Question

LeBron has wealth = $200 and has a 75% probability that he requires surgery, that costs $72. LeBron's utility curve over total wealth is given by U(W) = Square Root W.

(a) [1 point] is sam a risk- adverse , risk loving or risk neutral individual? Why?

(b) [1point] what is his expected loss

(c) [2 points] suppose an insurance company incurs an over head expense of $20 per additional client. Can the company profitably insure LeBron?

Answer 1

U = √W

1. LeBron is a risk-averse individual because his expected utility of wealth is less than that the utility of expected wealth i.e. U(E(W)) > E(U(W)). Following explanation proves this,

Here it is given that LeBron has wealth, W = $200 and there is a 75% probability that he requires a surgery, that costs $72.

Therefore his expected wealth = $200*25/100 + 128*75/100

                                                        = $146

Therefore utility of expected wealth, U(E(W)) = √146 = 12.08

And his expected utility of wealth, E(U(W)) = U(200)*25/100 + U(128)*75/100

                                                                             = √200*25/100 + √128*75/100

                                                                              = 12.02

Thus, U(E(W)) > E(U(W))

Therefore, we can see that Sam’s expected utility of wealth is less than utility of expected wealth; i.e. U(E(W)) > E(U(W)).

Therefore Sam is a risk averse person.

2. His expected loss = 72*75/100 = $54
3. The maximum amount of money that the insurance company can charge LeBron such that it will keep his utility same as his expected utility of wealth, E(U(W)) :

U = √W

Or, √W = 12.02

Or, W = 144.48

Therefore insurance company can charge a maximum amount of 200 – 144.48 = $55.52

Now, Company’s total cost = Expected loss of LeBron + Overhead expense of $20 = $54 + $20 = $74

But the maximum amount company can charge LeBron is $55.52

Therefore company will have a loss of $74 - $55.52 = $18.48

Therefore company can not profitably insure LeBron.