Inflation and indexation in Brazil

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Inflation and indexation in Brazil the influence on life insurance
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vi, 173 leaves : ill. ; 28 cm.
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Babbel, David F., 1949-
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Insurance -- Effect of inflation on   ( lcsh )
Indexation (Economics)   ( lcsh )
Inflation (Finance) -- Brazil   ( lcsh )
Life insurance -- Brazil   ( lcsh )
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Thesis--University of Florida.
Bibliography:
Includes bibliographical references (leaves 165-172).
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by David F. Babbel.
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Typescript.
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Vita.

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Full Text














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 index-linked 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 index-linked 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. Time-State 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 time-series 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 double-digit

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 index-linked 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 index-linked 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 index-linked 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 index-linked contracts is infeasible (e.g., no index-linked 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 index-linked 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 index-linked 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 long-term 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

double-digit 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 time-series 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 post-war 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
socio-demographic 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 twenty-one 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 index-linked 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 index-linked
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

index-linked 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 Time-State Preference

framework to derive definitive propositions with regard to consumer

demand for life insurance protection. The analysis is carried out for

both nonindexed and index-linked 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 time-series 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
real-valued 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, short-sighted 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 "Interest-Adjusted 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 Decree-Law 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 (1-DRa+j-1) 'DRa+n-i($ )
E_ ___, ( + ---.Rn^,
Ri=1 (i ) 1- 1 I- I t-





k kk k
T-"- lRa n TD TT (1 -)', CV (I-DI
+.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+n-1 is the conditional probability that an insured who survives
to age a+n-1 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 after-tax interest
rate selected by the individual representing his risk-free 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+it-l) 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 n-1 k
P --TT ( 1 -DRa+t- )D
k (I -DR -l)t=O a+DR (Cr$1000)
E NPCk (1-DR )t=0 a+n-] -
(1+ i) (1+ i)
n=l n=l


k
CVk --(1 DRa+t-l)

(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 n-i k
P 1 t(-DR a+j-) DRa+n-1(Cr$ 1000)
E NPCk] (l -DRa-l)t=O \ a I -

n=l n=l
k
CVk T (I1-DRa+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 n-1
dE PC = P 1 + r)-n1-n)(1 + )-n (1-DR
F J (1DR1-) TT-l+r)l(lI)( l-DR
a-t=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+t-l
t=l

Here it is seen that for k= 1, 2, 3,... the first term will be a non-posi-

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 counter-intuitive 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,
















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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' i--i ... ii

I I II

I I i i ,

I I
......l-i- 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 "Interest-Adjusted

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 cost-benefit 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 real-valued 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 risk-adjusted 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 Time-State 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 cost-benefit ratio is positive, a rise in antici-

pated inflation will cause an increase in the expected net present cost,

per (present-valued) 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 cost-benefit 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 paid-up

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 index-linked

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 (1-DRa+t_,) = the probability of survival through the end of
year n;


S n-I
sn-l TT (1-DRa+t_,) = the probability of survival through
n- (1-DRa-I) t=0 the end of year n-l;


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-
Pnsn-l
1 n-I
J o TT(Rt J)
n=l Ro0 t= 0
k '
dn(Cr$ 1000)
1 1
R2 J n-i
n=l R"Jo 7O(RtJt)
Rro t=

revised to include annual additional policies purchased,


becomes k n-i
EP s TT i
n n-1 TR t=o
1 n-1
E^pcJ ,n Rojo TTr (RtJt)
E [NPCZf n=1 R ol t=0
=- 1. (1
E[PV(BI k 1 n-I
d (Cr$1000) TT J
\ n Jo t=0 t
1 1
nn n-1
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)


Pnsn-l
n-I
Ro TT Rt
E[NPC] = n=l t= (1)

E PV(B)1 k
[ ] dn(Cr$1000)
11
RnJn n-1
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 cash-value 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 *... Jk-z, (14)

which reduces to

k-l1
n-1
sk CVk + CVk-nin 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 (present-valued) 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 k-1k
(16)

Clearly CVk> CVk-l> CVkZ so that for positive j's (16) will always









61

exceed (14).35 The conclusion follows that the effect of inflation on

the cash-value 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.

Index-Linked Policies

Attention now turns to index-linked 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

"index-linked" 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 (present-valued) 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 index-linked 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 index-linked 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 t-1 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 + CVkjk-11 'k-Z] (18)

sk CVk + CVk-lj + CVk-z 1 +***+ CV lk-ll k-2 (14)

sk [cvk + CV + CVkjZI1 +-...+ CVkjk-1 J1 k-2 + 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 1Jk-1 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
k-1
k-2
sk (CV -CV )j TT I
k Z k k-n 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, non-adjusting policy.) It is concluded

that an index-linked 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

index-linked 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 index-linked 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
t-4











(d
'-1
rd
in
0)







H-
>.








-I


iii i 1!~~j
-.i --
4--I-- t 4
-i r-- --i-~* i- I..

_______i i- -f-i-- i

4-. -----------i .- -
i :'.- i -i-
I__._. -1. r~ .t1
I-I. F
Ltij-I 'Ii L.i
i-mr -[-






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












Nonindexc-d Policy --,-, -
I i I ; I ,






*-^Hl- ----h-!--i-|---l--i --'-
_.: i : I ; ; i i j-i l : |


,L -. .....- -- I -i






|. Indexed Policy -- i i ..









-- I i :
- -- _-- I -- ,- ..... I,- .-I_- --
/ I I ii / i i ; i l
F-T;-F-ii-- -i .... ;- l .. f-i.. -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 cost-benefit ratios under expected
inflation rate assumptions of forty percent and zero percent.


Summary and Conclusions

The purpose of this chapter was two-fold: (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 by-products 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 index-linked 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 index-linked policy always results in lower

expected net present costs, per unit of (real-valued) 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 ill-designed 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 ill-contrived.
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 continuous-time 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 continuous-time 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 time-period 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), Jones-Lee (1975) and Klein (1975) have

approached the problem of optimal life insurance within the framework

of a two-period model (two time-points). Using mean-variance,

conditional expected utility, and time-state preference analyses,

respectively, each author assumes that the insured either dies irnm-e-

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 two-period (three time-points) 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 single-period 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 income-consumption 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. 629-30 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 income-consumption 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 Hofflander-Duvall 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 time-state 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 Time-State 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 -of-nature, the

utility for money in the various states and the likelihood of occurrence

of the particular states. Thus, unlike the expected utility hypothesis,

the Time-State 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.

Discrete-time analysis. The method of discrete-time 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 Time-State 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 discrete-time 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 time-point 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 Time-State 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 Time-States

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 Time-State Sequences 18




17
7The possibilities of a future incarnation or resurrection are
not considered in this model, although the time-state preference
framework may be sufficiently flexible to incorporate such events.

18
1The symbols preceding the time-state subscripts denote the
real value of disposable income received in that time-state deriving
from human capital (see pp. 89-91).









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, (1-TTld) 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 (l-TTld), 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

(after-tax) 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 risk-free (after-tax) 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 certainty-equivalent 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 after-tax 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 risk-free (after-tax) 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 = t-1



it t-1


and P is the premium payable at time t-l for a given nominal
t-l
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 certainty-equivalent 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 certainty-equivalent
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

certainty-equivalent 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 certainty-equivalent 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 certainty-equivalent 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. Single-period 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