Cummins, et al. - The Economic Role of Risk Classification
Principal result
The market may become unstable if insurers are unable to classify. There may be no
equilibrium set of contracts that will not eventually lose money. This will restrict
availability. In the absence of classification, even if an equilibrium exists, some policyholders
will be in a worse position than would be the case in a competitive market. Government may
need to regulate to help the operation of the market.
Adverse selection: the tendency of high risks
to be more likely to buy insurance or to buy
larger amounts than low risks.
Illustrated explanation
Consider a market of High & Low risks for 1
year term life ins. with probabilities of death
θH and θL respectively.
Simplifying assumptions (optional):
•
Equal amount of H & L risks
•
All have same financial profiles and
utility functions
•
Loss amount is Q* for both H & L
Demand is expressed by QH(p) and QL(p) and
increases as price of insurance per dollar of
coverage (p) declines.
Fair premiums per $ for H & L are θH and θL
respectively
Illustration 1: Adverse Selection in a Simple Insurance Market
Analysis of the illustration :
•
QH(p) > QL(p) from adverse selection definition above
•
If companies can identify H & L risks in offer coverage at fair rates
◦ Both group will demand Q*
◦ Premium = Losses = (θH + θL)·Q*
•
If companies can't classify risks (unable to or regulation doesn't permit it)
◦ Average premium per $ θ will be charged
◦ θ = λ·θL + (1 - λ)·θH where λ is the proportion of L risks in the market
◦ L risk will demand QL while H risk will still ask for Q*
◦ Losses on H risk is not completely offset by overcharges on L risk
◦ Plan will fail even if none of the L risks drop out of the market
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Cummins, et al. - The Economic Role of Risk Classification
◦ Plan might be financially sound if L risks are strongly risk averse and have a
different utility function than H risks because they might still buy full coverage.
However, historical examples show that L risks are not sufficiently risk averse
Adverse selection arises because of an informational asymmetry; buyers know their loss
probabilities but sellers either do not or are not permitted to use this knowledge.
Model of insurance markets
Notation
W : initial wealth
X : possible loss
θL : probability of loss for L
θH : probability of loss for H
NL : number of L
NH : number of H
QL : insurance amount for L
QH : insurance amount for H
pL : price per unit of coverage for L
pH : price per unit of coverage for H
PL : premium for L (pL ·QL)
PH : premium for H (pH ·QH)
Relationships
•
θH > θL
•
Q≤X
•
pi = θi @ fair premium rate
Insurance policy j is represented by Sji (Qji , Pji)
For H risks, area 1 is profitable while area 2 and
3 are unprofitable. Policy lying on the line PH
breaks even.
For L risks, area 1 and 2 are profitable while
area 3 is unprofitable. Policy lying on the line
PL breaks even.
Market is competitive. There is freedom of
entry and exit and there is no collusion.
Companies seek maximization of profits. They
are not concerned with the variance of profit.
Assuming no operational costs and adequate
resources, they offer any and all contracts
likely to make a profit. Each consumer can buy
only one policy.
Consumers buy insurance according to their
utility functions. Policy S is selected so that it
maximize the following equation.
j
j
Illustration 2: Fair Premium Lines for H and L Risks
j
j
EU i (S )=θ i U [W − X +(Q −P )]+(1−θ i )U [W −P ] , i ϵ( L , H )
Utility function U[W] properties are the usual U'[W] > 0 and U''[W] < 0, it increases at a
decreasing rate. Consumers all have the same U[W]. EU depends on the class of risk.
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Cummins, et al. - The Economic Role of Risk Classification
An indifference curve is a locus1 in the plane of policies S among which the consumer is
indifferent (constant utility).
The slope of the tangent line to the indifference curve at (P,Q) can be found by setting EU
differential to zero and finding dP/dQ. It's the fair premium when the tangent is the Pi line.
θ i U ' [W − X +(Q−P)]
dP
=
dQ θi U ' [W − X +(Q−P)]+(1−θi )U ' [W −P ]
Consumer choose the insurance contract that lies to the south-easternmost indifference
curve.
Equilibrium: A set of policies that, when offered, no firm has an incentive to change.
Perfect Information and Classification
Key results:
•
Equilibrium exists
•
All consumers purchase the full coverage amount X at a fair premium rate
•
Policies on indifference curve tangent to Pi at Q=X are selected and represent the
optimal equilibrium points
•
Companies will enter the market with lower price (still above fair price) until the
equilibrium is reached the expected profits is zero
Imperfect Information and Independent Firms
Insurance firms act independently and consumers know their loss probabilities but firms are
unable to classify due to regulation or inability to identify prior to issuing coverage.
Myopic: A firm that assumes that the set of
policies offered by other firms is independent of
its own actions.
An equilibrium achieved under these
assumptions is called a Nash Equilibrium.
When firm can't classify, they could charge the
average rate to all insureds and a pooling
equilibrium might exist. But if they act
independently, no pooling equilibrium exists. L
risks switch from S1 to S0. H risks would switch to
S0 because S1 would eventually be withdrawn.
Same shifting process will happen between S0
and S2. Firms do not foresee that by selling S2, S0
will have to be withdrawn.
Illustration 3: An Insurance Market without Risk Classification
1 A locus is a collection of points which share a property.
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Cummins, et al. - The Economic Role of Risk Classification
It is still possible but not guaranteed to reach
a self-selection or Nash separating equilibrium
under these assumptions. A necessary
condition is that buyers can buy only one
policy and that companies can monitor the
amount of insurance purchased from other
firms. Prices and coverage have to be set
correctly so that S1 and S2 constitute an
equilibrium position. S1 is a partial coverage
policy. A policy like S3 would be unprofitable in
the long run. Thus, firms have no incentive to
change their policy offers (S1, S2).
Effects of classification on insured welfare
H risks are in the same position with
classification as they are in a Nash separating
equilibrium with no classification (i.e. full
coverage at fair price). L risks are in a worst
position without classification (S4 with
classification vs S1 without, same rate, less
coverage).
Illustration 4: Self-Selection Equilibrium When Firms Price
Independently
Breaking the self-selection / Nash separating equilibrium
Since firms are myopic, a new entrant in the market could offer policies such that they are
both preferred by L and H risks respectively. The combination would eventually become
unprofitable because other firms would offer the profitable policy.
How Rothchild and Stiglitz avoid this problem and maintain the equilibrium
Each firm can offer only one contract (either selling to L or H but not both).
How Wilson avoid this problem and maintain the equilibrium
A firm expects the offers of other firms to remain unchanged except when it offers a policy
earning positive profits. If some of its policies do earn positive profits, other firms are
expected to respond by immediately including that policy into their own policy offers. Firms
cannot hope to balance losses on one policy with profits on another, and no contracts will
be offered that are expected to lose money.
Under their definition of independence, if firms act independently and the line PLH cuts the
L risks indifference curve implied by the self-selection equilibrium, equilibrium will not be
achieved. The smaller the proportion of H risks, the more unstable the self-selection
equilibrium is because there is a greater chance for PLH to cut the indifference curve for L. If
classification is forbidden for impairments affecting only a small portion of policyholders,
instability will be more likely if the model hold in real life.
Self-selection / Nash separating equilibrium are still inefficient in the sense that utility
would be higher for L risk under classification. If companies are unable to enforce limitation
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on the amount of coverage purchased, the market is still likely to be unstable.
Imperfect Information with Company Foresight
Wilson assumes that a company expect other firms to withdraw any policy as soon as it
become unprofitable. A firm must take into account the effect of a new set of policies offer
on existing policies. An equilibrium achieved under these assumptions is defined as a set of
policies such that each policy earns non-negative profits and there is no other set of policies
that could be offered which earn positive profits in the aggregate and non-negative profits
individually, after the unprofitable policies in the original set have been withdrawn in
response to the new policy offer. Such an equilibrium is called a Wilson equilibrium.
If firms anticipate their competitors response, a policy like S2 would not be offered and they
would stick to S0 because offering S2 would cause S0 to be withdrawn. Thus S0 represents a
Wilson pooling equilibrium. S1 and S2 would also represent a Wilson Equilibrium if S0 doesn't
emerge because of the relative position of EU1L to PLH.
Miyazaki and Spence have revised the analysis to find Pareto superior equilibriums. A Wilson
Equilibrium become a set of policies where each firm earns zero profits and if there exists
no new set of contracts which would make non-negative profits after the elimination of all
the existing sets of contracts thereby rendered unprofitable. Under M&S, no pooling
equilibrium can emerge. To find the Pareto optimal equilibrium :
EU L ( S L )=θ L U [W − X +(Q L −P L )]+(1−θ L )U [W − P L ]
Find
L
L
L
S =(Q , P ) and
H
H
S =(Q , P ) that maximize L risks utility subject to
EU H (S H )≥EU H (S L ) ,
(3.5)
EU H (S H )≥EU H (S HF ) ,
(3.6)
L
L
H
H
and λ (P −Q θ L )+(1−λ)( P −Q θ H )=0 .
S
HF
(3.4)
H
(3.7)
=( X ,θ H X ) is the full information equilibrium policy for H risks
For the equilibrium to be stable, the H risks should not have an incentive to purchase the
policy intended for the L risks (3.5). This is the separating condition. High should be offered
a policy as attractive as the full information policy (3.6). Third, firms should break even (3.7).
For the Rothschild-Stiglitz model, each policy as to break even, (3.7) becomes :
P L≥Q L θ L and P H ≥Q H θ H
(3.8a/b)
Under (3.7), a policy can be sold at a loss as long as the loss is counterbalanced by profits on
another policy.
Miyazaki and Spence general findings on Wilson equilibriums :
•
Pooling equilibrium will never emerge
•
An equilibrium exists and is unique, there is one solution to the optimization problem
defined by (3.4) through (3.7).
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Cummins, et al. - The Economic Role of Risk Classification
•
Equilibrium will either be equivalent to the Nash separating equilibrium or will
involve subsidization of high risks by low risks. The latter solution is called a Wilson
subsidizing equilibrium. Each firm in such an equilibrium offers a set of contracts,
including one for H risks and one for L risks, that breaks even on the average.
•
In any solution, the H risks will have full coverage and a premium rate no greater than
θH. The L risks will have less than full coverage and may have to pay a rate higher than
θL.
•
An analogous equilibrium exists when n groups are present in the market.
Under a subsidizing equilibrium, policy S3 will be
offered to both groups under competitive
pressure. Supplementary policies S4 and S5
offering coverage amount X-Q3 and Y-Q3 are sold
at fair rates. H risks are only subsidized in the
pooled policy S3. Both groups are better off
with (S8, S7) than (S1, S2). This outcome is possible
whenever there is some S3 for which a area
defined as (pink on illustration) :
•
above EU7H indifference curve
•
above line S3 with slope θL
•
below EU1L
Otherwise the equilibrium will be (S1, S2) the
Nash separating equilibrium. The subsidized
policy is offered to protect the profitability of
the policy intended for the L risks, that is, to
prevent H risks from buying it.
Illustration 5: A Wilson Subsidizing Equilibrium
An attempt to attract low risks by offering policies earning lower profits than S5 will fail and
cause the subsidized S4 to be dropped. Wilson firm will foresee this development and the
equilibrium will be stable.
Principal results of the analysis
1. If classification is permitted and firms have perfect information, equilibrium is attained
when all consumers purchase full coverage at the appropriate actuarial rates.
2. If firms act independently and classification is not permitted, no pooling equilibrium
exists.
3. With independent firms and no classification, a self-selection equilibrium may exist.
This is more likely to occur when proportion of H risks in the marker is relatively
large.
4. If firms behave with foresight, a separating equilibrium exists, which is the unique
solution to a constrained optimization problem. This equilibrium either is the same as
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the Nash separating equilibrium or involves the subsidization of H risks by L risks.
The critical factors in determining the existence and type of equilibrium are the ability to
classify and the expectations of firms regarding their competitors' behaviour.
Summary and Conclusion
The classification and full information equilibrium is Pareto optimal. L risks are always
worse off without classification. Regulation can be utilized to improve the position of both
groups if the market fails. One possibility is to enforce the separating equilibrium that would
be achieved under firm foresight. Another is to equalize the utility of individuals in two
groups by requiring everyone to buy full coverage at the pooled rate.
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