Question

Problem 2. Warren Buffy is an enormously wealthy investor who has built his fortune through his legendary investing acumen. He currently has been offered three major investments and he would like to choose one. The first one is a conservative investment that would perform very well in an improving economy and only suffer a small loss in a worsening economy. The second is a speculative investment that would perform extremely well in an improving economy but would do very badly in a worsening economy. The third is a counter-cyclical investment that would lose some money in an improving economy but would perform well in a worsening economy Warren believes that there are three possible scenarios over the lives of these potential investments: (1) an improving economy, (2) a stable economy, and (3) a worsening econ- omy. He is pessimistic about where the economy is headed, and so has assigned probabil- ities of 0.57, 0.29, and 0.14, respectively, to these three scenarios. He also estimates that his profits (in millions of dollars) under these respective scenarios are those given by the following table Investment Conservative Speculative Counter-cyclic Probability Improving StableWorsening conomy Economy Economy 10 10 20 10 10 0.57 10 20 0.14 al 0.29 Which investment should Warren make under each of the following criteria? (a) Maximin payoff criterion. (b) Expected payoff criterion. (c) Minimax regret criteiriorn (d) Warren believes that his preferences may be best represented by a utility function u(t) = c in t-| In t2, where c = 4.

Answer 1

(a)

Maximin payoff criterion is maximizing the minimum payoff.

The minimum payoff for Conservative investment is -30

The minimum payoff for Speculative investment is -10

The minimum payoff for Counter-cyclical investment is -20

The maximum of all minimum payoffs is -10 and is for Speculative investment.

Thus, the investment for Maximin payoff criterion is Speculative investment.

(b)

Expected payoff for Conservative investment is 0.57 * 20 + 0.29 * 10 - 0.14 * 30 = 10.1

Expected payoff for Speculative investment is 0.57 * 10 + 0.29 * 10 - 0.14 * 10 = 7.2

Expected payoff for Counter-cyclical investment is 0.57 * 10 + 0.29 * 15 - 0.14 * 20 = 7.25

The maximum expected payoff is for Conservative investment.

Thus, the investment for Expected payoff criterion is Conservative investment.

(c)

Regret for each outcome = Best Payoff - Payoff received

Regret table is,

	Improving economy	Stable economy	Worsening economy
Conservative	20 - 20 = 0	15 - 10 = 5	-10 + 30 = 20
Speculative	20 - 10 = 10	15 - 10 = 5	-10 + 10 = 0
Counter-cyclical	20 - 10 = 10	15 - 15 = 0	-10 + 20 = 10

The maximum regret for Conservative investment is 20

The maximum regret for Speculative investment is 10

The maximum regret for Counter-cyclical investment is 10

The minimum of all maximum regret is 10 and is for Speculative and Counter-cyclical  investment.

Thus, the investment for Minimax regret criterion is Speculative and Counter-cyclical  investment.

(d)

u(t) = 4/2 ln t² = 2 ln t²

Expected utility for Conservative investment is 0.57 * (2 ln 20² ) + 0.29 * (2 ln 10² ) + 0.14 * (2 ln (-30)² )= 11.40594

Expected utility for Speculative investment is 0.57 * (2 ln 10² ) + 0.29 * (2 ln 10² ) + 0.14 * (2 ln (-10)² )= 9.21034

Expected utility for Counter-cyclical investment is 0.57 * (2 ln 10² ) + 0.29 * (2 ln 15² ) + 0.14 * (2 ln (-20)² ) = 10.06884

The maximum expected utility is for Conservative investment.

Thus, the investment for Expected utility criterion is Conservative investment.