Download Full Insurance Theorem

Full Insurance Theorem
Risks to Wealth
Motives
 Practical
risk management
 Analysis: base case
 First example of general principles
Fair pricing I
 Fair
price = expected value of loss
 Law of large numbers says a firm can
survive with nearly fair pricing.
 Competition of insurers says it must do
so.
Fair pricing II
 Finance
theory says investors diversify
risk.
 They compete.
 Therefore they are not rewarded for
bearing insurance risks. They get only
the expected value.
 Details: insurance risks are not (very
much) correlated with the stock market.
Ignore interest for now
 Separate
it from risk.
 Absent risk, clients pay insurers first,
receive it back later.
 Insurers should pay some interest, just
like banks.
 They do, in fact.
The consumer
 Suppose
that a consumer
is
a risk-averse, expected utility maximizer
with utility independent of state
and the same subjective probabilities as
the insurer
 That
means
S
Utility    sU ( ws )
s 1
Two states
Utility   1U ( w1 )   2U ( w2 )
 The
pi’s are the same as for the insurer.
 The U’s are not of the form U s ( ws )
 as might happen if state s involved
illness or death.
Risk aversion
 Risk
aversion means the second
derivative of the vN-M utility function is
negative.
U ( w)  0
That means the first derivative is decreasing,
which is the same as decreasing marginal
utility of wealth.
Note on the use of risk
aversion
 Since
marginal utility is always
decreasing, each value of marginal
utility corresponds to exactly one value
of wealth.
MU
w
Insurers price fairly
 i.e..,
price of state-s wealth is the
probability of state-s
ps   s
Suppose state-s has probability = .5,
then
a dollar for state s costs fifty cents.
the world in the model
Time zero
Make insurance
contracts, i.e., trade
state-contingent claims
Time one
s=1
s=S
Execute the
contracts and
consume.
Full insurance theorem
 Suppose
that the consumer is a riskaverse, expected utility maximizer with
utility independent of state and having
the same subjective probabilities as the
insurer. Suppose further that the
insurer prices fairly. Then the optimum
insurance for the consumer is full
insurance
w2
equation of the budget
constraint:
p1w1  p2 w2  p1w1  p2 w2
p1
slope  
p2
( w1 , w2 ) = endowed risk
non loss state
w1
w2
slope =
MRS
( w1 , w2 )
non loss state
slope 
p1
1
 
p2
2
w1
Proof:
At the consumer optimum, MRS = price ratio.
Recall that MRS is the ratio of marginal utilities.
Specifically
 1U  ( w1 )
MRS 
 2U  ( w2 )
The price ratio is the ratio of probabilities,
implying
 U ( w ) 
MRS 
1
1

1
 2U ( w2 )  2
proof continued
Divide by the probability ratio on both sides.
Result:
U ( w )
1
U ( w2 )
1
Then U  ( w1 )  U  ( w2 )
which implies
w1  w2
which means full insurance. Q.E.D.