Question

Problem 4: Risk premium. Casey is considering buying a lottery ticket. The lottery pays $100 with probability and $0 with probability a) What is the "fair" price of the lottery ticket? In other words, what price of the lottery ticket provides the lottery owner an expected profit of zero? b) Suppose that a lottery ticket costs $15. If Casey is willing to buy the lottery ticket, what can we say about Casey's risk premium for this lottery? c) Suppose that Casey's utility function is U(C) = C. What is the highest price that Casey would be willing pay for a lottery ticket?

Answer 1

Given,

Lottery pays = $100 with probability 1/4 (= 0.25)

Lottery pays = $0 with probability 3/4 (= 0.75)

Now,

a) Fair Price of lottery = Expected value of lottery

We know,

$Expected\; value = \sum _{i}^{n} = p(x_{i})*x_{i}$

So, Expected Value of lottery = (0.25 * $100) + (0.75 * $0)

= $25 + $0

= $25

So, The fair price of lottery is $25

b) If lottery costs = $15

If Casey is willing to buy so the price of lottery for Casey = $15

So, Casey is certain of $15 return from the lottery as Casey is willing to buy

Now,

Risk Premium = Expected Price(or Fair price) - Price willing to pay (or Certainty Equivalent)

So, Risk Premium = $25 - $15 = $10

So, Casey's Risk Premium for the lottery is $10

c) If Casey's utility function is U(C) = √C

So, The Expected utility from the lottery

$Expected\; utility = \sum _{i}^{n} = p(x_{i})*U(x_{i})$

So, Expected Utility = (0.25 * u($100)) + (0.75 * u($0))

= (0.25 * √$100) + (0.75 * √$0)

= (0.25 * $10) + $0

= $2.5

We know Certainty Equivalent (C.E) is the level of return from the Expected Utility.

So, U(C.E) = $2.5

√C.E = $2.5

C.E = $2.5²

C.E = $6.25

As Certainty Equivalent represents the price at which she gets same utility as of playing the lottery. Hence, it is the maximum an individual will be willing to pay for lottery.

So, Highest price that Casey would be willing to pay for lottery ticket is $6.25

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