Suppose the typical Florida resident has wealth of $500,000 of which his or her

Question

Suppose the typical Florida resident has wealth of $500,000 of which his or her home is worth $100,000. Unfortunatly Florida is infamous for its hurricanes, and it is belived there is a 10% change of a hurricane that could totally destroy a house (a loss of $100,000). However, it is possible to retrofit the house with various protective devices (shutters, roof bolts) for a cost of $2,000. This reduces the 10% chance of a loss of $100,000 to a 5% chance of a loss of $50,000. The homeowner must decide whether to retrofit and thereby reduce expected loss. The problem for an insurance company is that is does not know whether the retrofit will be installed and therefore cannot quote a premium conditioned on the policyholder choosing this action. Nevertheless, the insurance company offers the following two policies from which the homeowner can shoose. 91) the premium for insurance covering total loss is $1,500. The typical homeover has a utility function equal to square root of wealth. Will the homeowner retrofit the house, and which insurance policy will the homeover buy?Will the insurance company make a profit (on average) given the homeownvers choice?

Explanation / Answer

Case 1: When owner go without retrofit the house, then the owner has a wealth of

W = 500000 with pr = 0.9 or 400000 with pr = 0.1

Expected wealth: W* = E(W) = 0.9 * 500000 + 0.1 * 400000 = 450000 + 40000 = 490000

Expected Utility: E[U(W)] = 0.9 * (500000)^0.5 + 0.1 * (400000)^0.5

= 0.9 * 707.107 + 0.1 * 632.46

= 636.4 + 63.25 = 699.65

Utility of expected wealth: U(W*) = 490000^0.5 = 700

Thus, U(W*) > E[U(W)] , thus owner will buy an insurance. Let the premium be 'P'

E[U(insurance)] = 0.9 * (500000 - P)^0.5 + 0.1 * (500000 - P)^0.5 = (500000 - P)^0.5

Equating it with E[U(W)], we get

(500000 - P)^0.5 = 699.65

P = 500000 - 489510.1225 = 10489.8775

Case 2: When owner retrofit the house, then the owner has a wealth of

W = 500000 - 2000 = 498000 with pr = 0.95 or W = 498000 - 50000 = 448000 with pr = 0.05

Expected wealth: W* = E(W) = 0.95 * 498000 + 0.05 * 448000 = 473100 + 22400 = 495500

Expected Utility: E[U(W)] = 0.95 * (498000)^0.5 + 0.05 * (448000)^0.5

= 0.95 * 705.7 + 0.05 * 669.33

= 670.41 + 33.47 = 703.88

Utility of expected wealth: U(W*) = 495500^0.5 = 703.92

Thus, U(W*) > E[U(W)] , thus owner will buy an insurance. Let the premium be 'P'

E[U(insurance)] = 0.95 * (498000 - P)^0.5 + 0.05 * (498000 - P)^0.5 = (498000 - P)^0.5

Equating it with E[U(W)], we get

(498000 - P)^0.5 = 703.88

P = 498000 - 495447.0544 = 2552.9456

But since insurance company is charging only $1500 whereas owner's maximum willingness to pay for insurance is $10489.8775 without retrofit the house and $2552.9456 with retrofit the house. Thus owner will go without retrofit the house as he saving more.

Now the expected profit of the insurance company (when owner go without retrofit the house) = 1500 - (0.9 * 0) - (0.1 * 100000) = 1500 - 10000 = (-)8500

and the expected profit of the insurance company (when owner go with retrofit the house) = 1500 - (0.95 * 0) - (0.05 * 50000) = 1500 - 2500 = (-) 1000

Thus insurance company is not making profit by charging only $1500 to cover the loss in both circumstances

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