J. D. Cummins, B. D. Smith, R. N. Vance, and J. L. VanDerhei, Chapter 3: "The Economic Role of Risk Classification,"

in Risk Classification in Life Insurance, pp. 27–49, 59–60.

OUTLINE

I. INTRODUCTION A. The Problems

What is the economic justification for risk classification and what are the economic effects when insurers are unable to classify risks?

B. The Answers

The need for risk classification is justified by the threat of adverse selection and the loss of welfare by low-risk insureds. Different equilibrium positions are possible, reflecting a variety of factors: whether firms are able to classify risks, whether firms have perfect information, and whether firms behave with foresight.

C. Definitions

1. Adverse selection – "the tendency of high risks to be more likely to buy insurance or to buy larger amounts than low risks"

2. Equilibrium – condition produced by a "set of policies that, when offered, no firm has an incentive to change"

3. Fair premium line – line on graph showing actuarially fair premiums "as a function of the amount of insurance purchased"

4. Indifference curve – "a locus in the plane of policies (premium and coverage pairs) among which the consumer is indifferent"

5. Modified Wilson equilibrium – equilibrium produced by a set of policies "if each firm earns zero profits after the elimination of all the existing sets of contracts" rendered unprofitable by a new policy offer

6. Moral hazard – situation where "insureds can affect the probability of loss θi or the loss amount X"

7. Myopic – adjective applied to a firm that "assumes that the set of policies offered by other firms is independent of its own actions"

8 Nash equilibrium – equilibrium produced when firms exhibit myopic behavior 9. Pooling equilibrium – equilibrium produced when firms charge the average rate

to all insureds 10. Self-selection equilibrium, a.k.a. Nash separating equilibrium – equilibrium

produced when high-risk insureds purchase a higher-priced full- coverage policy and low-risk insureds a lower-priced partial-coverage policy

11. Wilson equilibrium – equilibrium produced by "a set of policies such that each policy earns nonnegative profits and there is no other set of policies that could be offered which earn positive profits in the aggregate and nonnegative profits individually, after the unprofitable policies in the original set have been withdrawn in response to" a new policy offer

12. Wilson subsidizing equilibrium – equilibrium produced by a policy set that breaks even on average and under which low-risk insureds subsidize high-risk insureds

D. Symbols 1 EUi - expected utility for policies in group i

2. H - group of high-risk insureds 3. L - group of low-risk insureds 4. pi - price per unit of coverage for insureds in group i 5. Pi - premium for insureds in group i 6. Q - amount of insurance demanded 7. Qi(p) - demand curve for insureds in group i

8. Sji - policy j in group i

9. SHF - full information equilibrium policy for the high-risk group 10. W - amount of initial wealth 11. X - full-coverage amount, loss amount 12. θi - probability of loss for risk group i

13. θ_

- pooled rate 14. λ – proportion of low-rsik consumers in the market E. Equations 1. Expected utility of wealth EUi(S) = θi U[W − X + (Q − P)] + (1 − θi) U[W − P], where i = L, H

2. Slope of the tangent to the indifference curve

dPdQ =

θi U'[W − X + (Q − P)]θi U'[W − X + (Q − P)] + (1 − θi) U'[W − P]

3. Average fair premium P = [λθL + (1 − λ)θH]Q = θ

_ Q

4. Constrained utility of low-risk insureds in a Wilson subsidizing equilibrium

EUL(SL) = θL U(W − X + QL − PL) + (1 − θL) U(W − PL) subject to a. EUH(SH) ≥ EUH(SL) b. EUH(SH) ≥ EUH(SHF) c. λ(PL − QL θL) + (1 − λ)(PH − QH θH) = 0 5. Constraints of Rothschild and Stiglitz a. PL ≥ QL θL b. PH ≥ QH θH

II. ANALYSIS

A. Adverse Selection

1. Failure to distinguish between high- and low-risk insureds in pricing produces adverse selection and inadequate premium

2. Assessment societies a. In England and U.S., provide life insurance for one premium

b. Dropping out of younger members leads to collapse 3. Model for adverse selection

a. Assumptions 1) Two types of risks: θH and θL

2) Each group of the same size 3) Each member of a group has identical financial characteristics

and utility function 4) Same loss amount for each group: Q*

b. Demand for insurance curves 1) Curves reflect increased demand as price declines 2) High-risk curve QH(p) is to the right of the low-risk curve QL(p)

4. Actuarially fair prices a. Higher price for the high-risk group b. Produce same amounts of coverage demanded c. Produce premiums equal to loss costs

5. Absence of classification a. Causes

1) Inability to measure loss probabilities 2) Prohibited by legislation or regulation

b. Charging of average premium to all risks 1) Low-risk insureds cut back coverage 2) High-risk insureds purchase full coverage or overinsure 3) Inadequate premiums collected from high-risk insureds not

offset by the overcharging of low-risk insureds and the plan fails 4) Such a result even if no low-risk insureds drop out and if

different proportions of high- and low-risk insureds c. One possibility of success

1) Low-risk insureds have a different utility function than high-risk insureds and are so risk averse that they buy full coverage even at the average rate

2) Historical examples show that this does not occur 6. Causes of adverse selection

a. Informational asymmetry – insurers know less about loss probabilities than applicants or insurers prevented from using the knowledge

b. Moral hazards

B. A Model of Insurance Markets

1. Basic assumptions a. Two groups with different loss probabilities b. All have the same initial wealth c. Protection against loss by a policy whose payment amount < loss

2. Premium charges and policies a. Price per unit of coverage times coverage amounts b. Fair premium line relating actuarially fair premium and coverage

amounts has a slope equal to the probability of loss c. Comparison of points reflecting premium and coverage of actual policies

to the fair premium line shows which are profitable and which are not 3. Assumptions regarding the insurance market

a. Competitive, i.e., free entry and exit and no collusion b. Objective to maximize profits c. Companies will issue all policies likely to be profitable, i.e., are not

concerned with the variance of profits d. No cost for administration or for obtaining information

4. Assumptions regarding consumers a. Buy only one policy

b. Use utility theory to make decisions 1) Expected utility is based on two situations: loss and no loss 2) Prefer more wealth to less 3) Marginal value declines as wealth increases

c. Same utility function for all classes 5. Indifference curves

a. Obtain by setting expected utility equal to a constant and derive prices and quantities that produce it

b. Slope of a tangent line derived by setting the derivative of the utility curve equal to zero

c. Select an insurance contract on the curve with the greatest utility 6. Equilibrium

a. New set of policies has no effect b. Depends on other firm's expectations

c. Depends on information available to firms and consumers C. Perfect Information and Classification

1. Conditions

a. Firms and consumers have perfect information b. Firms permitted to classify based on information

2. Results a. Consumers seek to purchase a policy on the indifference curve that is

furthest to the right, i.e., has the lowest cost b. Insurers refuse to sell policies below the fair premium line for the class c. If policies on the fair premium line are offered, consumers should choose

that represented by the point of tangency of the curve and the line d. Companies prevented from offering policies above the line by

competition, which lowers price to that of the line e. Slope of the indifference curve equals slope of the fair premium line only

at the point of tangency when full coverage is purchased

D. Imperfect Information and Independent Firms

1. Conditions a. Firms act independently, i.e., are myopic b. Consumers know loss probabilities c. Firms do not classify because of regulation or lack of knowledge

2. Pooling equilibrium strategy a. Charging the average rate to all insureds fails to produce equilibrium if

firms lack knowledge and act independently b. Fair premium line for an average risk is a weighted average of those for

different groups c. Possible equilibrium at the point of tangency of the low-risk indifference

curve to the fair premium line for an average risk d. But this equilibrium undercut by the area between this curve and the fair

premium line for low-risk insureds, which is above the curve for high-risk insureds

1) Companies can lower price 2) Attract only low-risk insureds 3) Make a profit e. Further destabilization 1) Loss of low-risk insureds makes the original policy unprofitable

2) Its withdrawal from the market leads high-risk insureds to purchase a new lower-priced policy, even though above their curve

3) Second policy also becomes unprofitable f. Thus lack of information and the behavior of firms prevents a pooling

equilibrium 1) Second policy offered only because will not be bought by high-

risk insureds 2) Not foreseen that this will result in other firms withdrawing the

original policy, which is now unprofitable 3. Nash separating equilibrium

a. Basic characteristics 1) Offering of two policies a) Full coverage at a high unit price b) Partial coverage at a low unit price

2) If done correctly, may produce an equilibrium as high-risk insureds buy full coverage and low-risk insureds buy partial coverage

3) Requires consumer to purchase only one policy or company to monitor the coverage purchased from other firms

4) One policy not offered because the average fair premium line is above the low-risk indifference curve

b. Operation 1) Coverage offered at the actuarially fair rate for high-risk insureds

on the most favorable indifference curve 2) Competition forces the offering of a second policy represented

by the intersection of a) The same high-risk indifference curve

b) A low-risk indifference curve c) Low-risk fair premium line

3) Effects of the second policy a) Preferred by low-risk insureds b) High-risk insureds indifferent to it

4) Possibility of a third policy below the low-risk indifference curve and above the low-risk fair premium line

a) Preferred by low-risk insureds b) Not profitable if offered to both groups c) Inability to classify prevents its offering

d) Thus equilibrium produced in a two-policy situation 5) Comparison with the situation of classification

a) No change for high-risk insureds b) Negative effects on low-risk insureds

i) On a less-favorable indifference curve ii) Less than full coverage

c) Still requires restriction of coverage to high-risk insureds d) Negative information externality flows from high- to

low-risk insureds 4. Breakdown of the equilibrium a. Myopic behavior of one firm

1) Offering of more favorable policies to both groups 2) Unprofitable policy to high-risk insureds offset by the

profitability of the policy for low-risk insureds 3) Other firms react by offering only the profitable policy 4) Situation becomes unprofitable for the original firm 5) Ways to prevent such a breakdown

a) Firms sell only to one group of insureds (Rothschild and Stiglitz)

b) Independent firm expects that other firms will respond to profitable policies and thus no unprofitable policies will be offered (Wilson)

b. Intersection of the average fair premium line with the low-risk indifference curve that intersects the high-risk indifference curve and the low-risk fair premium line 1) In the earlier situation, the average fair premium line had been

above this low-risk indifference curve 2) Lower line reflects a smaller proportion of high-risk insureds 3) Incentive for one profitable policy to be offered for a lower price

to both groups 4) Eventually, one policy based on the average rate offered, but this

is not an equilibrium situation 5) Probability of market instability inversely related to the number

of high-risk insureds a) Policy must be preferred by both groups and be

profitable b) Low-risk insureds subsidize high-risk ones

c) Easiest to attract low-risk insureds if the price is close to the actuarially fair rate and the price difference is therefore overcome by the attraction of greater coverage

d) More likely if classification forbidden for impairments affecting only a small portion of the population

E. Imperfect Information with Company Foresight 1. Wilson's equilibrium a. Firms foresee the responses of competitors

b. Must consider the effects of new policies on the profits of existing policies

c. Only effect of a new policy offer is the withdrawal of unprofitable policies

2. Examples a. One policy offered to both groups produces a Wilson pooling

equilibrium if firms foresee that offering a second policy eventually produces an unprofitable situation

b. Nash separating equilibrium 3. Modified Wilson equilibriums (Miyazaki and Spence)

a. Unprofitable individual policies allowed to exist if, in the aggregate, policies have nonnegative profits

b. Able to identify equilibriums Pareto-superior to the two unmodified Wilson equilibriums

4. Maximization of the utility of low-risk insureds a. Information externalities reduce the welfare of low risks b. Pressures to improve their welfare face three constraints

1) No incentive for high risks to buy policies designed for low risks – constraint 3.5

2) Utility for high-risk insureds is at least equal to that in the full-information equilibrium situation – constraint 3.6

3) Firms must break even in the aggregate – constraint 3.7 c. Given constraints, determine policies for low- and high-risk insureds that

maximize the utility of the low-risk insureds

5. Characteristics of modified Wilson equilibriums a. Pooling equilibrium will not emerge

b. Unique equilibrium exists to maximize the utility for low-risk insureds subject to the constraints

c. Two solutions are possible 1) Nash separating equilibrium 2) Wilson subsidizing equilibrium

d. High-risk insureds have full coverage at a rate no greater than the actuarially fair one

e. Low-risk insureds have partial coverage at a rate possibly higher than the actuarially fair one

f. Situation applies to two or more groups 6. Operation of modified Wilson equilibriums

a. Competitive pressures to benefit low-risk insureds lead to a pooled contract 1) Subsidization of high-risk insureds 2) Small amount of coverage

b. Both groups able to purchase supplementary policies to provide more coverage

1) Different policies for each group 2) Both on the appropriate fair premium line

3) High-risk insureds subsidized in a pooled policy 4) Combined policies are on lower indifference curves for both

groups 5) Combined policy for low-risk insureds is on high-risk

indifference curve c. Requirements for this equilibrium – area exists bordered by

1) Original low-risk indifference curve 2) High-risk indifference curve lower than the original 3) Line through the point representing the pooled policy and

parallel to the low-risk fair premium line d. Resulting characteristics

1) Subsidization of high-risk insureds prevents the purchase of the policy intended for low-risk insureds, i.e., separation maintained

2) Coverage for low-risk insureds increased 3) Less profitable low-risk supplements not offered to attract more

low-risk insureds because firms that lose such will drop high-risk supplements and high-risks will then buy low-risk supplements

F. Summary of Alternatives

1. Perfect information and classification allowed – each group purchases full coverage at the actuarially fair rate

2. Information not available and firms lack foresight, i.e., act independently – no pooling equilibrium possible, but a Nash separating equilibrium is, especially with a large proportion of high-risk insureds

3. Information not available, firms have foresight, no unprofitable individual policies – Nash separating equilibrium or one policy at the average rate

4. Information not available, firms have foresight, no unprofitable sets of policies – Nash or other separating equilibrium involving the subsidization of high-risk insureds

5. Crucial determinants: the ability to classify and existence of foresight 6. Grossman's dissembling equilibrium (footnote 20)

a. Firms' behavior 1) Act independently 2) Offer coverage without limitation at the pooled rate 3) Policyholders requesting more coverage than provided for in the

pooled policy are charged the high-risk rather than the pooled rate

b. Effects 1) High-risk insureds only apply for coverage needed by low-risk

insureds 2) Invulnerable to competition 3) Will be dominated by a Nash separating equilibrium

I. Summary and Conclusions

1. Pareto-optimal market equilibrium achieved when risk classification allows the charging of actuarially fair rates for each class

2. If no classification and firms act with foresight, a unique separating equilibrium results, in which high-risk insureds are at least as well off as under classification and which may involve subsidization of high-risk insureds

3. Possible benefits of regulation if the market fails a. Enforce the separating equilibrium

b. Equalize the utility of individuals c. Impose a Pareto-superior allocation if firms behave independently

4. Need for caution when tampering with competitive markets