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INFLATION AND INDEXATION IN BRAZIL: THE INFLUENCE ON LIFE INSURANCE By DAVID F. BABBEL A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1978 TABLE OF CONTENTS Page ABSTRACT iv CHAPTER 1: INTRODUCTION 1 Background of the Study 1 The Problem 4 Purpose of the Study 4 Reasons for Studying the Brazilian Experience 5 Justification of the Study 6 Review of the Literature 8 Scope of the Study 14 Methodology and Format of the Study 15 CHAPTER 2: MEASURING THE COST OF LIFE INSURANCE 17 UNDER INFLATION An Overview 17 Money Illusion 20 Partial Money Illusion 23 Policy Illusion 25 No Illusion 47 Maintaining Real Values 54 Summary and Conclusions 71 CHAPTER 3: RATIONAL LIFE INSURANCE PURCHASING 75 AND INFLATION Survey of the Literature 75 Development of a Theoretical Model 82 A Methodological Note 106 CHAPTER 4: TESTS OF THE HYPOTHESES: 111 RESEARCH RESULTS The Model 111 The Dependent Variable 113 Explanatory Variables 114 Estimation Procedure 127 Comparison of Research Model and Results with 139 Previous Studies Summary and Conclusions 143 APPENDIX A: FORMAL RELATIONSHIP BETWEEN 150 INFLATION AND LIFE INSURANCE POLICY COST APPENDIX B: INFLATION AND THE COST OF 154 INSURANCE PROTECTION APPENDIX C: CALCULATION OF THE INDICES 158 APPENDIX D: DATA 162 BIBLIOGRAPHY 165 BIOGRAPHICAL INFORMATION 173 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INFLATION AND INDEXATION IN BRAZIL: THE INFLUENCE ON LIFE INSURANCE By David F. Babbel August 1978 Chairman: C. Arnold Matthews Cochairman: David A. Denslow Major Department: Finance Inflation is a problem that has plagued economies worldwide, and its effects upon some basic institutions are of widespread concern. One of the economic institutions that is most susceptible to the effects of inflation is life insurance. In an effort to mitigate the impact of in flation upon the values specified in life insurance contracts, several countries have adopted indexlinked life insurance policies. One coun try that has adopted such measures is Brazil. The investigation of this dissertation centers upon the effects of inflation on life insurance in Brazil, both before and after the advent of indexation. The study begins with an inquiry into the nature and extent to which inflation affects the cost of achieving financial protection. It is found that while the nominal cost of commodities, on average, accompanies the changes in the consumer price index, the nominal cost of life insur ance will grow at a more rapid pace. In real terms, life insurance costs increase with inflation, as long as insurance regulators are slow to allow adjustments of policy terms to higher expected rates of infla tion. The magnitude of this cost increase is shown to be extremely high in the case of Brazil, where policyowners do not participate in the insurance company profits. The real cost of life insurance protec tion through indexlinked policies is also shown to rise with inflationary expectations. After demonstrating the effect of inflation on life insurance costs, a theoretical model is developed to determine the influence of inflation upon consumer purchases of life insurance. TimeState Preference theory indicates that rational insurance purchases in real values will unambiguously be lower when inflation is anticipated, regardless of whether or not the policies feature indexing. The reason for the latter is that indexing in Brazil is not carried out on a continuous basis, but policy values are adjusted only annually, and on an ex post basis. Finally, a multivariate timeseries regression model is developed, and the Brazilian experience before and after indexing is tested to deter mine if inflation's impact on life insurance purchasing was negative, as hypothesized. The test results indicate that a negative relationship between inflation (expected or realized) and net life insurance in force per capital prevailed in both the preindexing and postindexing periods of Brazil's history. CHAPTER 1 INTRODUCTION Background of the Study Inflation is a problem that has plagued economies worldwide. Recently a growing number of countries have experienced doubledigit rates of inflation which have affected basic institutions such as life insurance. The effects of inflation on life insurance contracts is espe cially pronounced due to two factors: these contracts are usually specified in fixed, nominal currency units, and most of them are de signed to cover long periods of time. Because life insurance values are specified in fixed nominal currency units, they do not adjust to compensate for the value erosion produced by inflation, and because the contracts are generally long term, the accumulated erosive effects of inflation on the insurance values can be substantial. Thus, while life insurance products are designed to provide protection against the perils of longevity and premature death, inflation and a rising cost of living can undermine such protection. Faced with the prospect of chronic inflation, the insurance in dustry has three strategic alternatives: (1) it may exert its influence in attempting to eliminate or reduce inflation; (2) it may accept higher 1 rates of inflation as inevitable, and adapt its products to such an en vironment; or (3) in the absence of success in either or both of the above, it can watch as its functions in society are usurped by other institutions, presumably government institutions. For a number of years insurance industry officials have engaged in sporadic attempts aimed at influencing governmental officials to take measures to arrest inflation. In the United States, for example, some insurance companies engaged in advertising campaigns to edu cate consumers as to the consequences of inflation on their financial security, thereby inducing the public to bring pressure to bear upon the government for fiscal and monetary restraint. Other companies directly lobbied the executive and legislative branches of government to exercise more restraint. However, most countries have lacked the will to take measures necessary to eliminate inflation, measures whose harsh consequences upon employment and economic activity can become politically unfavorable if not intolerable. Having failed to make significant headway in controlling inflation, the insurance industry has turned toward developing insurance products 1To fill the void, one of the governmental responses might be an expansion of social insurance programs such as the social security system operating in the United States. However, unlike life insurance, social insurance is often an essentially unfunded operation. Thus, to the extent that social insurance leads consumers to save less, adverse consequences can occur in the process of capital formation. For a dis cussion of this point and an estimation of the impact of social security on savings in the U.S., see Feldstein (1974). better adapted to inflationary environments (Greene, 1974). One of the most promising alternatives is the indexlinked life insurance con tract. In such contracts, the nominal values of the insurance pre miums, cash values, and indemnification payments are linked to a price or cost of living index and adjusted periodically in accordance with changes that have transpired in the index. This approach has been taken in countries such as Brazil, Finland, and Israel.3 The feasibility of offering indexlinked policies depends upon the existence of an asset in which the insurer can invest its proceeds which will consistently yield a rate of return commensurate with the rate of change in the index utilized. Without such an asset the insurer could face problems of capital inadequacy. Another consideration is that the ability of the insured to pay the fluctuating premiums must be at least loosely related to changes in the index used for adjusting the premiums. This constraint is less restrictive than the first due to the flexibility in the average consumer's budget, which is generally capable of ab sorbing temporary aberrations between the rates of growth in wealth and in premium expenditure obligations. Other alternatives are discussed in the fourth section of Greene (1974). 3Finland and Israel have allowed partial inflation adjustments, whereas Brazil permits full inflation adjustments. The Problem Given an environment in which the existence of indexlinked con tracts is feasible, the insurer may ask if such contracts should be of ferred. If, on the other hand, conditions are such that the introduction of indexlinked contracts is infeasible (e.g., no indexlinked assets exist in which the insurer can invest insurance proceeds), the insurer may ask if its influence should be wielded in order to bring about the requisite conditions. The answers to these questions depend heavily upon the answers to three questions: (1) Does inflation affect the cost of life insurance protection? (2) Is the consumer sensitive to changes in life insurance cost wrought by inflation ? (3) Can indexlinked life insurance contracts mitigate any adverse ef fects of inflation on life insurance costs and values ? If the answer to any of these questions is in the negative, further inquiry into the area of indexing and life insurance is unwarranted. If, on the other hand, the responses to these questions are in the affirma tive, implications can then be drawn as to the advisability of developing and marketing indexlinked life insurance policies. Purpose of the Study The three questions listed above constitute the subjects of inves tigation in this paper. The purpose of the study is to resolve these questions and the issues they entail through theoretical analysis, where 5 possible, and, in cases where a question is not amenable to such an analytical solution, to provide and utilize empirical analysis to help resolve it. The data for the empirical portions of the study will be taken from the Brazilian experience (for reasons to be discussed in the next section). While conclusions resulting from the theoretical analyses will have general applicability, those based on the empirical data will have inferences with respect to the Brazilian population under study. However, an effort will be made to determine the impli cations of the Brazilian experience with regard to the perspectives and potential problems of indexing life insurance policies in general. Reasons for Studying the Brazilian Experience This study will focus upon the life insurance industry of Brazil for two reasons: Brazil's long history of inflation and its extensive experience with indexation. Almost all studies to date on the influence of inflation on life in surance demand have dealt primarily with the United States. The periods of study chosen have been ones of relatively low or unsustained moderate rates of inflation. There is possibly a threshold rate of in flation above which sales of longterm insurance are affected, i.e., where money illusion evaporates; if such is the case, empirical studies dealing exclusively with the United States may have overlooked or only marginally detected the disastrous effects that sustained inflation rates above this threshold can have on insurance markets. Brazil's endemic doubledigit inflation rates4 make it an excellent candidate for studying the economic impact of inflation on insurance markets. Since 1964, Brazil has undertaken systematic monetary correc tions ("indexing") in an attempt to cope with rampant inflation and the distortions fostered by it.5 In fact, indexing in Brazil has probably been more widespread than in any other country (Friedman, 1974). Thus, the Brazilian experience provides a rich base from which the effects that indexing can produce in actual practice can be investigated. It is anticipated that the Brazilian experience will provide some im portant insights into the benefits and problems of indexing for life in surance industries and for the consumers of life insurance in other countries suffering from inflation. Justification of the Study An inquiry into the influence of inflation and indexation on life insurance in general, and the Brazilian experience in particular, can be justified on three grounds. The issues involved are important: (1) to social welfare, (2) to capital formation, and (3) to the life insur ance industry. 4Brazil's prolonged experience with inflation is discussed by Buescu (1973). 5Fishlow (1974) describes in detail the system of indexing in Brazil. 7 The social significance of life insurance derives from its ability to protect covered members of society from fortuitous events that can produce disastrous aftermaths. Life insurance is used to provide pro tection against the financial consequences of premature death. It can also be used to accumulate a savings fund. In an environment of infla tion, protection and savings available through life insurance are eroded. Consumers of insurance are thus exposed to financial risk which in surance was designed to alleviate. If the consumers of insurance are unable to provide for their future, more pressure and responsibility may fall upon governments to do so. On the other hand, if indexing can restore life insurance as a viable instrument of saving and protec tion, people may wish to channel their resources in that direction. Life insurance can also be important to capital formation, stem ming largely from the fact that much of life insurance is sold in forms other than single year term insurance. These forms contain sizeable savings elements which provide funds to the insurance companies for financial investment. A large portion of these funds is generally chan neled into the markets for long term investments to hedge against the associated insurance risk, whose nature is generally long term. Accord ingly, life insurance has particular importance in the long term capital markets. Financial risk refers to the possibility that a desired or re quired pattern of returns (across different possible future states of nature) will not be achieved. 8 A paralyzed life insurance industry can have crippling effects upon capital markets whereas a healthy life insurance industry can con tribute significantly to their development. The extent to which inflation and indexation can paralyze and revitalize, respectively, a life insurance industry is therefore of great importance to capital formation. Finally, the effects of inflation and indexation upon the demand for life insurance are important to the life insurance industry itself. The survival of the industry depends on its ability to attract consumers to purchase life insurance policies. If inflation diminishes consumer in terest in the life insurance policies available through the industry, then the survival of the industry is in jeopardy.7 Review of the Literature These vital roles of life insurance, and the adverse impact which inflation may have upon them, have received attention in the literature. While the influence of inflation upon the cost8 of life insurance protec tion can be demonstrated analytically, no rigorous analytical treatments 7The survival of the industry is not only important to the share holders but also to those whose occupations are derived from the indus  try. The life insurance industry may employ large numbers of people, and through its investments provides employment for many more. As the industry loses vitality, it can no longer function effectively in this role. 8A thorough discussion of the concept of life insurance cost is presented in the next chapter. of this problem were encountered in the published literature. There is, however, a study (Fitzhugh and Greeley, 1974) which does treat the prob lem through a retrospective example using actual policy data and his  torical inflation rates. While no real attempt was made at measuring the cost of life insurance under inflation, the article offers important insights into the options available through participating life insurance policies that can help offset some of the value erosion incurred by in flation. The sensitivity of the consumer to changes in life insurance cost wrought by inflation can be examined through theoretical analysis, and resulting conclusions can then be tested empirically. To date there have been no rigorous theoretical treatments of the effect of inflation on con sumer demand for life insurance. However, an attempt was made by Hofflander and Duvall (1967). They utilized budget constraints and in difference curves in a graphical analysis which they contend demon strates that less life insurance (in real values) will be purchased if in flation is anticipated. Neumann (1968) correctly criticized their model and showed that the model can be used to demonstrate that purchases of life insurance protection could actually increase under inflationary expectations. While Neumann is correct in his criticism of the model, he is incorrect in using another version of the same model to substan tiate his own theoretical analysis. The problem is that the model em ployed by the three authors is a timeless microeconomic model incapable 10 of properly taking into account the inflation and protection factors, which necessarily occur over time. Thus, none of the authors' prop ositions in this regard were derived with appropriate rigor.9 There have been a large number of empirical studies conducted to ascertain the determinants of demand for life insurance.10 Some of the studies which used timeseries regression analysis have included anticipated inflation as an explanatory variable. Of special interest are five articles dealing specifically with the effects of inflation on life insurance. In an early study by David B. Houston (1960), the relationships between the price level and the pattern of savings through life insur ance11 in the United States were estimated through statistical tech niques. The period of investigation was from 1919 through 1958. Houston concluded that there was no simple long term relationship be tween the cost of living and the extent of savings through life insurance, and that there was no indication that the life insurance industry has suf fered as a result of the postwar inflation. 9Their analyses are examined at length in Chapter 3. 0For a compilation of life insurance demand analyses conducted prior to 1970, see Lee and Whitaker (1970). A more recent review is in Headen and Lee (1974). 11The Houston study related savings through life insurance to the price level and not savings to "changes in the price level," as he misstated. After indicating theoretically12 that sales of both permanent and term insurance may decrease if there are anticipations of price level increases, Alfred E. Hofflander and Richard M. Duvall (1967) used two multiple regression models to test the relationship between price level changes and sales of life insurance in the United States. Their study covered a twenty year period beginning in 1945. The authors found that large increases in the cost of living have been accompanied by rela tively smaller sales of term, as well as permanent life insurance. A doctoral dissertation by Seev Neumann (1967) considers the im pact of inflation on consumer savings through life insurance and arrives at a different conclusion. In his study of the period from 1946 until 1964, Neumann concludes that the data do not support the conclusion that consumer expectations of price changes13 had any discernible ef fect on saving through life insurance in the United States economy. He allows, however, that "creeping" inflation might have a cumulative ef fect that takes time to influence the slow process of social learning.14 12Misgivings with respect to the theoretical model used in the analysis have already been stated. 13In another article, Neumann (1969a) shows that theoretically it is anticipated inflation that has bearing on the problem and not the price level per se. 14Both the Hofflander/Duvall and the Neumann models use nomi nal, rather than real valued variables. In addition, the models are fraught with specification errors, to be discussed in detail later in this dissertation. 12 In a lengthy comment arising from the Neumann study, Peter Fortune (1972) produces a clearer and more precise theoretical dis cussion of the possible effects that inflation can have on savings through life insurance. After producing an alternative model,5 Fortune uses quarterly data covering the period from 1953 until 1968 to test the propositions he states. He finds that the expected rate of inflation does have a negative impact upon optimum policy reserves per dollar of in surance through its effect on the relative real yields of financial and real assets, but that this effect is offset by other effects induced by inflation. Fortune concludes that inflation actually increases the flows of funds into the life insurance sector of the United States, but no evi dence is presented with regard to the impact of inflation upon the flows into the life insurance sector relative to flows into other financial insti tutions. By such a relative measure the life insurance sector may be hurt by inflation. A more recent study by Fortune (1973) develops a theory of op timal life insurance. Although his model does not accommodate the in clusion of inflation in a theoretical context, his empirical work includes an explanatory variable that is closely related to anticipated inflation, 15The Fortune model uses financial variables, in contrast to the Hofflander/Duvall and Neumann models, which rely largely on sociodemographic variables. and this variable was shown to be highly significant. More will be said of the Fortune statistical tests later in this paper. An excellent review of these and other studies centering upon inflation and life insurance is given in a recently published booklet by Mark Greene (1974). Building on his article of twentyone years ear lier (1954), Greene discusses at length the many avenues through which inflation can and does affect the life insurance sector. Attention is first focused upon the impact of inflation upon life insurers, and then turns to inflation's effects upon the consumers of life insurance. In addition to the experience of the United States, Greene cites Canada and Colombia as indications that inflation has an adverse effect on life in surance demand. The extent to which indexlinked life insurance contracts can mitigate adverse effects of inflation on life insurance costs and values can be examined both theoretically and empirically. Economic litera ture is replete with articles on indexation.16 However, to date neither theoretical nor empirical studies have been encountered dealing spe cifically with indexation of life insurance contracts.17 1An OECD bibliography (1975) cites over 150 articles published until 1974 about indexation. 17 1Descriptive studies concerning the experience of indexlinked life insurance contracts of Finland are given in Junnila (1965) and Ingman (1971). Scope of the Study A theory of inflation's impact upon life insurance demand has not been rigorously developed in economic and financial literature. In this study a model is developed which is capable of facilitating an analysis of the effect of anticipated inflation on rational consumer de mand for life insurance protection. In addition, the theoretical reme dial properties of indexation in the context of life insurance will be elucidated. Statistical tests will be conducted to determine if the propo sitions derived from the theoretical discussions are substantiated in practice. The Brazilian case will be the basis for the tests, although tests already performed on the United States data base will also be examined for their relevance to the problems under investigation. The period covered in the Brazilian data base is 1950 through 1976, a period of elevated and highly fluctuating rates of inflation. This period has been subdivided into the pre 1968 period (before in dexing was implemented in life insurance contracts) and the post 1967 period (after indexing was adopted). This division is deemed to be appropriate for the purposes of the study, because the division isolates the issues under question for separate testing, while providing enough observations for statistical credibility. 15 Methodology and Format of the Study This study is divided into four chapters. Following this intro ductory chapter, a capital budgeting approach for determining the cost of life insurance in an environment of inflation is developed in the second chapter. It is shown how different kinds of money illusion are implicit in some of the popular methods of evaluating the cost of life insurance, and a method is suggested which overcomes this problem. Analytical techniques are employed in demonstrating the effect of in flation on the cost of life insurance available through nonindexed and indexlinked policy contracts. Finally, actual policy data obtained from a large Brazilian insurer are incorporated into the operational model to generate a sample array of life insurance cost data associ ated with various rates of anticipated inflation. In Chapter 3, a theoretical model is developed for analyzing the effect of anticipated inflation on rational life insurance purchasing. The Expected Utility Hypothesis is used in a TimeState Preference framework to derive definitive propositions with regard to consumer demand for life insurance protection. The analysis is carried out for both nonindexed and indexlinked policies. The outcome is a set of hypotheses concerning the relationship of anticipated inflation to con sumer demand for life insurance. 16 After a discussion of the major determinants of consumer demand for life insurance, a timeseries multiple regression model is developed in Chapter 4 for use in an empirical examination of the propositions derived in Chapter 3. The Brazilian experience, which serves as the data base for the statistical testing, is reviewed. Finally, the econometric model is used in testing the Brazilian data and the results of the tests are presented and interpreted. The chapter concludes with a summary of the major findings of this study, along with their policy implications, and suggests some areas for possible further research. CHAPTER 2 MEASURING THE COST OF LIFE INSURANCE UNDER INFLATION An Overview In this chapter a methodology appropriate for measuring life in surance costs in an environment of inflation is developed. The model is then used to examine (1) the effect of inflation on life insurance costs, and (2) the extent to which indexation of policies can mitigate any adverse effects of inflation on life insurance costs and values.1 To achieve unambiguous solutions to these problems will, at times, require some set of life insurance policy terms and provisions to be specified, as well as a knowledge concerning the influence of the insti tutional environment upon life insurance policy terms.2 Since the focus of this study centers on the Brazilian experience, these problems will be viewed as they apply to the life insurance provisions and regu latory environment exhibited in Brazil. Accordingly, analytical models developed and utilized in this chapter will be adapted, where neces sary, to the conditions prevailing in Brazil. These problems are presented as two of the three research questions listed on page 4. 2In the absence of such information, a general analysis may result in ambiguous solutions. 18 A number of authors have devised methodologies by which the costs of life insurance may be computed.3 Almost invariably, the methodologies have been developed and designed for use in comparing the costs of policies offered by differing companies. What is needed in this study is a method appropriate for measuring the changing cost of a given policy when subjected to an inflationary environment. In designing a procedure appropriate for measuring inflation's impact on the cost of life insurance from the consumers' point of view, it is instructive to consider first a number of procedures that might be used which are inappropriate.4 Such an approach is instructive in that the components of life insurance costing will be introduced in simpler forms, graduating in sophistication as a better understanding of the complexities of life insurance is gained. Moreover, this ap proach serves as a convenient vehicle for demonstrating the essential properties of an insurance costing procedure which adequately takes 3Most of these are conveniently summarized in the nontechnical Report of the Joint Special Committee on Life Insurance Costs (1970). See also Belth (1966). 4The purpose here is not to disparage the methods that have al ready been devised. Most of the methods are appropriate in the appli cations for which they have been employed comparing and ranking the costs of life insurance policies offered by different companies. In fact, there is some evidence (Kensicki, 1977) which suggests that all of the principal costing methods proposed in the literature yield similar rankings of policies. The purpose here is only to draw attention to the limitations of some of the methods if they are applied in estimating the costs of life insurance in an inflationary environment. into account the effects of inflation, while alerting the reader to the shortcomings of (mis)applying some of the conventional costing meth ods presently in use. In evaluating the effect of inflation upon the cost of life insurance, the consumer may be entrammeled by various degrees of money illu sion.5 The concerned consumer may estimate the (net) cost of life in surance in either nominal or real (or alternatively, present value) terms. Furthermore, he may link these costs to the nominal or real (or present) values of life insurance protection in force. Combining these alternatives leads to four general classifications of approaches which the consumer may take that are designated here as "money illusion," "partial money illusion," "policy illusion," and "no illu sion." The degrees of consumer awareness in appraising the cost of life insurance are presented in matrix form in the table below. TABLE 1 Consumer Approaches to Life Insurance Valuation Consumer's Primary Units of Units of Focus: Nominal Protection Real Protection Nominal (Net) Costs Money Illusion Partial Money Illusion Real (Net) Costs Policy Illusion No Illusion 5By "money illusion" it is meant that the consumer, to some de gree, bases his decisions on nominally valued economic data, rather than realvalued economic data. In the four sections that follow, each of these approaches is examined in detail, and methodologies which have been suggested in the literature are reviewed as they relate to these approaches. After an appropriate method for costing has been developed, it will be ap plied to determine the impact of inflation on the cost of life insurance, and the extent to which indexation of policy values can alleviate any adverse impact. Money Illusion A consumer suffering from money illusion may realize that in flation is occurring, but fail to recognize the impact of inflation on the real costs and values of his life insurance policy. "After all," he might remark, "the size of my premium has not gone up in spite of inflation." In fact, if insurance companies are able to lower the pre mium charge due to higher returns on investments from higher 6The features most often included in life insurance policies are premiums, death benefits, cash values, dividends, and terminal divi dends. The dividend features are available in "participating policies," but not in nonparticipatingg policies." All life insurance policies contain one or more of the three basic kinds of insurance: term, whole life, and endowment insurance. Term insurance features premiums and death benefits, with or without divi dends. Whole life and endowment policies feature premiums, death benefits, and guaranteed cash values, with or without dividends. Term insurance policies offer financial protection against the peril of pre mature death; whole life and endowment policies offer protection against the peril of premature death and also offer cash savings, which can be used in providing protection against the peril of outliving one's earning capacity. For further details, see Pfeffer and Klock (1974). 21 interest rates, a consumer suffering from this degree of money illu sion may even believe that the cost of life insurance is declining. His focus is on the nominal costs and nominal level of protection, and un less inflation affects these nominal values, the consumer does not recognize the impact of inflation on life insurance. There are a number of specific costing procedures such a con sumer might employ which may give rise to, or could serve to rein force his illusion. One such method, commonly called the "Tradi tional Method," which has long been in use and which continues to be popular among consumers of life insurance,7 shall be used here for illustrative purposes. Its procedure is to add together the insurance premiums for a number of years, usually twenty, and to subtract the sum of all policy dividends projected by the life insurance company for the period.8 From the resulting figure is subtracted the cash value at the end of the period, and the final amount is then divided by twenty (or by the length of the period if other than twenty years), and by the number of thousands of the amount insured. The result, which may be 7 Indeed, in a survey by the Institute of Life Insurance (1974), consumers of insurance in the United States identified the Traditional Method as being the most "preferred" method. 81n actuality, the calculations use an illustrative dividend scale; the scale does not represent an estimate of what a company will pay, but rather, the current scale paid out on existing policies. 22 positive or negative, is the insurance cost per year per thousand currency units of life insurance in force. The calculation procedure may be represented by the following formula: k k SPn Dn CVk NCk = n=l n=l (1) k where NCk is the average net cost of insurance per year, per thousand currency units of life insurance in force for an insured who surrenders his policy at the end of year k; Pn is the insurance premium payable at the beginning of year n, per thousand currency units of life insurance in force; Dn is the insurance dividend receivable at the end of year n, per thou sand currency units of life insurance in force; CVk is the guaranteed surrender cash value available to the insured in year k, per thousand currency units of life insurance in force; and k is the year of policy surrender. The Traditional Method could undergo refinements to reflect the probabilities of mortality and persistence, but the major drawback of the method is that it fails to give any recognition to the time when money is paid either by or to the policyholder. The focus is entirely on costs and coverage measured in nominal currency units. In none of the terms of equation (1) are inflation and interest even included as factors having bearing on the insurance values. 23 For years in the United States and elsewhere, shortsighted ap proaches to life insurance costing such as the Traditional Method were reinforced by some insurance agents who emphasized net cost calcula tions per thousand units of insurance in force to their clients. In coun tries with unstable currencies, such approaches can foster greatly dis torted views of the true cost of life insurance, since the cost and benefit flows, which occur over long periods of time, may exhibit large ranges of differing real values. To avoid this distortion, some states in the United States now require use of an "InterestAdjusted Method" to be used in calculating life insurance costs. More will be said about this in the section entitled "Policy Illusion." Partial Money Illusion While a consumer beset with money illusion would tend to believe that inflation has no effect or (if inflation leads insurers to lower the premium charges and/or increase the dividends and cash values) a re ducing effect on the cost of life insurance, a consumer having partial money illusion would tend to think that inflation causes the cost of life insurance to increase. A consumer who reaches this stage is somewhat less naive in his thinking. He realizes that indemnification or cash sur render value, when actually received, will exhibit a value that has been eroded by the inflation prevailing in the period intervening the date when the policy is purchased and the date when it ends at death, surrender, or maturity. This phenomenon has often led to criticism of life insurance prod ucts on the grounds that while premiums are paid in "good" money, benefits are received in "bad" money, i.e., money whose value has been eroded by inflation. The life insurance contracts in existence in Brazil before indexing was applied9 were among those subject to this criticism, as shown by the quote that follows. After the Second World War, galloping inflation created in the public a lack of interest for insurance in general and especially for life insurance. While premiums were effectively paid in strong currency, indemnification, in the case of death, or cash savings, at the end of the pol icy period, which were fixed in nominal terms at the be  ginning of the contract, had a purchasing power infinitely less than the same quantities represented when the pol icy first came into force. Life insurance was thus aban doned, and savings were channeled into real estate, stocks and treasury bills.10 A more careful analysis, however, discloses that since life in surance is generally paid for year by year over a long period of time, the value of the premiums paid by the insured also declines over time with inflation; thus, premiums are not all paid in "good" currency. In addition, some of the benefits of life insurance (such as dividends, where available, and death protection) are received during the policy period rather than in a single lump sum settlement at the end, and consequently are not all received in "bad" money. The application of indexation to insurance contracts was first authorized in Brazil by DecreeLaw No. 73 of November 21, 1966. 10The translation of this quote into English, from Chacel et al. (1970, p. 255), is that of the author. 25 A consumer who fails to recognize this may be led to making a distorted appraisal of the effect of inflation on the cost of life insur  ance. Although no models in the published literature were found that reinforce this brand of (partial) money illusion, the notion that pre  miums are paid in "good" money while benefits are received in "bad" money appears to be fairly common with the "man in the street."'1 Policy Illusion A more subtle error is made when the consumer is myopic in his perspective of life insurance, wherein he considers the net cost (in real or present value terms) of a policy offering a given number of units of insurance in force. His focus is incorrectly on the vehicle (i.e., policy contract) rather than on the design (i.e., protection) of the life insurance purchase. Hence, it is denoted "Policy Illusion" in this discussion, and is actually a special subset of money illusion. Because of the subtlety of this type of money illusion relative to the first two types, more space will be devoted to its discussion. In discussing this kind of money illusion, considerable care will be given to the elaboration of a capital budgeting procedure useful in measuring the impact of inflation on the costs and benefits associated 111n interviews with a number of Brazilian citizens, a common reaction was that insurance was a poor investment in an inflationary context because premiums are paid in "good" money, benefits in "bad" money. 26 with life insurance. The effort will be well spent since the model will also be used in the subsequent section of this chapter, after undergoing slight modification to remove the final element of money illusion from the valuation procedure. A Capital Budgeting Approach Several authors have advocated a capital budgeting approach to 12 the problem of consumer valuation of life insurance.2 While such an approach in isolation cannot theoretically justify the purchase of life insurance (Friedman and Savage, 1948), it is useful in analyzing the costs and benefits in monetary terms, and is well adapted to cost com parisons. The principal advantage in using a capital budgeting ap proach is in its ability to account for the opportunity costs of money over time. Since a life insurance contract typically involves streams of payments and benefits over long periods of time, a capital budgeting approach is especially well suited for measuring the values of these flows. Of the capital budgeting approaches available, the net present value (NPV) method is employed here because of its theoretical superi ority (Hirshleifer, 1958) and its mathematical efficiency in providing annual cost comparisons (Kensicki, 1974). The net present value of an 12The first study to treat the purchase of life insurance as a pure capital budgeting decision was Kensicki (1974). For a reader un familiar with capital budgeting techniques, see Brigham (1977, ch. 9,10). insurance policy, per thousand units of insurance in force, can be 13 estimated by the following formula:13 k kk k pn *o(1 DRa +1) IDn (1DRa+j1) 'DRa+ni($ ) E_ ___, ( + .Rn^, Ri=1 (i ) 1 1 I I t k kk k T" lRa n TD TT (1 )', CV (IDI +.0 1n ( k .l il+ i k + (it)) 13 1The formula appears in Babbel (1978) and reflects corrections of some theoretical and technical errors encountered in the Kensicki (1974) version. The model assumes the policyowner will surrender the policy in a particular year, given survival up to that point. While the model was designed for the valuation of a partici pating whole life insurance policy featuring a death benefit, a cash surrender value, and dividends, but not other options, the model may be extended to include options such as renewable and con vertible clauses, policy loan values, reduced paid up insurance, and extended term insurance. For a presentation of a capital bud geting analysis of these options, see Longstreet and Power (1970). The model may also be reduced to fewer terms when it is used for the valuation of nonparticipating whole life and term insurance policies. For a nonparticipating whole life policy, the dividend and terminal dividend expressions are simply eliminated, and for a term policy, the dividend, terminal dividend, and cash value expres sions are eliminated. Later in this chapter, a probabilistic approach will be taken with respect to the uncertain timing of policy surrender. Aggregate lapse rates will be incorporated into the formula and k, the year of surrender, will no longer be viewed as the only year of policy surrender. where E [NPV is the Expected Net Present Value of the insurance policy, per thousand units of insurance in force, for the insured who plans to surrender the policy in year k; Pn is the premium payable at the beginning of year n, per thousand units of insurance in force; CVk is the cash value at the end of year k, per thousand units of insurance in force; k is the year of surrender; DRa+n1 is the conditional probability that an insured who survives to age a+n1 will die before reaching age a+n where the insured's attained age as the policy goes into force is represented by the letter a; it (or in) represents the opportunity cost for the time value of money in year t (or year n) and serves as a basis for determining the present value of any stream of future costs and benefits. In operational terms, it can be viewed as the aftertax interest rate selected by the individual representing his riskfree rate of return in year t (or n); Dn is the dividend payable at the end of year n, per thousand units of insurance in force; and TDn is the terminal dividend at the end of year n, per thousand units of insurance in force. Formula (2) shows the basic cash flows in a participating whole life policy: premiums, death benefit, cash surrender value, and divi dends. The first expression represents the expected present value of the premiums payable. This outflow is weighted at each step to re  flect the possibility that the premiums will not be paid due to death of 14 1 n the insured. Pn is discounted by l+i n(l+ it1) because pre miums are payable at the beginning of the year. The second expression represents the present value of the dividends that are expected to be received by the insured, weighted according to the probability that the insured will survive to receive n them. This expression is discounted by TT1(1+it) since dividends are receivable at the end of each policy year. The third expression represents the death benefit, which is the amount of insurance in force (one thousand units) multiplied by the probability that it is received (i.e., the probability that the insured n will die). The death benefit is discounted by l+in T (l+itl) due to 1+io t=l the availability of the death benefit uniformly throughout the year. The fourth and fifth expressions represent the expected present value of the terminal dividend. The terminal dividend appears twice because this cash flow is payable to the insured when the policy ma tures by death or surrender. The discount factors in the expressions are based on the assumption that the policy will terminate due to death or surrender at the end of a policy year.15 14Later in this chapter the possibility that the insured may not persist with the policy in a given year will be incorporated into the model. (See footnote 13.) 15Although the terminal dividend is not necessarily received at the end of the policy year in the event of death, the discount factor for 30 The final expression represents the expected present value of the cash surrender value for year k, the year of surrender. The probability that death will terminate the policy before the projected year of surrender is included in the numerator, and the denominator gives the discount fac tor associated with cash flows occurring in the year of surrender. Since, for the purpose of this study, it is the life insurance policies of Brazil that are of concern, some of the expressions of equation (2) (along with their problems of estimation) can be eliminated. In Brazil, participating life insurance policies are not available; thus, the dividend and terminal dividend expressions may be removed. The currency unit for measuring insurance in force for a Brazilian policy may now be iden tified as the cruzeiro (Cr$). It will be convenient (but not necessary) to simplify equation (2) further by assuming that i = i for all t. Then, by multiplying each term by a negative one, the equation for a level premium whole life policy reduces to: k 1 n1 k P TT ( 1 DRa+t )D k (I DR l)t=O a+DR (Cr$1000) E NPCk (1DR )t=0 a+n]  (1+ i) (1+ i) n=l n=l k CVk (1 DRa+tl) (1+i)1k the "mortuary dividend" is calibrated as if the dividend is paid at the end of the policy year. The model assumes that if the mortuary dividend (whose value, unlike the death benefit, is not fixed in the contract) is paid out during a policy year, its nominal value will be reduced so that its present value will remain the same. where E NPC is the expected net present cost of the insurance policy, per thousand cruzeiros of life insurance in force, for the in sured who plans to surrender the policy in year k, and the rest of the variables are defined as before. Inflation and the Cost of Life Insurance Policies To be able to utilize equation (3) in examining the effect of anticipated inflation on the NPC of whole life insurance, the relatibn ship between the nominal required rate of return, i, used by the indi vidual in his discount factor, and the expected rate of inflation must be specified. Along the lines of Irving Fisher's (1930) monumental work on the inflation/interest rate issue, their relationship is speci fied as the following: (1+i) = (l+r) (1+j), (4) where r is the real rate of return required on a riskless investment17 and j is the annual rate of inflation expected to prevail during the period of concern to the insured. In the analysis that follows, it is assumed that the real required rate of return is expected to vary independently 1As previously noted, anticipated inflation, as opposed to realized inflation, is the economically relevant factor in consumer valuation of life insurance. For an explanation, see Neumann (1967, 1969). 17 7Life insurance is considered to exhibit characteristics similar to a riskless investment when risk is defined in terms of the proba bility of payment default. of the expected inflation rate.18 Substituting equation (4) into (3) yields the following formulation: k ni k P 1 t(DR a+j) DRa+n1(Cr$ 1000) E NPCk] (l DRal)t=O \ a I  n=l n=l k CVk T (I1DRa+j1 t=j (1+ rk(l+j)k (5) 18While this assumption is often employed in models of a world with perfect certainty, it has been the subject of considerable debate. The hypothesis that real discount rates are unrelated to the rate of in flation, of course, goes back to Fisher (1896, 1930). Mundell (1963) has argued that nominal interest elastic demand or supply of money can lead to a reduction in the real rate of interest due to inflation expecta tions. However, Mussa (1975) and Enders (1976) separately have con tended that a more appropriate macroeconomic model specification leads to conditions that do not necessarily result in a real interest rate decline with increased inflation expectations. Mundell's result or its inverse depends on whether cash balances and capital are com plimentary or substitute assets. The response of interest rates to inflation expectations has been examined by numerous economists. The research is almost entirely based on the response in one national economy, that of the United States. Hess and Bicksler (1975) have shown that the real rate of in terest has not been stable, whereas Fama (1975, 1976) and Feldstein and Eckstein (1970) have found that the real rate of interest has been stable. The empirical justification for the use of this assumption in the Brazilian case is provided in Silveira (1973). The propositions derived in this paper will assume an indepen dent relationship, but will also hold true even if inflation expectations are positively or negatively related to the real rate of discount. The only condition that must be met is that if anticipated inflation rises, the nominal discount rate must also increase. 33 To determine the likely impact of a change in the expected rate of inflation on the cost of life insurance, the method of differential calculus may be employed such that dE [NPCk] dE NPC = dj dj . dj Returning to the level premium whole life policy (equation (5)) and differentiating19 k n1 dE PC = P 1 + r)n1n)(1 + )n (1DR F J (1DR1) TTl+r)l(lI)( lDR at=0 a+j n=l k (Cr$1000) DRa+n(l+ r)n(n)( +)n n=l k CVk (+r)k(k)(l+j)k TT (DRa+t) dj. (6) k a+tl t=l Here it is seen that for k= 1, 2, 3,... the first term will be a nonposi tive expression while the second and third terms (including their 19In addition to assuming independence between the real required rate of return and the expected inflation rate, the model implicitly assumes that (1) mortality rates are independent of the inflation rate, and (2) policy terms are independent of the inflation rate. While the first assumption may be a close approximation to reality (except, per haps, in the case of the fixed income recipient, whose anxiety level increases with inflation, thereby contributing to earlier death), the sec ond assumption may not hold if markets are perfectly free to adjust policy terms in accordance with the rates of inflation. In Brazil, not only are the mortality assumptions permissible in actuarial calculations specified by the regulatory agencies, but the capitalization rate utilized preceding signs) will be positive.. The sum of these three expres sions, whether positive or negative, specifies the direction of impact of a change in the expected rate of inflation on the expected net present cost of a given amount of life insurance coverage. If the sum is posi tive (negative), an increase in the expected rate of inflation will pro duce a rise (decline) in the expected net present cost of the policy. In Appendix A it is shown that for the insured who plans to surrender at the end of the first policy year, an increase in the expected rate of inflation will unambiguously lead to an increase in the expected net present cost of a term policy. It is also shown that if insurance is priced fairly (where present values of expected costs and benefits are equal), the same pattern will hold true for an insured who plans to surrender at the end of the second policy year. In general, however, determining the impact of expected inflation on the cost of life insur ance will require more information with regard to the levels of costs in relation to benefits, the time horizon, and the mortality rates. When such information is provided, it was found for the Brazilian case that the expected net present cost of life insurance increases in the calculations is also regulated. This rate has changed only twice in the past 25 years remaining within a range of four to six percent, far below the rate that would be indicated by the endemic inflationary environment of Brazil. with anticipated inflation for surrender in year one, but decreases with anticipated inflation for surrender thereafter.20 Cost Comparisons Over Time Under Differing Inflation Rate Assumptions To illustrate the potential magnitude of the influence inflation may have on policy costs, two life insurance policies commonly availa ble in Brazil were analyzed. The results of these analyses are pre sented in the tables that follow. The insurance values of a term policy, calculated in accordance with the valuation equation (5) given earlier, are shown in Table 2.21 A real rate of return required on the insurance investment equal to four percent (r= .04) was used in all calculations. Anticipated levels of inflation, j, selected for comparison include zero percent, five per cent and thirty percent. Thus, recalling the Fisher condition ZOThe author has extended the analysis to a leading participating whole life insurance policy offered in the United States, resulting in similar cost patterns. 21Because a term insurance policy produces no cash values, the final term of equation (5) may be ignored. In the equation, actual policy data obtained from one of the largest insurers in Brazil were used. The mortality assumptions employed in the estimations are given in Moura (1976). These are the most recently calculated mor tality statistics available in Brazil. Unfortunately, the data reflect the death experience of group insurers only; using them here is at best an approximation of the mortality rates that apply to individual policies. Another assumption is that the representative individual possesses mortality probabilities similar to the population of indi vidual life insurance policyholders as a whole. ((1+i) = (l+r)(l+j)), the discount factor (l+i) becomes 1.04, 1.092 and 1.352, respectively. In Table 2, column one indicates the policy year under consid eration. Columns two, three and four give the expected present value of the future premiums due, calculated for the assumed anticipated inflation rates of zero, five and thirty percent, respectively. Columns five, six and seven indicate the expected present value of the death benefit available during each policy year under each of the differing inflationary assumptions. Finally, in columns eight, nine and ten, the expected net present cost of the policy for each of the inflation rate assumptions is tabulated. An examination of columns eight, nine and ten reveals that in year one, the expected net present cost of the term policy increases with increasing inflation. This is precisely in accordance with the a priori analysis.22 In year two, however, this trend is reversed and the expected net present cost of the term policy decreases with increasing inflation. Upon extending the examination to the third, fourth and fifth years, it is found that at higher rates of inflation the expected net present cost of the term policy continues to be lower than that of the same policy without inflation. 2See Appendix A. 37 TABLE 2 Five Year Term Insurance Policy: Expected Present Valuces* Present Value Present Value Expected Net Present Yr. YFuture Premiunms Death Benefits Cost j=.00 j=.05 j.30 j=.00 j=.05 j=.30 j=.00 j=.05 j=.30 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 8.35 18.35 18.35 3.49 3.41 3.06 14.86 14.94 15.29 2 35.93 35.09 31.87 7.09 6.76 5.49 28.84 28.33 26.38 3 52.77 "50.36 41.83 10.81 10.05 7.42 41.96 40.31 34.41 4 68.90 64.29 49.17 14.66 13.29 8.96 54.24 51.00 40.21 5 84.34 76.99 54.58 18.65 16.49 10.1 j65.69 60.50 43.77 Policy data for male, age 35; premium of Cr$ 18.35 per Cr$ 1000 insurance in force. Next, the effects of differing anticipated rates of inflation upon the cost of an ordinary life policy over a twenty year period are viewed.23 The valuation formula used is givenby equation (5). In all of the compari sons, the required real rate of return, r, is constant throughout all time periods at four percent. Anticipated levels of inflation selected for com  parison are the same as before.Z4 In Table 3, column one indicates the policy year under considera tion. Columns two, three and four give the expected presentvalues of the future premiums tobe paid, as computed under the assumed rates of an ticipated inflation of zero, five and thirty percent, respectively. Column 23Policy data is from a policy marketed by a leading Brazilian company. Mortality assumptions are the same as before. 24While thirty percent may seem a high anticipated rate of infla tion, this was the average realized rate of inflation in Brazil for the twenty year period from 1948 through 1967. five lists the values of the death benefit in nominal terms for each of the twenty years under consideration. Columns six, seven and eight indicate the present values of these accumulated benefits under their respective inflation rate assumptions. Column nine lists the cash values of the policy at the end of each year, guaranteed in the contract. The three columns that follow indicate the present values of these guaranteed cash values under each of the inflation rate assumptions, adjusted for mortality. Columns thirteen, fourteen and fifteen show the expected net present costs of the life insur ance policy that relate to each of the inflation rate assumptions for the twenty possible years of policy surrender. A comparison of the last three columns reveals that for year one, as with the term policy, the expected net present cost of the ordinary life policy increases with increasing inflationary expectations. In year two and thereafter, increasing inflationary expectations tend to decrease the net present cost of the policy. These cost patterns may seem counterintuitive at first blush, but can readily be explained through examining the interaction of two opposing forces deriving from the timing of cost and benefit flows, and the magnitudes of these flows. The timing of cost and benefit flows leads to an increase in the expected net present cost of an insurance policy, when inflation is introduced into the model. It will be noted that the bene fit flows are always discounted at higher rates than the premium flow, 0, 0 p I V C M~ c I I .o. U ! ~C) d 3 in .cq C ) .' I) >4 0 g a 0 rd o *: i *? G6 >~ I S ^ 4 >< c^ C) N CO n 1 r O 0m oo a'0 In t co 1I0 I I 1o O N0 0 O n n oo co 0s' 0' 0' acOO 000 oo o^ acoo * * t 4 co o ON cn co0o rc oo * *O'O [^ 0sO 00 ION 0 10 O r4 4 0'o 0 s0 In t 14 o i O* 00 'to sr Ln 1 4 NN 00 coa , * *~c .I r t .4 4  NNN C) fl o 000 0n 00 0 (o co aN 00 cO (0 oo c~ 0 0\ ~c^ o o in .o IN r   ' 0 r in * * r1 O*I I o N iC oo4 .i ml o n 0 IN in co N N N In [n N o o 2 00 o o 00 60O O cr C)N 4 * 1 14 NS . in v0 * * (M in 00 14 0 t o 1 r4 A  R\ r i O Mn m rN n t NN" N N NC o In o 4 *4 4 h]< <  ooN  0co * *4 Nfl oo N '00 N4 t cO ocn 0 1 om c' tO O tO co ox a* N 0o N 4  04 ^0 4  0 in co o4 00 4  CONO ' CI o 0' M CO Lo Ln cOg0  r4 ,4 4 sO il m "4 < '* 00 ' co co o t r ** oCo Ol 0 N *~r Or co O 2  ,4  ON I oCo NCo 00 CO 0' ,4  In (I' coo'14 00 C0 00 Ot 1 0 ..0 O0 '.. 0o Do CO c  r4 nn oIn c to 4 a' to r >o.0 N C O o N" N, 1'O 0' '.0 N Co 0 n oo 0'  0 O 0oo 0 *c r .Ir l * 1 __1~__) since they occur later within each time period, producing a positive effect upon the expected net present cost of the policy. The magnitudes of the cost and benefit flows exert a negative in fluence on the expected net present cost of an insurance policy when in flation is introduced into the model, if costs exceed benefits (the usual case). An example will serve to illustrate this point. If two unequal quantities are discounted equally in percentage terms, the larger of the two quantities will decline more in absolute terms. Thus, since costs are of larger magnitude than benefits, they will fall more rapidly in absolute terms with inflation, thereby decreasing the net present cost. It appears in the case of the two Brazilian insurance contracts analyzed, the effect of inflation via the magnitudes of cost and benefit flows dominated the opposing effect of inflation deriving from the timing of the flows in all but year one. In the foregoing methodology, uncertainty with respect to cash out flows and inflows was restricted to uncertainty about the time of death. (The nominal values of the premiums, death benefit, and cash values are fixed in the policy contract and thus are not subject to fluctuations.) The approach was designed to answer the question: "What is the ex pected net present cost of a life insurance policy that will be surren dered, given survival, at the end of year k?" The model may be generalized by extending uncertainty not only to time of death, but also to time of policy surrender. The assumption of the former model that the insured persists with the policy, given sur vival, up until year k, at which time he surrenders with probability of unity, is relaxed. Instead, policy surrender is viewed as possible in any year, with varying degrees of probability in each year.2 Rather than producing a matrix of net cost data according to year of surrender and expected rate of inflation, this probabilistic approach produces a vector of net cost data that vary only according to the ex pected rate of inflation. The probable duration of the contract is al ready taken into account in the calculations. To illustrate this method, the ordinary life policy used in the pre vious example is recalled and the appropriate mortality and surrender rate assumptions are applied as follows: (1) The premiums due outflow is weighted at each step by the proba bility that the premiums will be paid, that is, the joint probability that the insured survives and persists. (2) The surrender cash values are weighted by the probability that they will be received, that is, the joint probability of surviving and per sisting until the end of each year and surrendering at the end of the year. 2The inclusion of persistence rates was suggested by Ferrari (1968) and Belth (1969). The persistence rates employed in the calcula tions that follow were based on the experience of one of the largest in surers in Brazil. According to the actuary of the company, the rates have not varied significantly over time, in spite of varying inflation rates. (3) The expected death indemnification cash flow is equal to the face value of the policy multiplied by the probability of receiving it, which is given each year by the probability that the insured has sur  vived and persisted up until that year and then dies during the year. Having properly weighted the annual cost and benefit flows ac cording to their probability of occurrence, their present values are cal culated (for various rates of expected inflation) and summed over the possible life of the contract. The sum of expected benefits (in present cruzeiros) is subtracted from the sum of expected costs (in present cru zeiros) resulting in the expected net present cost (which by reversing the sign can be viewed as the expected net present value) of the policy. In Figure 1 the expected present values of the cost and benefit flows per Cr$1000 insurance in force are shown, calculated for various inflation rate assumptions ranging from negative four percent to positive forty percent per year. The expected net present cost of the ordinary life policy over the same range of inflation rates is illustrated in Figure 2, which is a derivative of Figure 1. The curve in Figure 2 is derived by taking the difference between the top two curves in Figure 1. One notable characteristic of the particular ordinary life policy analyzed is that the expected net present cost reaches a maximum when stable prices are expected to prevail. As inflation is expected to rise, the ex pected net present cost of the life policy declines. This finding is 240 200 160 120 80 40 0 10 20 30 4( Rate of Expected Inflation (in percent) FIGURE 1: Expected Present Values of Life Insurance Cost and Benefit Flows under Differing Rates of Anticipated Inflation E[PV] Premiums 5 0 E[NPC] per Cr$1000 in surance in force in Cr$ 110 100 90 80 70 60 50 I Ij SII i i I SI i S ". :, : : I * i i i I i i I 1 II I i .. . I I I I II I I I' ~ j I .!I. I * I I t i I II I 1 i I I i, IN, I I I i ] I I I.. I. __.~.._ i  "! I .. .. I _.[ i Ii' ii ... ii I I II I I i i , I I ......li i ! 1 Inflation Rate (in percent) Expected for Duration of Contract FIGURE 2: Expected Net Present Cost of a Whole Life Policy under Inflation c contingent, of course, upon the lapse rates used to weight the compo nents in the streams of costs and benefits.26 The problem encountered in comparing expected (net) present costs of a life insurance policy under differing inflation rate assump tions, as performed in the analyses of this section, is that not only do the costs change, but the product also changes. Therefore, any com parison yields about as much useful information as comparing the cost of apples under one inflation rate assumption with the cost of peanuts under another inflation rate assumption, hoping thereby to infer the ef fect of inflation on the cost of apples. Under inflation, the real protec tion achieved through a policy purchase declines, while indemnification varies in real terms according to the date of death. Since the nominal terms of the policy remain the same, it is tempting to compare the ef fects of inflation on the cost of policies, rather than on life insurance protection. However, the policy is merely the vehicle through which the objective (i.e., protection) is sought. Thus, attention is more properly centered upon the effects of inflation on the cost of protection. To demonstrate more concisely the problem associated with using the Expected Net Present Cost model, employed above, in 26The lapse rates applicable depend upon the particular consumer involved in the purchase of the policy. In the calculations performed for the graph, aggregate lapse rates were used and it was assumed that a representative individual manifests probabilities of voluntary policy sur render similar to the group. determining the effect of inflation on the cost of life insurance, the for mula is given in simplified notation below: E[NPC/per 1000 units = E iV(C/per 1000 units E V(B]/per 1000 units (7) ins. in force ins. in force ins. in force In the above formula, C and B represent the cost and benefits associated with each one thousand currency units of insurance in force. The E and PV are expectations and present value operators, respectively. Unfor tunately, the numeraire to which the benefits and costs are attached is a poor choice. The one thousand nominal units of insurance in force rep resent different levels of protection under differing inflation rate assump tions. Hence, such a model is inappropriate for use by a consumer free from policy illusion in determining the cost of life insurance. Regrettably, many of the life insurance cost computation methods presently in use incorporate elements that tend to induce policy illu sion on the part of the user. For example, the "InterestAdjusted Method," which was developed to overcome defects in the "Tradi tional Method" and has been recommended by the Joint Special Com mittee on Life Insurance Costs (1970), and required by law in some states of the United States, makes explicit use of a nominally valued numeraire (one thousand units of insurance in force). Accordingly, when the discount factors are adjusted to reflect inflationary expec tations, the change in the cost index that results will only indicate the influence of inflation on the real cost of life insurance policies, and not on the real cost of protection available through the policies; hence the term "policy illusion." No Illusion It has been shown that if a consumer is befuddled with money il lusion, partial money illusion, or policy illusion, he is likely to view the influence of inflation as producing unchanged, higher or lower costs, respectively, of life insurance. To arrive at a costing methodology in which money illusion, in any of its forms, is absent, a good starting point is formula (7), copied below for convenience. E N /per000 units = E 'V(C/per 1000 units E L(B]/per 1000 units ins. in force ins. in force ins. in force The nominal numeraire, which gives rise to policy illusion, can be elim inated in alternative ways. For the Expected Net Present Cost method, considered in detail earlier in the chapter, a simple and effective ap proach is to simply divide each term in the above formula by the last term that appears on the right hand side. The resulting equation is PC/pr 00 uit(%:1 E E [V(BI ENPC]/per 1000 units E V /per 1000 units /per 1000 units ins. in force ins, in force ins. in force (8) E PV(B*/per 1000 units E PV(B /per 1000 units E EV(B]/per 1000 units ins. in force ins. in force ins. in force which simplifies to E PC E [PV(CG1 (9) E PV(BI) E PV(B]) Note that in the above expression, the numeraire of one thousand nominal units of insurance in force has been cancelled out of all the terms. What remains is the expected net cost (in present value terms) per unit of expected benefits (also in present value terms). It is noted that the above expression differs from the expected costbenefit ratio by an amount equal to unity. In economic terms what has been done is that the price of insurance has been deflated by the actuarially fair price of insurance. This is appropriate because a fair price is "cost less" to the individual in the sense that his expected expenditure is equal to his expected benefits (Ehrlich and Becker, 1972). Three problems immediately arise in operationalizing the gen eral formulation given by (9). First, the matter of what should be con sidered a cost and what should be considered a benefit of life insurance. This problem was not encountered in the horizontal outlay of the Ex pected Net Present Cost method, but the vertical nature of the cost benefit ratio raises questions of what properly constitute the components of the ratio. Subtracting a particular item from the cost component rather than adding the item to the benefit component will affect the ratio. Although insurance premiums and death benefits are unquestionably costs and benefits, respectively, of a life insurance policy, other items such as cash surrender values, and dividends (where available) can 49 either be viewed as reductions in the cost or as additional benefits (Belth, 1969). Still another approach is to remove such elements al together from insurance cost calculations and to determine separately the costs of these benefits.27 A second problem arises if uncertainty is extended to include the inflation rate, as well as the intraperiod moment of death. Up to this point it has been assumed that if death occurs in a given period, it oc curs, on the average, about midway through the period. Another implicit assumption was that there is no uncertainty regarding rates of infla tion that will prevail in the future. Relaxing either of these assumptions will result in an additional source of randomness in the values of cost and benefit flows. Relaxing both assumptions will serve to reinforce the possible deviations in realized flows from expected flows. For any insured who is concerned with more than just the first moments of the cost and benefit distributions, the existence of uncer  tainty will affect the consumer's perception of the cost of insurance protection. For example, if the insured is risk averse with respect to the value of his bequest in the event of death, the fact that the expected present value of his bequest is adequate may not quell his concern. For such an insured, the unit of account expected present value units of Z7Examples of this approach can be found in Levy and Kahane (1970), Ferrari (1968), Schwarzschild (1967, 1968), Linton (1964), and Belth (1961, 1966, 1968). benefits will no longer serve as an adequate numeraire. Expecting to leave "on the average" a given realvalued bequest in the event of death is not the same as leaving with certainty the same real valued be quest in the event of death. Since death is a once in a lifetime occur rence to the (typical) insured, the law of large numbers is of little con solation to the risk averse individual or to the heirs for whom he is seeking to ensure financial security. There is a third problem in using the model. Different expected cash flows may elicit differing discount rates, even in a model such as the one presented here where uncertainty is limited only to survival (Hirshleifer, 1970). If the model is used in isolation as a decision criterion to buy or not to buy insurance it should be recognized that, even though the expected cash flows associated with insurance may not exhibit covariance with other financialassets in the insured's invest ment portfolio, the death benefit has negative covariance with the return on human capital portion of the insured's portfolio and thus may elicit a lower or even negative discount rate.Z8 28Whether or not differing discount rates may properly be ap plied to separate cash flow streams that are part of a "package" is a debatable issue. Arditti (1974) has demonstrated that the procedure of using a discount factor for cash flows exhibiting complete certainty dif ferent from that applied to uncertain flows is appropriate. Other cash flows, he contends, should be discounted by a similar riskadjusted rate, because they are not separable. Whether these conclusions are appliable to the life insurance product is debatable. The policy can be arranged to include any desirable cash flow provisions, and is cancellable at any moment. 51 Having identified three potential problems in using formula (9) for the valuation of life insurance, we proceed to deal with each. In the remainder of this chapter, the first problem will be resolved by treating the surrender cash value as a benefit, rather than a reduction in cost; hence, it is included in the denominators of each term in for mula (9). The second and third problems both evoke individual utility considerations in their resolution; these problems are deferred until the next chapter, where they are handled in the more powerful analyti cal framework of TimeState Preference. In defense of the methodology employed in this chapter, there is merit in devising a cost index that is independent of specific utility functions. The index is then universal in application, and provides an unbiased measure of insurance costs in expected monetary (present) values. This information can then be evaluated in conjunction with other factors (e.g. other moments of the cost and benefit distributions) which may receive varying levels of im  portance, in accordance with the individual who is appraising the insur  29 ance product. To determine the impact of anticipated inflation upon the cost of life insurance (as measured by the expected net present cost per unit of expected benefit in present value) the differential of equation (9) may be 29The Expected Monetary Value method is discussed at length in Mao (1969), and is evaluated in conjunction with the variance associated with the resulting value. taken as follows: E PV(C E [NPCJ d [P d = E PV(B]J dj E LPV(B) dj If the derivative of the costbenefit ratio is positive, a rise in antici pated inflation will cause an increase in the expected net present cost, per (presentvalued) unit of expected benefit. In Appendix B it is demonstrated mathematically that as long as an increase in antici pated inflation is associated with an increase in the discount rate, the sign of the derivative of the costbenefit ratio is positive; hence, an increase in expected inflation will be associated with an increase in the cost of life insurance. To illustrate the potential magnitude of the effect of inflation on life insurance cost, actual policy data may be substituted into formula (9). For expository purposes, the same ordinary life policy analyzed in the previous section will be subjected to the analysis procedure specified by formula (9). Returning to Figure 1, the pro cedure simply involves selecting an inflation rate and dividing the expected cost by the expected benefits (top two curves) at that point, and subtracting one from the resulting quotient. Below are presented the expected net present costs, per unit of benefit, at various levels of expected inflation. 53 Rate of Inflation PV Expected Cost Expected Net Present 1 Anticipated (r= 4%) PV Expected Benefits Cost per Unit of Benefit  3.85% 237.83/136.76 1 0.74 0.0 % 181.53/75.42 1 1.41 + 5.0 % 138.24/38.97 1 2.55 + 10.0 % 112.64/22.84 1 3.93 + 15.0 % 96.35/14.92 1 5.46 + 25.0 % 77.46/8.19 1 8.46 + 30.0 % 71.56/6.55 1 9.93 + 40.0 % 63.34/4.68 1 12.53 An unambiguous upward trend in expected cost is shown in the above figures for life insurance protection as anticipated inflation rises. In particular, a rise in anticipated inflation from zero to forty percent is shown to be associated with a rise in expected net present cost per unit of benefit of 791 percent. Although the precise magnitudes of cost increases shown are obviously dependent upon the conditions of the par ticular policy analyzed, including the surrender and mortality rate as  sumptions utilized, the upward trend is not subject to these qualifica tions. Increasing rates of inflation always lead to higher costs of life insurance protection available through conventional contracts. The only exception to this rule is when the nominal terms of the insurance contract (specified premiums and benefits) adjust to fully compensate the consumer for changes in inflation rates. Unfortunately, this is seldom the case. Life insurance industries are often constrained by governmental regulators, and are generally very slow to adjust to in flation.30 Maintaining Real Values By reducing the realvalue of protection provided by life insurance, inflation thwarts the primary design of the insurance contract, i.e., to provide protection against the perils of premature death and of outliving one's earning capacity. Thus, it may be argued that a more meaning ful measure of inflation's impact upon the valuation of life insurance will reflect the cost of achieving a desired pattern of real protection over time (as opposed to simply providing for a desired expected real value of protection). Whether the insured desires increasing, decreas ing, constant, or some other pattern of real protection over time is a behavioral question and beyond the scope of this study. For expository purposes, it will be assumed in the following two subsections of this paper that the insured wishes to maintain a con stant real level of protection over time against the peril of premature 30See explanation in footnote 19. The blame does not necessarily lie with the regulators. Uncertainty regarding the future interest rates makes it irresponsible for a regulator to permit the full expected rate of interest (and inflation) to be incorporated in the actuarial calculations, because a negative deviation between the expected and realized rates could create problems of insolvency. Furthermore, the insurance in dustry may deem it more profitable not to adjust their rates to incorpo rate interest and inflation rate expectations, especially if insurance de mand is price inelastic. 55 31 death. Protection from outliving earning capacity will be adjusted in nominal terms according to steps taken by the insured aimed at main taining constant a real level of protection against the peril of premature death; this adjustment, however, will not necessarily result in main taining constant a real level of protection against the peril of outliving earning capacity. Additional Insurance Purchases The insured may seek to maintain a real level of protection in several ways. Insurance policies are often equipped with provisions for increasing coverage periodically through purchases of paidup additions, term riders, etc. Another option available to the insured is to periodically purchase additional policies to compensate for the value of the protection eroded in his existing policy(ies). If indexlinked policies are available, still another option is possible for the insured. It is assumed in the following analysis that the insured seeks to maintain a real level of protection through annual purchases of addi tional life insurance policies (hereafter abbreviated AAP). Whether these purchases are made to cover real protection reduced by realized 31 31The analysis is only slightly more difficult for alternative pa terns of desired protection. It should be noticed, moreover, that in order to achieve the goal of maintaining constant a desired real level of protection, policy values would need to adjust constantly to compensate for changes in the purchasing power of money. Thus, constant real levels of protection will not actually be maintained, but will only be ap proximated by strategies listed in this section. or anticipated inflation is unimportant for the present analysis as both yield similar results. In this section it will be assumed that adjustments are made each year to recapture protection value lost due to realized inflation.32 As with the previous analysis, policy fees incurred will be ignored.33 However, later in this chapter the direction of their ef fect under inflation, when included in the analysis, will be shown. Examined first is the effect of inflation upon an insured who seeks a constant real level of protection through term insurance. Term in surance offers protection against the peril of premature death, but it offers no cash values which could be used to offset expenses incurred after earning capacity is outlived. Hence, it may be inferred that the insured seeks only to protect himself from the first peril through the medium of life insurance if he patronizes exclusively term insurance products. At the end of each year, the insured increases the coverage of his death benefit by the amount of inflation realized during the 32In a world of perfect certainty, anticipated rates of inflationwill be realized. Alternatively, the insured may opt to adjust his insurance coverage so as to obtain his desired real level of protection at the end of each period. This strategy entails carrying (and paying for) more real protection than is desired during the period. The results of either strategy yield similar effects upon the net present costs. 3In addition to ignoring policy fees, it is further assumed that life insurance coverage canbe purchased in any amount at a fixed percentage costperCr$ 1000 coverage. In reality,there are restrictions onthemini mum amount purchasable per policy (and the amounts insurable are usually multiples of Cr$ 1000). Furthermore, a policyholder's health may render him ineligible to continually purchase new life insurance policies. period, which shall be designated j. This adjustment will incur premium costs to the insured that will rise by the same percentage that coverage rises, namely j. The effect of these adjustments upon the expected net present cost of the insurance can be determined by returning to equation (9) (omitting the cash surrender value expression, which is not relevant in term insurance policies) and making the indicated revisions. To facili tate the presentation notationally, the following abbreviated notation will be employed: n s = TT (1DRa+t_,) = the probability of survival through the end of year n; S nI snl TT (1DRa+t_,) = the probability of survival through n (1DRaI) t=0 the end of year nl; d = DR = the probability of death in year n, given survival up through the beginning of the year; Rt = (+rt) = the real discount factor; and t = (l+jt) = the inflationary adjustment to the discount factor. Substituting these expressions into equation (9) yields:34 34Unlike the analysis of insurance encountered earlier in this paper, the analyses of this section do not assume level premiums; on the contrary, Pn reflects the premium required to insure during each year n, with no surpluses in earlier years carried forward to cover increasing mortality rate expenditures in later years. If the insured may only purchase level premium term policies of (say) k years, the strategy of additional policy purchases would serve to increase the net present costs even further by increasing the real value of premiums charged. E NPC]_ E [PV(Bj which, when k V Pnsnl 1 nI J o TT(Rt J) n=l Ro0 t= 0 k ' dn(Cr$ 1000) 1 1 R2 J ni n=l R"Jo 7O(RtJt) Rro t= revised to include annual additional policies purchased, becomes k ni EP s TT i n n1 TR t=o 1 n1 E^pcJ ,n Rojo TTr (RtJt) E [NPCZf n=1 R ol t=0 = 1. (1 E[PV(BI k 1 nI d (Cr$1000) TT J \ n Jo t=0 t 1 1 nn n1 n=l TT (RtJt) Rojo t=O Equation (11) simplifies further by noting that most of the terms de noting inflation adjustments cancel out of the ratio, leaving 0) 1) Pnsnl nI Ro TT Rt E[NPC] = n=l t= (1) E PV(B)1 k [ ] dn(Cr$1000) 11 RnJn n1 n= 1 =TT Rt R0 t=O If the AAP strategy were able to completely overcome the value erosion produced by inflation, the valuation equation would not retain any adjustments for inflation, since adjustments for payments and I 59 benefits would exactly be canceled by adjustments in the discount rates. However, as shown in formula (12) above, the AAP strategy does result in the retention of one of the inflationary adjustment terms: 1 J2 remains in the denominator of the (lower) death benefit expression. n Therefore, for any positive Jn' the expected cost of the strategy will be increased. Hence it may be concluded that the expected net present cost, per unit of benefit, of a term insurance policy under stable price level conditions will always be less than the expected net present cost of maintaining a real level of protection through additional policy pur  chases under inflationary conditions. Next, the same strategy (AAP) is examined, but with regard to investment life insurance (i.e., policies exhibiting cash values). This will require the inclusion of the cash surrender benefit added to the denominator of equation (12). Since the effect of inflation on the pre mium and death benefit cash flows has already been examined, only the surrender cash value expression remains to be analyzed. It is noted a priori that if inflation has either a neutral or negative effect upon the value of this component, the overall effect of inflation on an investment life insurance policy will be one of increasing its expected net present cost. It is helpful to observe at the outset that the cashvalue compo nent will remain neutral to inflation in real terms only if the AAP strate gy achieves adjustments in the nominal surrender value that exactly k compensate for inflation. More formally, CVk must rise by TT Jt t=l so that: k CVksk iT J CVksk k k j t k K (13) k k TT RJ TT Rt t=l tt t=1 Unfortunately, the AAP strategy does not achieve this result. The nominal cash value expected to be received upon surrender in year k is given by sk CVk + CVk 11 + CVk jZi2 + ... +CVjk 1 *... Jkz, (14) which reduces to kl1 n1 sk CVk + CVknin 17t (15) n=1 Whether (15) is greater than, equal to, or less than the numerator of (13) determines if inflation will subtract from, render unchanged, or add to the expected net present cost per (presentvalued) unit of benefit of an investment life insurance policy. The l.h.s. numerator of (13) can be expanded and rewritten as sk CVk + CVkjl + CVkjJ1 + ... + CVkjklJ... Jk + CVkjkl k1k (16) Clearly CVk> CVkl> CVkZ so that for positive j's (16) will always 61 exceed (14).35 The conclusion follows that the effect of inflation on the cashvalue component contributes to an increase in the cost of a life insurance policy with a savings component (as measured by the expected net present cost per unit of benefits). Once again, stable price levels result in a higher valuation to life insurance than rising price levels. IndexLinked Policies Attention now turns to indexlinked life insurance policies, as a strategy to mitigate the adverse effects of inflation upon the value of life insurance. In particular, the analysis focuses upon the Bra zilian system of indexing which was authorized for insurance contracts beginning November 21, 1966. Brazilian life insurance contracts are offered in a variety of "indexlinked" packages. A contract may be opted for in prefixed indexes, or may be periodically adjusted for realized inflation rates. Policies adjusted for realized inflation rates may feature partial or full adjustments of premiums, death benefits and cash values.36 Of special interest are the policies fully adjusted for realized inflation, since these will best help the insured maintain a real level of protection. 35Cash values increase in nominal terms over the life of the contract. 6For instance, a policyowner may elect to have his policy premiums and benefits adjusted for 50, 75 or 100 percent of the index utilized in "correcting" the values of life insurance contracts. In Brazil a fully indexed37 life insurance policy has the follow ing characteristics: (1) Annual premiums are adjusted at the beginning of each year in accordance with the rate of inflation realized during the course of the 38 preceding year. (2) Death benefits are adjusted at the beginning of each year to re cover the protection lost due to the erosion of inflation from the pre vious year. The effect of adjusting the death benefit on an annual basis, (where the death benefit is available uniformly throughout the year), differs from the effect of adjusting premiums due on an annual basis, since premiums are an annual cash flow. In an inflationary environ ment, the latter maintains the real value of the cash flows, whereas with the former the real protection is being constantly eroded. In other words, the costs are maintained constant in real terms but the death benefits are losing value throughout the year. For instance, if an insured dies ten months into the policy year, his beneficiaries re ceive the same amount in nominal terms as if he had died at the begin ning of the year. Where inflation rates are high, there can be a sub stantial difference between the real value of the payments. 37Although a policy may be 100 percent linked to the index, the index may not necessarily reflect 100 percent of the realized infla tion. For an explanation of the formulation and modification of the in dices used for life insurance contracts, see Appendix C. 3Because of difficulties of a technical nature in preparing the indices, the value actually used may reflect another period (see Appendix C). (3) Surrender cash values, for policies that feature them, are tied in fixed nominal terms to the value of insurance in force. Because the nominal value of insurance in force is only readjusted at the beginning of each year, and thus, the cash value received at the end of the year will not reflect any adjustment for inflation occurring during the year. The effects of inflation are easily seen when the above charac teristics are reduced to mathematical form. The formula for the ex pected net present cost per (presentvalued) unit of benefit of an index linked term policy is precisely the same as formula (12), where the in sured purchases new policies on an annual basis to compensate for the effects of inflation realized during each year. Note that both strategies result in higher insurance costs than those obtained where price levels are stable. (If prices were stable, the appropriate valuation formula 1 would be identical to formula (12) after the J2 term is removed.) n The equivalent formulas for determining expected net present cost ratios may lead one to conclude that an indexed term policy has equal merits to a simple annual routine of additional policy purchases. By relaxing some of the assumptions, however, it becomes readily ap parent that there are significant differences: (1) Earlier it was assumed that the insured could purchase additional term insurance policies at will. In practice, the future state of the insured's health is uncertain, and the insured may not be able to always qualify for new insurance policies. (2) Upon relaxing the assumption that the insured could purchase ad ditional term insurance in any quantities, the insured may be forced to purchase either more or less coverage than he desires. Ifthemini mum amount of insurance purchasable is high (say Cr$250,000) this may lead to extended periods where the insured is under protected. (3) Medical examinations are often required for new policy purchases, and this recurring cost can amount to a substantial sum over time. (4) Policy fees and other expenses are often fixed costs, independent of the level of coverage desired. Buying many small policies peri odically will incur these costs more frequently, if they are added directly to the premium charged. (On the other hand, these costs added to the premium would automatically increase with indexlinked con tracts in accordance with the rate of inflation. It can be presumed, however, that costs incurred by taking the first approach would be greater unless the inflation rate is in excess of 100 percent.) (5) Making annual new insurance policy purchases involves a commit ment of time and energy above that involved in purchasing an index linked policy. When the above factors are taken into consideration, it is concluded that in an inflationary environment, an index linked term insurance policy will result in lower expected net present costs, per unit of benefit, than will a series of term policy purchases aimed atmaintaining a real level of pro tection. 65 Will index linking an investment policy produce similar benefits ? The analysis for the premium and death benefit cash flows is the same as that given for a term policy. That leaves only the surrender cash value to investigate. Previously it was noted that to remain neutral to the effects of inflation, the nominal value of the surrender cash values must increase at the same rate as inflation (as per equation 13). Because cash values in Brazilian indexlinked policies are only adjusted to recapture the real value lost to realized inflation at the beginning of the next year, their expected present values are given by k CVkk s i TT J CVksk CVkk tJ 1 t1 kk (17) k k TT (RtJt) k TT Rt t=l t=l This formulation clearly shows that under positive rates of infla tion (Jk>0), the expected present value of the cash value will be less than if there were no inflation. Expanding the numerator of the left side enables a more precise comparison of the cash values under the vari ous alternatives. Formula (18) shows the nominal cash value expected for an indexed policy. Formulas (14) and (16), representing the ex pected cash values obtained through annual additional policy purchases and through a hypothetical policy which maintains inflation neutrality, respectively, are reproduced here to allow for convenient comparison. 66 Sk CVk + CV CVkjZ +*V + CVkjk11 'kZ] (18) sk CVk + CVklj + CVkz 1 +***+ CV lkll k2 (14) sk [cvk + CV + CVkjZI1 +...+ CVkjk1 J1 k2 + CVkJkIl k (16) Because these nominal cash values are all discounted by the same fac tor, namely TT RtJt, a direct comparison of the above values is admis sible. A comparison of the above terms reveals that an inflation neutral policy will return ((16)(18)) or skCVkkJlk 1Jk1 in nominal expected cash value more than a Brazilian "fully indexed" policy will return. However, a Brazilian "fully indexed" policy will return ((18)(14)) or k1 k2 sk (CV CV )j TT I k Z k kn nt=l t n=l more than would be achieved through purchases of additional policies. (It is interesting to note here that removing the CVk term at the be ginning of each of the above three expressions gives in nominal terms the additional cash value that each alternative is expected to return over the purchase of a single, nonadjusting policy.) It is concluded that an indexlinked investment life insurance policy will yield superior returns (or result in lower expected net present costs) than those ob tained through additional policy purchases, not only for the five 67 reasons stated earlier in the term policy section, but also because an indexlinked policy will result in higher cash values. In Table 4 the mathematical determination of the present values of the insurance cost and benefit flows is summarized, according to the mode of policies purchased.39 Note that while in the first three rows the present values of the costs are maintained in real terms, the present values of the benefits are not, except when there is no inflation. A graphical representation (Figure 3) follows the summary table, in which the present values of actual policy data (as determined by the expressions in Table 4) are combined with mortality and persistence data characteristic of the insurance industry. The resulting net pres ent cost ratios for a typical nonindexed whole life policy and an index linked whole life policy are represented by the upper and lower curves of the figures, respectively. It is worthwhile to note the dimensions of the cost changes as inflation increases. Neither the indexed nor the nonindexed policies are insulated fully from the effects of inflation, as both curves slope upward to the right. However, the indexlinked policy exhibits a much more gradual slope upward, indicating less cost sen sitivity to inflation. The data from which the curves were plotted is provided in Tables 5 and 6. 39The formula components shown in Table 4 do not include the sk and sn_ multipliers in the premiums and cash value columns, because each of the insurance modes uses the same survival rates; however, this omission was only for the purpose of convenient comparisons, and the factors are included in calculations used in constructing the figures. nl in   o 0 ,t 4 1 +) + Uk I o U )o S, . 6 1 + II o c 4 oo 4 o 3 0 I0 1 C S. o o o so ra  H > 1 0 d P 0 5 U (d 4 4U) Ic *4 r o N In 0 1.H 0 4 i i 1 C r e4 0 0 ( > 0 gc uU A01 4 0 p 4 a P0 P., U) o 0 ,i 4d U C) U) 0 Lo 4 (d l U) '4 t4 (d '1 rd in 0) H >. I iii i 1!~~j .i  4I t 4 i r i~* i I.. _______i i fi i 4. i .  i :'. i i I__._. 1. r~ .t1 II. F LtijI 'Ii L.i imr [ I 'i ]1 I i I J I I 4 4 0 10 20 30 40 Inflation Rate (in percent) Expected for Duration of Contract FIGURE 3: Expected Net Present Cost per (PV) Unit of Expected Benefits Ratio E [NPC] E rV(B r J_ 4 :r I:: I 7i  I .. I .. [ .. . 1 .i i  i I ''II Nonindexcd Policy ,,  I i I ; I , *^Hl h!ili ' _.: i : I ; ; i i ji l :  ,L . .....  I i . Indexed Policy  i i ..  I i :   _ I  , ..... I, .I_  / I I ii / i i ; i l FT;Fii i .... ; l .. fi.. I l I ..* .I.. .. l  I .... i i it  i / : / , ?1 '! .. ... /t .. ....   .. i . .. . .I .      TABLE 5 Nonindexed and Indexed Policy Values Expected Rate of Inflation' 3.9% 0.0% 5.0% 10.0% 15.0% 25.0% 30.0% 40.0% Nonindexed Policy E [PV E[PV] E PV] Premiums Indemnification Cash Surrender Value 237.83 181.53 138.24 112.64 96.35 77.46 71.56 63.34 52.86 34.26 21.34 14.52 10.60 6.61 5.48 4.07 83.90 41.16 17.63 8.32 4.32 1.58 1.07 .61 Indexed Policy 3.9% 0.0% 5.0% 10.0% 15.0% 25.0% 30.0% 40.0% 181.53 181.53 181.53 181.53 181.53 181.53 181.53 181.53 34.94 34.26 33.43 32.67 31.95 30.64 30.05 28.95 42.81 41.16 39.20 37.42 35.79 32.93 31.66 29.40 * Real rate of interest used in all calculations was four percent. 71 TABLE 6 Life Insurance Cost Benefit Ratios with Inflation * Real rate of interest used in all calculations was four percent. *: "R" designates the ratio of costbenefit ratios under expected inflation rate assumptions of forty percent and zero percent. Summary and Conclusions The purpose of this chapter was twofold: (1) to develop a model capable of measuring the cost of life insurance, and (2) to utilize the model in measuring the effects of inflation and indexation on the cost of life insurance in Brazil. 72 En route to accomplishing these objectives, a number of inter esting byproducts emerged. Among the more important were (1) a classification scheme for identifying various degrees of money illusion on the part of the consumer; and (2) the observation that some of the insurance costing procedures serve to reinforce consumer money il lusion in one of its forms when employed in an inflationary context. Finally, a method appropriate for determining the cost of life insurance was introduced. The method was capable of measuring changes in the cost of life insurance incurred by inflation. The model was then applied in determining how the cost of insurance would change under different rates of anticipated inflation. It was demonstrated mathematically that when insurance terms are slow to adjust to the realities of inflation (perhaps due to regulatory constraint), the net cost of insurance would rise in real terms. Actual policy data were then substituted into the mathematical model, and the analytical con clusions were corroborated. In one example it was shown that with an expected inflation rate of forty percent (somewhat below the realized inflation rate of the past three years in Brazil), the net cost per unit of insurance benefit was almost nine times higher than the cost of similar protection under stable prices. Next, the case of the insured who through various methods attempts to maintain the real value of his insurance was examined. Although there are several approaches the insured could take, investigation was 73 limited to two of these: purchasing indexlinked policies or purchase  ing additional policies. These two approaches were selected for fur  ther investigation for three reasons: (1) Both approaches are attempts to maintain a real level of protec tion and are more likely to approximate this goal than other approaches. (2) The two approaches have been viewed as equivalent by some writers. (3) Each approach has been said to neutralize the value reduction produced by inflation. In summary, the investigation resulted in the following findings: (1) Neither of the two approaches maintains constant a real level of protection against either the peril of premature death or the peril of outliving earning capacity. (2) Both approaches help in better achieving a desired real level of protection than could be accomplished by the purchase of a single, nonadjusting policy, but at higher premium outlays. (3) The purchase of an indexlinked policy always results in lower expected net present costs, per unit of (realvalued) benefit, than the purchase of additional policies on an annual basis, under positive in flation. Conceivably, an indexed policy could be designed to maintain con stant the real levels of protection (via continuous indexing), but until 74 such a product is marketed, it can be concluded that a world with stable prices will best promote the objectives of an insured desiring to maintain constant a real level of protection against the perils of dying prematurely or outliving earning capacity. CHAPTER 3 RATIONAL LIFE INSURANCE PURCHASING AND INFLATION Survey of the Literature A number of writers have examined various aspects of rational insurance purchasing, but most of their studies have not dealt explic itly with inflation as an explanatory factor. This is understandable, in part, because inflation rates at the time when many of these studies were published were mild relative to their more recent levels, at least in the United States. Perhaps another reason that insurance purchasing has not been considered in the context of an inflationary environment is that the models used have often been illdesigned for 1 that purpose. For example, models for rational insurance purchasing are given by Smith (1968), Mossin (1968) and Ehrlich and Becker (1972). Al though the models are oriented toward insurance in general, and prop erty insurance in particular, their applicability to the problem of life insurance is straightforward. However, in none of these analyses is 1This, of course, is not to say that the models were illcontrived. Each model sheds additional light on the theory of optimal life insurance, and all have greatly contributed to the present writer's understanding. The approach set forth in this chapter has bene fited from and incorporated components of several of the models. 75 76 the dimension of time included (or when included, there is no assumed time preference for wealth i.e., the rate of interest is set at zero). Since inflation is a phenomenon which, by definition, occurs over time, its significance is effectively precluded from the analyses. Another set of articles, which are oriented specifically toward life insurance, include the dimension of time. Yaari (1965) and Richard (1977) propose continuoustime models to consider the problem of consumption and portfolio choices when lifetime is uncertain and life insurance is available. The life insurance offered is of an instan taneous term variety where new insurance contracts, which remain in force for an infinitesimally short period of time, are continually bought at each moment in time. While the continuoustime frame work is intuitively appealing for analyzing other aspects of their models (such as consumption), the instantaneous term variety of insurance could conceivably result in the prospective consumer spending twenty four hours of each day at the insurance office applying for new insur ance policies, leaving little time for consumption related activities. The loss of the purchasing power of insurance benefits resulting from the inflation that transpires during the interval between premium payment ZIf each policy is accompanied by a fixed policy fee (to help offset transaction costs the usual case) the transaction costs involved could be phenomenal. The closest kind of policy actually offered to the instantaneous term variety is flight insurance, which expires in hours. An ingenious analysis of economic aspects of the purchase of flight insurance is given in Eisner and Strotz (1961). 77 and insurance settlement is of little or no consequence when the time interval approaches zero. The omission of inflation in their analyses was therefore justified in light of the variety of insurance that was examined. In a third group of articles life insurance demand is treated in discrete timeperiod analyses. Hakansson (1969) and Fischer (1973) propose multiperiod models in which the utility functions exhibit con stant relative risk aversion. Many of their more specific results turn strongly on their choices of utility functions, and although the models conceivably could have permitted inflation to be included as an explanatory factor, this aspect of the problem was not explicitly considered. Fortune (1973), JonesLee (1975) and Klein (1975) have approached the problem of optimal life insurance within the framework of a twoperiod model (two timepoints). Using meanvariance, conditional expected utility, and timestate preference analyses, respectively, each author assumes that the insured either dies irnme diately after the payment of the initial (and only) life insurance premium, at which time an insurance benefit is paid to the insured's benefici aries, or that the insured dies (or retires) at the beginning of the second period, at which time he either receives the cash value (savings por tion) of the policy or receives nothing, depending on the features of the policy purchased. Under these formulations, inflation's possible im pact is restricted to its effect upon the savings that are sought through 78 the vehicle of life insurance, since in the event of premature death, the ink will not have dried on the policy contract before death occurs. This obviously does not allow time for inflation to affect the level of real protection provided against the peril of premature death. Finally, Razin (1976) presents a twoperiod (three timepoints) modelto high light the effect of lifetime uncertainty on the optimal investment in human capital with and without markets for life insurance. In this case, life insurance is of the singleperiod term variety, and death occurs either at the end of the first period, or at the end of the second period. This formulation allows for the inclusion of inflation as an explanatory variable, but it is not explicitly introduced intothe analy sis. Moreover, the particular model he presents treats life insurance in terms of "percentage coverage" instead of absolute coverage, and is thus unable to result in a demand for insurance function (when infla tion is allowed to influence the price of insurance coverage) that yields unambiguous predictions with respect to changes in anticipated inflation. The only theoretical model encountered where inflation was ex plicitly posited as an explanatory factor for desired levels of life in  3 surance protection3 is that in Hofflander and Duvall (1966). For this reason, more space willbe usedhere to scrutinize their presentation. 3 In this chapter the phrase "life insurance protection" is used to designate protection against the peril of premature death; protection against the peril of outliving earning capacity will be referred to as savings. Hofflander and Duvall employed the technique of indifference curve analysis in examining the relationship between the amount of (real valued) life insurance protection purchased and the rate of anticipated inflation. Specifically, they assumed that (1) there were expectations of price level increases in the future time periods; (2) the amounts of other goods and services which can be purchased during the present time period with a given budget remained unchanged, as well as the nominal level of protection; (3) the consumer expects and acts as if the price of real protection has increased; (4) real protection is not an inferior good; and (5) income, employment, and population remain at a given level. From these assumptions their analysis leads them to conclude that expectations of rapidly increasing price levels in the future will lead life insurance consumers to decrease their purchases of life insurance. Their ideas are illustrated below in Figure 4, where 11 is identified as an indif Other I ooQds ant I ference curve representing com sercs a 2 C binations of life insurance pro tection and other goods and ser H F vices which give the individual the same level of satisfaction. I1 is a similar curve but represents a lower  level of satisfaction, and the line FIGURI 4 connecting points A and C is the 80 individual's budget line, representing attainable combinations of life insurance protection and other goods and services. To reach the high est indifference curve, the individual would purchase OD of insurance and OF of other commodities. Hofflander and Duvall claim that if the individual anticipates a higher rate of inflation in the future, the price of real protection is increased and therefore the budget constraint rotates downward to BD. The new optimal combination will be OE of insurance and OH of other goods and services. Thus, if insurance is a normal good, anticipations of inflation will lead to lowered purchases of real protection. Neumann (1968) disputes their conclusions on theoretical grounds. He notes that "during an inflationary period not only the price of real protection increases, but also the price of real goods and services goes up... therefore unless the form of the incomeconsumption curve is known no conclusions can be reached, a priori, as to the direction in which purchases of life insurance SL Nom i not e ir ,once would change.5 He illustrates his P Pi' c lIcel claim with Figure 5, at right, where F represents the indifference curve, / AA is the initial budget constraint, and R is the optimum combination 0 B A l 5See pp. 62930 for a more de cFU 5 tailed discussion of these points. FIRE 5 81 point of real protection and real consumption. If prices increase with no change in monetary income, the household suffers losses in both real income and real protection. The budget line shifts toward the origin to BB and the household suffers a preference loss. The new optimum point will be positioned according to the form of the household's incomeconsumption curve. Possible shapes of these curves are illustrated by I, II, and III. Therefore, the direction in which life insurance purchases would change depends upon which form of income consumption curve is appropriate for the family. In the opinion of this author, Neumann could have made his point simply by recognizing that the HofflanderDuvall model was a timeless model and hence incapable of furnishing a suitable frame work for the analysis of the effect of inflation, which by its very na ture occurs over time, upon the demand for protection, which is also received over time. To extend an impotent analysis to its "logical" end, and then to conclude that the implications of the Hofflander Duvall study are incorrect because they are at variance with those derived by Neumann, who uses a similar timeless model, only com pounds the error. Clearly what is required is a model which can take into account time, and its accompanying uncertainty, as well as inflation. 82 Development of a Theoretical Model In this section a theoretical model is developed for examining rational life insurance purchasing under inflationary conditions.6 In accomplishing this objective, the expected utility hypothesis is in voked in a timestate preference framework. The model facilitates a rigorous investigation of the issues of major importance, while laying groundwork for inquiry into other interesting aspects of the life insurance purchase decision. Background Information The expected utility model is based on a theorem derived from axioms concerning individual behavior. The theorem on which the model is based dates back to the endeavors of two eighteenth century mathematicians, Daniel Bernoulli and Gabriel Cramer, to resolve the St. Petersburg Paradox.7 In general terms, the expected utilitytheorem states that when faced with a set of mutually exclusive actions, each involving its own probability distribution of "outcomes," the individual behaves as if he attaches numbers which are called "utilities"8 to each Recall that in the previous chapter, models were constructed to show inflation's effect uponthe cost of life insurance. Nothing was implied about rational purchases. 7The St. Petersburg Paradox and the solutions posited by Bernoulli and Cramer are discussed in Levy and Sarnat (1972, Ch. 6). 8The use of the term "utilities" is unfortunate in that it gives rise to confusion as to its actual meaning. An excellent clarification of the term is given by Friedman and Savage (1952). 83 outcome and then chooses that action whose associated probability 9 distribution of outcomes provides maximum expected utility. Sub sequent to the publication of the Bernoulli and Cramer solutions to the St. Petersburg Paradox, John von Neumann and Oskar Morgenstern (1947) provided a rigorous axiomatic justification for the use of ex pected utility to explain choices under conditions of uncertainty. In essence, Neumann and Morgenstern demonstrated that if a decision maker acts in a rational and consistent manner, the expected utility theorem leads to optimal results under conditions of uncertainty.10 An extension of the expected utility hypothesis which provides a powerful analytical framework for decision making under uncertainty is the TimeState Preference approach. Evolving from the pioneering works of Arrow (1964), Debreu (1959), and Hirshleifer (1965), this ap proach assumes that the present values of uncertain future returns de pend on the pattern of returns across various states ofnature, the utility for money in the various states and the likelihood of occurrence of the particular states. Thus, unlike the expected utility hypothesis, the TimeState Preference model explicitly allows for the possibility 9To compute the expected utility of a given probability distribution of outcomes, the utility of each possible outcome is multiplied by the probability of the outcome, and the sum of these products over all pos sible outcomes is the expected value of utility for the probability distri bution of outcomes. 10An elegant derivation of the model is in Hernstein and Milnor (1953). 84 that the value (utility) of income received at a future point in time depends not only upon the length of time between now and then but also the circumstances of the individual when the income is received.I Specification of the Model The general properties of the model to be employed in the analysis having been stated, the components of the model are now specified. Discretetime analysis. The method of discretetime analysis lends itself readily to the task at hand, as insurance premiums are paid in lump sums at points in time. For purposes of analysis, time is organized into "periods" based upon the natural decision time junc tures associated with the periodic incurrence of premiums (to pay or not to pay, and in the case of term insurance extending beyond a single period, to continue or not to continue paying).12 These decision points are assumed to occur over regular intervals (periods) of time.13 Events will be assumed to occur at the beginning or the end of each 1In technical terms, the axiom of uniqueness used in deriving the expected utility hypothesis is relaxed under the TimeState Pref erence approach. For an excellent discussion of the two approaches and their differences, see Hirshleifer (1970). Premiums may be incurred on a monthly, quarterly, semi annual, or an annual basis, and the time periods could be defined to accommodate any of these arrangements. Although it is less common in practice, premiums may be incurred at even lengthier intervals (e.g. single payment life insurance). 13See Haley and Schall (1973, Chapter 1) for a more thorough description of discretetime analysis. 85 period, unless otherwise indicated. These assumptions are illustrated in Figure 6 below. Period Period Period 1 2 5 I I i l T  Time "me 0 Time I TIe 2 Time 5 The decision maker is now at time 0. Period 1 extends from time 0 to time 1; period 2 extends from time 1 to time 2; etc. Thus, in ac cordance with the assumptions stated above, events for period 1 occur either at timepoint 1 or 2. Throughout this chapter, a subscript will be attached to events of concern to indicate at which point in time they occur. States of nature. In the TimeState Preference approach, the concept of "states" or "states of nature" is central. Other charac teristics of the model, such as utility functions (or more accurately, preference scaling functions) of individuals, returns on assets, and probabilities, are all based on the definition of states. The individual is assumed to have in mind a set of possible states of nature in which each state is a particular sequence of events occurring from the present to a future point in time where the state is defined. In other words, if state s is said to occur at time t, the definition of state s includes a descrip tion of relevant events which have happened up to that point. . only one state can occur at a given point in time (states are mutually exclusive and exhaustive).14 14See Haley and Schall (1973, p. 192). 86 In the present discussion, two time states will be explicitly con sidered at each point in time.15 Uncertainty of lifetime is the primary concern here, and other uncertainties with which a consumer must normally cope are ignored. Thus, the uncertainty of future earnings, for example, will enter the model only insofar as the flow of earnings stops when the consumer dies. SA multiperiod model will be used to examine rational life insur ance purchasing under inflation. It is assumed that the individual will die sooner or later, where "sooner" will correspond to death during the first period, and "later" will correspond to death during a sub sequent period. There is a maximum number of periods, T, for which the individual can live. If he is alive at time T, he will be dead at T + 1.16 The individual will be referred to as the "breadwinner," denoting primary responsibility for the financial welfare of his depend ents. If the breadwinner dies during period 1, his heirs will not re ceive from the estate that value which would have accrued to them had the breadwinner survived that (and perhaps subsequent) periodss. It is this potential loss against which the breadwinner may desire protec tion for his heirs. 15These states need not always be distinct from each other. 16The letter T can be set to represent any length of time necessary (say 200 years) to ensure that the assumption approximates reality for all practical purposes. The possible time path sequences considered in the model are pictured in Figure 7. Alive Dead TimeStates 0 T+1 T+ld yTa Td Ta, Td Ta Td y4a 0 4a, 4d Y4a 04d 3a 03d 3a, 3d Y2a 2d Za, 2d Yla 10 1d la, Id y 0 Yo Sequence (0, Id, 2d, 3d, 4d, ) will be denoted State TI; Sequence (0, la, 2d, 3d, 4d,... ) will be denoted State TZ; Sequence (0, la, 2a, 3d, 4d, ... ) will be denoted State T3; and so on. FIGURE 7: Tree of TimeState Sequences 18 17 7The possibilities of a future incarnation or resurrection are not considered in this model, although the timestate preference framework may be sufficiently flexible to incorporate such events. 18 1The symbols preceding the timestate subscripts denote the real value of disposable income received in that timestate deriving from human capital (see pp. 8991). 88 Probabilities. Probabilities are specified in terms of the likeli hood of a particular state occurring at a given point in time. Because states are mutually exclusive and exhaustive, the sum of probabilities over all possible state sequences is equal to unity. In the present model, the probabilities associated with the occur rence of each state are dependent upon the likelihood of breadwinner mortality at a given point in time. The notation'ITld will denote the probability of death occurring to the breadwinner during the first period, and TTla will denote the probability of surviving the first period (such that the breadwinner is alive at the second decision point, time 1). Since it is assumed that the individual will either die in period one or will not die in period one, (1TTld) will be equal to TTla. Thus, the probability of sequence (0, Id, Zd, .) isTd, while the probabilities of all other sequences of states occurring, when summed over all possible sequences, will equal (lTTld), or simply Tla' Wealth status. The resources of the breadwinner consist of non human (physical) capital and human capital, each of which can generate future income streams. Physical capital is inheritable, and its value will be considered independent of breadwinner lifetime. The present value of the income generated by the physical capital with which the individual is endowed at time 0, where uncertainty is limited to length 89 of life, will be equal to the value of the assets, and will be de noted A0.19 Human wealth. The valuation procedure applicable for human wealth is somewhat more complicated, because of the uncertainty that surrounds lifetime. In this model, the potential income from human capital (which may consist of salary, wages, pensions, unemployment compensation, and so forth) may possess any pattern over time, but it is assumed to be known in advance and to terminate upon death of the breadwinner.Z0 It is further assumed that the real rate of return on human capital is independent of the rate of inflation.21 The rate of interest is presumed to be known but may have any pattern over time. If the individual is alive at the decision point t, he will be paid the (aftertax) installment pertaining to the period t + 1 at the beginning of that period (i.e., at time t). The real value of this installment will be denoted yt, where Yt10. If he is not alive at that point, no income will be received. Under these conditions, the present value of the in dividual's potential disposable income stream deriving from human 191t is assumed that the nominal return on physical capital adjusts to fully compensate for any inflation that may transpire. 20 ZThe word "potential" is used here because the income is con tingent upon the survival of the individual, whose lifetime is uncertain. Z1This assumption is not as strong as it may appear; in Brazil, wages are linked to a price level index to help neutralize inflation's impact on real values. capital, at the present decision point (time 0), is given by the 22 formula:2 P Y1Ji Yz2 Z YT J1 '" JT Y = Yo + + + ... + (19) R1 J R1R2J1JZ R1. .RTI ". JT which simplifies to p Y Y2 YT Y + + +. + (20) R1 RIRZ RI. .RT where Rt = (1 + rt) Jt = (1 + jt rt = the real riskfree (aftertax) rate of interest expected to prevail during period t (i.e., the time interval between time t 1 and time t); and jt = the expected rate of inflation associated with period t. Yaari (1965), Fischer (1973), and Richard (1977) have shown that when insurance is available, there is an amount of human capital, in dependent of risky market opportunities and preferences, which is the present certaintyequivalent value for the individual's future (human capital) earnings stream which is assumed to be sure if the individual is alive. This human capital term is calculated by discounting the 22The nominal level of the aftertax installment in any future period t, Yt 'r Jk adjusts so that it exactly compensates for in Sk=l flation's effect upon the corresponding discount rate. future earnings stream until the maximum time of death at a discount rate equal to the product of (1) the riskfree (aftertax) rate of in terest plus one, It = l+it = (1+rt) (1+jt), and (2) the insurance rate applicable at that time plus one, Xt = +xt, where P x = t1 it t1 and P is the premium payable at time tl for a given nominal tl amount of insurance to remain in force throughout the tth period, INSt. (The actual amount of insurance in force, if any, is immaterial if the insurance rate per unit of insurance in force is invariant with regard to the number of units purchased.)23 Thus, if the breadwinner is alive at time t, his present certaintyequivalent value (at time t) of dis posable income to be received is given by ce Yt+1 t+1 Yt+2 t+1t+2 tYta = Yt + ++ ... (22) It+lXt+1 It+l t+ Xt+ Xt+2 which simplifies to ce Yt 21 Yt+2 tYta = t + l + + (23) Rt+l t+l Rt+l Rt t+l t+2 23Note that the determination of the present certainty equivalent value of income does not require that any insurance be actually purchased. 92 where the subscript preceding the Y term indicates the.time back to which income is discounted, and the subscript appearing after Y, ta, indicates individual is alive at point t. The superscript ce indicates certainty equivalent. Since the individual is permitted to choose whether or not to (con tinue to) insure at each decision point, and how much to insure, his concern at any point in time is in providing coverage up until the next decision point. In the model developed here, the certaintyequivalent ce value that will be of particular importance is Yl since this corre 1 la spends to the amount of income forfeited if death occurs during the 24 first period. This value, expressed in currency units valued at time 1, can be discounted back to the present (time 0) to facilitate comparison with the present value of physical capital. The present certaintyequivalent value (at time 0) of all future25 returns on human capital, conditional upon breadwinner survival of the first period, is 26 given by 241f the individual survives the first period, he will have another chance to adjust his insurance levels for future periods. 25Recall that the breadwinner receives y immediately and with certainty. Z6Because this certaintyequivalent value is conditional upon sur vival of the first period, the discount factors for yt do not reflect the insurance factor for first period income, X1. 93 ce c Yla Y1 Y2 3 ce Yc Y = +  + (24) aR1 RI R1R 2X2 RR2R3X2X3 This particular value will figure importantly in the model shortly to be presented. All of the possible sequences of return on human capi tal associated with states T2, T3, TT+ 1 can be collapsed into a ce single present certaintyequivalent value, Yla, which will be added to endowed physical wealth, Ao, and first period income, yo, condi tional upon breadwinner survival of the first period. The probability e that the individual's endowed wealth, Wa will result in this sum is Tla. If the individual does not survive the first period, his endowed e wealth, Wd, will include only the endowed physical wealth and the first 27 period income.7 The probability of his endowed wealth amounting to this sum is'fld. These are the two conditional outcomes with which the breadwinner is concerned and for which he must make provision at decision point 0. Insurance. It is assumed that claims to consumption provided by e wealth in states T2, T3, TT+1 Wa, can be traded for claims to e 28 consumption provided by wealth in state Tl, Wd at the fixed rate 27 7The model could be easily modified to allow for additional future claims derived from social insurance programs by adding a term to that state's wealth. This extension is not undertaken here, although the implications of the existence and expansion of such programs will be discussed later in the chapter. 28The superscript "e" indicates "endowed" wealth. dWa dW x1, (25) dWd where x1, defined as before, can be called the insurance rate applicable for period one. The kind of insurance incorporated in the model is term insur ance. Singleperiod term insurance was selected for analysis for three reasons: (1) there is no loss of generality in using term insur ance since all available life insurance is a linear combination of one period (year) term insurance and a savings plan of some sort (Richard, 1977); (2) there are persuasive arguments that other types of insur ance (i.e., investment insurance) may be suboptimal (see, for exam ple, Aponte and Denenberg, 1968, and Klein, 1975); and (3) the effect of anticipated inflation on saving through life insurance has already 29 been examined (Neumann, 1969, Fortune, 1972) . Preference scaling functions for contingent wealth. In addition to being endowed with consumption claims (wealth) over different states, and having opportunities for transforming his endowed bundle 29The economics of this problem can be summarized with a state ment by George Stigler (1966, p. 57) made in another context, by sub stituting phrases relating to life insurance, where appropriate: In ad dition to the yield on life insurance savings, one can explain the par ticipation in a life insurance "forced saving" plan by introducing another item of preference: a desire of people to protect themselves against a future lack of will power. If we stopped the analysis with this explanation, we would turn utility into a tautology: a reason, we would be saying, can always be found for whatever we observe a man 