Optimal level of deductible in insurance contracts

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Optimal level of deductible in insurance contracts

• Lecture 2 - Economics of insurance
• EOCN6053 Selected topic in financial economics
• Raymond Yeung, PhD
• Honorary Assistant Professor
• 8 February 2007

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Demand for insurance - reference

• Mossin, J. (1968) Aspects of rational insurance
  purchasing, Journal of Political Economy 79,
  553-568
• Raviv, Artur (1979), The design of an optimal
  insurance policy, American Economic Review,
  84-96
• Schlesigner, H. (1981) The optimal level of
  deductibility in insurance contracts? Journal of
  Risk and Insurance 48, 465-481

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1. Deductible
Loss to the insured
Examples automobile insurance, medical insurance
D
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Loss x
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1. Deductible policy

• In a deductible policy, the claim payments
  received by the insured could be
• Insurance company sets the premium in accordance
  to the amount of deductible, given a loading
  factor ?
• The last constraint ensures no gambling, i.e.
  indemnity principle

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2. Deductible policy

• With initial wealth W, the final wealth of the
  insured is
• Under the deductible contract, the expected
  utility of the insured is
• The optimal D is the one that maximizes E(U(D))

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2. Deductible policy

• There is an assumption that the contract is
  offered actuarially is
• Making use of this assumption, the first
  derivative is

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2. Deductible policy

• If D 0, the first derivative is
• The first derivative is positive as long as the
  loading factor is positive. That is, zero
  deductible is not optimal in the presence of
  loading factor.

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2. Deductible policy

• The second derivative is

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2. Deductible policy

• Whenever the first derivative is non-negative,
  the second derivative is negative. Two
  possibilities for the shape of E(u(D)) as D
  increases
• a) D goes to infinity as E(u(D)) is monotonically
  increasing, i.e. as D increases, demand for
  coverage drops
• b) E(u(D)) has a maximum at a finite value of D
  at which

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2. Effects of wealth

• By differentiating D by W, we get
• where Z is a complicated, negative term in the
  second derivative expression

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2. Effects of wealth

• Making use of the risk aversion parameter. For
  xltD, W-P-X gt W-P-D, their risk aversion will have
  reverse relationship

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2. Effects of wealth

• This term can be shown as zero and implies dD/dW
  gt 0, i.e. if risk aversion decreases, the optimal
  deductible is larger the larger the wealth

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3. Extension and subsequent research

• Optimality of the above model is simply for the
  consumer to decide on the optimal deductible
  level that maximizes the expected utility
• The above model is restrictive in a sense that
  the policy premium is computed as actuarial
  value. That is, the insurance company is risk
  neutral
• In the real world, insurance companies are
  reluctant to accept unlimited liability. There
  would be an upper limit (M) for claim above the
  deductible level (for x gt D)
• Arrow (1971) shows that if insurers are risk
  averse, the optimal policy involves coinsurance
  for payment above D

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3. Extension and subsequent research

• Raviv (1979) extends the model to include risk
  averted insurance
• The equilibrium level is about sharing of risk
  between two risk aversed parties, with the
  following results
• a) the Pareto optimal insurance policy involves
  deductible and coninsurance of losses
• b) the optimal deductible is positive if and only
  if the cost of insurance is a function of
  insurance coverage. In other words, if the
  loading is fixed, optimal deductible should be
  zero.
• c) a policy with upper limit on coverage is not
  Pareto optimal

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3. Extension and subsequent research

• The presence of risk averted insurer adds another
  constraint
• where S is initial wealth of the insurer (e.g.
  capital) c(y) is the amount of loading in
  general form and k is a constant and k v(S)
• Raviv (1979) shows that if P is fixed, the
  optimal contract is either a deductible provision
  coupled with coinsurance of loss or full coverage
  of losses up to a limit with coinsurance above
  which.

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1. Deductible
Loss to the individual
Policies with a deductible
y(x)
D
y(x)
Policies with upper limit
45 degree
Loss x
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3. Extension and subsequent research

• In both cases, the marginal coverage satisfies
  the optimal risk sharing rule
• R(.) denotes absolute risk aversion

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3. Extension and subsequent research

• P has been assumed fixed, to determine the
  optimal policy the next step is to determine the
  optimal rule for P(.)
• Raviv (1979) proceeds and finds that
• a) even when the insurer is risk neutral, the
  optimal deductible is positive as long as the
  loading is present. If c is fixed, optimal
  deductible is zero full coverage.
• b) coinsurance is observed if insurer is risk
  averse or if the c is a convex function of
  coverage
• c) under a special form of premium formula
  (linked to actuarial value), upper limit policies
  can be an optimal

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4. Numerical example

• Open the excel file