, at a cost of $P, the investment in the riskfree asset is:
$(50,000 – P)
Yearend wealth will be certain (since you are fully insured) and equal to:
[$(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 onehalf 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 endofyear 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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b.
With insurance coverage for the full value of the house, costing $200, endof
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 endofyear 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 underinsurance.
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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 Spring '13
 Ohk
 Standard Deviation, Utility, CML, Risk premium

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