condition F 1 (x) F 2 (x) we know that, for any 0 < ε < 1 and all large x, say x y 0 for some y 0 > 0, (1 ε)f 2 (x) F 1 (x) (1 + ε)f 2 (x). (5.42) It is not difficult to verify that since F i (y 0 ) > 0 for all y 0 > 0, i = 1, 2, we have by the definition of the class R for i = 1, 2, that lim sup x + y0 0 G(x/y)F i (dy) G(x/y)F i (dy) + y 0 lim sup x + lim sup x + = 0 y0 0 G(x/y)F i (dy) G(x/y)F i (dy) + 2y 0 G(x/y 0 )(F i (y 0 ) F i (0)) G(x/2y 0 )F i (2y 0 ) and hence that for i = 1, 2 F i G(x) = y0 0 + y 0 G (x/y) F i (dy) + G (x/y) F i (dy) = G (x/y 0 ) F i (y 0 ) + Substituting (5.42) to the above leads to + y 0 x/y0 0 G (x/y) F i (dy) F i (x/y)g(dy). (1 ε)f 2 G(x) F 1 G(x) (1 + ε)f 2 G(x). Hence, relation (5.41) follows from the arbitrariness of 0 < ε < 1.

247 5.6. Proofs 221 Then, we proceed with the proof of Theorem 12. Proof. Clearly, it holds that [ ] ( ( ( ))) ] Pr a i Z i > x = Pr [Y 1 a1 + Y 2 a Y n 1 an 1 + a n Y n > x. Since F Y L and a n > 0, we have that Pr [a n 1 + a n Y n > x] Pr [a n Y n > x]. Hence, applying Lemma 11 we obtain that Pr [Y n 1 (a n 1 + a n Y n ) > x] P [a n Y n 1 Y n > x]. Repeatedly applying Lemma 11, we finally obtain that ( ( ( ))) ] Pr [Y 1 a1 + Y 2 a Y n 1 an 1 + a n Y n > x Pr [a n Y 1 Y 2 Y n 1 Y n > x]. For the remainder of the proof it suffices to verify that the probabilities Pr [a i Z i > x], i = 1, 2,, n 1, on the right-hand side of (5.20) can be neglected when compared with the probability Pr [a n Z n > x]. Since the class R is closed under product convolution, we have that the d.f. of the product i j=1 Y j belongs to the class R for each i = 1, 2,.... Hence, we verify that for each i = 1, 2,..., n 1, and some 0 < v < 1, [ ] lim sup x + Pr Pr lim sup x + = = 0 a i i j=1 Y j > x [ ] a n n j=1 Y j > x [ ] a i i j=1 Y j > x Pr [ Pr a i i j=1 Y j > vx, an 1 Pr [ an ai n j=i+1 Y j > 1/v ] n a i j=i+1 Y j > 1/v [ ] Pr a i i j=1 ] Y j > x lim sup [ ] x + Pr a i i j=1 Y j > vx. This proves that (5.20) holds.

248 222 Chapter 5 - Approximation techniques for sums of r.v. s Theorem 13 To prove the theorem, we first state three lemma s. Lemma 12. Let X and Y be two independent r.v. s, where X is supported on (, + ) with a d.f. F, and Y is strictly positive with a d.f. G. Let V = XY and denote by H the d.f. of V. If F D L and G R, then H D L S and H(x) F (x). Proof. This lemma can easily be proved by Lemma 3.8 and Lemma 3.10 of Tang & Tsitsiashvili (2003). Lemma 13. If F D and G R, then there exists some ε > 0 such that G ( x 1 ε) = o ( F (x) ). Proof. This lemma can be proved by Lemma 3.7 of Tang & Tsitsiashvili (2003). Lemma 14. Let F = F 1 F 2, where F 1 and F 2 are two d.f. s supported on (, + ). If F 1 S, F 2 L, and F 2 (x) = O ( F 1 (x) ), then F S and F (x) F 1 (x) + F 2 (x). Proof. This result can be obtained by fixing γ = 0 in Lemma 3.2 of Tang & Tsitsiashvili (2003). We are now ready to prove Theorem 13. Proof. First we prove (5.21), which says that Pr [(a 1 + X 1 )Y (a n 1 + X n 1 )Y n 1... Y 1 + +(a n + X n )Y n Y n 1... Y 1 > x] Pr [(a 1 + X 1 )Y 1 > x] Pr [(a n 1 + X n 1 )Y n 1... Y 1 > x] +Pr [(a n + X n )Y n Y n 1... Y 1 > x].

249 5.6. Proofs 223 Applying Lemma 12, we have that the product (a n + X n )Y n is subexponentially distributed and Applying Lemma 14, we have that Pr [(a n 1 + X n 1 ) + (a n + X n )Y n > x] Pr [(a n + X n )Y n > x] F (x). (5.43) Pr [(a n 1 + X n 1 ) > x] + Pr [(a n + X n )Y n > x]. Since, by Lemma 13, there exists some ε > 0 such that G ( x 1 ε) = o ( F (x) ), we have that Pr [(a n 1 + X n 1 )Y n 1 + (a n + X n )Y n Y n 1 > x] ( x 1 ε ) + = + Pr [(a n 1 + X n 1 )y + (a n + X n )Y n y > x] dg(y) = 0 x 1 ε 0 x 1 ε 0 x 1 ε [ Pr (a n 1 + X n 1 ) + (a n + X n )Y n > x ] dg(y) + o ( F (x) ) y ( [ Pr (a n 1 + X n 1 ) > x ] [ + Pr (a n + X n )Y n > x ]) dg(y) y y +o ( F (x) ) ( + + ) ( [ = Pr (a n 1 + X n 1 ) > x ] 0 x 1 ε y [ +Pr (a n + X n )Y n > x ]) dg(y) + o ( F (x) ) y = Pr [(a n 1 + X n 1 )Y n 1 > x] + Pr [(a n + X n )Y n Y n 1 > x] + o ( F (x) ) Pr [(a n 1 + X n 1 )Y n 1 > x] + Pr [(a n + X n )Y n Y n 1 > x]. Furthermore, by application of Lemma s 12 and 14, it follows that (a n 1 + X n 1 )Y n 1 + (a n + X n )Y n Y n 1 is subexponentially distributed and that Pr [(a n 1 + X n 1 )Y n 1 + (a n + X n )Y n Y n 1 > x] F (x). Simply repeating the procedure above and observing that (a n 2 + X n 2 )Y n 2 + (a n 1 + X n 1 )Y n 1 Y n 2 + (a n + X n )Y n Y n 1 Y n 2 = [(a n 2 + X n 2 ) + (a n 1 + X n 1 )Y n 1 + (a n + X n )Y n Y n 1 ] Y n 2,

250 224 Chapter 5 - Approximation techniques for sums of r.v. s we obtain that Pr [(a n 2 + X n 2 )Y n 2 + (a n 1 + X n 1 )Y n 1 Y n 2 + +(a n + X n )Y n Y n 1 Y n 2 > x] Pr [(a n 2 + X n 2 )Y n 2 > x] + Pr [(a n 1 + X n 1 )Y n 1 Y n 2 > x] +Pr [(a n + X n )Y n Y n 1 Y n 2 > x]. Hence, repeating the procedure above n 1 times yields the announced result (5.21). The proof of (5.22), can be given completely analogously to the above, since the distribution of a i X i satisfies and is subexponential. Corollary 3 Pr [a i X i > x] = F (x/a i ) F (x) Proof. Using (5.43), one can easily verify that n ] Pr[ (a i + X i )Z i > x lim inf [ x + n 1 ] > 1, Pr (a i + X i )Z i > x and that lim inf x + n ] Pr[ (a ix i )Z i > x [ n 1 ] > 1. Pr (a ix i )Z i > x Hence, we can prove (5.23) and (5.24) by substituting (5.21) and (5.22) into the left-hand-side of (5.23) and (5.24), respectively. Corollary 4 Proof. Given the asymptotic results (5.21) and (5.22), the proof of this corollary follows immediately from a well-known result, which was referred by Cline (1986) to Proposition 3 of Breiman (1965). Theorem 14 In case the conditions 1 and 2 of Theorem 13 are replaced by the conditions 1, 2 and 3 of Theorem 14, the proof of (5.21) can be established completely analogously to the proof of Theorem 13 using the following three

251 5.6. Proofs 225 lemma s, which are the analogs of Lemma 12, Lemma 13 and Lemma 14, respectively: Lemma 15. Let X and Y be two independent lognormally distributed r.v. s with σ Y < σ X. Furthermore, let V = XY and denote by H the d.f. of V. Then V follows a lognormal law and F (x) = o(h(x)). Lemma 16. If both F and G are lognormal laws with σ G < σ F, then there exists some ε > 0 such that G ( x 1 ε) = o ( F (x) ). Lemma 17. Let F = F 1 F 2, where F 1 and F 2 are two lognormal laws. Then F S and F (x) F 1 (x) + F 2 (x). Proof. This is a special case of Corollary 1 of Cline (1986) and moreover is a special case of Lemma 14. We are now ready to proof Theorem 14. Proof. The proof of (5.22) can be given analogously, since the distribution of a i X i is again lognormal with Var[log(a i X i )] = Var[log(X i )] = σ 2 X. Finally, we prove (5.23) and (5.24). By application of Lemma 11 and the same reasoning as in the proof of Theorem 12, we have for each n = 1, 2,..., and some 0 < v < 1 that

252 226 Chapter 5 - Approximation techniques for sums of r.v. s = = n [(a Pr i + X) ] i j=1 Y j > x lim inf x + [ n 1 Pr (a i + X) ] i j=1 Y j > x [ Pr (a n + X) ] n j=1 Y j > x lim inf x + [ n 1 Pr (a i + X) ] i j=1 Y j > x 1 n 1 lim sup i Pr[(a i +X) j=1 Y j>x] x + n 1 lim sup x + n 1 lim sup x + = + > 1. Pr[(a n+x) n j=1 Y j>x] 1 i Pr[(a i +X) j=1 Y j>x] Pr[(a n+x) i j=1 j>vx]pr[ Y n 1 i Pr[X j=1 Y j>x] i Pr[X j=1 j>vx]pr[ Y n j=i+1 Y j>1/v] j=i+1 Y j>1/v] and = lim inf x + lim inf x + n [(a Pr i X) ] i j=1 Y j > x [ n 1 Pr (a i X) ] i j=1 Y j > x [ Pr (a n X) ] n j=1 Y j > x [ n 1 Pr (a i X) ] i j=1 Y j > x 1 n 1 lim sup x + = + > 1. i Pr[(a i X) j=1 Y j>x] Pr[(a nx) n j=1 Y j>x] Hence, we can prove (5.23) and (5.24) by substituting (5.21) and (5.22) into the left-hand-side of (5.23) and (5.24), respectively.

253 Samenvatting in het Nederlands (Summary in Dutch) Inleiding In deze thesis bekijken we de reserveringsproblematiek in de verzekeringswereld van naderbij. Een reserveringsstudie komt in grote lijnen neer op de bepaling van de huidige waarde van de toekomstige schade-uitkeringen. De deskundigheid en nauwkeurigheid waarmee dit onzeker bedrag tot stand komt is dan ook cruciaal voor een maatschappij en haar polishouders. De intrinsieke onzekerheden die hiermee gepaard gaan, mogen bovendien geen excuus zijn om van een sterk wetenschappelijk onderbouwde analyse af te zien. Belangen en prioriteiten kunnen verschillen tussen al diegenen die te maken krijgen met reserveschattingen. Voor het management moet deze schatting betrouwbare informatie verschaffen om de leefbaarheid en de winstgevendheid van de maatschappij te maximaliseren. Voor de controle instantie, die zich bezighoudt met de solvabiliteit, moeten de reserves conservatief bepaald worden om de kans op een faillissement te reduceren. Voor de fiscus moeten de reserves de werkelijke betalingen zo goed mogelijk weergeven. De polishouder ten slotte wil dat de reserves voldoende zijn om verzekerde schadegevallen te kunnen betalen, maar wil niet beboet worden onder de vorm van een te hoge premie voor die garantie. Het voornaamste doel van het reserveringsproces kan eenvoudig als volgt beschreven worden. Vanaf een bepaalde, vooraf overeengekomen, dag is een verzekeraar verantwoordelijk voor alle opgelopen claims. Kosten die dit schadegeval met zich meebrengen worden opgedeeld in twee categorieën: 227

254 228 Samenvatting in het Nederlands (Summary in Dutch) diegene die reeds betaald zijn en diegene die nog niet (volledig) betaald zijn. Het voornaamste doel van het reserveringsproces is nu het schatten van die kosten die nog niet betaald zijn door de maatschappij. De verdeling van mogelijke geaggregeerde onbetaalde schadegevallen kan voorgesteld worden als een kansdichtheidsfunctie. Er is reeds veel geschreven over de statistische verdelingen die geschikt zijn bij de studie van risico s en verzekeringen. In de praktijk kan men niet beschikken over de volledige informatie van de onderliggende verdelingen. Daarom moet men zich dikwijls beroepen op beperkte informatie, zoals bv. schattingen van de eerste momenten van de verdeling. Niet enkel de basisrisicomaten maar ook meer gesofisticeerde maten (zoals scheefheidsmaten, extreme percentielen van de verdeling,... ) die een dieper inzicht in de onderliggende verdeling vereisen, zijn erg van belang. De berekening van de eerste momenten kan gezien worden als een eerste poging om meer te weten te komen over de eigenschappen van een verdeling. Bovendien is de variantie niet de meest geschikte risicomaat om de solvabiliteitsvereisten van een verzekeringsportefeuille te bepalen. Als tweezijdige risicomaat houdt deze zowel rekening met de positieve als met de negatieve tekortkomingen hetgeen tot onderschatting van de reserve zal leiden in geval van een scheve verdeling. Bovendien benadrukt deze maat niet de staarteigenschappen van de verdeling. In dit geval lijkt het meer geschikt de VaR (het p-de kwantiel) te gebruiken of zelfs de TVaR (hetgeen in essentie neerkomt op een gemiddelde van alle kwantielen boven een voorgedefinieerd niveau p). Ook risicomaten gebaseerd op stop-loss premies (bv. de verwachte shortfall) kunnen in deze context aangewend worden. Het verkrijgen van de verdeling waarvan dan allerlei maten kunnen berekend worden is het uiteindelijke doel. Deze trends worden ook aangehaald in de huidige bank- en verzekeringsvoorschriften (Basel 2 en Solvency 2) die de risico-gebaseerde benadering in ALM benadrukken. Dit vereist een nieuwe methodologische aanpak die toelaat meer gesofisticeerde informatie over de onderliggende risico s te verkrijgen. In de huidige actuariële wetenschappelijke literatuur vinden we weinig terug over de geschikte berekeningsmethode van de verdeling van reserveuitkomsten. Verscheidene methoden bestaan om efficiënt de verdeling van sommen van onafhankelijke risico s te benaderen (zoals Panjer s recursie, convolutie,...). Als bovendien het aantal risico s in een portefeuille groot genoeg is, kan men gebruik maken van de Centrale Limiet Stelling om de geaggregeerde claims via de normale verdeling te benaderen. Zelfs indien deze onafhankelijkheidsveronderstelling niet voldaan is (wanneer bv. de

255 Inleiding 229 aanname van onafhankelijkheid op basis van statistische testen verworpen wordt) wordt deze benadering veel gebruikt in de praktijk omwille van de mathematische eenvoud. In een aantal praktische toepassingen wordt deze onafhankelijkheidsveronderstelling nochtans geschonden, hetgeen tot een significante onderschatting van het risico van de portefeuille kan leiden. Dit is onder meer het geval wanneer het actuarieel technische risico gecombineerd wordt met het financiële investeringsrisico. In tegenstelling tot in het bankwezen, is het concept van stochastische interestvoeten pas recent aan de oppervlakte gekomen in het verzekeringswezen. Traditioneel vertrouwen actuarissen op deterministische interestvoeten. Een dergelijke vereenvoudiging laat toe efficiënte risicomaten (zoals het gemiddelde, de standaarddeviatie, bovenkwantielen,...) van financiële contracten te bepalen. Door een hoge onzekerheid over toekomstige investeringsresultaten worden actuarissen nochtans gedwongen conservatieve aannames te doen om verzekeringspremies en wiskundige reserves te berekenen. Dit heeft tot gevolg dat de diversificatie-effecten van returns in verschillende investeringsperioden niet in rekening kunnen worden gebracht. Hiermee bedoelen we dat slechte investeringsresultaten in bepaalde perioden gewoonlijk gecompenseerd worden door zeer goede resultaten in andere perioden. Deze bijkomende kosten worden ofwel naar de verzekerden doorgerekend, die hogere premies moeten betalen, ofwel naar de aandeelhouders, die meer economisch kapitaal moeten voorzien. Het belang van de introductie van modellen met stochastische interestvoeten is daarom goed begrepen in de actuariële wereld. Ook de laatste bank- en verzekeringsvoorschriften (Basel 2, Solvency 2) onderstrepen dit belang. Deze voorschriften leggen de nadruk op de risico-gebaseerde benadering om economisch kapitaal te bepalen. Het projecteren van cash flows met stochastische returns is ook belangrijk in de prijsbepaling van verzekeringstoepassingen zoals de embedded value (de huidige waarde van cash flows voortgebracht door de van kracht zijnde polissen) en de appraisal value (de huidige waarde van cash flows voortgebracht door de van kracht zijnde polissen en door polissen die in de toekomst zullen onderschreven worden). Een wiskundige beschrijving van het aangehaalde probleem kan als volgt samengevat worden. Zij X i (i = 1,..., n) een stochastisch bedrag dat betaald moet worden op tijdstip t i en zij V i de verdisconteringsfactor over de periode [0, t i ]. We beschouwen dan de huidige waarde van toekomstige

256 230 Samenvatting in het Nederlands (Summary in Dutch) betalingen, die geschreven kan worden als een scalair produkt van de vorm S = X i V i. (N.1) De stochastische vector X = (X 1, X 2,..., X n ) kan bv. het verzekeringsof kredietrisico weergeven, terwijl de vector V = (V 1, V 2,..., V n ) het financiële/investeringsrisico weergeeft. In het algemeen veronderstellen we dat deze vectoren onderling onafhankelijk zijn. In praktische toepassingen kan deze onafhankelijkheidsaanname wel eens geschonden zijn bv. door een inflatiefactor met een sterke invloed op betalings- en investeringsresultaten. Men kan dit probleem echter aan pakken door sommen van de volgende vorm te beschouwen S = X i Ṽ i, waarbij Xi = X i /Z i en Ṽi = V i Z i de aangepaste waarden zijn uitgedrukt in reële termen (Z i is een inflatiefactor over de periode [0, t i ]). Daarom is de onafhankelijkheidsveronderstelling tussen het verzekeringsrisico en het financiële risico in vele gevallen realistisch en kan zij efficiënt aangewend worden om verschillende grootheden te verkrijgen die het risico in financiële instituten beschrijft (bv. verdisconteerde claims of de embedded/appraisal waarde van een maatschappij). Deze verdelingsfuncties zijn typisch complex en niet voor de hand liggend omwille van twee belangrijke redenen. Eerst en vooral behoort de verdeling van een som van stochastische veranderlijken met marginale verdelingen in dezelfde verdelingsklasse in het algemeen niet tot deze verdelingsklasse. Ten tweede verhindert de stochastische afhankelijkheid tussen de elementen in de som het gebruik van convolutie en maakt het geheel aanzienlijk ingewikkelder. Bijgevolg worden benaderingsmethoden om functies van sommen van afhankelijke variabelen te berekenen noodzakelijk. In vele gevallen kan men natuurlijk Monte Carlo simulatie gebruiken om empirische verdelingsfuncties te verkrijgen. Dit is echter typisch een tijdrovende benaderingsmethode, in het bijzonder indien men staartkansen wenst te benaderen hetgeen een groot aantal simulaties vereist. Daarom moet men opzoek gaan naar nieuwe alternatieve methoden. In deze thesis bestuderen en evalueren we de meest frequent gebruikte benaderingstechnieken voor verzekeringstoepassingen.

257 Inleiding 231 Het centrale idee in dit werk is het comonotoniciteitsconcept. We stellen voor het hierboven uiteengezette probleem op te lossen door onderen bovengrenzen voor de som van afhankelijke variabelen te berekenen gebruikmakend van de beschikbare informatie. Deze grenzen zijn gebaseerd op een algemene techniek voor het berekenen van het onder- en bovengrenzen van stop-loss premies van een som van afhankelijke variabelen, zoals uiteengezet in Kaas et al. (2000). De eerste benadering voor de verdelingsfunctie van de verdisconteerde reserve wordt afgeleid door de afhankelijkheidstructuur tussen de betrokken stochastische veranderlijken te benaderen door een comonotone afhankelijkheidsstructuur. Op deze manier wordt het meerdimensionale probleem gereduceerd tot een tweedimensionaal probleem hetgeen opgelost kan worden door te conditioneren en gebruik te maken van eenvoudige numerieke technieken. Deze benadering is plausibel in actuariële toepassingen aangezien het leidt tot voorzichtige en conservatieve waarden van de reserves en solvabiliteitsmarges. Indien de onderliggende afhankelijkheidsstructuur sterk genoeg is, geeft deze bovengrens in convexe orde bevredigende resultaten. De tweede benadering, die afgeleid wordt door voorwaardelijke verwachtingswaarden te beschouwen, neemt een deel van de afhankelijkheidsstructuur in beschouwing. Deze benedengrens in convexe orde is zeer nuttig om de kwaliteit van de bovengrens als benadering te evalueren en kan ook gebruikt worden als een benadering van de onderliggende verdeling. Alhoewel deze keuze niet (actuarieel) voorzichtig is, doet de relatieve fout van deze benadering significant beter dan de relatieve fout van de bovengrens. Daarom zal de ondergrens verkozen worden in toepassingen waarbij een hoge nauwkeurigheid van de toegepaste benaderingen vereist wordt (zoals het prijzen van exotische opties of strategische portefeuille selectie problemen). Deze thesis is als volgt ingedeeld. Het eerste hoofdstuk herhaalt de basis van de actuariële risicotheorie. We definiëren enkele veel gebruikte afhankelijkheidsmaten en de belangrijkste risico-orderelaties voor actuariële toepassingen. We introduceren verder verscheidene welbekende risicomaten en de relaties die onderling gelden. Verder geeft het eerste hoofdstuk een theoretische achtergrond voor de concepten van comonotoniciteit en herhaalt het de belangrijkste eigen-

258 232 Samenvatting in het Nederlands (Summary in Dutch) schappen van comonotone risico s. In Hoofdstuk 2 herhalen we hoe de convexe grenzen kunnen afgeleid worden en illustreren we de theoretische resultaten aan de hand van een toepassing met betrekking tot verdisconteerde reserves. Het voordeel van te werken met een som van comonotone variabelen ligt in de eenvoudige berekening van de betrokken verdeling. In het bijzonder is deze techniek zeer nuttig om betrouwbare schattingen te verkrijgen van bovenkwantielen en stoploss premies. In praktische toepassingen is de bovengrens enkel nuttig indien de afhankelijkheid tussen opeenvolgende termen van de som sterk genoeg is. Maar zelfs dan zijn deze benaderingen voor stop-loss premies niet bevredigend. In dit hoofdstuk stellen we een aantal technieken voor om meer efficiënte bovengrenzen voor stop-loss premies te bepalen. We gebruiken hiervoor enerzijds de conditioneringsmethode zoals in Curran (1994) en in Rogers & Shi (1995) en anderzijds de traditionele onder- en bovengrenzen voor stop-loss premies van sommen van afhankelijke stochastische veranderlijken. We tonen ook hoe deze resultaten kunnen toegepast worden in het speciale geval van lognormale stochastische veranderlijken. Dergelijke sommen komt men vaak in de praktijk tegen, zowel in de actuariële als in de financiële wereld. We leiden comonotone benaderingen af voor het scalaire produkt van stochastische vectoren van de vorm (N.1). Een algemene procedure voor het berekenen van accurate schattingen van kwantielen en stop-loss premies wordt uiteengezet. We bestuderen de verdelingsfunctie van de huidige waarde van een serie van stochastische betalingen in een stochastisch financiële omgeving beschreven door een lognormaal verdisconteringsproces. Dergelijke verdelingen komen frequent voor in een breed spectrum van verzekerings- en financiële toepassingen. We verkrijgen nauwkeurige benaderingen door onder- en bovengrenzen in convexe orde te ontwikkelingen voor dergelijke huidige-waarde-functies. We beschouwen verscheidene toepassingen voor verdisconteerde schadeprocessen onder de Black & Scholes setting. In het bijzonder analyseren we in detail de gevallen waarbij de stochastische veranderlijken X i verzekeringsschades voorstellen gemodelleerd door lognormale, normale (meer algemeen elliptische) en gamma of invers Gaussische (meer algemeen gematigd stabiele) verdelingen. Door middel van een reeks numerieke illustraties tonen we dat de methode zeer nauwkeurige en eenvoudig te verkrijgen benaderingen verschaft voor

259 Inleiding 233 verdelingsfuncties van stochastische veranderlijken van de vorm (N.1). In Hoofdstuk 3 en 4 passen we de verkregen resultaten toe op twee belangrijke reserveringsproblemen in het verzekeringswezen en illustreren we de benaderingen zowel numeriek als grafisch. In Hoofdstuk 3 beschouwen we een belangrijke toepassing in het domein van de levensverzekeringen. We trachten conservatieve schattingen te bekomen voor kwantielen en stop-loss premies van een annuïteit en een ganse portefeuille van annuïteiten. Gelijkaardige technieken kunnen aangewend worden om schattingen te verkrijgen van meer algemene verzekeringsprodukten in de sector leven. Onze techniek laat toe personal finance problemen zeer nauwkeurig op te lossen. Het geval van een portefeuille van annuïteiten is reeds uitgebreid onderzocht in de wetenschappelijke literatuur, maar enkel in het grensgeval voor homogene portefeuilles, wanneer het sterfterisico volledig gediversifieerd is. De toepasbaarheid van deze resultaten in de verzekeringspraktijk kan echter in vraag gesteld worden: in het bijzonder hier, aangezien een typische portefeuille niet genoeg polissen bevat om te spreken over volledige diversificatie. Daarom stellen we voor het aantal actieve polissen in de opeenvolgende jaren te benaderen gebruikmakend van een normal power verdeling en de huidige waarde van de toekomstige uitkeringen te modelleren als een scalair produkt van onderling onafhankelijke vectoren. Hoofdstuk 4 focust op het schadereserveringsprobleem. Het correct schatten van het bedrag dat een maatschappij opzij moet zetten om tegemoet te komen aan de verplichtingen (schadegevallen) die zich in de toekomst voordoen, is een belangrijke taak voor verzekeringsmaatschappijen om een correct beeld van haar verplichtingen te krijgen. De historische data die nodig zijn om schattingen te bekomen voor toekomstige betalingen worden meestal weergegeven als incrementele betalingen in driehoek-vorm. De bedoeling is deze schadedriehoek te vervolledigen tot een vierkant en eventueel tot een rechthoek indien schattingen nodig zijn die behoren tot afwikkelingsjaren waarvan geen data in de driehoek opgenomen zijn. Hiervoor kan de actuaris gebruik maken van een aantal technieken. De intrinsieke onzekerheid wordt beschreven door de verdeling van mogelijke uitkomsten en men zoekt steeds naar de beste schatting van de reserve. Schadereservering heeft te maken met de bepaling van de onzekere huidige

260 234 Samenvatting in het Nederlands (Summary in Dutch) waarde van een ongekend bedrag van toekomstige betalingen. Aangezien dit bedrag zeer belangrijk is voor een verzekeringsmaatschappij en haar polishouders zijn de intrinsieke onzekerheden geen excuus om een wetenschappelijke analyse links te laten liggen. Opdat de reserveschatting werkelijk de beste schatting van de actuaris zou weergeven, moet zowel de bepaling van de verwachte waarde van niet-betaalde schadegevallen alsook de geschikte verdisconteringsvoet de beste schatting van de actuaris weergeven (hiermee bedoelen we dat deze niet opgelegd moet worden door anderen of door de wetgeving). Aangezien de reserve een provisie is voor toekomstige betalingen van niet-afgehandelde schadegevallen, geloven we dat de geschatte schadereserve de tijdswaarde van geld moet weergeven. In vele situaties is de verdisconteerde reserve nuttig, bv. in een dynamisch financiële analyse, winstbepaling en het prijs zetten, risicokapitaal, schadeportefeuille transfers,.... Idealiter zou de verdisconteerde reserve ook aanvaardbaar moeten zijn voor rapportering. De huidige wetgeving laat het echter meestal niet toe. Niet-verdisconteerde reserves bevatten in feite een zekere risicomarge afhankelijk van het niveau van de interestvoet. In dit hoofdstuk beschouwen we de verdisconteerde IBNR reserve en leggen we een impliciete marge op gebaseerd op een risicomaat van de verdeling van de totale verdisconteerde reserve. We modelleren de schadebetalingen gebruikmakend van lognormale lineaire modellen, loglineaire locatie-schaal modellen en veralgemeende lineaire modellen en leiden accurate comonotone benaderingen af voor de verdisconteerde reserve. De bootstraptechniek heeft bewezen zeer nuttig te zijn in vele statistische toepassingen en kan in het bijzonder interessant zijn om de variabiliteit van de schadevoorspellingen te bepalen en bovendien om bovengrenzen te construeren met een geschikt betrouwbaarheidsniveau. Haar populariteit is te wijten aan een combinatie van rekenkracht en theoretische ontwikkeling. Een voordeel van de bootstrapbenadering is dat de techniek op elke dataset kan toegepast worden zonder een onderliggende verdeling te veronderstellen. Bovendien kan de meeste software omgaan met zeer grote aantallen bootstrapiteraties. In Hoofstuk 5 leiden we andere methoden af om benaderingen te verkrijgen voor S. We herhalen en evalueren ook kort enkele reeds bestaande technieken. In de eerse sectie van dit hoofdstuk, herhalen we twee bekende moment gebaseerde benaderingen: de lognormale en de inverse gamma benadering. Mensen uit de praktijk gebruiken vaak een moment gebaseerde

261 Inleiding 235 lognormale benadering voor de verdeling van S. Deze benaderingen zijn zo gekozen dat de eerste twee momenten samenvallen met de corresponderende momenten van S. Alhoewel de comonotone benaderingen in convexe orde bewezen hebben goede benaderingen te zijn in geval de onderliggende variabiliteit klein is, doen ze het een stuk minder wanneer de variantie toeneemt. Daarom kijken we hier naar benaderingen voor functies van sommen van afhankelijke variabelen door gebruik te maken van asymptotische resultaten. Alhoewel asymptotische resultaten geldig zijn op oneindig, kunnen ze ook nuttig zijn als benaderingen in de buurt van oneindig. We leiden enkele asymptotische resultaten af voor de staartkans van een som van zwaarstaartige afhankelijke variabelen. Sedert 1990 kent het toegepaste Bayesiaanse onderzoek een enorme groei bij de statistici. Deze explosie heeft weinig te maken gehad met de groeiende interesse van statistici en econometrici voor de theoretische basis van de Bayesiaanse analyse of met een plotselinge bewustwording van de voordelen van de Bayesiaanse aanpak ten opzichte van de frequentistische methoden, maar heeft vooral een pragmatische grondslag. De ontwikkeling van krachtige rekeninstrumenten (en de bewustwording dat bestaande statistische tools nuttig kunnen zijn om Bayesiaanse modellen te fitten) heeft een groot aantal onderzoekers aangetrokken om de Bayesiaanse benadering te gebruiken in de praktijk. Het gebruik van dergelijke methoden laat onderzoekers toe ingewikkelde statistische modellen te schatten, die gebruikmakend van standaard frequentistische technieken redelijk moeilijk zijn, al dan niet onmogelijk. In deze sectie schetsen we vrij algemeen de basiselementen van de Bayesiaanse berekening. Bayesiaanse gevolgtrekking komt neer op het fitten van een kansmodel op een dataset en het resultaat samenvatten door middel van een kansverdeling op de modelparameters en op niet-waargenomen grootheden zoals predicties voor nieuwe observaties. Er bestaan eenvoudige simulatiemethoden om een steekproef te nemen van de posterior- en predictieverdeling, waarbij onzekerheid in de modelparameters automatisch meegenomen wordt. Een voordeel van de Bayesiaanse aanpak is dat we steeds, gebruikmakend van simulatie, de posterior predictieverdeling kunnen berekenen zodat we niet veel energie moeten steken in het schatten van de steekproefverdeling van teststatistieken. Uiteindelijk vergelijken we deze benaderingen met de comonotone benaderingen uit het vorig hoofdstuk in de context van de schadereserveringsproblematiek. In geval de onderliggende variantie van het statistische

262 236 Samenvatting in het Nederlands (Summary in Dutch) en financiële gedeelte van de verdisconteerde IBNR reserve groter wordt, presteren de comonotone benaderingen slecht. We illustreren dit aan de hand van een eenvoudig voorbeeld en stellen de asymptotische resultaten uit het vorig hoofdstuk als een alternatief voor. We vergelijken al deze resultaten ook met de lognormale moment gebaseerde benaderingen. Tenslotte bekijken we ook de verdeling van de verdisconteerde reserve wanneer we de data in de schadedriehoek modelleren met behulp van een veralgemeend lineair model en vergelijken de resultaten van de comonotone benaderingen met de Bayesiaanse benaderingen.

263 Bibliography [1] Ahcan A., Darkiewicz G., Goovaerts M.J. & Hoedemakers T. (2005). Computation of convex bounds for present value functions of random payments, Journal of Computational and Applied Mathematics, to appear. [2] Albrecher H., Dhaene J., Goovaerts M.J. & Schoutens W. (2005). Static hedging of Asian options under Lévy models: The comonotonicity approach, The Journal of Derivatives, 12(3), [3] Antonio K., Beirlant J. & Hoedemakers T. (2005). Discussion of A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving by Richard Verrall, North American Actuarial Journal, to be published. [4] Arnold L. (1974). Stochastic Differential Equations: Theory and Applications, Wiley, New York. [5] Artzner P. (1999). Application of coherent risk measures to capital requirements in insurance, North American Actuarial Journal, 3(2), [6] Artzner P., Delbaen F., Eber J.M. & Heath D. (1999). Coherent measures of risk, Mathematical Finance, 9, [7] Barnett G. & Zehnwirth B. (2000). Best estimates for reserves, Proceedings of the Casualty Actuarial Society, 87(2), [8] Beekman J.A. & Fuelling C.P. (1990). Interest and mortality randomness in some annuities, Insurance: Mathematics & Economics, 9(2-3),

264 238 Bibliography [9] Beekman J.A. & Fuelling C.P. (1991). Extra randomness in certain annuity models, Insurance: Mathematics & Economics, 10(4), [10] Beekman J.A. & Fuelling C.P. (1993). One approach to dual randomness in life insurance, Scandinavian Actuarial Journal, 76(2), [11] Bellhouse D.R. & Panjer H.H. (1981). Stochastic modeling of interest rates with applications to life contingencies - Part II, Journal of Risk and Insurance, 48(4), [12] Beirlant J., Goegebeur Y., Segers J. & Teugels J. (2004). Statistics of Extremes: Theory and Applications, Wiley, New York. [13] Bingham N.H., Goldie C.M. & Teugels J.L. (1987). Regular Variation, Cambridge University Press, Cambridge. [14] Black F. & Scholes M. (1973). The pricing of options and corporate liabilities, Journal of Political Economy, 81, [15] Blum K.A. & Otto D.J. (1998). Best estimate loss reserving: an actuarial perspective, Casualty Actuarial Society Forum Fall 1998, [16] Boyle P.P. (1976). Rates of return as random variables, Journal of Risk and Insurance, 43(4), [17] Bowers N.L., Gerber H.U., Hickman J.C., Jones D.A. & Nesbitt C.J. (1986). Actuarial Mathematics, Schaumburg, Ill.: Society of Actuaries. [18] Breiman L. (1965). On some limit theorems similar to the arc-sin law, Theory of Probability and Its Applications, 10(2), [19] Bühlmann H., Gagliardi B., Gerber H.U. & Straub E. (1977). Some inequalities for stop-loss premiums, ASTIN Bulletin 9, [20] Cesari R. & Cremonini D. (2003). Benchmarking, portfolio insurance and technical analysis: a Monte Carlo comparison of dynamic strategies of asset allocation, Journal of Economic Dynamics and Control, 27,

265 Bibliography 239 [21] Christofides S. (1990). Regression models based on log-incremental payments, Claims Reserving Manual, 2, Institute of Actuaries, London. [22] Cline D.B.H. (1986). Convolution tails, product tails and domains of attraction, Probability Theory Related Fields, 72, [23] Cohen A.C. & Whitten B.J. (1988). Parameter estimation in reliability and life span models, Marcel Dekker, Inc., New York. [24] Cordeiro G.M. & McCullagh P. (1991). Bias correction in generalized linear models, Journal of the Royal Statistical Society B, 53(3), [25] Curran M. (1994). Valuing Asian and portfolio options by conditioning on the geometric mean price, Management Science, 40(12), [26] Darkiewicz G., Dhaene J. & Goovaerts M.J. (2005a). Risk measures and dependencies of risks, Brazilian Journal of Probability and Statistics, to appear. [27] Darkiewicz G. (2005b). Value-at-Risk in Insurance and Finance: the Comonotonicity Approach, PhD Thesis, K.U. Leuven, Faculty of Economics and Applied Economics, Leuven. [28] Davison A.C., & Hinkley D.V. (1997) Bootstrap Methods and their Application, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press. [29] De Alba E. (2002). Bayesian estimation of outstanding claims reserves, North American Actuarial Journal, 6(4), [30] Dȩbicka J. (2003). Moments of the cash value of future payment streams arising from life insurance contracts, Insurance: Mathematics & Economics, 33(3), [31] Decamps M., De Schepper A. & Goovaerts M.J. (2004). Pricing exotic options under local volatility, Proceedings of the second International Workshop on Applied Probability (IWAP), Athens.

266 240 Bibliography [32] Deelstra M., Liinev J. & Vanmaele M. (2004). Pricing of arithmetic basket options by conditioning, Insurance: Mathematics & Economics, 34(1), [33] Denuit M. & Dhaene J. (2003) Simple characterizations of comonotonicity and countermonotonicity by extremal correlations, Belgian Actuarial Bulletin, 3, [34] Denuit M. & Dhaene J. (2004) Dependent risks, Encyclopedia of Actuarial Science, Wiley, Vol. I, [35] Devroye L. (1986). Non-Uniform random variate generation, Springer-Verlag, New York. [36] De Vylder F. & Goovaerts M.J. (1979). Proceedings of the first meeting of the contact group Actuarial Sciences, K.U.Leuven, nr 7904B, wettelijk Depot: D/1979/23761/5. [37] De Vylder F. & Goovaerts M.J. (1982). Upper and lower bounds on stop-loss premiums in case of known expectation and variance of the risk variable, Mitt. Verein. Schweiz. Versicherungmath., [38] Dhaene J. (1989). Stochastic interest rates and autoregressive integrated moving average processes, ASTIN Bulletin, 19(2), [39] Dhaene J. (1990). Distributions in life insurance, ASTIN Bulletin, 20(1), [40] Dhaene J., Wang S., Young V. & Goovaerts M.J. (2000). Comonotonicity and maximal stop-loss premiums, Mitteilungen der Schweiz. Aktuarvereinigung, 2000(2), [41] Dhaene J., Denuit M., Goovaerts M.J., Kaas R. & Vyncke D. (2002a). The concept of comonotonicity in actuarial science and finance: Theory, Insurance: Mathematics & Economics, 31(1), [42] Dhaene J., Denuit M., Goovaerts M.J., Kaas R. & Vyncke D. (2002b). The concept of comonotonicity in actuarial science and finance: Applications, Insurance: Mathematics & Economics, 31(2),

267 Bibliography 241 [43] Dhaene J., Goovaerts M.J. & Kaas R. (2003). Economical capital allocation derived from risk measures, North American Actuarial Journal, 7(2), [44] Dhaene J., Vanduffel S., Tang Q., Goovaerts M.J., Kaas R. & Vyncke D. (2004). Solvency capital, risk measures and comonotonicity: a review, Research Report OR 0416, Department of Applied Economics, K.U.Leuven. [45] Dhaene J., Vanduffel S., Goovaerts M.J., Kaas R. & Vyncke D. (2005). Comonotonic approximations for optimal portfolio selection Problems, Journal of Risk and Insurance, 72(2), [46] Doray L.G. (1994). IBNR reserve under a loglinear location-scale regression model, Casualty Actuarial Society Forum 1994, 2, [47] Doray L.G. (1996). UMVUE of the IBNR reserve in a lognormal linear regression model, Insurance: Mathematics & Economics, 18(1), [48] Dufresne D. (1990). The distribution of a perpetuity with applications to risk theory and pension funding, Scandinavian Actuarial Journal, 9, [49] Dufresne D. (2002). Asian and basket asymptotics, Research Paper No. 100, Centre for Actuarial Studies, University of Melbourne. [50] Dufresne D. (2004). Stochastic life annuities, Research Paper, Centre for Actuarial Studies, University of Melbourne. [51] Efron B. (1979). Bootstrap methods: another look at the jackknife, Ann. Statist., 7, [52] Efron B. & Tibshirani R.J. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York. [53] Embrechts P., Klüppelberg C. & Mikosch T. (1997). Modelling Extremal Events for Insurance and Finance, Springer, Berlin. [54] England P.D. & Verrall R.J. (1999). Analytic and bootstrap estimates of prediction errors in claim reserving, Insurance: Mathematics & Economics, 25(3),

268 242 Bibliography [55] England P.D. & Verrall R.J. (2001). A flexible framework for stochastic claims reserving, Proceedings of the Casualty Actuarial Society, 88(1), [56] England P.D. & Verrall R.J. (2002). Stochastic claims reserving in general insurance, British Actuarial Journal, 8(3), [57] Fang K.T., Kotz S. & Ng K.W. (1990) Symmetric Multivariate and Related Distributions, Chapman & Hall, London. [58] Feller W. (1971). An Introduction to Probability Theory and Its Applications, Wiley, New York. [59] Frees E. (1990). Stochastic life contingencies with solvency considerations, Transactions of the Society of Actuaries, 42, [60] Gilks W.R., Richardson S. & Spiegelhalter D.J. (1996) Practical Markov Chain Monte Carlo, Chapman and Hall, London. [61] Goovaerts M.J., Kaas R., Van Heerwaarden A.E. & Bauwelinckx T. (1990). Effective Actuarial Methods, North-Holland, Amsterdam. [62] Goovaerts M.J. & Redant H. (1999). On the distribution of IBNR reserves, Insurance: Mathematics & Economics, 25(1), 1 9. [63] Goovaerts M.J., Dhaene J. & De Schepper A. (2000). Stochastic upper bounds for present value functions, Journal of Risk and Insurance, 67(1),1 14. [64] Goovaerts M.J., Kaas R., Dhaene J., & Tang Q. (2003). A unified approach to generate risk measures, ASTIN Bulletin, 33(2), [65] Goovaerts M.J., Kaas R., Dhaene J., & Tang Q. (2004). Some new classes of consistent risk measures, Insurance: Mathematics & Economics, 34(3), [66] Heerwaarden A.E. van (1991). Ordering of Risks: Theory and Actuarial Applications, Thesis Publishers, Amsterdam. [67] Hoedemakers T., Beirlant J., Goovaerts M.J. & Dhaene J. (2003). Confidence bounds for discounted loss reserves, Insurance: Mathematics & Economics, 33(2),

269 Bibliography 243 [68] Hoedemakers T. & Goovaerts M.J. (2004). Discussion of Risk and discounted loss reserves by Greg Taylor, North American Actuarial Journal, 8(4), [69] Hoedemakers T., Beirlant J., Goovaerts M.J. & Dhaene J. (2005). On the distribution of discounted loss reserves using generalized linear models, Scandinavian Actuarial Journal, 2005(1), [70] Hoedemakers T., Darkiewicz G. & Goovaerts M.J. (2005). Approximations for life annuity contracts in a stochastic financial environment, Insurance: Mathematics & Economics, to be published. [71] Hoedemakers T., Darkiewicz G., Deelstra G., Dhaene J. & Vanmaele M. (2005). Bounds for stop-loss premiums of stochastic sums (with applications to life contingencies), Research Report OR 0523, Department of Applied Economics, K.U.Leuven. [72] Huang H., Milevsky M.A. & Wang J. (2004). Ruined moments in your Life: how good are the approximations?, Insurance: Mathematics & Economics, 34(3), [73] Hürlimann W. (1996). Improved analytical bounds for some risk quantities, ASTIN Bulletin, 26(2), [74] Hürlimann W. (1998). On best stop-loss bounds for bivariate sums by known marginal means, variances and correlation, Mitt. Verein. Schweiz. Versicherungmath., [75] Ibbotson Associates (2002). Stocks, Bonds, Bills and Inflation: , Chicago, IL. [76] Jansen K., Haezendonck J. & Goovaerts M.J. (1986). Upper bounds on stop-loss premiums in case of known moments up to the fourth order, Insurance: Mathematics & Economics, 5(4), [77] Jeffreys H. (1946). An invariant form for the prior probability in estimation problems, Proc. Roy. Soc. London Ser. A, 196, [78] Kaas R., Van Heerwaarden A.E. & Goovaerts M.J. (1998). Ordering of Actuarial Risks, Caire Education Series 1, Caire, Brussels.

270 244 Bibliography [79] Kaas R., Dhaene J. & Goovaerts M.J. (2000). Upper and lower bounds for sums of random variables, Insurance: Mathematics & Economics, 27(2), [80] Kaas R., Goovaerts M.J., Dhaene J. & Denuit M. (2001). Modern Actuarial Risk Theory, Kluwer Academic Publishers. [81] Kalbfleisch J.D. & Prentice R.L. (1980). The Statistical Analysis of Failure Time Data, Wiley, New York. [82] Karatzas I. & Shreve S.E. (1991). Brownian Motion and Stochastic Calculus, Springer-Verlag, New York. [83] Kass, R.E. & Wasserman L. (1996). The selection of prior distributions by formal rules, Journal of the American Statistical Association, 91, [84] Kremer E. (1982). IBNR-claims and the two-way model of ANOVA, Scandinavian Actuarial Journal, [85] Laeven R.J.A, Goovaerts M.J. & Hoedemakers T. (2005). Some asymptotic results for sums of dependent random variables with actuarial applications, Insurance: Mathematics & Economics, to be published. [86] Landsman Z. & Valdez E.A. (2003). Tail conditional expectations for elliptical distributions, North American Actuarial Journal, 7, [87] Lawless J.F. (1982). Statistical Models and Methods for Lifetime Data, Wiley, New York. [88] Lehmann E. (1955). Ordered families of distributions, Ann. Math. Statist., 26, [89] Lowe J. (1994). A practical guide to measuring reserve variability using: Bootstrapping, operational time and a distribution free approach, Proceedings of the 1994 General Insurance Convention, Institute of Actuaries and Faculty of Actuaries. [90] Mack T. (1991). A simple parametric model for rating automobile insurance or estimating IBNR claims reserves, ASTIN Bulletin, 22(1),

271 Bibliography 245 [91] Mack T. (1993). Distribution free calculation of the standard error of chain ladder reserve estimates, ASTIN Bulletin, 23(2), [92] Mack T. (1994). Measuring the variability of chain-ladder reserve estimates, Casualty Actuarial Society Forum Spring 1994, 1, [93] McCullagh P. & Nelder J.A. (1992). Generalized Linear Models, 2nd edition, Chapman and Hall, New York. [94] Merton R. (1971). Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory 3, [95] Merton R. (1990). Continuous Time Finance, Cambridge, Blackwell. [96] Michael J.R., Schucany W.R. & Haas R.W. (1976). Generating random variates using transformations with multiple roots, The American Statistician, 30, [97] Milevsky M.A., Ho K. & Robinson C. (1997). Asset allocation via the conditional first exit time or how to avoid outliving your money, Review of Quantitative Finance and Accounting, 9(1), [98] Milevsky M.A. (1997). The present value of a stochastic perpetuity and the Gamma distribution, Insurance: Mathematics & Economics, 20(3), [99] Milevsky M.A. & Posner S.E. (1998). Asian options, the sum of lognormals, and the reciprocal gamma distribution, Journal of Financial and Quantitative Analysis, 33(3), [100] Milevsky M.A. & Robinson C. (2000). Self-annuitization and ruin in retirement, North American Actuarial Journal, 4(4), [101] Milevsky M.A. & Wang J. (2004). Stochastic annuities under exponential mortality, Research paper, York University and The IFID Centre. [102] Nielsen J.A. & Sandmann K. (2003). Pricing bounds on Asian options, Journal of Financial and Quantitative Analysis, 38(2),

272 246 Bibliography [103] Norberg R. (1990). Payment measures, interest and discounting. An axiomatic approach with applications to insurance, Scandinavian Actuarial Journal, 73, [104] Norberg R. (1993). A solvency study in life insurance, Proceedings of the Third AFIR International Colloquium, Rome, [105] O Hagan A. (1994). Bayesian Inference, Kendall s Advanced Theory of Statistics, Arnold, London. [106] Panjer H.H. & Bellhouse D.R. (1980). Stochastic modeling of interest rates with applications to life contingencies, Journal of Risk and Insurance, 47, [107] Panjer H.H. (1998). Financial economics: With applications to investments, insurance and pensions, Schaumburg, Ill.: Society of Actuaries. [108] Parker G. (1994a). Moments of the present value of a portfolio of policies, Scandinavian Actuarial Journal, 77(1), [109] Parker G. (1994b). Stochastic analysis of portfolio of endowment insurance policies, Scandinavian Actuarial Journal 77(2), [110] Parker G. (1994c). Limiting distribution of the present value of a portfolio, ASTIN Bulletin, 24(1), [111] Parker G. (1994d). Two stochastic approaches for discounting actuarial functions, ASTIN Bulletin, 24(2), [112] Parker G. (1996). A portfolio of endowment policies and its limiting distribution, ASTIN Bulletin, 26(1), [113] Parker G. (1997). Stochastic analysis of the interaction between investment and insurance risks, North American Actuarial Journal, 1(2), [114] Pinheiro P.J.R., Andrade e Silva J.M. & de Lourdes Centeno M. (2003). Bootstrap methodology in claim reserving, Journal of Risk and Insurance, 70(4),

273 Bibliography 247 [115] Renshaw A.E. (1989). Chain ladder and interactive modelling (claims reserving and GLIM), Journal of the Institute of Actuaries, 116(III), [116] Renshaw A.E. (1994). On the second moment properties and the implementation of certain GLIM based stochastic claims reserving models, Actuarial Research Paper No. 65, Department of Actuarial Science and Statistics, City University, London. [117] Renshaw A.E. (1994b). Claims reserving by joint modelling, Actuarial Research Paper No. 72, Department of Actuarial Science and Statistics, City University, London. [118] Renshaw A.E. & Verrall R.J. (1994). A stochastic model underlying the chain-ladder technique, Proceedings XXV ASTIN Colloquium, Cannes. [119] Rogers L.C.G. & Shi Z. (1995). The Value of an Asian option, Journal of Applied Probability, 32, [120] Schoutens W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives, Wiley, New York. [121] Shaked M. & Shanthikumar J.G. (1994). Stochastic orders and their applications, Academic Press. [122] Simon S., Goovaerts M.J. & Dhaene J. (2000). An easy computable upper bound for the price of an arithmetic Asian option, Insurance: Mathematics & Economics, 26(2-3), [123] Tang Q. & Tsitsiashvili G. (2003). Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavytailed insurance and financial risks, Stochastic Processes and their Applications, 108, [124] Tang Q. & Tsitsiashvili G. (2004). Finite and infinite time ruin probabilities in the presence of stochastic return on investments, Advances in Applied Probability, 36, [125] Taylor G.C. & Ashe F.R. (1983). Second moments of estimates of outstanding claims, Journal of Econometrics, 23,

274 248 Bibliography [126] Taylor G.C. (1996). Risk, capital and profit in insurance, SCOR International Prize in Actuarial Science. [127] Taylor G.C. (2000). Loss Reserving: An Actuarial Perspective, Kluwer Academic Publishers. [128] Taylor G.C. (2004). Risk and discounted loss reserves, North American Actuarial Journal, 8(1), [129] Valdez E. & Dhaene J. (2004). Bounds for sums of dependent logelliptical risks, Working Paper, University of New South Wales. [130] Vanduffel S., Hoedemakers T. & Dhaene J. (2004). Comparing approximations for sums of non-independent lognormal random variables, Research Report OR 0418, Department of Applied Economics, K.U.Leuven. [131] Vanduffel S. (2005). Comonotonicity: From Risk Measurement to Risk Management, PhD Thesis, University of Amsterdam, Faculty of Economics and Econometrics, Amsterdam. [132] Vanmaele M., Deelstra G. & Liinev J. (2004a). Approximation of stop-loss premiums involving sums of lognormals by conditioning on two random variables, Insurance: Mathematics & Economics, 35(2), [133] Vanmaele M., Deelstra G., Liinev J., Dhaene J. & Goovaerts M.J. (2004b). Bounds for the price of discrete arithmetic Asian options, Journal of Computational and Applied Mathematics, to appear. [134] Verrall R.J. (1989). A state space representation of the chain-ladder linear model, Journal of the Institute of Actuaries, 116, [135] Verrall R.J. (1991). On the unbiased estimation of reserves from loglinear models, Insurance: Mathematics & Economics, 10, [136] Verrall R.J. (2004). A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving, North American Actuarial Journal, 8(3), [137] Vyncke D. (2003). Comonotonicity: the Perfect Dependence, PhD Thesis, K.U. Leuven, Faculty of Sciences, Leuven.

275 Bibliography 249 [138] Vyncke D., Goovaerts M.J. & Dhaene J. (2004). An accurate analytical approximation for the price of a european-style arithmetic Asian option, Finance (AFFI), 25, [139] Wang S. & Young V.R. (1998). Ordering risks: Expected utility theory versus Yaari s dual theory of risk, Insurance: Mathematics & Economics, 22, [140] Waters H.R. (1978). The moments and distributions of actuarial functions, Journal of the Institute of Actuaries, 105, [141] Wedderburn R.W.M. (1974). Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method, Biometrika, 61, [142] Wilkie A.D. (1976). The rate of interest as a stochastic process: Theory and applications, Proceedings of the 20th International Congress of Actuaries, Tokyo 1, [143] Wolthuis H. & Van Hoek I. (1986). Stochastic models for life contingencies, Insurance: Mathematics & Economics, 5(3), [144] Wright T.S. (1990). A stochastic method for claims reserving in general insurance, Journal of the Institute of Actuaries, 117, [145] Yaari M.E. (1987). The dual theory of choice under risk, Econometrica, 55, [146] Zehnwirth B. (1989). The chain-ladder technique - A stochastic model, Claims Reserving Manual, 2, Institute of Actuaries, London.