FINC 6022 Behavioural Finance Workshop Questions 1 1. Keith is considering purchasing insurance. His utility function is given by U(w) = w0.5, where w is his wealth, equal to $25,000. There is a 5% chance that Keith will fall ill, which will cost him $5,000. a) What is Keith’s expected utility from not purchasing any insurance? b) What is the actuarially fair premium for the health insurance in this case? c) Assuming that the premium is actuarially fair, what is the expected utility of buying full insurance (ie. insuring all $5,000 of potential loss)? What is the expected utility of insuring only half the potential loss (i.e. pay half the premium in (b) but receive only $2,500 coverage)? 2. Suppose a (fair) coin is tossed, and a gambler receives $100 if it lands “heads” and $0 otherwise. a) If the gambler has utility function U(w) = w-0.5, what is the utility of playing this coin tossing game. b) What is the equivalent amount that could be offered to the gambler in (a) for which he would be indifferent between playing the coin tossing game and taking the gamble (that is, the certainty equivalent)? c) Show that this utility function is from the family of Constant Relative Risk Aversion utility functions (isoelastic utility) d) Suppose that the utility function is changed to U(w) = w-0.25. What is the new certainty equivalent for the gambler? 3. Now suppose that an unfair coin is tossed, where P(Heads) = 0.52. For each dollar invested, the bettor will receive $α = $2 if the coin lands heads, and $0 otherwise. For a gambler with a CRRA utility function, the optimal decision will be to wager a fraction f of his bankroll on the outcome Heads. Suppose that the bettor’s utility function is U(w) = ln(w). The bettor’s problem is to work out f such that E[U(w)] = pH ln(1 + (α - 1)f) + pT ln(1 – f) which reduces to E[U(w)] = pH ln(1 + f) + pT ln(1 – f) when α = 2. Find the value of f that will maximise the gambler’s Expected Utility. Note that you can either derive the value of f using calculus or you can use a spreadsheet with solver to find the solution.

4. Consider the following data: x 1 2 3 4 5 6 7 8 9 10 TOTAL Frequency (f) 1 0 3 4 4 2 3 0 2 1 20 Frequency (g) 2 5 1 5 7 0 0 0 0 0 20 PDF (f) 0.05 0 0.15 0.2 0.2 0.1 0.15 0 0.1 0.05 1 PDF (g) 0.1 0.25 0.05 0.25 0.35 0 0 0 0 0 1

Show that lottery f (first-order) stochastically dominates lottery g. Draw a diagram of the cumulative density functions of f and g.