An Example of Risky Project . Two individuals (numbered 1 and 2) are considering a real-estate development project. Suppose that they have an option to buy a tract of land Individual Risk Tolerances for $125,000, after which they would then need to spend an additional $40,000 on improvements before . Suppose that each of these two individuals they could sell the land in subdivided lots. evaluates risky incomes using a utility function The total revenue that they could then earn from with constant risk tolerance, where individual selling these lots would be uncertain: it has an expected value of $200,000 and a standard deviation 1 has risk tolerance T, = $20,000, and of $25,000. individual 2 has risk tolerance T, = $30,000. So the net returns from this real estate project next year will have expected value . They must decide whether to undertake this H = 200,000 - (125,000 + 40,000) = $35,000 real estate project, and if so, how to divide the and standard deviation o = $25,000. returns among themselves. Certainty Equivalent : Individual 1 Certainty Equivalent: Individual 2 Recall that when an individual with constant risk . So if individual 2 were to undertake this project tolerance T has a gamble that will pay a random himself, his certainty equivalent would be amount of money with mean u and standard deviation 3, his certainty equivalent for the gamble is CE = H - (0.5/T,)* q' CE = H - (0.5/T)* o = 35000 - (0.5/30000)*(25000') So if individual 1 were to undertake this project = 35000 - 10417 = $24,583. himself, his certainty equivalent would be . That is, the option to buy this land and undertake this CE, = H - (0.5/T,)* 02 project would be worth $24,583 to individual 2, if he = 35000 - (0.5/20000) *(25000?) had to undertake all the risks of the project alone. = 35000 - 15625 = $19,375. So if individual 1 had the option to buy this land, then That is, the option to buy this land and undertake this individual 2 would be willing to pay up to $24,583 to project would be worth $19,375 to individual 1, if he buy the option from him, and individual 1 would be had to undertake all the risks of the project alone. glad to sell the option for any price above $19,375. Optimal Risk Sharing Rule Certainty Equivalents with Risk Sharing When individual 1 takes a 40% share, his expected monetary But even though individual 2 is strictly more risk value is 0.40 35000 = $14,000 and his standard deviation is tolerant than individual 1, the project could be even 0.40*25000 = $10,000, and so his certainty equivalent is more valuable to these individuals if they undertake CE(1) = 14000 - (0.5/20000)*(10000) the project as partners, with individual 1 taking a = 14000 - 2500 = $11,500 positive share of the project's risks. When individual 2 takes a 60% share, her expected monetary What is the optimal sharing rule? value is 0.60 35000 = $21,000 and her standard deviation is A person's optimal shares of the risk is equal to her 0.60 25000 = $15,000, and so her certainty equivalent is CE(2) = 21000 - (0.5/30000)*[150001) share of the total risk tolerance of the group = 21000 - 3750 = $17,250 Since individual 1 has risk tolerance T, = $20,000, and When they plan to share the risks in this way, their total individual 2 has risk tolerance T, = $30,000, thus 1's certainty equivalent of the project is share should be 20000/(20000+30000)=40% CE(1) + CE(2) = 11500 + 17250 = $28,750 That is, individual 1 takes a 40% share, while individual . This total $28,750 is the maximal sum of certainty equivalents 2 takes a 60%% share. that the partners can achieve by sharing the profits of this Q4: In the real estate project discussed in class, now suppose individual 1 has risk tolerance ri = $10,000, and individual 2 has risk tolerance r = $40,000. (1) What is the optimal risk sharing rule? (2) What are their respective CE? (3) What is the maximal total CE with joint risk tolerance