Question

Willingness to Pay for Insurance. Frank Black has finally quit playing guitar for the Pixies, and has decided to pursue a lifelong dream of his, generating power through small-scale electricity generation plants. Suppose Frank has acquired the rights to a technology that allows for the sustainable generation of electric power, and the license to sell it onto the Arizona electricity-grid (no small feat on its own). Table 1 shows his projections for the first 5 years of operating profit (in millions of dollars). Despite being a rock-star, however, Frank is relatively risk averse, so his coefficient of risk aversion is 0.4. Use the data in table 1 to answer the questions that follow.

Table 1. Expected Utility of Profit

Year Profit U(profit) U(E[profit])

1 10

2    12

3    8

4 16

5 14

Average = 12

Assume the coefficient of risk aversion given above. Find the missing values for the utility of profit, for each year, in the above table, and find the expected (average) utility of profit. Next, find the utility of expected (average) profit. Compare these two values to show how you know Frank is risk averse.

Frank is a pretty shrewd businessman, and would be willing to sell his power-generation firm to a willing investor. Assume Frank would accept his certainty equivalent (CE) value as a fair price for his firm. Find the CE value, and interpret what the CE value means in intuitive terms.

Now suppose Frank is shopping for an insurance company that would be willing to insure his profit stream over time. Suppose his expected loss, or the amount his profit falls below average, is $2.0 million. What is his total willingness to pay for insurance, including his expected loss and his risk premium? Explain how this relates to the Bernoulli Hypothesis.

Answer 1

Year	Profit (x)		()2	P (Probability )	Utility of Profit profit - 0.5 X Risk Averse Ratio X volatility	Expected Utility of Profit
1	10	-2	4	= 10/60 = 0.1666	= 10 - 0.5 X 0.4 X 7.4833= 8.5034	=8.5034 X 1.666= 14.1666
2	12	0	0	=12/60 = 0.2	=12 - 0.5 X 0.4 X 7.4833=10.5034	=10.5034 X .2 =2.1006
3	8	-4	16	=8/60 = 0.1333	=8 - 0.5 X 0.4 X 7.4833=6.5034	=6.5034 X 0.1333= 0.8669
4	16	4	16	=16/60 = 0.2666	=16 - 0.5 X 0.4 X 7.4833=14.5034	=14.5034 X .2666 = 3.8666
5	14	2	4	= 14/60 = 0.2333	=14 - 0.5 X 0.4 X 7.4833=12.5034	= 12.5034 X 0.2333= 2.9170
			56		54.0136	23.9177

Average () = 60/5 =12

= 7.4833

Average Utility of Profit = 54.0136/5 = 10.8027

Average Expected Utility of Profit = 23.9177 /5 =4.7835

Frank is a pretty shrewd businessman, and would be willing to sell his power-generation firm to a willing investor. Assume Frank would accept his certainty equivalent (CE) value as a fair price for his firm. Find the CE value, and interpret what the CE value means in intuitive terms.

Here, Certainty Equivalent Value to be calculated = Total Profit for 5 years / (1 + ( Risk Premium)

Risk premium is return earned over risk free return ,Here it is = (12- 10.8027) /12 = 0.0997 or say 9.97%

Certainty Equivalent Value = 60/(1+9.97%) = $ 54.5603 million

Now suppose Frank is shopping for an insurance company that would be willing to insure his profit stream over time. Suppose his expected loss, or the amount his profit falls below average, is $2.0 million.

His total willingness to pay for insurance, including his expected loss and his risk premium = same premium

Explain how this relates to the Bernoulli Hypothesis : It considers loss and profit as both are at risk and should be treated equally. which is as per Bernoulli Hypothesis.

In this Insurance it covers profit and loss both. Hence it considers and follow Bernouli Hypothesis for Insurance premium Paid by Frank