Economics of Insurance

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Economics of Insurance
Lecture 9 A
Optimal Insurance
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INSURANCE TRADING LINE
Suppose the deal was that the insurance company
would pay 9 net compensation for every 1 of
premium paid. That is the NET PREMIUM RATE 1/9
0.111 or 11.1 pence per cover. We also would
say that the NET COMPENSATION RATE was 9.
Say PREMIUM 500, NET COMP would be 4500
Car worth 10,000 undamaged, 5000 damaged
We have a CHOICE OF TWO PROSPECTS INSURED
PROSPECT, NOT INSURED PROSPPECT
(9500,9500P1,P2)
(10000, 5000P1, P2).
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Equation for insurance trading line
The general form of any straight line equation
gives us
X2 -rX1 D
We know r is the insurance company's net
compensation rate which equals 9  
so X2 -9X1 D
The line must go through the initial prospect
(10000,5000, P1,P2)
so
5000    -910000    D
so
D    95000          
X2    95000  -  9 X1
This is the equation for the insurance trading
line
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Insurance budget constraint
X2    95000  -  9 X1
  X2    D  -  rX1
This equation and the line it represents is a
BUDGET CONSTRAINT
X2
D 95000
SLOPE -r -net compensation rate -1/net
premium rate -9
insured
9500
Net comp
5000
No insurance
X1
10000
9500
premium
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Trading insurance line as a budget constraint
A smaller premium payment , say 200, would buy a
lower amount of compensation
Moving the individual less far up the trading line
X2
Cant insure to any point to the right of the
trading line, initial prospect not big enough
Previous insurance
9500
A smaller amount of insurance
lower net comp 1800
5000
No insurance
X1
10000
9500
Smaller premium - 200
The trading line thus shows the range of deals
possible at any one price
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Indifference curves and optimum state contingent
assets
X2
We have reached the optimum
Of course, this is the TANGENT between the SCIC
and the constraint
still some feasible preferable points
Anywhere on this part of the budget line is
preferable to the initial (uninsured prospect)
For example here ?
No longer any reachable preferable points
But anywhere on this section of the constraint is
STILL preferable
What about here ?
Initial (uninsured) prospect
Value if damaged uninsured
X1
Value if undamaged and uninsured
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Indifference curves and optimum state contingent
assets
X2
We have reached the optimum
Of course, this is the TANGENT between the SCIC
and the constraint
This is where the client wants to be
They will insure to reach this point
Net compensation
Initial (uninsured) prospect
Value if damaged uninsured
X1
Value if undamaged and uninsured
premium
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Clients Optimal Insurance
Before using the following method (or derivative
of it) always check to see if INSURANCE CONTRACT
IS FAIR i.e.  NCR odds as perceived by the
customer
If so, optimisation methods are unnecessary
clients will choose FULL insurance. On the
diagram they will insure up to the certainty line.
However insurance is seldom fair, so optimality
implies either partial or over insurance (see
previous lecture). Optimisation methods must be
used to understand clients choices.
Diagrammatically, the task is to find the tangent
of the SCIC to the budget constraint, as in the
diagram below-
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Mathematical condition for optimum insurance
Mathematically, the problem is one of constrained
maximisation - maximising expected utility
subject to the budget constraint ie the insurance
trading line-
Maximise E(U) subject to X2 C - r X1
Set up a Lagrangian
L P1.U(X1 ) P2.U(X2) - l( X2 - C r X1 )
Utility is maximised when all dL/ dXi 0 and d
L/ dl 0
If U'(Xi) dU/ dXi
and Xi is the optimised value of Xi
(P1/P2 ). U'(X1)/ U'(X2) r
dL/ dX1 P1 U'(X1) - l r 0

gt
OR
dL/ dX2 P2 U'(X2) - l 0
U'(X1)/ U'(X2) r / (P1/P2 )
i.e. slope of indifference curve net
compensation rate
This is the point of tangency on the diagram
ratio of marginal utilities of money in each
state of the world ratio of net compensation
rate to odds against bad event
OR