Fundamentos da Matemática Atuaria: Estruturas e Modelos

ACTUARIAL Mathematics Authors Biographies xv Authors Introductions and Guide to Study xvii Introduction to First Edition xvii Life Tables 58 Introduction to Second Edition xxi Life Insurance 93 Guide to Study xxii Life Annuities 133 The Economics of Insurance 1 168 Modified Preliminary Term 521 Introduction 1 166 Modified Reserve Methods 515 Utility Theory 3 167 Full Preliminary Term 519 Insurance and Utility 7 Appendix 5 Some Mathematical Formulas Useful in Actuarial Mathematics 701 Elements of Insurance 15 APPENDIX 6 Bibliography 705 Optimal Insurance 16 Appendix 7 Answers to Exercises 749 Notes and References 18 AUTHORS INTRODUCTIONS AND GUIDE TO STUDY Appendix 19 Introduction to First Edition This text represents a first step in communicating the revolution in the actuarial profession that is taking place in this age of highspeed computers During the short period of time since the invention of the microchip actuaries have been freed from numerous constraints of primitive computing devices in designing and managing insurance systems They are now able to focus more of their attention on creative solutions to societys demands for financial security To provide an educational basis for this focus the major objectives of this work are to integrate life contingencies into a full risk theory framework and to demonstrate the wide variety of constructs that are then possible to build from basic models at the foundation of actuarial science Actuarial science is ever evolving and the procedures for model building in risk theory are at its forefront Therefore we examine the nature of models before proceeding with a more detailed discussion of the text Intellectual and physical models are constructed either to organize observations into a comprehensive and coherent theory or to enable us to simulate in a laboratory or a computer system the operation of the corresponding fullscale entity Models are absolutely essential in science engineering and the management of large organizations One must however always keep in mind the sharp distinction between a model and the reality it represents A satisfactory model captures enough of reality to give insights into the successful operation of the system it represents The insurance models developed in this text have proved useful and have deepened our insights about insurance systems Nevertheless we need to always keep Exercises 20 up the idea that real insurance systems operate in an environment that is more complex and dynamic than the models studied here Because models are only approximations of reality the work of model building is never done approximations can be improved and reality may shift It is a continuing endeavor of any scientific discipline to revise and update its basic models Actuarial science is no exception Actuarial science developed at a time when mathematical tools probability and calculus in particular the necessary data especially mortality data in the form of life tables and the socially perceived need to protect families and businesses from the financial consequences of untimely death coexisted The models constructed at the genesis of actuarial science are still useful However the general environment in which actuarial science exists continues to change and it is necessary to periodically restate the fundamentals of actuarial science in response to these changes We illustrate this with three examples 1 The insurance needs of modern societies are evolving and in response new systems of employee benefits and social insurance have developed New models for these systems have been needed and constructed 2 Mathematics has also evolved and some concepts that were not available for use in building the original foundations of actuarial science are now part of a general mathematics education If actuarial science is to remain in the mainstream of the applied sciences it is necessary to react basic models in the language of contemporary mathematics 3 Finally as previously stated the development of highspeed computing equipment has greatly increased the ability to manipulate complex models This has farreaching consequences for the degree of completeness that can be incorporated into actuarial models This work features models that are fundamental to the current practice of actuarial science They are explored with tools acquired in the study of mathematics in particular undergraduate level calculus and probability The proposition guiding Chapters 114 is that there is a set of basic models at the heart of actuarial science that should be studied by all students aspiring to practice within any of the various actuarial specialities These models are constructed using only a limited number of ideas We will find many relationships among those models that lead to a unity in the foundations of actuarial science These basic models are followed in Chapters 1521 by some more elaborate models particularly appropriate to life insurance and pensions While this book is intended to be comprehensive it is not meant to be exhaustive In order to avoid any misunderstanding we will indicate the limitations of the text Mathematical ideas that could unify and in some cases simplify ideas presented but which are not included in typical undergraduate courses are not used For example moment generating functions but not characteristic functions are used in developments regarding probability distributions Stieltjes integrals which could be used in some cases to unify the presentation of discrete and continuous cases are not used because of this basic decision on mathematical prerequisites The chapters devoted to life insurance stress the randomness of the time at which a claim payment must be made In the same chapters the interest rates used to convert future payments to a present value are considered deterministic and are usually taken as constants In view of the high volatility possible in interest rates it is natural to ask why probability models for interest rates were not incorporated Our answer is that the mathematics of life contingencies on a probabilistic foundation except for interest does not involve ideas beyond those covered in an undergraduate program On the other hand the modeling of interest rates requires ideas from economics and statistics that are not included in the prerequisites of this volume In addition there are some technical problems in building models to combine random interest and random time of claim that are in the process of being solved Methods for estimating the parameters of basic actuarial models from observations are not covered For example the construction of life tables is not discussed This is not a text on computing The issues involved in optimizing the organization of input data and computation in actuarial models are not discussed This is a rapidly changing area seemingly best left for readers to resolve as they choose in light of their own resources Many important actuarial problems created by longterm practice and insurance regulation are not discussed This is true in sections treating topics such as premiums actually charged for life insurance policies costs reported for pensions restrictions on benefit provisions and financial reporting as required by regulators Ideas that lead to interesting puzzles but which do not appear in basic actuarial models are avoided Average age at death problems for a stationary population do not appear for this reason This text has a number of features that distinguish it from previous fine textbooks on life contingencies A number of these features represent decisions by the authors on material to be included and will be discussed under headings suggestive of the topics involved Probability Approach As indicated earlier the sharpest break between the approach taken here and that taken in earlier English language textbooks on actuarial mathematics is the much fuller use of a probabilistic approach in the treatment of the mathematics of life contingencies Actuaries have usually written and spoken of applying probabilities in their models but their results could be and often were obtained by a deterministic rate approach In this work the treatment of life contingencies is based on the assumption that timeuntildeath is a continuoustype random variable This admits a rich field of random variable concepts such as distribution function probability density function expected value variance and moment generating function This approach is timely based on the availability of highspeed Individual Risk Models for a Short Term 27 Integration with Risk Theory Risk theory is defined as the study of deviations of financial results from those expected and methods of avoiding inconvenient consequences from such deviations The probabilistic approach to life contingencies makes it easy to incorporate longterm contracts into risk theory models and in fact makes life contingencies only a part but a very important one of risk theory Ruin theory another important part of risk theory is included as it provides insight into one source the insurance claims of adverse longterm financial deviations This source is the most unique aspect of models for insurance enterprises Introduction 27 Introduction to Second Edition Actuarial science is not static In the time since the publication of the first edition of Actuarial Mathematics actuarial science has absorbed additional ideas from economics and the mathematical sciences At the same time computing and communications have become cheaper and faster and this has helped to make feasible more complex actuarial models During this period the financial risks that modern societies seek to manage have also altered as a result of the globalization of business technological advances and political shifts that have changed public policies It would be impossible to capture the full effect of all these changes in the revision of a basic textbook Our objective is more modest but we hope that it is realistic This edition is a step in an ongoing process of adaptation designed to keep the fundamentals of actuarial science current with changing realities Models for Individual Claim Random Variables 28 Guide to Study The reader can consider this text as covering the two branches of risk theory Individual risk theory views each policy as a unit and allows construction of a model for a group of policies by adding the financial results for the separate policies in the group Collective risk theory uses a probabilistic model for total claims that avoids the step of adding the results for individual policies This distinction is sometimes difficult to maintain in practice The chapters however can be classified as illustrated below Sums of Independent Random Variables 34 Individual Risk Theory Collective Risk Theory 1 2 3 4 5 6 7 8 9 10 11 15 16 17 18 21 12 13 14 19 20 Approximations for the Distribution of the Sum 39 LongTerm Insurances ShortTerm Insurances Life Insurance Pensions 1 2 12 13 14 3 4 5 6 7 8 9 10 9 10 11 11 15 16 17 18 21 19 20 21 Applications to Insurance 40 The following diagram illustrates the prerequisite structure of the chapters The arrows indicate the direction of the flow For any chapter the chapters that are upstream are prerequisite For example Chapter 6 has as prerequisites Chapters 1 2 3 4 and 5 Notes and References 46 are not essential to an understanding of the material They may be excluded from study at the readers discretion Exercises associated with these appendices should also be considered optional Third general appendices appear at the end of the text Included here are numerical tables for computations for examples and exercises an index to notation a discussion of general rules for writing actuarial symbols reference citations answers to exercises a subject index and supplemental mathematical formulas that are not assumed to be a part of the mathematical prerequisites Fourth we observe two notational conventions A referenced random variable X for example is designated with a capital letter This notational convention is not used in older texts on probability theory It will be our practice in order to indicate the correspondence to use the appropriate random variable symbol as a subscript on functions and operators that depend on the random variable We will use the general abbreviation log to refer to natural base e logarithms because a distinction between natural and common logarithms is unnecessary in the examples and exercises We assume the natural logarithm in our computations Fifth currencies such as dollar pound lira or yen are not specified in the examples and exercises due to the international character of the required computations Finally since we have discussed prerequisites to this work some major theorems from undergraduate calculus and probability theory will be used without review or restatement in the discussions and exercises Exercises 47 1 THE ECONOMICS OF INSURANCE 11 Introduction Each of us makes plans and has expectations about the path his or her life will follow However experience teaches that plans will not unfold with certainty and sometimes expectations will not be realized Occasionally plans are frustrated because they are built on unrealistic assumptions In other situations fortuitous circumstances interfere Insurance is designed to protect against serious financial reversals that result from random events intruding on the plans of individuals We should understand certain basic limitations on insurance protection First it is restricted to reducing those consequences of random events that can be measured in monetary terms Other types of losses may be important but not amenable to reduction through insurance For example pain and suffering may be caused by a random event However insurance coverages designed to compensate for pain and suffering often have been troubled by the difficulty of measuring the loss in monetary units On the other hand economic losses can be caused by events such as property set on fire by its owner Whereas the monetary terms of such losses may be easy to define the events are not insurable because of the nonrandom nature of creating the losses A second basic limitation is that insurance does not directly reduce the probability of loss The existence of windstorm insurance will not alter the probability of a destructive storm However a welldesigned insurance system often provides financial incentives for loss prevention activities An insurance product that encouraged the destruction of property or the withdrawal of a productive person from the labor force would affect the probability of these economically adverse events Such insurance would not be in the public interest Several examples of situations where random events may cause financial losses are the following The destruction of property by fire or storm is usually considered a random event from which the loss can be measured in monetary terms Survival Distributions and Life Tables 51 A damage award imposed by a court as a result of a negligent act is often considered a random event with resulting monetary loss Prolonged illness may strike at an unexpected time and result in financial losses These losses will be due to extra health care expenses and reduced earned income The death of a young adult may occur while longterm commitments to family or business remain unfulfilled Or if the individual survives to an advanced age resources for meeting the costs of living may be depleted These examples are designed to illustrate the definition An insurance system is a mechanism for reducing the adverse financial impact of random events that prevent the fulfillment of reasonable expectations It is helpful to make certain distinctions between insurance and related systems Banking institutions were developed for the purpose of receiving investing and dispensing the savings of individuals and corporations The cash flows in and out of a savings institution do not follow deterministic paths However unlike insurance systems savings institutions do not make payments based on the size of a financial loss occurring from an event outside the control of the person suffering the loss Another system that does make payments based on the occurrence of random events is gambling Gambling or wagering however stands in contrast to an insurance system in that an insurance system is designed to protect against the economic impact of risks that exist independently of and are largely beyond the control of the insured The typical gambling arrangement is established by defining payoff rules about the occurrence of a contrived event and the risk is voluntarily sought by the participants Like insurance a gambling arrangement typically redistributes wealth but it is there that the similarity ends Our definition of an insurance system is purposefully broad It encompasses systems that cover losses in both property and humanlife values It is intended to cover insurance systems based on individual decisions to participate as well as systems where participation is a condition of employment or residence These ideas are discussed in Section 14 The economic justification for an insurance system is that it contributes to general welfare by improving the prospect that plans will not be frustrated by random events Such systems may also increase total production by encouraging individuals and corporations to embark on ventures where the possibility of large losses would inhibit such projects in the absence of insurance The development of marine insurance for reducing the financial impact of the perils of the sea is an example of this point Foreign trade permitted specialization and more efficient production yet mutually advantageous trading activity might be too hazardous for some potential trading partners without an insurance system to cover possible losses at sea Introduction 51 Probability for the AgeatDeath 52 We now study another approach to explain why a decision maker may be willing to pay more than the expected value At first we simply assume that the value of utility that a particular decision maker attaches to wealth of amount w measured in monetary units can be specified in the form of a function uw called a utility function We demonstrate a procedure by which a few values of such a function can be determined For this we assume that our decision maker has wealth equal to 20000 A linear transformation uw a uw b a 0 yields a function uw which is essentially equivalent to uw It then follows by choice of a and b that we can determine arbitrarily the 0 point and one additional point of an individuals utility function Therefore we fix u0 1 and u20000 0 These values are plotted on the solid line in Figure 121 Determination of a Utility Function uw The Survival Function 52 We now ask a question of our decision maker Suppose you face a loss of 20000 with probability 05 and will remain at your current level of wealth with probability 05 What is the maximum amount G you would be willing to pay for complete insurance protection against this random loss We can express this question in the following way For what value of G does u20000 G 05 u20000 05 u0 050 051 05 If he pays amount G his wealth will certainly remain at 20000 G The equal sign indicates that the decision maker is indifferent between paying G with certainty and accepting the expected utility of wealth expressed on the righthand side TimeuntilDeath for a Person Age x 52 If people could foretell the consequences of their decisions their lives would be simpler but less interesting We would all make decisions on the basis of preferences for certain consequences However we do not possess perfect foresight At best we can select an action that will lead to one set of uncertainties rather than another An elaborate theory has been developed that provides insights into decision making in the face of uncertainty This body of knowledge is called utility theory Because of its relevance to insurance systems its main points will be outlined here One solution to the problem of decision making in the face of uncertainty is to define the value of an economic project with a random outcome to be its expected value By this expected value principle the distribution of possible outcomes may be replaced for decision purposes by a single number the expected value of the random monetary outcomes By this principle a decision maker would be indifferent between assuming the random loss X and paying amount EX in order to be relieved of the possible loss Similarly a decision maker would be willing to pay up to EY to participate in a gamble with random payoff Y In economics the expected value of random prospects with monetary payments is frequently called the fair or actuarial value of the prospect Although the method of eliciting and using a utility function may seem plausible it is clear that our informal development must be augmented by a more rigorous chain of reasoning if utility theory is to provide a coherent and comprehensive framework for decision making in the face of uncertainty If we are to understand the economic role of insurance such a framework is needed An outline of this more rigorous theory follows CurateFutureLifetimes 54 In Section 12 we outlined utility theory for the purpose of gaining insights into the economic role of insurance To examine this role we start with an illustration Suppose a decision maker owns a property that may be damaged or destroyed in the next accounting period The amount of the loss which may be 0 is a random variable denoted by X We assume that the distribution of X is known Then EX the expected loss in the next period may be interpreted as the longterm average loss if the experiment of exposing the property to damage may be observed under identical conditions a great many times It is clear that this longterm set of trials could not be performed by an individual decision maker Force of Mortality 55 Suppose that an insurance organization insurer was established to help reduce the financial consequences of the damage or destruction of property The insurer would issue contracts policies that would promise to pay the owner of a property a defined amount equal to or less than the financial loss if the property were damaged or destroyed during the period of the policy The contingent payment linked to the amount of the loss is called a claim payment In return for the promise contained in the policy the owner of the property insured pays a consideration premium Contents vii Jensens inequalities require the existence of the two expected values Proofs of the inequalities are required by Exercise 13 A second proof of 132 is almost immediate from consideration of Figure 131 as follows Formally we say a decision maker with utility function uw is risk averse if and only if uw 0 A decision makers utility function is given by uw eαw The decision maker has two random economic prospects gains available EuY Ee5Y MY5 e56252 e125 Therefore EuX 1 EuY e125 A decision makers utility function is given by uw w13 The decision maker has wealth of w 10 and faces a random loss X with a uniform distribution on 0 10 What is the maximum amount this decision maker will pay for complete insurance against the random loss The probability that a property will not be damaged in the next period is 075 The probability density function pdf of a positive loss is given by fx 025001e001x x 0 The owner of the property has a utility function given by uw e0005w Calculate the expected loss and the maximum insurance premium the property owner will pay for complete insurance payment is EX2 1250 Calculate the maximum premium that the property owner will pay for this insurance 14 Elements of Insurance Individuals and organizations face the threat of financial loss due to random events In Section 13 we saw how insurance can increase the expected utility of a decision maker facing such random losses Insurance systems are unique in that the alleviation of financial losses in which the number size or time of occurrence is random is the primary reason for their existence In this section we review some of the factors influencing the organization and management of an insurance system An insurance system can be organized only after the identification of a class of situations where random losses may occur The word random is taken to mean along with other attributes that the frequency size or time of loss is not under the control of the prospective insured If such control exists or if a claim payment exceeds the actual financial loss an incentive to incur a loss will exist In such a situation the assumptions under which the insurance system was organized will become invalid The actual conditions under which premiums are collected and claims paid will be different from those assumed in organizing the system The system will not achieve its intended objective of not decreasing the expected utilities of both the insured and the insurer 15 Optimal Insurance The ideas outlined in Sections 12 13 and 14 have been used as the foundation of an elaborate theory for guiding insurance decision makers to actions consistent with their preferences In this section we present one of the main results from this theory and review many of the ideas introduced so far A decision maker has wealth of amount w and faces a loss in the next period This loss is a random variable X The decision maker can buy an insurance contract that will pay Ix of the loss x To avoid an incentive to incur the loss we assume that all feasible insurance contracts are such that 0 Ix x We make the simplifying assumption that all feasible insurance contracts with EIX β can be purchased for the same amount P The theorem is proved in the Appendix to this chapter Theorem 151 is an important result and illustrates many of the ideas developed in this chapter However it is instructive to consider certain limitations on its applicability First the ratio of premium to expected claims is the same for all available contracts In fact the distributions of the random variables IX can be very different and the provision for risk in the premium usually depends on the characteristics of the distribution of IX Second in Theorem 151 it is assumed that the premium P is fixed by a budget constraint Alternatives to amount P are not considered In Exercise 122 relaxation of the budget constraint is considered Third while the theorem indicates the form of insurance it does not help to determine the amount P to spend In the theorem P is fixed Appendix Lemma If uw 0 for all w in a b then for w and z in a b uw uz w zuz 1A1 Proof The lemma may be established with the aid of Figure 131 Using the point slope form a line tangent to uw at the point z uz has the equation y uz uzw z and is above the graph of the function uw except at the point z uz Therefore uw uz uzw z Figure 131 shows the case uw 0 The same argument holds for uw 0 In Exercise 120 an alternative proof is required Proof of Theorem 151 Let Ix be associated with an insurance policy satisfying the hypothesis of the theorem Then from the lemma uw x Ix P uw x Irx P Ix Irxuw x Irx P 1A2 In addition we claim Ix Irxuw x Irx P Ix Irxuw d P 1A3 To establish inequality 1A3 we must consider three cases Case I Irx Ix In this case equality holds 1A3 is 0 on both sides Case II Irx Ix In this case Irx 0 and from 151 x Irx d Therefore equality holds with each side of 1A3 equal to Ix Irxuw d P Case III Irx Ix In this case Ix Irx 0 From 151 we obtain Irx x d and Irx x P d P Therefore uw x Irx P uw d P since the second derivative of ux is negative and ux is a decreasing function Therefore in each case Ix Irxuw x Irx P Ix Irxuw P d establishing inequality 1A3 Now combining inequalities 1A2 and 1A3 and taking expectations we have Euw X Ix P Euw X IrX P EIx Irxuw d P β βuw d P 0 Therefore Euw X Ix P Euw X IrX P and the expected utility will be maximized by selecting Irx the stoploss policy a Show that the probability function pf of N is given by fn 12ⁿ n 1 2 3 b Find EN and VarN c If a reward of X 2ⁿ is paid prove that the expectation of the reward does not exist d If this reward has utility uw log w find EuX Section 13 Jensens inequalities a Assume uw 0 EX μ and EuX exist prove that EuX uμ Hint Express uw as a series around the point w μ and terminate the expansion with an error term involving the second derivative Note that Jensens inequalities do not require that uw 0 a Calculate EX and VarX b Consider a proportional policy where Ix kx 0 k 1 and a stoploss policy where Idx 0 x d x d x d Determine k and d such that the pure premium in each case is P 125 c Show that VarX Ix VarX Idx Appendix 120 Establish the lemma by using an analytic rather than a geometric argument Hint Expand uw in a series as far as a second derivative remainder around the point z and subtract uz ii The decision maker elects to retain loss X IX such that VarX IX V 0 This requirement imposes a risk rather than a budget constraint The constant is determined by the degree of risk aversion of the decision maker Fixing the accepted variance and then optimizing expected results is a decision criterion in investment portfolio theory iii The decision maker selects Ix to minimize fVarIX The objective is to minimize the security loading the premium paid less the expected insurance payments Confirm the following steps a VarIX V VarX 2 CovX X IX b The Ix that minimizes VarIX and thereby fVarIX is such that the correlation coefficient between X and X IX is 1 c It is known that if two random variables W and Z have correlation coefficient 1 then PrW aZ b where a 0 1 In words the probability of their joint distribution is concentrated on a line of positive slope In part b the correlation coefficient of X and X IX was found to be 1 Thus X IX aX b which implies that IX 1 aX b To be a feasible insurance 0 Ix x or 0 1 aX b x These inequalities imply that b 0 and 0 1 a 1 and 0 a 1 d To determine a set the correlation coefficient of X and X IX equal to 1 or equivalently their covariance equal to the product of their standard deviations Thus show that a V VarX and thus that the insurance that minimizes fVarX is IX 1 V VarXX 2 INDIVIDUAL RISK MODELS FOR A SHORT TERM 21 Introduction In Chapter 1 we examined how a decision maker can use insurance to reduce the adverse financial impact of some types of random events That examination was quite general The decision maker could have been an individual seeking protection against the loss of property savings or income The decision maker could have been an organization seeking protection against those same types of losses In fact the organization could have been an insurance company seeking protection against the loss of funds due to excess claims either by an individual or by its portfolio of insureds Such protection is called reinsurance and is introduced in this chapter The theory in Chapter 1 requires a probabilistic model for the potential losses Here we examine one of two models commonly used in insurance pricing reserving and reinsurance applications For an insuring organization let the random loss of a segment of its risks be denoted by S Then S is the random variable for which we seek a probability distribution Historically there have been two sets of postulates for distributions of S The individual risk model defines S X1 X2 Xn 21 where Xi is the loss on insured unit i and n is the number of risk units insured Usually the Xis are postulated to be independent random variables because the mathematics is easier and no historical data on the dependence relationship are needed The other model is the collective risk model described in Chapter 12 The individual risk model in this chapter does not recognize the time value of money This is for simplicity and is why the title refers to short terms Chapters 411 cover models for long terms 22 Models for Individual Claim Random Variables First we review basic concepts with a life insurance product In a oneyear term life insurance the insurer agrees to pay an amount b if the insured dies within a year of policy issue and to pay nothing if the insured survives the year The probability of a claim during the year is denoted by q The claim random variable X has a distribution that can be described by either its probability function pf or its distribution function df The pf is fXx PrX x 1 q x 0 q x b 0 elsewhere 221 and the df is FXx PrX x 0 x 0 1 q 0 x b 1 x b 222 From the pf and the definition of moments EX bq EX2 b2q 223 and VarX b2q1 q 224 These formulas can also be obtained by writing X Ib 225 where b is the constant amount payable in the event of death and I is the random variable that is 1 for the event of death and 0 otherwise Thus PrI 0 1 q and PrI 1 q the mean and variance of I are q and q1 q respectively and the mean and variance of X are bq and b2q1 q as above The random variable I with its 0 1 range is widely applicable in actuarial models In probability textbooks it is called an indicator Bernoulli random variable or binomial random variable for a single trial We refer to it as an indicator for the sake of brevity and because it indicates the occurrence I 1 or nonoccurrence I 0 of a given event where X is the claim random variable for the period B gives the total claim amount incurred during the period and I is the indicator for the event that at least one claim has occurred As the indicator for this event I reports the occurrence I 1 or nonoccurrence I 0 of claims in this period and not the number of claims in the period PrI 1 is still denoted by q Let us look at several situations and determine the distributions of I and B for a model First consider a 1year term life insurance paying an extra benefit in case of accidental death To be specific if death is accidental the benefit amount is 50000 For other causes of death the benefit amount is 25000 Assume that for the age health and occupation of a specific individual the probability of an accidental death within the year is 00005 while the probability of a nonaccidental death is 00020 More succinctly PrI 1 and B 50000 00005 and PrI 1 and B 25000 00020 Summing over the possible values of B we have PrI 1 00025 and then PrI 0 1 PrI 1 09975 The conditional distribution of B given I 1 is PrB 25000I 1 PrB 25000 and I 1 PrI 1 00020 00025 08 PrB 50000I 1 PrB 50000 and I 1 PrI 1 00005 00025 02 Let us now consider an automobile insurance providing collision coverage this indemnifies the owner for collision damage to his car above a 250 deductible up to a maximum claim of 2000 For illustrative purposes assume that for a particular individual the probability of one claim in a period is 015 and the chance of more than one claim is 0 PrI 0 085 PrI 1 015 this probability mass is 01 Furthermore assume that claim amounts between 0 and 2000 can be modeled by a continuous distribution with a pdf proportional to 1 x2000 for 0 x 2000 In practice the continuous curve chosen to represent the distribution of claims is the result of a study of claims by size over a recent period Summarizing these assumptions about the conditional distribution of B given I 1 we have a mixed distribution with positive density from 0 to 2000 and a mass at 2000 This is illustrated in Figure 221 The df of this conditional distribution is 0 PrB xI 1 09 1 1 x2000² 1 x 2000 We see in Section 24 that the moments of the claim random variable X in particular the mean and variance are extensively used For this automobile insurance we shall calculate the mean and the variance by two methods First we derive the distribution of X and use it to calculate EX and VarX Letting Fₓx be the df of X we have Fₓx PrX x PrIB x PrIB xI 0 PrI 0 PrIB xI 1 PrI 1 For x 0 Fₓx 0085 0015 0 For 0 x 2000 Fₓx 1085 09 1 1 x2000²015 For x 2000 Fₓx 1085 1015 1 This is a mixed distribution It has both probability masses and a continuous part as can be seen in its graph in Figure 222 Corresponding to this df is a combination pf and pdf given by PrX 0 085 PrX 2000 0015 with pdf fₓx Fₓx 0000135 1 x2000 0 x 2000 0 elsewhere Moments of X can then be calculated by EX 0 PrX 0 2000 PrX 2000 ²⁰⁰⁰₀ x fₓx dx specifically EX 120 and EX² 150000 Thus VarX 135600 There are some formulas relating the moments of random variables to certain conditional expectations General versions of these formulas for the mean and variance are EW EEWV Then EVarXI σ²EI σ²q Substituting 2219 2220 and 2224 into 2212 and 2213 we have EX μq and VarX μ²q1 q σ²q Let us now apply these formulas to calculate EX and VarX for the automobile insurance in Figure 222 Since the pdf for B given I 1 is fBIx1 000091 x2000 0 x 2000 0 elsewhere with PrB 2000I 1 01 we have μ 2000 0 00009 x 1 x2000 dx 012000 800 EBI 1 2000 0 00009 x² 1 x2000 dx 012000² 1000000 and σ² 1000000 800² 360000 Finally with q 015 we obtain the following from 2225 and 2226 EX 800015 120 and VarX 800²015085 360000015 135600 There are other possible models for B in different insurance situations An example let us consider a model for the number of deaths due to crashes during an airlines year of operation We can start with a random variable for the number of deaths X on a single flight and then add up a set of such random variables over the set of flights for the year For a single flight the event I 1 will be the event of an accident during the flight The number of deaths in the accident B will be modeled as the product of two random variables L and Q where L is the load factor the number of persons on board at the time of the crash and Q is the fraction of deaths among persons on board The number of deaths B is modeled in this way since separate statistical data for the distributions of L and Q may be more readily available than are total data for B We have X ILQ While the fraction of passengers killed in a crash and the fraction of seats occupied are probably related L and Q might be assumed to be independent as a first approximation 23 Sums of Independent Random Variables In the individual risk model claims of an insuring organization are modeled as the sum of the claims of many insured individuals The claims for the individuals are assumed to be independent in most applications In this section we review two methods for determining the distribution of the sum of independent random variables S X Y with the sample space shown in Figure 231 Event X Y s The line X Y s and the region below the line represent the event S X Y s Hence the df of S is FSs PrS s PrX Y s For two discrete nonnegative random variables we can use the law of total probability to write 231 as FSs Σ all ys PrX Y sY y PrY y Σ all ys PrX s yY y PrY y When X and Y are independent this last sum can be written FSs Σ all ys FXs yfYy FSs 0 PrX s yY y fYy dy FSs 0 FXs yfYy dy fSs all ys fXs yfYy When either one or both of X and Y have a mixedtype distribution typical in individual risk model applications the formulas are analogous but more complex For random variables that may also take on negative values the sums and integrals in the formulas above are over all y values from to In probability the operation in 233 and 236 is called the convolution of the pair of distribution functions FXx and FYy and is denoted by FX FY Convolutions can also be defined for a pair of probability functions or probability density functions as in 234 and 237 To determine the distribution of the sum of more than two random variables we can use the convolution process iteratively For S X1 X2 Xn where the Xis are independent random variables Fi is the df of Xi and Fk is the df of X1 X2 Xk we proceed thus F2 F2 F1 F2 F1 F3 F3 F2 F4 F4 F3 Fn Fn Fn1 Example 231 illustrates the procedure using probability functions for three discrete random variables For illustrative purposes we include column 6 the df for column 1 column 7 which can be derived directly from columns 2 and 6 by use of 233 and column 8 derived similarly from columns 3 and 7 Column 5 can then be obtained by differencing column 8 illustrated in the figure for five values of s For each value the line intersects the yaxis at s and the line x 2 at s 2 The values of FS for these five cases are If X1 X2 ldots Xn are independent then the expectation of the product in 238 is equal to EX1 EX2 cdots EXn Example 235 The inverse Gaussian distribution was developed in the study of stochastic processes Here it is used as the distribution of B the claim amount It will have a similar role in risk theory in Chapters 1214 The pdf and mgf associated with the inverse Gaussian distribution are given by fxx α 2πβ x32 expβx α² 2βx x 0 Mxt expα 1 1 2t β 24 Approximations for the Distribution of the Sum The central limit theorem suggests a method to obtain numerical values for the distribution of the sum of independent random variables The usual statement of the theorem is for a sequence of independent and identically distributed random variables X1 X2 with EXi µ and VarXi σ² For each n the distribution of S X1 X2 Xn has mean 0 and variance 1 Example 251 A life insurance company issues 1year term life contracts for benefit amounts of 1 and 2 units to individuals with probabilities of death of 002 or 010 The following table gives the number of individuals nk in each of the four classes created by a benefit amount bk and a probability of claim qk The policyholders of an automobile insurance company fall into two classes Again the probability that total claims exceed the amount collected from policyholders is 005 We assume that the relative security loading θ is the same for the two classes Calculate θ A life insurance company covers 16000 lives for 1year term life insurance in amounts shown below PrS 275 825 PrS 550 Pr S ES VarS 550 ES VarS Pr S ES VarS 25 so σS 5086 26 Notes and References The basis of the material in Sections 22 23 and 24 can be found in a number of postcalculus probability and statistics texts Mood et al 1974 prove the theorems given in 2210 and 2211 They also provide an extensive discussion of properties of the moment generating function For a discussion of the advanced mathematical methods for deriving the distribution function that corresponds to a given moment generating function see Bellman et al 1966 Methods are also available to obtain the pf of a discrete distribution from its probability generating function see Kornya 1983 Let S X₁ X₂ X₃