At a cost of p the investment in the risk free asset

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, at a cost of $P, the investment in the risk-free asset is: $(50,000 – P) Year-end wealth will be certain (since you are fully insured) and equal to: [$(50,000 – P) × 1.06] + $200,000 Solve for P in the following equation: [$(50,000 – P) × 1.06] + $200,000 = $252,604.85 P = $372.78 This is the most you are willing to pay for insurance. Note that the expected loss is “only” $200, so you are willing to pay a substantial risk premium over the expected value of losses. The primary reason is that the value of the house is a large proportion of your wealth. 2. a. With insurance coverage for one-half the value of the house, the premium is $100, and the investment in the safe asset is $49,900. By year end, the investment of $49,900 will grow to: $49,900 × 1.06 = $52,894 If there is a fire, your insurance proceeds will be $100,000, and the probability distribution of end-of-year wealth is: Probability Wealth No fire 0.999 $252,894 Fire 0.001 $152,894 For this distribution, expected utility is computed as follows: E[U(W)] = [0.999 × ln(252,894)] + [0.001 × ln(152,894)] = 12.4402225 The certainty equivalent is: W CE = e 12.4402225 = $252,766.77
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6-13 b. With insurance coverage for the full value of the house, costing $200, end-of- year wealth is certain, and equal to: [($50,000 – $200) × 1.06] + $200,000 = $252,788 Since wealth is certain, this is also the certainty equivalent wealth of the fully insured position. c. With insurance coverage for 1½ times the value of the house, the premium is $300, and the insurance pays off $300,000 in the event of a fire. The investment in the safe asset is $49,700. By year end, the investment of $49,700 will grow to: $49,700 × 1.06 = $52,682 The probability distribution of end-of-year wealth is: Probability Wealth No fire 0.999 $252,682 Fire 0.001 $352,682 For this distribution, expected utility is computed as follows: E[U(W)] = [0.999 × ln(252,682)] + [0.001 × ln(352,682)] = 12.4402205 The certainty equivalent is: W CE = e 12.440222 = $252,766.27 Therefore, full insurance dominates both over- and under-insurance. Over- insuring creates a gamble (you actually gain when the house burns down). Risk is minimized when you insure exactly the value of the house.
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