Question

Terry’s utility of wealth is given by: u(w) = ln(w).

Suppose Terry has $1 million in his bank account and a beach house worth $2 million. With probability 1/3, his beach house will get destroyed by a hurricane.

(a) Is Terry risk-averse, risk-neutral, or risk-loving? Verify your answer using calculus.

(b) Determine the actuarially fair premium for an insurance plan that will compensate him $2 million if his beach house gets destroyed by a hurricane.

(c) Write out the two expressions you would compare to determine if Terry will purchase this insurance if the premium is actuarially fair (do not evaluate them).

Answer 1

to no Terry is risk overse, UC ECws] > E[vcws] o Terg in risk ventral, U[ECw] = E[vcw)] 9 Terry iis risk loving, U[E(W] < E[Uw)] when ULE (w)] is the utility of expected wealth E[vcws) in the expected utility of wealth E [urwo] - ** U(42m) + x 0(43) NIN x 14.914 = 13.816 3 = 14.548 O[E60] - V[ x $imn + x $ 3mn] = 14.663 Since U[ECCO] SE[vcws] risk averse

(b) W Actuarially fair premiums allows the insurang companied to just break even; die they don't make ang profits. Actuavilly fair premium = enpected costs = x( loss of hurricane strikes) - $* 2,000,000 + 2x0 = $ 666, 666667 (c) equat © We take the difference between the level a wealth if insurance is taken 3mm) and the level y wealth which provides utility the utility q expected wealth when insuranu is not taken. We compare this value with the actuarially fair premium and if the actor actuarially fan premium is less than this value me Terry will purchase the insurance wly misurance to tako) - o'[E6w]]] >premier then Terry bugs the insurance