(Our thanks to David Pyle, University of California, Berkeley, for providing this problem.) Mr. Casadesus's current wealth consists of his home, which is worth $50,000, and $20,000 in savings, which are earning 7% in a savings and loan account. His (one-year) homeowner's insurance is up for renewal, and he has the following estimates of the potential losses on his house owing to fire, storm, and so on, during the period covered by the renewal:
Value of Loss ($)Probability
0.............................. .98
5.000......................... .01
10.000.........................005
50,000.........................005
His insurance agent has quoted the following premiums:
Amount of Insurance ($)Premium ($)
30.0..........................................30 + AVL1
40.0..........................................27 + AVL2
50.0..........................................24 + AVL3
where AVL = actuarial value of loss = expected value of the insurer's loss.
Mr. Casadesus expects neither to save nor to dissave during the coming year, and he does not expect his home to change appreciably in value over this period. His utility for wealth at the end of the period covered by the renewal is logarithmic; that is, U(W) = ln(W).
(a) Given that the insurance company agrees with Mr. Casadesus's estimate of his losses, should he renew his policy (1) for the full value of his house, (2) for $40,000, or (3) for $30,000, or (4) should he cancel it?
(b) Suppose that Mr. Casadesus had $320,000 in a savings account. Would this change his insurance decision?
(c) If Mr. Casadesus has $20,000 in savings, and if his utility function is
U(W) = -200,000-1,
should he renew his home insurance? And if so, for what amount of coverage?