UTILITY ANALYSIS HANDOUTS

Transcription

1 UTILITY ANALYSIS HANDOUTS 1

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3 UTILITY ANALYSIS Motivating Example: Your total net worth = $400K = W 0. You own a home worth $250K. Probability of a fire each yr = Insurance cost = $1K. Question: Should you buy insurance? Analysis: Let W = end-of-yr wealth. Option 1: Don t buy insurance: E[W] = 0.999(400,000) (250,000) = 399,750. Option 2: Buy insurance: E[W] = 0.999(399,000) (399,000) = 399,000. Conclusion: Buy insurance! Lottery perspective: Lottery 1: 99.9% chance of winning 400K or 0.1% chance of winning 150K. Lottery 2: 100% chance of winning 399K. Which would you choose? Most people would buy the insurance. This is because people are risk-averse. Consequently, people do not use the Expected Monetary Value (EMV) criterion to make their decisions. Need a decision-making theory that accommodates risk-averseness, which can be used to help decisionmakers make decisions best for them. Example 2: Suppose I flip a fair coin. If it comes up heads, you win $100; otherwise, you win $0. a. What is the maximum amount you would be willing to pay to play this game? b. Suppose the coin had an 80% chance of coming up heads. Now how much would you pay? 3

4 Lotteries: L = { (w 1, p 1 ), (w 2, p 2 ),, (w N, p N ) } w k = a wealth amount p k = probability of realizing wealth w k Without loss of generality we assume that w 1 < w 2 < < w N. p k 0 and p 1 + p p N = 1. (We allow zero probabilities.) Assume that there are numbers W Lower and W Upper such that W Lower < w 1 < w 2 < < w N < W Upper. A trivial lottery involves only one outcome. It will be denoted by its outcome w. A simple lottery involves only two outcomes. Key Assumption: Investors (decision-makers) prefer more wealth to less: Assume w i < w j. Given a choice between L A = {(w i, 1-p A ), (w j, p A )} and L B = {(w i, 1-p B ), (w j, p B )} with p A < p B, investors will choose lottery B. We write L A > L B. For example, suppose L A = {(0, 0.5), (100, 0.5)} and L B = {(0, 0.20), (100, 0.80)}? Two lotteries L A and L B are said to be equal if the investor is indifferent between them. Notation: L A ~ L B. 4

5 Key Assumption: Assume for each p ε [0, 1] the investor can assign a unique W(p) ε [W Lower, W Upper ] such that L p := {( W Lower, 1-p), (W Upper, p)} ~ W(p) = L W(p) := {(W(p), 1)}. W(p) is called the Certainty Equivalent (CE). Also assume that for each W ε [W Lower, W Upper ] the investor can assign a unique p := U(W) such that L U(W) := {( W Lower, 1-U(W), (W Upper, U(W))} ~ W = L W := {(W, 1)}. Note that. As p ranges from 0 to 1, the values of W(p) range from W Lower to W Upper. The values of W(p) are non-decreasing. As W ranges from W Lower to W Upper, the values of U(W) range from 0 to 1. The values for U(W) are also non-decreasing. The assumptions imply that U(.) is a continuous, non-decreasing function such that U(W Lower ) = 0 and U(W Upper ) = 1. The function U(.) is called a utility function. Existence? Example: Consider the motivating example with U(W) = W ½. 5

6 Extension of Motivating Example: Suppose our investor has a utility function U(W) = W ½ but now must choose between these two lotteries: L A = {(49, 0.4), (64, 0.6)} or L B = {(25, 0.3), (81, 0.7)}. Which one should he choose? Analysis: Consider Lottery A: Case (a): Coin is flipped and our investor wins 49. Right before receiving the 49, we point out that the investor is indifferent between receiving this 49 for sure and the lottery L = {(0, 1 U(49)), (100, U(49))} = {(0, 0.3), (100, 0.7)}. Case (b): Coin is flipped and our investor wins 64. Right before receiving the 64, we point out that the investor is indifferent between receiving this 64 for sure and the lottery L = {(0, 1 U(64)), (100, U(64))} = {(0, 0.2), (100, 0.8)}. So we can imagine that lottery A can be played as a two-stage game (or compound lottery): Stage 1: Flip a coin with a 40% chance of winning 49, 60% chance of winning 64. Stage 2: Replace the winnings after stage 1 with its equivalent lottery and flip the appropriate coin. Conclusion: In the two-stage game associated with lottery A, the final outcomes are either 0 or 100. The chance of winning 100 is 0.4(0.7) + 0.6(0.8) = Thus, lottery A is equivalent to the lottery {(0, 0.24), (100, 0.76)}. Consider Lottery B: Case (a): Coin is flipped and our investor wins 25. Right before receiving the 25, we point out that the investor is indifferent between receiving this 25 for sure and the lottery L = {(0, 1 U(25)), (100, U(25))} = {(0, 0.5), (100, 0.5)}. Case (b): Coin is flipped and our investor wins 81. Right before receiving the 81, we point out that the investor is indifferent between receiving this 81 for sure and the lottery L = {(0, 1 U(81)), (100, U(81))} = {(0, 0.1), (100, 0.9)}. So we can imagine that lottery A can be played as a two-stage game (or compound lottery): Stage 1: Flip a coin with a 30% chance of winning 25, 70% chance of winning 81. Stage 2: Replace the winnings after stage 1 with its equivalent lottery and flip the appropriate coin. Conclusion: In the two-stage game associated with lottery B, the final outcomes are either 0 or 100. The chance of winning 100 is 0.3(0.5) + 0.7(0.9) = Thus, lottery A is equivalent to the lottery {(0, 0.22), (100, 0.78)}. Now which lottery should our investor choose? Key Assumption: Okay to substitute lotteries as we did. 6

7 Key Observation: Notice that 0.76 = 0.4(0.7) + 0.6(0.8) = 0.4U(49) + 0.6U(64) = 0U(25) + 0.4U(49) + 0.6U(64) + 0U(81) := E[U(W A )], where W A is the random variable whose outcomes are {0, 49, 64, 81} with probabilities {0, 0.4, 0.6, 0}. Similarly, 0.78 = 0.3(0.5) + 0.7(0.9) = 0.3U(25) + 0.7U(81) = 0.3U(25) + 0U(49) + 0U(64) + 0.7U(81) := E[U(W B )], where W B is the random variable whose outcomes are {0, 49, 64, 81} with probabilities {0.3, 0, 0, 0.7}. By allowing zero probabilities both W A and W B have the same set of outcomes; they only differ by their respective distributions on this outcome space. Since 0.78 > 0.76, our investor should prefer lottery B. But this is equivalent to the statement that E[U(W B )] > E[U(W A )]. Expected Utility Criterion for Selecting Among Risky Alternatives: Under our assumptions an investor can choose between two risky alternatives (lotteries) by simply computing each lottery s Expected Utility. The lottery with the higher expected utility is chosen. This is called the Expected Utility Criterion. Key Fact: Given a utility function U(.), let V(W) = au(w) + b, where a > 0. Now E[V(W B )] = E[aU(W B ) + b] = a E[U(W B )] + b. E[V(W A )] = E[aU(W A ) + b] = a E[U(W A )] + b. Thus, E[V(W B )] - E[V(W A )] = E[U(W B )] - E[U(W A )]. We know that lottery B is preferred to lottery A if and only if E[U(W B )] > E[U(W A )] if and only if E[V(W B )] > E[V(W A )]. Thus, we can use the function V(.) in lieu of U(.) and we will make the same recommendations. Conclusion: A utility function is unique up to (positive) linear transformation. It does not have to be associated with a probability. 7

8 General Assumptions on the Utility Function: Nondecreasing and concave. Concavity models risk-averse preferences (and diminishing marginal utility). Definition: U(.) is concave if for each w 1 and w 2 and p ε [0, 1], U((1-p)w 1 + pw 2 ) (1-p)U(w 1 ) + pu(w 2 ), i.e., U(average w) is at least as large as the average of the U values. When U(.) is twice-differentiable, then U(.) is concave if and only if its second derivative is nonpostive. Remark: U(.) is convex if the inequality above is reversed. An investor with a convex utility is a risk-taker! Commonly used parametric forms: Logarithmic: ln(w), W > 0. Power: W a, 0 < a 1. Special case: a = 1 corresponds to risk-neutral behavior. Only means relevant! Exponential: -exp(-λw), λ > 0. Quadra tic: aw 2 + bw, a < 0, W -b/2a. Note: lim (W a 1)/a = lim (ln W) W a /1 = ln W. (L Hopital s Rule.) Examples: 1. Let s go back to the motivating example. Assume the investor s utility function is U(W) = -exp(-2*10-5 *W). (The 10-5 converts wealth in units of 100,000.) The E[U(W)] for buying insurance = -exp(-2*3.99) = , and its CE is obviously 399,000. The E[U(W)] for not buying insurance = exp(-2*1.5) exp(-2*4) = Certainty equivalent CE satisfies = -exp(-2*ce), which implies CE = 393,125. Better to buy insurance! 2. Suppose U(W) = ln W. Initial wealth W 0 = L = {(-1000, 0.5), (1000, 0.5)}. E[U(W)] = 0.5U(1000) + 0.5U(3000) = U(CE) = , which implies that CE = exp(7.4571) = So, netting out for initial wealth, the investor would be willing to pay 268 to avoid this gamble. 3. Jones has an opportunity to invest in a business with this distribution of payoffs: {(0.1, -70,000), (0.2, -20,000), (0.3, 100,000), (0.4, 500,000)}. W 0 = 80,000. U(W) = ln W. a. How much money to offer Jones so that he would prefer this to the business opportunity? b. Suppose there is a second business opportunity {(0.75, 0), (0.25, M)}. What is the smallest value of M for which Jones would prefer the second business opportunity to the first? Stochastic Dominance: Analysis of the distributions may be sufficient to make decision without knowledge of U(.)! 8

9 Utility Homework Problems I. Lotteries and Certainty Equivalents 1. Consider an individual with zero initial wealth and a utility function U(W) = 1 exp[ w]. Find the certainty equivalent for each of the alternatives below. Alternative -10, ,000 20,000 30, Consider an individual with zero initial wealth and a utility function U(W) = 1 exp[ w]. The individual is choosing between a certain 20,000 and a lottery with probability p of winning 30,000 or probability (1-p) of winning 10,000. There is no cost for either alternative. a. Find the probability p so that the individual is indifferent if the initial wealth is 10,000. b. Find the probability p so that the individual is indifferent if the initial wealth is 20, Consider an individual with initial wealth of 20,000 and a quadratic utility U(W) = W W 2 /100,000. The individual faces a lottery with probability 0.50 of winning 10,000 and a probability of 0.50 of winning 20,000. Find the certainty equivalent of the lottery. 4. Smith has an initial wealth of $125,000 and a utility function U(W) = 3ln (W) Smith has the option of playing the following lottery: There is a 5% chance of losing 100,000, a 75% chance of winning 10,000, and a 20% chance of winning 500,000. There is no cost for the lottery. Determine the minimum amount of money Smith would have to be offered so that Smith would prefer to take this money and not play lottery A. 5. Smith has an initial wealth of $50,000 and a utility function U(W) = ln (W). Smith has the option of playing the following lottery A: { (0.10, -$40,000), (0.70,$10,000), (0.20, $100,000) }. There is no cost for the lottery. a. What is the minimum amount of money you would have to offer Smith so that Smith would prefer to take this money and not play lottery A? b. Smith also has the option of playing lottery B: with probability 0.50 Smith will win $M and with probability 0.50 Smith will win nothing. There is no cost for the lottery. Determine the smallest value of M for which Smith would prefer lottery B to lottery A. 9

10 II. Insurance Analysis 6. Consider a homeowner with utility function U(W) = 1 exp[ w] who is deciding on whether or not to buy fire insurance. There is a 1% chance of a fire in which case it will cost the homeowner 60,000. The insurance premium is 700 per year, and the deductible amount on the loss is D. (In the case of a fire the homeowner will pay D in addition to the already paid premium of 700.) Determine the deductible amount that would make the homeowner indifferent between buying and not buying insurance. The homeowner s initial wealth is 80, Consider a homeowner with utility function U(W) = ln(w) who is deciding on whether or not to buy fire insurance. There is a 1% chance of a fire in which case it will cost her 75,000. The homeowner s initial wealth W 0 = 100,000. If she chooses to purchase an insurance policy, she will pay an insurance premium P up front, i.e., the cost of the policy, and she will pay an additional deductible amount D should a fire occur. She has spoken with two insurance companies, and each company has offered her a policy, as follows: Company 1: Premium P = Deductible D = 3,000. Company 2: Premium P = Deductible D = 8,000. a. Determine the best option for her. b. For the option you recommend in part (a), determine its Certainty Equivalent and associated Risk Premium. 8. Consider a homeowner with utility function U(W) = ln(w) who is deciding on whether or not to buy fire insurance. The homeowner s initial wealth W 0 = 200,000. There is a 1% chance of a fire in which case it will cost her 150,000. If she chooses to purchase an insurance policy, she will pay an insurance premium P up front, i.e., the cost of the policy, and she will pay an additional deductible amount D should a fire occur. She has spoken with two insurance companies, and each company has offered her a policy, as follows: Company 1: Premium P = Deductible D = 6,000. Company 2: Premium P = Deductible D = 16,000. a. Determine the best option for her. b. Determine the Certainty Equivalent and Risk Premium for the option you recommend in part (a). 9. Consider a homeowner with utility function U(W) = ln(w) who is deciding on whether or not to buy fire insurance. There is a 1% chance of a fire in which case it will cost her 75,000. The homeowner s initial wealth W 0 = 100,000. If she chooses to purchase an insurance policy, she will pay an insurance premium P up front, i.e., the cost of the policy, and she will pay an additional deductible amount D should a fire occur. She has spoken with two insurance companies, and each company has offered her a policy, as follows: Company 1: Premium P = Deductible D = 3,000. Company 2: Premium P = Deductible D = 8,000. a. Determine the best option for her. b. For the option you recommend in part (a), determine its Certainty Equivalent and associated Risk Premium. 10

11 III. Stochastic Dominance 10. Consider the following two mutually exclusive investment opportunities. PV pdf f pdf g F = cdf f G = cdf g F^ = F G^ = G -10, , , , a. Fill out the above table. b. Which opportunity would you recommend? Explain your reasoning. 11. Consider the following two mutually exclusive investment opportunities. PV pdf f pdf g F = cdf f G = cdf g F^ = F G^ = G 5, , , , , a. Fill out the above table. b. Which opportunity would you recommend? Explain your reasoning. 12. For the problems below, we consider two risky alternatives, X and Y. The probability density function associated with X is f(x) = 1/6 if 0 x 1, f(x) = 1/6 if 4 x 9, and f(x) = 0 everywhere else. The probability density function associated with Y if g(y) = 1/3 if 1 y 4 and g(y) = 0 everywhere else. a. Smith prefers more wealth to less. He has been known to exhibit risky behavior. Can first-order stochastic dominance be used to know which risky alternative, X or Y, Smith prefers? Explain. b. Jane s utility function is U(w) = w 0.5. Which alternative, X or Y, does Jane prefer? (Assume Jane s initial wealth is zero.) c. What is Jane s certainty equivalent for the risky alternative X? 11

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13 Utility Homework Problem Solutions 1. U(W) = 1 - exp[-.0001w]. Cash Amount -10, ,000 20,000 30,000 Utility Alternative E[U(W)] CE = U -1 [E(U(W))] = 10,000 ln [1/ (1-U)] , , , , , a. Initial wealth is 10,000. U(W) = 1 - exp[-.0001w]. Seek the value of p for which U(30,000) = pu(40,000) + (1-p)U(20,000). Therefore: = p[ ] p = b. Initial wealth is now 20,000. U(W) = 1 - exp[-.0001w]. Seek the value of p for which U(40,000) = pu(50,000) + (1-p)U(30,000). Therefore: = p[ ] p = (as before) NOTE: In general, we seek the value of p for which U(W 0 + CE) = U(W 0 + X 1 )p + U(W 0 + X 2 )(1-p). Let V(W) = 1-U(W). The equality holds with V(W), too. Since V(W) is of the form exp(aw), it follows that V(x + y) = V(x)*V(y), which means we can divide both sides of the equality by V(W 0 ). We have now shown that the answer for p must be independent of the value for W U(W) = W - W 2 /100,000. W 0 = 20,000. Begin by finding the value Y for which: 0.50U(30,000) U(40,000) = U(Y). Y satisfies the quadratic equation Y 2 /100,000 - Y + 22,500 = 0, whose roots are 34,189 and 65,811. Clearly, Y = 34,189. Now Y = W 0 + CE, which means that CE = 14, EU(W) = 0.05U(25,000) U(135,000) U(625,000) = Thus ln(w) = , which implies that W = 168,586.70, the certainty equivalent of the lottery. Since initial wealth W 0 = 125,000, Smith would have to be offered 43, not to play. Note that ln(w) in lieu of 3ln(W) + 10 would have worked just fine, too. 5. a. E[U(A)] = 0.10[ln(10,000)} [ln(60,000)] [ln(150,000)] = = ln(c). Thus C = 60,245, which means that break-even point for Smith is 10,245. b. E[U(B)] = 0.50[ln(50,000)] [ln(50,000 + M)] = E(U(A)] = Thus, the break-even point for Smith is M = 22, We seek the deductible D for which the expected utility of not buying insurance equals the expected utility for buying insurance. U(W) = 1 - exp[ w]. If you do not buy insurance: E[U(W)] =.99U(80,000) +.01U(20,000) = If you do buy insurance: E{U(W)] =.99U(79,300) +.01U(79,300 - D) Therefore D = 11, Option 1: E[U(W)] = 0.99ln(97,000)+0.01ln(94,000)= Option 2: E[U(W)] = 0.99ln(98,000)+0.01ln(90,000)=

14 No Insurance: E[U(W)] = 0.99ln(100,000)+0.01ln(25,000)= Best option. CE of no insurance option = exp( ) = 98,623. Expected wealth of no insurance option = 0.99(100,000)+0.01(25,000) = 99,250. Risk premium = 99,250-98,623 = Option 1: E[U(W)] = 0.99ln(194,000)+0.01ln(188,000)= Option 2: E[U(W)] = 0.99ln(196,000)+0.01ln(180,000)= No Insurance: E[U(W)] = 0.99ln(200,000)+0.01ln(50,000)= Best option. CE of no insurance option = exp( ) = 197,247. Expected wealth of no insurance option = 0.99(200,000)+0.01(50,000) = 198,500. Risk premium = 198, ,247 = Option 1: E[U(W)] = 0.99ln(97,000)+0.01ln(94,000)= Option 2: E[U(W)] = 0.99ln(98,000)+0.01ln(90,000)= No Insurance: E[U(W)] = 0.99ln(100,000)+0.01ln(25,000)= Best option. CE of no insurance option = exp( ) = 98,623. Expected wealth of no insurance option = 0.99(100,000)+0.01(25,000) = 99,250. Risk premium = 99,250-98,623 = PV pdf f pdf g F = cdf f G = cdf g F^ = F G^ = G -10, , , ,000 2,000 20, ,000 10,000 30, ,500 18,500 A risk-averse investor would select g due to 2 nd degree stochastic dominance. PV pdf f pdf g F = cdf f G = cdf g F^ = F G^ = G 5, , ,500 40, ,000 24,500 95, ,500 63, , ,500 68,.000 By second-degree stochastic dominance, a risk-averse investor would prefer the density f since G(.) F(.). (If the investor is not necessarily risk-averse, then we cannot say.) 12. a. No. The cdfs F X (.) and F Y (.) cross. b. E[U(X)] = U(x)f(x)dx = 20/9. E[U(Y)] = U(y)g(y)dy = 14/9. Jane prefers alternative X. c. (20/9) 2 =

15 Utility Homework Problem Solutions 1. U(W) = 1 - exp[-.0001w]. Cash Amount -10, ,000 20,000 30,000 Utility Alternative E[U(W)] CE = U -1 [E(U(W))] = 10,000 ln [1/ (1-U)] , , , , , a. Initial wealth is 10,000. U(W) = 1 - exp[-.0001w]. Seek the value of p for which U(30,000) = pu(40,000) + (1-p)U(20,000). Therefore: = p[ ] p = b. Initial wealth is now 20,000. U(W) = 1 - exp[-.0001w]. Seek the value of p for which U(40,000) = pu(50,000) + (1-p)U(30,000). Therefore: = p[ ] p = (as before) NOTE: In general, we seek the value of p for which U(W 0 + CE) = U(W 0 + X 1 )p + U(W 0 + X 2 )(1-p). Let V(W) = 1-U(W). The equality holds with V(W), too. Since V(W) is of the form exp(aw), it follows that V(x + y) = V(x)*V(y), which means we can divide both sides of the equality by V(W 0 ). We have now shown that the answer for p must be independent of the value for W U(W) = W - W 2 /100,000. W 0 = 20,000. Begin by finding the value Y for which: 0.50U(30,000) U(40,000) = U(Y). Y satisfies the quadratic equation Y 2 /100,000 - Y + 22,500 = 0, whose roots are 34,189 and 65,811. Clearly, Y = 34,189. Now Y = W 0 + CE, which means that CE = 14, EU(W) = 0.05U(25,000) U(135,000) U(625,000) = Thus ln(w) = , which implies that W = 168,586.70, the certainty equivalent of the lottery. Since initial wealth W 0 = 125,000, Smith would have to be offered 43, not to play. Note that ln(w) in lieu of 3ln(W) + 10 would have worked just fine, too. 5. a. E[U(A)] = 0.10[ln(10,000)} [ln(60,000)] [ln(150,000)] = = ln(c). Thus C = 60,245, which means that break-even point for Smith is 10,245. b. E[U(B)] = 0.50[ln(50,000)] [ln(50,000 + M)] = E(U(A)] = Thus, the break-even point for Smith is M = 22, We seek the deductible D for which the expected utility of not buying insurance equals the expected utility for buying insurance. U(W) = 1 - exp[ w]. If you do not buy insurance: E[U(W)] =.99U(80,000) +.01U(20,000) = If you do buy insurance: E{U(W)] =.99U(79,300) +.01U(79,300 - D) Therefore D = 11, Option 1: E[U(W)] = 0.99ln(97,000)+0.01ln(94,000)= Option 2: E[U(W)] = 0.99ln(98,000)+0.01ln(90,000)=

16 No Insurance: E[U(W)] = 0.99ln(100,000)+0.01ln(25,000)= Best option. CE of no insurance option = exp( ) = 98,623. Expected wealth of no insurance option = 0.99(100,000)+0.01(25,000) = 99,250. Risk premium = 99,250-98,623 = Option 1: E[U(W)] = 0.99ln(194,000)+0.01ln(188,000)= Option 2: E[U(W)] = 0.99ln(196,000)+0.01ln(180,000)= No Insurance: E[U(W)] = 0.99ln(200,000)+0.01ln(50,000)= Best option. CE of no insurance option = exp( ) = 197,247. Expected wealth of no insurance option = 0.99(200,000)+0.01(50,000) = 198,500. Risk premium = 198, ,247 = Option 1: E[U(W)] = 0.99ln(97,000)+0.01ln(94,000)= Option 2: E[U(W)] = 0.99ln(98,000)+0.01ln(90,000)= No Insurance: E[U(W)] = 0.99ln(100,000)+0.01ln(25,000)= Best option. CE of no insurance option = exp( ) = 98,623. Expected wealth of no insurance option = 0.99(100,000)+0.01(25,000) = 99,250. Risk premium = 99,250-98,623 = PV pdf f pdf g F = cdf f G = cdf g F^ = F G^ = G -10, , , ,000 2,000 20, ,000 10,000 30, ,500 18,500 A risk-averse investor would select g due to 2 nd degree stochastic dominance. PV pdf f pdf g F = cdf f G = cdf g F^ = F G^ = G 5, , ,500 40, ,000 24,500 95, ,500 63, , ,500 68,000 By second-degree stochastic dominance, a risk-averse investor would prefer the density f since G(.) F(.). (If the investor is not necessarily risk-averse, then we cannot say.) 12. a. No. The cdfs F X (.) and F Y (.) cross. b. E[U(X)] = U(x)f(x)dx = 20/9. E[U(Y)] = U(y)g(y)dy = 14/9. Jane prefers alternative X. c. (20/9) 2 =