Question

2. Risk Premium

a. Suppose the utility function is given by the equation ?(?) = ln(? + 1). Graph utility at the points $0, $80, $100.

b. Suppose there is a lottery where you win $100 with 20% chance, $80 with 30% chance, and $0 with 50% chance. What is the expected winnings (in dollars)? What is the expected utility (in utils)? Add this point to the graph.

c. What is the utility at the expected dollar winnings (i.e., what is the utility if the expected winning amount were instead guaranteed)? Add this point to the graph.

d. What is the most one would be willing to pay for a lottery ticket (i.e., what is the dollar amount that, when paid with certainty, gives you the same utility as the expected utility)?

Answer 1

a) For x=0, u = ln (1) = 0

For x=80, u = ln (81) = 4.39

  For x=100, u = ln (101) = 4.61

b) Expected Winning = 0.2(100) + 0.3(80) + 0.5 (0) = 20+ 24 = 44

Expected utility = (0.2) (ln (101)) + (0.3) (ln (81)) + (0.5) (ln (1)) = 0.923 + 1.318 + 0 = 2.241

c) utility of expected return = ln (45) = 3.80

d) U'(r) = U'() = e^2.241 = x+1 [ This is the inverse function of utility]

or 9.40 = x+1

or. x = 8.340 This is the willingness to pay amount.