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# Real world risk distribution

Suppose a population consists of two equally-sized groups:
- Type 1 individuals have a 15% annual chance of experiencing a catastrophic health care event thaat will reduce their wealth by \$40,000.
- Type 2 individuals have a 25% annual chance of experiencing a catastrophic health care even that will reduce their wealth by \$40,000.

Suppose an insurance company is willing to offer a single catastrophic health care "full insurance" policy that is actuarially fair, as computed across the population as a whole.
a) Suppose that all individuals in the population (both Type 1 and Type 2 possess an initial wealth of \$80,000 and a VN-M utility function given by U(W)=W^(1/2). Will Type 1 individuals purchase an insurance policy? Will Type 2? Show your work.
b) Repeat your work from part a) but not assume that all individuals possess an initial wealth of \$40,000
c) What do your results indicate about the risk attitude associated with this particular utility function, at different income levels? As part of your explanation, refer to (and calculate) the Pratt coefficient for this function form.
d) In "real life" insurance markets, what differences (income levels, risk attitudes, etc.) would you expect to observe between "safer" and "riskier" populations? Would there differences make it more or less likely that a "pooling equilibrium" (in which a one-size-fits-all insurance policy would appeal to all subgroups) would prevail?

#### Solution Preview

a) W0 = 80000, U = W^0.5
The insurance price will be:
P = (0.15+0.25)40000 / 2 = 8000
Type 1 person’s expected utility without insurance is :
E(U1) = 15%*U(loss) + 85%*U(no loss)
E(U1) = 15%*(80000-40000)^0.5 + 85%*80000^0.5 = 270.4
However, the expected wealth 80000-8 = 72,000
Then the utility with insurance is constantly is
U(W1) = 72000^0.5 = 268.3
As E(U1) > U(W1)), type 1 persons will NOT purchase insurance

Type 2 person’s expected utility without insurance is :
E(U1) = 25%*U(loss) + 75%*U(no loss)
E(U1) = 25%*(80000-40000)^0.5 + ...

#### Solution Summary

This solution examines world risk distribution by examining the two different insurance demographics and analyzing them against real life insurance markets and pooling equilibriums.

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