Risk categorization

1
Risk categorization

• Lecture 4 - Economics of insurance
• EOCN6053 Selected topic in financial economics
• Raymond Yeung, PhD
• Honorary Assistant Professor
• 1 March 2007 (draft)

2
Background

• We have presented a general adverse selection
  model in which the insurer could not tell the
  risk type of an individual client
• The only piece of information the insurer knows
  is the proportion of risk type in the population
• In reality, there are variables that can
  imperfectly inform insurers about their risk
  type, e.g. smoking status, driving record,
  occupation etc. We do see that insurance
  companies utilize these variables to rate their
  clients risk class
• Studies subsequent to Rothschild Stigliz (1976)
  analyzes the efficiency property of risk
  categorization

3
1. Basic model

• The two-state model can be written in a
  simplified form as
• where E is endowed wealth level and
• Recall that actuarially fair offers locate along
  the line where
• so that high risk type will be faced with a
  flatter price line

4
1. Basic model
W1 W2
W2
Actuarially fair W1 - p(C/L)L, W2 - p(C/L)L
L C
W1-P, W2-P
W1,W2
W1-pC, W2-pC-LC
45o
W1
5
1. Basic Model

• When a pooling contract is offered, the
  proportion of risk type determines the slope of
  the offer line
• When the proportion of high-risk type is
  sufficiently small, the slope of the offer line
  moves towards the low-risk offer line
• In our numerical example, we have shown that a
  pooling contract can persist in this case

6
1. Basic model
VH2

• In Rothschild Stigliz (1976), an equilibrium is
  defined in a very restrictive sense that insurers
  do not consider market reaction to their own
  decision
• As long as they see positive profit, they enter
  the market
• This will reduce a huge set of potential
  equilibrium points

pooled
VH1
a
f
aL
VL2
aH
VL1
E
45o
7
1. Basic model

• In what RS called Nash equilibrium, firm will
  offer f that will make positive profit and
  low-risk will prefer it to a. If a is not
  offered, high-risk will also shift to f. The
  mixing with high-risk will cause a loss to the
  insurer.
• Wilson (1976) relaxes this and introduces perfect
  foresight to insurance companies so that all
  firms will not offer f because they know the
  unhealthy competition
• Under this interpretation, the optimization
  problem can be specified as to maximize the
  expected utility for the low-risk subject to an
  incentive constraint for the high risk and
  zero-profit for firms

8
2. Pareto efficiency

• Pareto efficient allocation of contract can be
  specified as the solution of the following
  maximization problem
• subject to
• saying the expected wealth after insurance must
  equal the level of the original endowment

9
2. Pareto efficiency

• and also incentive constraint
• that is, you are not interested in taking other
  types contract
• For a Pareto Efficient Optimization, there is an
  additional constraint to maintain the initial
  welfare of the high-risk
• that is, high-risk should not be a loser in the
  new allocation

10
2. Pareto efficiency
VH

• Allocations along the green line are potentially
  Pareto efficient resource permissible and
  incentive compatible. But the tangency point is
  the optimal solution to the PE problem specified
  in Crocker Snow (1986)

VH1
WLPE
WHPE
VL
WL1
WH1
45o
11
2. Pareto efficiency

• Crocker Snow (1986) (Theorem 1) shows that
  Pareto efficient allocations must involve
• 1) Full insurance for the high-risk
• 2) The high-risk does not take low-risk contract
• 3) The high-risk welfare is no less than their
  initial position
• The pooled contract at could also be
  efficient but may not be the most Pareto
  efficient. The high-risk could be a loser

12
3. Categorization

• The literature focuses on a policy question of
  whether risk categorization is Pareto efficient
  because for equity concern, government may choose
  to regulate the market by prohibiting insurers to
  discriminate their clients based on observable
  characteristics
• Subsequent sections of the Crocker and Snow
  (1986) paper establishes that the risk
  categorization (costless) will only make things
  better off. The gain from additional information,
  through a redistribution policy, is large enough
  to compensate the loser of the (re)allocation
  after categorization

13
3. Categorization

• Observable characteristic is informative and we
  know pA gt pB
• If the observable is perfect, the table below
  will be diagonal

14
3. Categorization

• There are winners and losers between and after
  categorizations
• Recall that under Wilson (1978) regime, the
  initial equilibrium before categorization can
  either be
• (i) a separating equilibrium involving full
  insurance for the high risk and underinsurance
  for the low risk
• (ii) a pooling case where the low-risk subsidizes
  the high-risk while insurers make zero-profit
• Whether (i) or (ii) depends on the proportion of
  high-risk in the population. We can illustrate
  this by altering the proportion of high risk in
  adverse.xls

15
3. Categorization

• After categorization, separating contract is
  offered to high-risk group as there are
  sufficient level of high-risk type. People
  assigned to the high-risk class will find worse
  off, compared to initial condition

VH
VL
E
45o
16
3. Categorization
VH

• If the initial equilibrium is a pooling contract,
  people assigned to the low-risk class will only
  be better off people assigned to the high-risk
  class will find worse off

VL
WH
WL
E
45o
17
3. Categorization

• If the initial equilibrium is separating,
  categorization may allow the insurer to identify
  low-risk type and offer lower premium pooling
  contract in group B, given pB is sufficiently
  small

VHVL
VH1
VL1
WL1
WH1
45o
18
3. Categorization

• Theorem 2 of Crocker Snow (1978) supports that
  in the costless case, government can implement a
  policy to tax away the profit earned in the
  low-risk class to subsidize all contracts sold to
  the high-risk class

VH1
WL
WH
WL1
WH1
45o