Question

# LeBron has wealth = \$200 and has a 75% probability that he requires surgery, that costs...

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?

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.

1. His expected loss = 72*75/100 = \$54
2. 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.

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