Question

(Mathematical Question) Suppose you have a utility function   where W  is the level of wealth you end up with. You are currently in possession of 81  dollars. There is a  1/3 chance that a miserable event will happen and cost you all 81  dollars. Assume your preference over uncertainty is characterized by the expected utility.

a. (5pts) Define the gamble

b. (5pts) What is the expected value of this gamble?

c. (10pts) Find the certainty equivalent of the gamble. Then find the insurance premium you will be willing to pay.

d. (10pts) Suppose now that you are wealthier and are in procession of 225 dollars, instead of 81 dollars. The same miserable event might happen with 1/3 chance and cost you 81 dollars. (You may want to define the gamble and calculate the expected value of the gameble yourself). Find the certainty equivalent and the insurance premium in this case.

Answer 1

a.

Gamble is defined in terms of probability of uncertainty in the future. The gambler plays the gamble in order to acheive higher pay-off than what the current possession he/she has. There are two states of the world: good and bad. In good state, you win the gamble and in the bad state, some amount of money is lost from the current possession. The probability to each state is assigned and then decisions are made based on the aggregates like expected value or expected utility.

b.

U = sqrt(w)

where w = wealth/income

in good state, income = 81
In bad state, loss of $81
So income in bad state = 81 - 81 = $0
Bad state probability = 1/3 = 0.33, good state P = 2/3 = .67
Thus EU (expected utility) = .67*sqrt(81) + .33 *sqrt(0)

EU = 6.03

Expected Value is the sum of each probability times the expected pay-off.

EV = .67 ( 81) + .33 (0)

EV = 54.27

c.

Certainty Equivalent (CE) is the guaranteed return to the person in the present as compared to the unexpected return to be received in the future.

For CE:

u(CE) = EU

sqrt (CE) = EU = 6.03

CE = square(6.03)

CE = $36.36

Insurance Premium is the difference between Expected Value and Certainty Equivalent

Risk Premium = EV - CE

RP = 54.27 - 36.36

RP = $17.91

d.

in good state, income = 225
In bad state, loss of $81
So income in bad state = 225 - 81 = $144
Bad state probability = 1/3 = 0.33, good state P = 2/3 = .67
Thus EU = .67*sqrt(225) + .33 *sqrt(144)

EU = 10.05 + 3.96

EU = 14.01

EV = .67 ( 225) + .33 (144)

EV = 150.75 + 47.52 = $198.27

New CE:

u(CE) = EU

sqrt (CE) = EU = 14.01

CE = square(14.01)

New CE = 196.28

New Insurance Premium (Risk Premium) = EV - CE

RP = 198.27 - 196.28

New RP = $1.99 (roughly $2)

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