Belinda faces uncertainty about the value X_Bof her production.

With probability 0.6, the value of X_Bis 600. With probability 0.4, the value of X_Bis 200.

Belinda is risk-averse and her utility function is u(x) = 15(x + 150)^2/3, where x denotes the monetary payoff.

It is recommended to round numbers to 2 decimal places.

(a) Compute the expected utility of X_B.

Show that Belinda prefers trading X_Bfor a fixed deterministic amount that yields the same

returns in expectation.

***

An insurer can offer partial coverage of Belinda's risk. At a price C = 70, the insurer can provide a financial compensation of 160 when the value of X_Bis low. The deal is thus equivalent to an asset of value Z_Bthat pays 530 with probability 0.6 and 290 with probability 0.4.

(b) Find out whether Belinda benefits from accepting the deal, or equivalently from trading X_Bfor Z_B.

Show that the insurer is also making positive profits in expectations.

***

Belinda has another alternative to obtain some form of insurance. She can pool her income with her best friend Matt who faces the same uncertainty. Matt also has an asset of value X_Mthat produces 600 with probability 0.6 and 200 with probability 0.4. The productions of Matt and Belinda are independent.

By pooling their income, Belinda gets an asset of value, which yields 600 with probability 0.36, 200 with probability 0.16 and 400 with probability 0.48.

(c) Explain in detail how we obtain the value Y as defined just above.

Does Belinda prefer to pool income with Matt or to buy the insurance contract?

***

Define Z_Mas the value of buying insurance for Matt, which is defined like Z_B(same

probabilities and same conditional returns). A fourth option is for Belinda and Matt to

buy insurance, each one getting Z_Band Z_Mand thenpool their income.

(d) Findout the best option for Belinda. Explain your findings carefully.

(e) The insurer thinks about its pricing strategy, so that Z_Bdepends on C. Without solving

any equation, explain how we can determine the minimum price the insurer can propose.

How can the insurer determine the maximum price they can fix? Is the possibility of Belinda

pooling her income good or bad news for the insurer?