Demand for insurance

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Demand for insurance

• Lecture 1 - Economics of insurance
• EOCN6053 Selected topic in financial economics
• Raymond Yeung, PhD
• Honorary Assistant Professor
• 1 February 2007

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Major learning point

• Expected utility theory is the hard core of
  conventional theory of insurance economics
• Based on EU theory, Mossins model (1968) implies
  that people buy full coverage only if premium
  equates its average loss (actuarial value)
• The theory contradicts our observation that
  insurance companies impose a loading and yet
  people do demand for full insurance
• Arrow (1971,1974) generalized the premium formula
  that includes a loading factor proportional to
  expected claims
• Based on Arrows model, subsequent research
  studies the features of insurance demand e.g.
  changes initial wealth, risk aversion, accident
  probabilities etc.

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1. Historical development of the theories

• Insurance is a product in response to the
  presence of uncertainty
• People demand for a contract involving prepaid
  elements ahead of the occurance of an event(s)
• P the premium paid b the insured when the
  contract is concluded
• x the compensation which the insured receives
  if specific events occur when the contract is in
  force
• where F(x) is a probability distribution of the
  random variable x
• If F(x) is unknown or does not exist, the
  insurance product cannot be priced

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1. Historical development of the theories

• Adam Smiths Wealth of Nation premium must be
  sufficient to compensate the common losses, to
  pay the expense of management, and to afford such
  a profit as might have been drawn from an equal
  capital employed in any common trade
• Austrian School Eugen Bohm-Bawerk in 1881 shows
  that values, or certainty equivalents could be
  computed for contingent claims
• Walras (1874) saw insurance a device to remove
  uncertainty inherent in all other economic
  activities, that can be interpreted as cost of
  capital in business economics
• Marshall (1890) reckoned businessmen are willing
  to pay a value above the actuarial value of risk,
  pointing to the presence of risk aversion

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2. Preference ordering of risk

• An uncertain loss may be characterized as risk
  which can be described by a probability
  distribution
• If a person covers a part of the whole of the
  risk by insurance priced at P, the actual
  distribution of x can be altered from F1 to F2
• People can rank their preference of F1 to F2

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2. Preference ordering of risk

• For any distributions such that
  , and any scalar it follows
  that
• With the assumptions of consistency, completeness
  of and continuity of preference ordering
• (1) There exists a function u(x) such that
  implies
• (2) The function u(x) is unique up to a positive
  affine transformation, i.e. the functions u(x)
  and
• where Agt0 represent the same preference ordering

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2. Demand for insurance

• Consider a person with initial wealth W. Assume
  he is exposed to a risk of loss x with density
  f(x). He is willing to pay a premium P only if
• There is clearly an upper limit to the premium,
  say so that the equality sign will hold
• Recall Jensens inequality. If u(x) is concave,
  it follows that
• For some , the buyer is willing to
  pay more than E(x) for insurance

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1. Bernoulli principle

• The fair value of a gamble was the mathematical
  expectation of the gain
• St. Peterburg Paradox is an counter example by
  Bernoulli (1738) if the first head appears at
  the nth toss, a prize of 2n is paid

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1. Bernoulli principle

• Bernoulli argued there is a moral value to x, or
  ln(x)
• In fact, any concave function u(x) will do. From
  Jensens inequality, it follows that a person
  will pay less than the mathematical expectation
  for the right to play a gamble
• Bernoulli hypothesis was proved as a theorem only
  until 1947 by von Neumann and Morgenstern, and
  Arrows optimal allocation of risk bearing in 1953

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2. Demand for insurance

• The supply side consider now an insurer with
  utility u1(x) and initial wealth W1.
• He is willing to assume a contract to cover the
  risk against a premium P only if
• An insurance contract exists only if there are
  values P which are consistent with the conditions
  above

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2. Mossin theory of partial cover

• Assume a person can pay a premium qP and receive
  a compensation of qx. This is a fixed
  proportional contract without any loading.
• The expected utility to the person is
• The decision problem is to find the value of q
  which maximizes his expected utility. The FOC is

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2. Mossin theory of partial cover

• For full insuance (q1),
• If E(x) lt P, U(1) is negative, violating the
  FOC. It appears full insurance is only observed
  if premium is set at actuarially fair rate.
• The SOC is always satisfied (even when q1), as

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2. Mossin theory of partial cover (cont)

• But this result contradicts our observations.
  When a person insures his house, the sum insured
  is usually full rather than partial value of the
  property.
• To explain this deviation, one may question
  whether expected utility theory is applicable to
  insurance analysis, or the assumption of concave
  u(.). However, we do observe the norm of risk
  aversion.
• A simple way out is to add a loading factor in
  formulating the insurance contract P qE(x)
  c
• Please work it out the FOC in the case with
  loading and its implications.

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2. Mossin theory example

• Assume an individual has initial wealth W0 and
  will suffer a loss L with probability p. She
  faces a prospect
• An insurance is available with premium P pC,
  where p is premium rate and C is the level of
  coverage.
• With insurance, the prospect becomes

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2. Mossin theory example

• In the case of full insurance, L C
• The consumer will buy insurance if and only if a
  C exists such that the expected utility of being
  insured is higher than the expected utility of
  being uninsured

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2. Mossin theory example

• This problem is solved by
• The FOC is that the probability-weighted marginal
  utility is equal for both states of the world

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2. Mossin theory example

• Suppose Bernoulli Principle holds premium rate
  is set at its actuarially fair level. It is
  easily to show that p p
• The FOC becomes
• The only condition that allows this to happen is
  CL, full coverage.

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2. Demand function

• Forget about our assumption on p at this moment.
  Assume log utility function, u(.) ln x. The FOC
  becomes
• Let assume L 1 for simplicity. Solving this
  equation for C, the amount of coverage

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2. Demand function

• Effect of changes in initial wealth
• If insurance is actuarially fair, changes initial
  wealth do not affect insurance demand. If there
  is a loading, the relationship is negative
  wealthier demand for less coverage

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2. Demand function

• Effect of changes in risk probability
• The numerator is negative as WgtL. The denominator
  is negative. It means there is a positive
  relationship between risk factor and demand for
  coverage

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2. Demand function

• Effect of changes in loss severity
• There is positive relationship between C and L

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2. Demand function

• What about price elasticity
• If the numerator is positive, insurance would be
  a Giffen goods, depending on the relative
  strength of income and substitution effect

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2. Arrows model Choice of contracts

• The other way is to generalize the contract form
  into a contingent benefit function y(x) when
  event x occurs.
• Premium P in turn depends on how claims are paid,
  i.e. P(y)
• The buyers problem is then
• Arrow (1963,1974) makes use of this format in
  analyzing insurance with loading and deductible

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2. Arrows model Choice of contracts

• Arrow (1963) assumes that
• The solution is an insurance contract with
  deductible

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2. Arrows model Choice of contracts

• Assume the choices are P(D) with different D
• The FOC is
• Expected claim payment under this contract is

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2. Arrows model Choice of contracts

• If P is set in Mossin fashion with expense c,
• The FOC is satisfied if D 0
• The model above implies the optimal choice is
  either full cover, or no cover at all, depending
  on the size of c
• How premiums are formulated affect the model
  solution but the presence of c seems critical to
  the existence of a solution

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2. Arrows model Choice of contracts

• With deductible D, the net premium is
• The minimum premium which the company can quote
  to customers will then be
• Hence the loading will be increasing with D

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Class exercise

• Suppose W0100, L50, p0.5, p0.5
• Assume log utility functions, substitute the
  parameters to the FOC
• Derive the demand for coverage
• Work out the case of increasing loading to 0.6
• Suppose an increase in wealth to 120 and p 0.6.
  What is C?
• If L increases to 60, what is C?
• Finally, suppose p0.6. What is C?
• Discuss this case in relation to PC insurance