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Pricing Insurance Contracts

Chapter · January 1998

DOI: 10.1007/978-1-4615-6187-3_9

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 PRICING INSURANCE 9
CONTRACTS

Objective
This chapter identifies the requirements and procedures used to determine
insurance prices. Unlike other products, the production cost of an insurance
contract is not known in advance. The ultimate cost will be known only in some
future date: an inversion of the production cycle.
Correct pricing of insurance is therefore the foundation of the existence of
insurance contracts and special techniques and methods are developed to price the
different insurance products.

Definitions
A "rate" is the price an insurer charges for each unit of a loss exposure that is
covered under a specific insurance contract. The premium an insured pays for
coverage is the price per unit of protection multiply by the number of exposure
units. In property and liability insurance the exposure unit is generally 100
monetary units (dollars, ecus, pounds, francs, pesos, etc.) while in life insurance
the exposure unit is 1,000 monetary units.
The pricing of an insurance contract may be described as the process of
calculating the expected claims to be paid and expenses involved in assuming the
risk of a defined loss exposure. The prediction of claims is accurate only if there
is a proper identification of the risk and a large number of similar loss exposures
(characteristics of an ideal insurable risk) to be used to calculate the expected
frequency and severity of the losses.
Ignoring transactions costs, an insurer that would charge a premium equal to
the expected value of the loss to a large number of policyholders, would expect
no gain or no loss from the operation. This premium is called a pure premium
and it is expected to be a "fair premium" (or actuarially fair premium).
The expense portion of a rate or a premium is an important element of the
pricing of insurance contracts. All expenses an insurer is expected to incur must
be considered, including selling costs like commissions to the intermediaries, and
all the administrative expenses. The "competitiveness" of a particular
 2

"commercial premium" will depend upon the inclusions for expenses and any
margins for surplus and profit.1
The term "loading" is used to describe the process used to adjust the pure
premium to take into account all the costs of administrating an insurance contract
and eventually to provide a profit to the insurer.
Insurance premiums are generally payable in advance on contracts written for
periods of one or three years. The amount of premium written does not become
fully earned until the contract has expired. The earned premium represents in an
annual contract the same proportion to the total premium written that the period
elapsed from the date of issuance of the contract to the end of the year.

Major Requirements for Prices

The rates determined by the insurance company are obviously affected by many
subjective and anticipatory aspects but they should reflect not only some
principles of fairness but also some business considerations.

Principles Considerations
The rate should be adequate, reasonable and equitable. The application of these
principles by the insurance companies is often supervised by regulatory bodies in
many countries. Their nature is based partly on the requirements of good business
judgment and partly on social considerations.

Rate adequacy requires that a realistic estimation of the losses and expenses is
reflected in the price. A low price may give the insurer a competitive edge on the
short run but may affect the financial soundness of the insurance operation on a
longer period of time. This requirement is therefore intended to help minimize
the risk of insolvency of insurers by preventing irresponsible price competition.

Rate reasonableness requires that a "reasonable" profit margin should be

expected by the insurer. To be sure that a rate is adequate the insurer could
determine a price that is excessive in comparison to the expected losses and
expenses. It should be remembered, however, that because of market competition,
it is unlikely that rates would remain excessive. If competition is not operative,
for any reason, in rate determination, then excessive rates may have a negative
impact on the demand for insurance coverage by individuals and firms.

Rate equity refers to the fair treatment of individual insureds. This standard is
often stated as "not unfairly discriminatory." Insurance rates are unfair if the
insured is overcharged for the loss exposure in comparison with another similar
loss exposure. Of course, this requirement is very subjective as the insurer will
always have good arguments to justify its pricing decisions. Today, a new concept
 3

of equity has arisen in many individual types of insurance coverage, that might be
termed "social equity."

Business Considerations
Business considerations also suggest that rates should be workable or simple to
apply, stable, responsive to changes, and should encourage loss control activities.
Some of these objectives, however, are conflicting each others.

Rate simplicity means that the pricing procedures should be as simple as possible.
From an insurer's point of view, the system should be workable and relatively
inexpensive to apply. Some accuracy will inevitably be sacrified to attain this
goal. From the insured's point of view, the rates should be understandable.
Finally, simplicity but also full information available, will make it difficult for
both parties, the insurer and the insured, to manipulate the rates to their advantage.

Rate stability is necessary in order to maintain consumer satisfaction. If

insurance rates fluctuate or change rapidly, the individual policyholders or the
firms will be facing new uncertainty concerning the future price of insurance. This
will certainly have an impact on the financing decision of the firms. Consumer
behavior in reaction to price changes may also generate more regulatory controls
on the rates.

Rate responsiveness is important to make sure that the price is sensitive enough
to any new information concerning the frequency or severity of the risk. The rates
should also, and at the same time, be responsive to long term trends and responsive
to competitive tendencies. This is a good example of conflicting objectives which
affect the pricing decisions of insurance companies.
Rates should also encourage or promote loss control activities. This objective
is fundamental to minimize the effects of morale hazards. The rate will be reduced
by a fixed amount or percentage if some loss control equipment and activities (
for example automatic sprinkler systems) are implemented by the firm to reduce
the frequency and severity of the risk.

Principal Pricing Methods

According to Williams and Heins (1989, p. 641), insurance pricing methods can
be classified into three categories: (1) individual or judgment rating, (2) class
rating, and (3) merit rating. The last category refers to methods usually reserved
for business firms or large organizations such as schedule rating, experience rating
and retrospective rating.
 4

Individual or Judgment Rating

Individual risk rates are supposed to reflect directly the loss experience (i.e. based
on the historical loss severity and loss frequency) of a particular insured. The
theory underlying individual rating is that the number of loss exposure units is
large enough (how large is large enough?) for the price to reflect adequately the
actual characteristics of the risk.
The opposite assumption would be made when no information is available
and the price is only based upon the judgment of the person deciding on the rating
of the risk (the example of the Monster of the Loch Ness). In most cases, however,
the judgment of the rate-maker is based on a long experience of the factors to be
considered to determine the price (examples are common in marine insurance).

Class Rates
The derivation of class rates (also called manual rates) for specific types of
insurance coverage involves all the problems associated with the identification of
risk categories and the determination of specific frequency and severity
distributions. The classification of the risks (and consequently of the
policyholders) is done according to a few important factors that have a significant
impact on the frequency and severity of the expected losses. This procedure
incorporates the pooling and averaging of risks which is the foundation of
insurance.
Determining the significant factors and the number of classes for similar
types of loss exposure, however, may involve some subjective judgment. For
most of the individual types of coverages, the statistical experience available to
the company is large enough to permit the use by actuaries (the rate-makers) of
more sophisticated statistical tools. This method is used to set most individual life
insurance and health insurance rates, worker's compensation rates, automobile
insurance rates, fire insurance rates for homeowners.
The principal advantage of the method is its simplicity of application. The
list of generally accepted criteria for classification is provided in table 9.1 below.

Table 9.1
Criteria for Classification
• Causal relationship to claim exposure
• Homogeneity of the risks within the class
• Incentive ( insured's ability to control the risk)
• Adequacy of the class size
• Overall practicability
 5

Public policy considerations may also be important. For example if the

reduction of rates on individual life insurance for non-smokers is admitted by the
public, more and more countries now prohibit the use of sex or marital status as
a criteria of classification in the pricing of individual insurance contracts (mainly
automobile insurance). From an equity point of view, a criteria for an acceptable
classification system is based on probability distributions. Classes should be
sufficiently homogeneous and different to warrant the establishment of separate
and fair premium rates ( Figure 9.1).

Merit Rating
Merit rates are, in a sense, a compromise between class rates and individual rates.
The principal objective of merit rates is to encourage loss control activities and
match the premium more precisely to the insured's loss experience.

Schedule Rating is one of the earliest forms of individual risk rating. The method
uses a schedule which lists the average characteristics for a given type of risk.
Credit (deficiency) points or percentages are given where the loss exposure is
better (worse) than average. The total evaluation determine the extent to which
the appropriate class rate will be modified to reflect the specific loss exposure and
risk. The professional experience and proper judgment of the rate-maker are
necessary conditions for the application of this method. The main advantage of
schedule rating, like in commercial fire insurance, is to recognize the effect of loss
control activities.

Experience Rating uses the past experience of a particular insured to determine

the rate to be paid for future loss exposures. For this reason, it is sometimes called
prospective experience rating. By comparing the actual losses with the losses the
firm was expected to incur, a percentage reduction (or increase) is applied to the
a priori determined rate. Experience rating is commonly used in group insurance
for employees of large firms.

Retrospective rating uses the results observed during the period of the contract
to determine the final rate to be applied for that period. In other words,
retrospective rating requires a deposit premium at the beginning of the period of
coverage but the final price or premium is adjusted at the end of the period
according to the loss experience during the period. It may take several years after
the policy period before the final premium is known.
Usually, the final premium is subject to a stipulated minimum and maximum
(See Appendix 9.2). Retrospective rating is very close to self-rating by the firm
and is very often used in large firms primarily for casualty risks or group benefits
plans. Insurers employ retrospective rating as a method to underwrite accounts
that are likely to have adverse loss experience or when loss experience is not
 6

predictable. Buyers use retrospective rating to benefit from the time value of
money through deferred premium payments.

Adverse Selection (Anti-Selection)

Adverse selection was first considered in connection with life insurance contracts.
Insurers concerned with the state of health of people buying life insurance would
require a medical examination. Adverse selection can clearly occur in any kind of
insurance because the potential buyer has prior-knowledge on the risk (own
health, living habits, driving behavior, quality of the maintenance of equipments)
that the insurer does not have. There is "asymmetric information" between the two
parties.
Seminal work on asymmetric market information was done by
Akerlof(1970). He offered a theoretical reason for the lack of health insurance
owned by many of the elderly. Because it is impossible or prohibitively expensive
for an insurer to distinguish between "high" and "low" risks, the price is fixed at
an average value that attracts only whose who are worse than average risks (See
Appendix 9.1).2 Similarly Dahlby(1983) found that adverse selection leads to
reduced insurance consumption by low risks in the automobile insurance market.
When the insurance buyer makes his decision, he inevitably conveys some
information to the insurer. If the buyer prefers not to buy full insurance coverage,
it is interpreted by the insurer as a "signal" that he represents a lower risk than a
buyer requesting full insurance coverage. Of course the signal may prove wrong
(moral hazard) and it may takes time for the insurer to realize the erroneous
classification of the risk.
From the insurer point of view, the danger of anti-selection comes from the
possibility of misclassifying a risk or a policyholder. Insurance companies are
aware of such adverse selection and try to protect themselves both by adding
policy provisions to the contract (exclusion of some specific perils, coinsurance
clause, monetary and time deductibles).
The selection procedures usually used by life insurance companies to identify
people with poor health, and by automobile insurers to identify bad drivers, are
mainly intended to address this problem.
 7

______________________________________________________
DISCUSSION:
GENCO is a small general insurance company writing
automobile insurance in a Canadian province. The management
of the company decided in 1982 to implement an aggressive
marketing strategy to increase its business. The rate-making
department was using the classification system of drivers
suggested by the Association of Automobile Insurers and based
on 38 classes established from a multi criteria which included
the following factors:
- Characteristics related to the driver (age, sex, occupation
and driving experience)
- Characteristics related to the vehicle (power, usage,
mileage )
- Territory in which the vehicle operates.

The decision was taken to simplify the rating system and to

reduce the number of classes to 20 by amalgamating some
classes. For example class A and B formed a single new class.
Observing that in a national sample of 100,000 policyholders,
80% belong from class A and only 20% belong from class B,
the new average premium was fixed at $180.

Figure A ( a-priori classification) Figure B ( ex-post class )

After only two years of operation, although the company

had increased its business as expected, the overall financial
results were disastrous as the company was losing money in
several classes of business. For example, contrary to the
national average, a sample of policyholders from the new class
shown above revealed that about 67% were in fact
policyholders belonging from original class B. The company
should have charged an average premium estimated at $250.
______________________________________________________
 8

Bonus-Malus Systems
The insurer can eliminate the moral hazard problem by choosing an appropriate
no-claims discount strategy. Experience rating provides a mechanism that enables
the parties to the contract to mitigate or eliminate the moral hazard inefficiencies
like in a repeated games model or strategy in which a player makes a move in a
given period without full knowledge of the previous moves of the other player . 3
No claims bonus schemes are an important feature of some insurance
contracts, most notably those for motor (automobile) insurance. The policyholder
has to decide whether the magnitude of any accident (loss), is sufficiently great to
justify a claim, since making a claim, necessarily involves a future loss of
discount. It works like a self-applied franchise.
In practice, there is a variety of discount schemes. A particurly simple bonus-
malus scheme is termed the "one up/one down" scheme. It rewards individuals for
each claim-free period by a reduction in premium to the next lowest category and,
each claim generates an increase in premium to the next highest category. Some
schemes are defined as "two up/one down."
In health insurance, to reduce the over-utilisation of health care providers,
German insurers have introduced experience-rated bonuses for no-claims
(Zweifel, P., 1992). However, bonuses could conceivably induce policyholders to
defer or even forgo necessary medical treatment, jeopardizing their health. The
negative consequences of bonuses would leave their traces on medical care over
time.
In automobile insurance, bonus-malus system are popular in European
countries (Lemaire, J., 1985). The first bonus-malus system was introduced in
Europe in Switzerland in 1963 and consists of 22 classes. It has the following
feature: no claim = -1 class, one claim = +3 class.
 9

The Swiss Bonus-Malus System:

Class Premium Level
21 270
20 250
19 230
18 215
17 200
16 185
15 170
14 155
13 140
12 130
11 120
10 110
9 Class of Entry 100 Basic Rate
8 90
7 80
6 75
5 70
4 65
3 60
2 55
1 50
0 45

The Concept of Pure Premium

The pure premium method can be defined as the average loss per exposure unit,
or more specifically, the product of the average severity and the average frequency
of loss. In statistics, this combination is referred to as a convolution.
The average frequency of loss (F) is the number of losses incurred (NL)
divided by the number of exposure units (NE) in the relevant class. This is a mean
concept in the sense that it is the average number of losses for all insured. The
average severity of a loss (S) is the monetary amount of all losses ($L) divided by
the number of losses incurred (NL). It indicates, given that a loss has occurred,
how large it is likely to be.
If the average frequency of loss exposure is multiply by the average size loss,
the pure premium is determined and reflects the average insured's loss
expectation. The result is the amount, before commissions and administrative
expenses, which each insured of a class of business (or risk) must pay if all losses
are to be met.
 10

Pure Premium (PP) = (NL/NE) x ($L/NL) = ($L/NE)

These concepts are mean concepts but they do not take into account the
dispersion around the mean. The probability distribution of total losses for a
particular class of business is usually referred to as the pure premium distribution.
A measure of the variability inherent in the population is the variance denoted by:

 PP2 = ∑ ( PP -  ) / (n -1)
were  is the hypothetical mean of the pure premium distribution.

However, the rate-maker does not observe the entire pure premium
population, but only a sample of it. Thus, when he attempts to estimate  a
sampling error or variation from the true value is expected.
If it is assumed that the rate-maker takes a random sample from the basic pure
premium distribution, it can be shown that the average losses for a sample of n
exposure units will follow a normal distribution because of the central limit
theorem. In other words, if random samples were repeatedly taken, the sample
mean pure premium would follow a normal distribution. Thus, the standard error
of the sample mean pure premium distribution, m, is the standard deviation of
the population pure premium distribution adjusted by the number of exposure
units, that is:

m = PP / √n

It is important to calculate the standard error of the pure premium distribution

and to increment the average pure premium by a risk factor to compensate for the
expected variations in the result.4
Moreover, the accuracy of the estimate increases as the number of exposure
units increases since the standard error of the sample mean pure premium
decreases as the sample size increases. However, it should be emphasized that, at
the same time, the total risk faced by the insurer increases ( Figure 9.2).
The general formula includes a mean pure premium and a risk charge given
a confidence interval Z (normal distribution), that is: PP ± Z.m . By
transforming the sample mean pure premium into a standard normal variable, an
insurance rate can be derived. The insurer can be statistically confident
(depending of the Z value) that by applying this rate he will not become
insolvent.5
 11

Of course, this result implicitly recognizes that insurance companies are

required to have some minimum capital and surplus to avoid the small probability
of becoming insolvent. Even with a large number of exposure units, the
possibility of insolvency can never be eliminated completely. Theoretically, any
attempt to decrease the probability of insolvency to zero would imply the rate,
other things being equal, would have to approach infinity. 6
Pfeffer (1956, p. 43) recognized that no line of insurance meet the complete
set of assumptions because "The insurance experience is constantly changing with
the economic and social environment. This means that the result gleaned from
the past no longer has the same measure of relevance for the present."

_____________________________________________________
DISCUSSION:
The rate-maker wants to use the information concerning the
frequency and severity distributions for BEST-RENT-A-CAR
Company to estimate the pure premium ( see Appendix 1,
Chapter 5).

Frequency Distribution Severity Distribution

Number of losses Probability Average Dollar Amount Probability
0 0.70 1,000 0.30
1 0.20 2,000 0.50
2 or more 0.10 3,000 0.20

The statistics of the pure premium distribution are the

following: Mean = $ 760.
Standard Deviation = $1335.81.
Assuming the total number of loss exposures (cars) to be
10,000 m = 1335.81 / √10,000 = $13.36
Assuming the rate-maker desires to be 99 percent confident
that the actual losses do not exceed the rate charged, then the
appropriate rate, excluding expenses, should equal:
760 ± (2.33 x 13.36) = $ 729 to $791.
The insurer would become insolvent in only one percent of
the cases, other things being equal.
______________________________________________________
 12

The Expected Loss Ratio Method

The loss ratio is calculated as the ratio of losses incurred (often including loss
adjustment expenses) to premiums earned. In concept, the loss ratio represents the
percent of the premium returned to the insurance buyer as benefits.
Because the value of incurred losses is only knowed after a certain time, the
value is only an estimate of the loss ratio that would occur during some
representation period of time. The expected loss ratio is compared to a "target loss
ratio" which would have occured if an optimal amount had been allocated to the
coverage of the risk.
The loss ratio method is used when loss exposures are not reported because
of the complexity of the rating system. In marine insurance for example, the nature
of the risks makes such reporting impractical. The difficulty of the method is
associated with the proper estimation of the expected loss ratio. Any industry
figures should be adjusted to reflect the individual company's characteristics and
selected loss ratio should be reviewed regularly as to their continued
appropriateness and necessary adjustment made.
This method produces estimated ultimate loss ratio. The difficulty associated
with the expected loss ratio method lies in the estimation of the expected loss
ratio.

Trend and Projection Factors

Rates based upon past experience are valid only if the conditions that prevailed in
the past remain unchanged during the period of the contract. Because of economic
or social changes in liability risks for example, the importance of trend factors has
become very apparent. Inflation trend factors are certainly the most common ones
and the rate maker has to determine the most suitable index to be used.

______________________________________________________
DISCUSSION:
For the construction of rates based on expected loss ratio the
rate-maker will use the following formula:
Modification factor = 1 + (Expected -Target) / Target x Trend.
Past information available on incurred loss ratio:

EARNED Annual INCURRED Annual LOSS

YEAR PREMIUMS Increase LOSSES Increase RATIO
% % %
1987 24,005 - 18,028 - 75.1
1988 26,117 8.8 20,450 13.4 78.3
1989 27,976 7.1 22,353 9.3 79.9
 13

1990 31.022 10.9 25,537 14.2 82.3

1991 34,997 12.8 29,011 13.6 82.9
1992 37,759 7.9 31,922 10.0 84.5
1993 41,342 9.5 35,298 10.6 85.4
1994 44,649 8.0 38,906 10.2 87.1

The target loss ratio is 83%,

The losses increase on average by 10% each year
Factor = 1 + (87.1- 83.0)/83.0 x 1.10 = 1.1543

1995 estimated 51,540 15.43 42,797 10.0 83.0

Would this result be realistic when compared to the average growth of earned
premiums of 8.5% over the past 3 years?
______________________________________________________

The Credibility Theory

The extent to which a particular insured's experience is used in the rate-making
process is referred to as credibility. Credibility is the amount of confidence the
rate-maker has that the available statistics accurately indicate the losses to be
anticipated in the future. Thus, 60 percent credibility indicates that the future rate
is calculated by giving a weight of 60 percent to the experience of the firm and a
weight of 40 percent to the class rate.
In accordance with the law of large numbers, other things being equal, the
larger is the number of exposure units ( for example the number of employees for
group health insurance), the higher should be the credibility associated with the
insured's experience. The question is: how many exposure units are needed before
the actual losses will be acceptably near the expected losses for an acceptable
percentage of time?
The answer is usually given by the following general type of formula: 7

PPacceptable = C. PPi + (1-C). PP*

where PPi is the pure premium derived from the insured's
experience
PP* is the pure premium that would be derived
from the population real experience
C is the credibility factor, 0 ≤ C ≤ 1.

The organization of rating bureaus by insurance companies can be viewed

as a mean to gather and analyze a larger set of statistical data than that which is
 14

available to any single insurer. The rates generated by this information are usually
advisory only. However, in some countries and for certain lines of insurance
business they may become compulsory.

Loading Factors

The loading for transaction expenses (selling commissions and expenses for the
administration of the contract) and the loading for a profit margin are usually
calculated as a constant percentage. Acquisition cost is usually distinguished
from management expenses. What is usually referred to as acquisition costs are
the commissions paid to the intermediaries such as the brokers and agents and
sometimes some other incentive expenses.
It seems that different practices exist in the payment of commissions to
intermediaries. In most countries, there are no tariff or agreements prescribing
the rates of commissions to be paid. Every insurance company is free to grant the
commissions to their brokers, general agents and agents. Naturally, the
intermediary who not only solicits the business but also issues the policy and
service the clients are given higher rates of commission.
Insurance regulators have recently focussed not only on traditional issues
concerning economic objectives of regulation but also on areas of public policy
emphasis including insurance cost disclosure to policyholders and prohibition of
unfair price discrimination. In this context, fixing commission rates in insurance
premium-rate structure has been the subject of the approval or strict control by
insurance supervisory bodies.
In determining a gross premium rate, the loading factor is established as a
percentage of the gross rate itself. If we disregard credibility factors ( or assuming
100% credibility to the company loss experience), the following formula
illustrates the nature of a gross premium-rate structure based on the pure premium
method:

Gross premium = Pure premium / (1 - loading factor).

As an example, the usual commission rates for automobile insurance

business ranges from 10% to 25% of the gross premium and, on average, we can
assume a loading of 20% for administrative costs and anticipated profit (or
surplus). As shown in Table 9.2, it is evident that under a fixed commission rate
(in the example, 15%), the amount of commission being paid will increase when
the pure premium increases.
 15

Table 9.2
Rating with a Fixed Commission Rate
Average Pure Loading Gross Commission
Frequency & Severity Premium Ratio Premium Rate $Amount

Impact of Severity:
.10 2,000 200 35% 308 15% 46
.10 2,500 250 35% 385 15% 58
.10 3,000 300 35% 462 15% 69
.10 3,500 350 35% 538 15% 81

Impact of Frequency:
.10 2,200 220 35% 338 15% 51
.15 2,200 330 35% 508 15% 76
.20 2,200 440 35% 677 15% 102
.25 2,200 550 35% 846 15% 127

The Fairness of the Rating System

The above example leads to question the basis of variation in the commissions
perceived by the intermediaries. Commission differences should reflect cost
differences. The assumption is that administrative expenses generated by each
class of policyholders vary in proportion to the average claim cost, i.e. in
proportion to the pure premium. However, from the insurer's viewpoint the
commission which is a fixed percentage of the gross premium is also considered
a variable expense. The expense loading formula does not allow for fixed costs.
There is no evidence that sales activities are variable with the risk
classification. In fact, common sense suggests that a wider choice of insurers is
available only to lower-risk insureds and therefore the search-cost for the
intermediary increase for his lower-risk clients. If higher claim frequency results
in higher claim handling expenses then a higher commission and a higher gross
premium are fair. However, a higher pure premium resulting merely from higher
expected claim severity may not justify a higher commission.
A simulation in Table 9.3 shows how the commission can be increased
according to the average frequency and still remain unrelated to the level of
severity. In fact the decreasing commission rate structure, in this example, is
totally independent of the average frequency. The fee-for-service approach that
sometimes has been proposed as an alternative approach to commissions
 16

correspond exactly to this experimentation. It would result in a need to determine

the fee related to the service.
For practical reasons and in order not to increase the complexity of the rating
system in individual lines of insurance, insurers usually file a rating system which
considers commissions a uniform percentage of the gross premium-rate.
Theoretically and practically a flexible commission structure could be adopted
and would be more fair.

Table 9.3
Cost Varies with Frequency but not Severity
Average Pure Loading Gross Commission
Frequency & Severity Premium Ratio Premium Rate $Amount

Impact of Severity:
.15 2,000 300 35.0% 462 15.0% 69
.15 2,500 375 32.4% 555 12.4% 69
.15 3,000 450 30.6% 649 10.6% 69
.15 3,500 525 29.3% 743 9.3% 69
Average Pure Loading Gross Commission
Frequency & Severity Premium Ratio Premium Rate $Amount

Impact of Frequency:
.20 2,000 400 35.0% 616 15.0% 92
.20 2,500 500 32.4% 740 12.4% 92
.20 3,000 600 30.6% 865 10.6% 92
.20 3,500 700 29.3% 990 9.3% 92

Summary
The reader should remember that most insurance prices rely on claims data. The
connection may be direct through probability estimates of frequency and severity
through formulas that recognize loss experience relative to the policyholder.
The loading process adjusts estimated claim costs to account for the cost of
administrative expenses, acquisition expenses and services associated with
insurance.
Concepts: Rate, Rating, Class Rates, Merit Rates, Adverse Selection, Bonus-
Malus, Pure Premium, Credibility, Commission Rate.

________________________________________________________________
 17

Notes
 18

________________________________________________________________

Suggestion for Additional Reading

Michelbacher, G.F. and N.R. Roos, Multiple-Line Insurers: Their Nature and
Operation, New York: McGraw-Hill Book Co, 2nd ed., 1970.

References
Akerlof, G.A., "The Market for Lemons: Quality Uncertainty and the Market
Mechanism," Quarterly Journal of Economics, vol. 84, 1970, pp. 488-500.
Borch, Karl, Economics of Insurance , Amsterdam: North-Holland, 1990.
Cresta, J.P., Théorie des Marchés d'Assurance , Paris: Economica, 1984.
Dahlby, B.G., "Adverse Selection and Statistical Discrimination, " Journal of
Public Economics, vol. 20, 1983, pp.121-130.
Denenberg, H.S., Eilers, R.D., Melone, J.J. and R.A. Zelten, Risk and Insurance
, Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1974.
Dionne, G. and S. Harrington, Foundations of Insurance Economics ,
Dordrecht, Netherlands: Kluwer Academic Pub., 1992.
Lemaire, Jean, Automobile Insurance: Actuarial Models, Boston: Kluwer
Academic Publishers, 1985.
Lereah, David A., Insurance Markets: Information Problems and Regulation ,
New York: Praeger Publishers, 1985.
Pfeffer, Irving, Insurance and Economic Theory , Homewood, Illinois,
R.D. Irwin, Inc., 1956.
Williams, C. A. and R.M. Heins, Risk Management and Insurance ,
New York: McGraw-Hill Book Co., 6th ed., 1989.
Zweifel, P., Bonus Options in Health Insurance, Boston: Kluwer Academic
Publishers, 1992.
 19

Appendix 9.1
Information Problems in Pricing Insurance
Insurance firms operate in an environment with inherent information problems
because, although the contract is supposed to be of “utmost good faith," they are
unable to distinguish among consumers according to their probabilities of
incurring losses. The problem relates to the asymmetric information that exists
between the insurer and the insured.1 The later knows his own behavior and will
have a higher propensity to insure if his probability of incurring losses is higher
than the average of the group to seek insurance. This has been defined earlier as
adverse selection. It will lead to greater losses than are expected unless the insurer
can identify the risks he is assuming.
A very simple two-state, contingent-claim model is summarized in Figure
9.3. 2 Consider an individual whose initial endowment is at point E where his
wealth is W if no loss occurs and W-L if a loss occurs. Any point on the 45˚ line
represents a situation of certainty , the level of wealth is the same in both states of
nature, and corresponds to an actuarially fair insurance contract. The line EL is
the fair price line.
Assume that two risk group are facing the same loss L with corresponding
probabilities of loss P(H) for the high-risk group and P(L) for the low-risk group.
If individuals are assumed to have risk averse utility functions UH and UL, they
will fully insure when faced with actuarially fair price for insurance. An
individual will partially insure (will be to the right of the certainty line) when
faced with a price greater than the fair price.3
Asymmetric Information exists when the firm is not able, or it is prohibitively
expensive, to differentiate between low and high risk individuals and high risk
individuals purchase insurance at low risk price. Rothschild and Stiglitz
concluded that a market equilibrium with contracts H , L depends on the
proportion of high risks individuals in the market.4
If an insurer was offering contracts priced on the price line of high risk
individuals, then low risk individuals would signal themselves by asking for a
contract  with a deductible.5 However, there exists a pooling contract ß, offered
with a fixed deductible and priced on the market fair price line EF (assuming a
low proportion of high risk individuals), that will be attractive for both groups.
Low and high risk individual indifference curves are passing through point ß as
shown in figure 9.4(a) .
 20

If a pooling equilibrium exists, the low-risk group, in effect, subsidizes the

high risk group, since both groups purchase insurance at the same price. Also,
from an insurer point of view, it becomes always optimal to offer contracts with
a deductible clause. Such an equilibrium (a Wilson equilibrium) exists only if the
firms in the market agree not to offer a contract priced in favour of the low risk
group. Such a contract would attract low risk individuals (the point is on a higher
utility curve) and the firm offering ß would incur losses since this contract would
serve only high risk individuals. Ultimately, the price competition would result
in replacing the contract ß by a contract  and all the marketed contracts would
become unprofitable.
Consequently, if there is an equilibrium, it has to be a separating one in which
different types of risks are offered different types of contracts. Graphically, as
shown in figure 9.4(b), CL and CH correspond to a separating equilibrium in
which high risk individuals buy full insurance coverage ( C H is on the 45° line)
and low risk individuals buy only partial insurance coverage. Unfortunately at this
point, it is impossible to improve the welfare situation of all individuals in each
class of risk.6

Appendix 9.2
Retrospective Rating: An Example

Retrospective rating is a popular method of determining a premium range when

loss experience is not predictable. Also, if the loss experience is favorable, the
insured will benefit from a reduced premium.
The premium that is subject to retrospective rating is the standard fixed
premium to be charged at the beginning of the coverage period. The final premium
to be paid after the expiration is subject to final losses incurred and loss adjustment
expenses to compensate the insurer for the claims handling function and
settlement procedures. A portion of the premium represents the fixed cost to the
insurer to issue the policy and the final premium is subject to a minimum and a
maximium.
In the following example, a manager wants to compare a retrospective rating
arrangement to a basic standard premium of $1000,000 covering workmen's
compensation for the company. The analysis is done assuming no tax rate.
The administrative expenses represents 15%,
the loss adjustment expenses are charged 10%,
the expected loss ratio is 60%,
the minimum premium is $ 300,000 (30%)
the maximum premium is $ 1300,000 (130%).
 21

Retrospective premium = (1000,000 x .15) + (600,000 x 1.10) = $ 810,000

Charges for Expected losses and
administration loss adjustment expenses

The graph of the premium compared to losses is represented by the broken line
ABDE in figure 9.5.
Level of losses to reach the maximum premium = (1300,000 - 150,000) / 1.10
= $ 1045,545
At this level (D), whatever additional losses, the premium will remain at the
maximum.
Level of losses to reach the minimum premium =( 300,000 - 150,000) / 1.10
= $ 136,364
If this level (B) of losses is not realistic, the risk manager may request a higher
minimum for a lower maximum.
Breack-even point (C) when comparing with the standard premium
= (1000,000 - 150,000) / 1.10
= $ 772,727
At this level, the risk manager is indifferent between a retroactive rate or a
standard rate.
Slope of BD = (1300,000 -300,000) / (1045,545 - 136,364) = 1.10
Which correspond to the charges for loss adjustment expenses.

0 136,364 772,727 1,045,545

It is easy to simulate the effect of a different charge rate for loss adjustment
expenses.
To know the effect of taxes on the slope it is sufficient to multiply by the tax rate.
The fixed costs in the contract are the administrative expenses and the loss
adjustment expenses attached to the minimum premium. The variable cost
depends on the slope of BD. For every $1 in losses, premiums increase by 10%.
If loss forecast by the risk manager are higher than predicted by the insurer, the
risk manager will trade off a higher minimum premium for a lower maximum
premium.
 22

Appendix 9.3
Mortality Tables and the Price of Life Insurance
Mortality tables show the death rates an insurer may reasonably anticipate among
a particular group of insured persons at certain ages. The term expected mortality
is used in life insurance to mean the number of deaths which should occur
according to the mortality table assumed by the insurer.
A table usually begins with a group of 10 million people at age zero and by
substracting each year the number of deaths from the original figure, it is possible
to calculate the mortality rate from age zero to age 99, at which age the last people
in the original group of 10 million can be expected to die. Some mortality tables
are developed beyond the age of 99. Mortality tables also show that, on average,
women live longer than men. In some countries, companies calculate the rates
based on unisex mortality tables.
The age patterns of mortality for lower mortality countries are J shaped, but
show a steep rise in the curve at ages 15 to 20 (specially for males), followed by
a flattenning and then an exponential increase. In less developed countries, due to
the relatively high mortality in the early childhood ages, mortality tables are
developed only after age 10. A comparison of mortality rates for the United States
of America, The Netherlands, the ASEAN countries and CIMA is shown for
selected ages in table 9.4.
An example of a mortality table developed at age 10 and older is shown in
table 9.5. It is extrapolated from data published by the ASEAN Insurance
Commissioners for a male population of one million for the period 1977-1983.

Table 9.4
A Comparison of Mortality Rates per 1,000 (males)
Age USA Netherlands Malaysia Singapore Thailand Africa
CSO1980 1985-90 1977-83 1977-83 1977-83 CIMA*

10 0.73 0.21 1.03 0.13 2.98 0.39

20 1.90 0.69 1.53 1.23 2.45 1.35
30 1.73 0.79 1.10 0.57 2.68 1.82
40 3.02 1.55 1.88 1.31 4.05 3.90
50 6.71 4.86 5.63 4.51 7.81 8.94
60 16.08 13.99 17.33 15.44 16.78 21.11
70 39.51 38.98 45.14 40.21 40.08 50.12
80 98.84 95.27 112.64 101.12 88.41 117.24

* CIMA includes 14 sub-tropical French speaking countries ( Benin, Burkina Faso,

Cameroon, CentralAfrica, Comores, Congo, Cote d'Ivoire, Chad, Equatorial Guinea,
Mali, Niger, Senegal, Togo).
 23

Table 9.5
Mortality Table - Male Experience
Age at the Number living at Number dying Yearly death
Beginning beginning during probability
of the year of the year the year per 1000
x lx dx qx

10 1000,000 888 0.88

11 999,120 880 0.88
12 998,240 928 0.93
13 997,312 1,017 1.02
14 996,295 1,086 1.09
15 995,209 1,194 1.20
16 994,015 1,282 1.29
17 992,733 1,370 1.38
18 991,363 1,437 1.45
19 989,926 1,495 1.51
20 988,431 1,532 1.55
21 986,899 1,539 1.56
22 985,360 1,517 1.54
23 983,843 1,486 1.51
24 982,357 1,444 1.47
25 980,913 1,393 1.42
26 979,520 1,352 1.38
27 978,168 1,320 1.35
28 976,848 1,299 1.33
29 975,549 1,288 1.32
30 974,261 1,296 1.33
31 972,965 1,304 1.34
32 971,661 1,312 1.35
33 970,349 1,329 1.37
34 969,020 1,366 1.41
35 967,654 1,413 1.46
36 966,241 1,488 1.54
37 964,753 1,582 1.64
38 963,171 1,714 1.78
39 961,457 1,875 1.95
40 959,582 2,063 2.15
41 957,519 2,288 2.39
42 955,231 2,541 2.66
43 952,690 2,820 2.96
44 949,870 3,135 3.30
45 946,735 3,484 3.68
46 943,251 3,867 4.10
47 939,384 4,274 4.55
48 935,110 4,666 4.99
 24

49 930,444 5,062 5.44

Age at the Number living at Number dying Yearly death

Beginning beginning during probability
of the year of the year the year per 1000
x lx dx qx

50 925,382 5,460 5.90

51 919,922 5,897 6.41
52 914,025 6,416 7.02
53 907,609 7,061 7.78
54 900,548 7,853 8.72
55 892,695 8,820 9.88
56 883,875 9,908 11.21
57 873,967 11,082 12.68
58 862,885 12,296 14.25
59 850,589 13,524 15.90
60 837,065 14,732 17.60
61 822,333 15,929 19.37
62 806,404 17,144 21.26
63 789,260 18,413 23.33
64 770,847 19,749 25.62
65 751,098 21,166 28.18
66 729,932 22,635 31.01
67 707,297 24,140 34.13
68 683,157 25,659 37.56
69 657,498 27,161 41.31
70 630,337 28,624 45.41
71 601,713 30,019 49.89
72 571,694 31,317 54.78
73 540,377 32,493 60.13
74 507,884 33,500 65.96
75 474,384 34,298 72.30
76 440,086 34,850 79.19
77 405,236 35,118 86.66
78 370,118 35,068 94.75
79 335,050 34,674 103.49
80 300,376 33,915 112.91
81 266,461 32,783 123.03
82 233,678 31,287 133.89
83 202,391 29,446 145.49
84 172,945 27,299 157.85
85 145,646 24,901 170.97
86 120,745 22,322 184.87
87 98,423 19,636 199.51
88 78,787 16,930 214.89
89 61,857 14,287 230.97
 25

Age at the Number living at Number dying Yearly death

Beginning beginning during probability
of the year of the year the year per 1000
x lx dx qx

90 47,570 11,784 247.72

91 35,786 9,485 265.05
92 26,301 7,440 282.88
93 18,861 5,679 301.09
94 13,182 4,212 319.53
95 8,970 3,032 338.01
96 5,938 2,120 357.02
97 3,818 1,437 376.37
98 2,381 944 396.47
99 1,437 599 416.84
100 838 391 466.59
101 447 258 577.18
102 189 189 1000.00

Life Insurance Rate Computation

The simplest form of life insurance is term insurance which provides coverage for
a determined period of time. If a policyholder buys a term insurance contract for
one year, the premium rate should reflect the probability of death at the age of the
policyholder.
According to the mortality table presented in table 9.5, the mortality rate at
age 40 is 2.15 per 1,000. If 100,000 persons are insured for $1,000 each, an insurer
would expect to pay 215 claims for a total disbursement of $215,000. In this case,
the company would have to collect $2.15 from each insured person.
Assume the insurer collects the premium at the beginning of the year and
pays the death benefits at the end of the year, therefore, the insurer has use of the
funds for a full year for investment. At 5% interest the present value of $215,000
is $204,762, thus the insurer could charge at the beginning of the year only $2.05
per insured person to accumulate a fund sufficient to meet the 215 deaths.
If a policyholder buys a term insurance contract for 5 years, each year the
premium rate increases to reflect the annual rise in death rates as age advances.
Another method of purchasing insurance is to pay for the policy with a single
premium. Based on the mortality table, the insurer calculates the fund necessary
at the beginning of the period to meet all the death claims during the 5 years.
Assuming $1,000 insurance coverage at age 40, according to the mortality
table, 12,847 persons would die during the 5 year period and an accumulated fund
 26

of $11,011,220 would be necessary to pay all the death benefits (see table 9.6).
A total of 959,582 persons being alive at age 40, the insurer will charge a premium
rate of $11.475 per person (11,011,220 / 959,582) at the beginning of the coverage
period.

Table 9.6
Calculation of a Single Premium
5 year term insurance, age 40, 5% interest

Age Benefits paid Present value Present value

(Nb people dying) factor at 5% of the fund

40 2,063 x 1,000 .9524 1,964,801

41 2,288 x 1,000 .9070 2,075,216
42 2,541 x 1,000 .8638 2,194,916
43 2,820 x 1,000 .8227 2,320,014
44 3,135 x 1,000 .7835 2,456,273
---------- ----------------
total 12,847 11,011,220

Another method would be to pay the same premium during five years, a level-
premium calculated by using the number of people living at the beginning of
each year. Assuming each person pays $1, the present value of the premiums
collected would be $4,342,310 and therefore to meet the necessary fund of
$11,011,220, each person would have to pay $2.54 each year (table 9.7).

Table 9.7
Calculation of a Level-Premium
5 year term insurance, age 40, 5% interest

Age Nb of people living Present value Present value

at beginning of year factor at 5% of premiums

40 959,582 x$1 1.0 959,582

41 957,519 x$1 0.9524 911,941
42 955,231 x$1 0.9070 866,395
43 952,690 x$1 0.8638 822,934
44 949,870 x$1 0.8227 781,458
----------------
total 4,342,310
 27

A life insurance policy for the whole of life (ordinary/whole-life insurance)

can be viewed as a series of yearly renewable term insurances continuing to the
end of the mortality table. The fundamental idea of the level-premium plan is that
the company can accept the same premium each year, provided that the total
premiums collected are the mathematical equivalent of the corresponding yearly
premiums.
As a result, the level-premiums collected in the yearly years of the contract,
will be much higher than the expected death benefits but will become less than
adequate to meet death claims that occur in later years. This leveling results in the
creation of a fund (a mathematical reserve) from premiums paid in the early policy
years.

________________________________________________________________

Suggestion for Additional Reading

Black, K., Jr. and H.D. Skipper, Jr., Life Insurance, 12th ed., Englewood Cliffs,
NJ: Prentice Hall, 1994.
 28

1
See Denenberg et al., 1974, Chap. 26, p. 512.
2
Similarly Dahlby found that adverse selection leads to reduced insurance consumption by low risks in the automobile insurance
market and Browne found the same result in the individual health insurance market.
Dahlby, B.G., "Adverse Selection and Statistical Discrimination," Journal of Public Economics, vol.20, 1983, pp.121-130.
Browne, M.J., "Evidence of Adverse Selection in the Individual Health Insurance Market," Journal of Risk and Insurance, vol.59,
March 1992, pp. 13-33.
3
See A. Rubinstein and M.E. Yaari, "Repeated Insurance Contracts and Moral Hazard," Journal of Economic Theory, vol. 30,
1983, pp. 74-97 and George Dionne, "Adverse Selection and Repeated Insurance Contracts," Geneva Papers on Risk and
Insurance, vol. 8, 1983, pp. 316-333.
4
See David B. Houston, "Risk, Insurance, and Sampling," Journal of Risk and Insurance, vol. 31, Dec. 1964, pp. 511-538, and
Robert C. Witt, "Pricing and Underwriting Risk in Automobile Insurance: A Probabilistic View," Journal of Risk and Insurance,
vol. 40, Dec. 1973, pp. 509-531.

5
It is important to specify the nature of the probability distribution. In property insurance a Poisson distribution is often assumed
while in Life insurance a Binomial distribution is used. Assuming the same number of loss exposures and the same probability of
loss, the standard error for a Binomial distribution would be smaller . Therefore the risk faced by insurers will often be smaller in
life insurance than in property and liability insurance.
6
See Robert C. Witt, "Pricing, Investment Income, and Underwriting Risk: A Stochastic View," Journal of Risk and Insurance,
vol. 41, March 1974, pp. 109-133.
7
Houston, op. cit., represented credibility as an example of the general problem of statistical inference to which Bayes' Theorem
provides an answer. The a posteriori rate could be derived from the a priori rate and any new set of information.

1
An important body of literature exists on insurance market behavior in the context of information problems. See for example:
M. Spence and R. Zeckhauser, "Insurance, Information and Individual Action," American Economic Review, vol. 61, May
1971, pp. 380-397;
M. Pauly, "Over Insurance and Public Provision of Insurance: The Roles of Moral Hazard and Adverse Selection," Quarterly
Journal of Economics, vol. 88, Feb. 1974, pp. 44-62;
M. Rothschild and J. Stiglitz, "Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect
Information," Quarterly Journal of Economics, vol. 90, Nov. 1976, pp. 629-650;
C. Wilson, ,"A Model of Insurance Markets with Incomplete Information," Journal of Economic Theory, vol. 12, Dec. 1977,
pp. 167-207;
H. Miyazaki, "The Rat Race and Internal Labor Markets," The Bell Journal of Economics, vol. 8, 1977, pp. 394-418.
M. Hoy, "Categorizing Risks in the Insurance Industry," Quarterly Journal of Economics, vol. 97, May 1982, pp. 321-336.
M.J. Browne, "Evidence of Adverse Selection in the Individual Health Insurance Market," Journal of Risk ans Insurance, vol.
59, March 1992, pp. 13-33.
2
The reader could refer to the presentation by D. Lereah (1985, Chap. 2 & 3) developed for accident insurance. It is following
the work of Hirshleifer, J. and Riley, J., "The Analytics of Uncertainty and Information: An Expository Survey," Journal of
Economic Literature, vol. 17, Dec. 1979.
3
This result has been demonstrated by J. Mossin, "Aspects of Rational Insurance Purchasing," Journal of Political Economy,
vol. 79, 1968, pp. 553-568.
4
Most of the recent work on problems of informational asymmetries in insurance markets are extensions of the model of
Rothschild and Stiglitz (op. cit.).
A collection of classic articles including the article of Rothschild and Stiglitz, is presented in Dionne and Harrington (Eds.),
Foundations of Insurance Economics, Kluwer Pub., 1992.
5
K. Borch ( 1990, p. 322) notes that it is possible for the individual to send a false signal by asking for a deductible. A
comprehensive survey of the implications of asymmetric information can also be found in Cresta (1984).
6
See a selective survey on this issue by Dionne, G. and N. Doherty, "Adverse Selection in Insurance Markets: A Selective
Survey," in G. Dionne, ed., Contributions to Insurance Economics, Boston: Kluwer Academic Pub., 1992.

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