Unformatted text preview: University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain Practice Problems 23 Solutions Please help improve the course by sending me an email about typos or suggestions for improvements Question 1 The actuarially fair price of insurance was derived using the argument that if an insurance company is to ensure that it has sufficient funds on its books to cover payout then: Funds on books = Expected Payout Insurance premium on books = Expected insurance claims (Price per dollar of insurance)(Insurance amount) = (Probability of loss)(Insurance amount) Price per dollar of insurance = Probability of loss In this question you will derive the same result through a different approach. Suppose all customers of an insurance company purchase an insurance policy worth $X. That is, if the "event" (such as an accident, fire or death) occurs, the insurance company will pay the customer $X. Let the probability of the event be p. Suppose the insurance policy premium is $P (don't confuse P and p, the probability of loss). That is, $P is what the customer pays the insurance company to purchase the insurance policy $X. This implies that the price per dollar of insurance is $P = price per dollar of insurance * $X Denote the price per dollar of insurance as "price/$". (a) What are the expected profits of the insurance company? Answer: The expected profits of the insurance company are calculated as follows: if the customer does 1 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain not have an "accident" (which happens with probability 1 p) the company collects the premium. Similarly, if the customer has an "accident" (which happens with probability p) the company collects the premium but then pays the insurance claim: E = (Probability of no event)*(Premium) + (Probability of event)*(Premium Claim) E = (1 p) $P + p ($P $X) E = $P p $P + p $P p $X E = $P p $X (b) Suppose the insurance industry is perfectly competitive. What are the expected profits in the long run? Prove that the price per dollar of insurance is the probability of loss. Answer: If the insurance company is competitive, in the long run, expected profits must be 0. Thus: E = $P p $X = $0 $P = p $X But $P = (price per dollar of insurance) $X (price per dollar of insurance) $X = p $X price per dollar of insurance = p This is actuarially fair insurance: your price per dollar of insurance is equal to your probability of an "event". When we say your probability of an "event", it really means, the probability of others like you. For example, if you are a hormonally over laden teenager, the insurance company uses the frequency data for all teenagers to attach a probability that you, a teenager, will get into (say) a car accident. Thus if 10% of all teenagers have a car accident, it is fair to say that you have a 10% chance of getting into a car accident. The insurance company will charge you $0.10 per dollar of insurance. By the way, observe how you're being "priced" on the basis of which "group" you belong to. If this sounds like discrimination, that's because it is. In fact, it is legal to discriminate on the basis of groupings when it comes to insurance, but not when it comes to (say) hiring. Question 2 (20072008 Test 2 question) Mr. S. Hussein, a distant relative of Saddam Hussein, owns a very expensive home worth $1.5m 2 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Saddam left Mr. S. Hussein a vast sum of money). Mr. A. Hussain, unrelated to Saddam Hussein, owns a cheap home (more like a hut actually) worth $10,000. A newspaper reporter creates a scandal by reporting that the price per dollar of insurance is the same for Mr. S. Hussein and Mr. A. Hussain. She argues that the insurance company is discriminating against Mr. A. Hussain. As the economist lawyer representing the insurance company, can you give a simple explanation for why Hussein and Hussain may be paying the same price per dollar of insurance? A two sentence answer suffices. Answer: Under actuarially fair pricing, the price per dollar of insurance is the probability of loss. Thus, Hussein and Hussain may be paying the same price per dollar of insurance because their probability of loss is the same. Question 3 (20082009 Final Exam Question) As you walk through Toronto's Pearson airport to catch a flight to Paris, you see a booth selling flight insurance for $12. If you die on the flight, the insurance company will pay your family $200,000. As of 2009 the probability of dying flying was 1 in 1.1 million. What is the price per dollar of insurance? Should you buy the policy? State any assumptions. Answer: The price per dollar of insurance is $12/$200,000 = 0.00006. If the insurance policy is actuarially fair, it should equal the actual probability of dying flying. The actual probability is 1/1,100,000 = 0.0000009. Since the price/$ of insurance > probability of death, you should not buy the insurance policy. Question 4 (20082009 Final Exam Question) When booking a one way flight on Air Canada, you have the option of purchasing a travel insurance policy "On My Way" for $25. Here is the description on Air Canada's website: For $25, get extra protection in case of flight delays or disruptions that are beyond the airline's responsibility or control. On My Way offers aroundtheclock priority rebooking service, a hotel if needed, and much more for only a small fee. Under actuarially fair insurance, what is the "benefit" of the On My Way travel insurance program for a passenger traveling one way from Toronto to NYC? According to Flightstats.com, 86% of flights from Toronto to NYC are on time. Answer: Let the benefit of "On My Way" be $X. If you purchase insurance, the price per dollar of insurance is 25/X. Under actuarially fair insurance this must equal the probability of the "event" which in this case is delay or cancellation which happens with probability 14%. Under actuarially fair insurance: 3 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain 25/X = 0.14 X = 25/0.14 $179 Thus, a traveler who purchases the "On My Way" program thinks the benefit will be $179. Thus, Air Canada is valuing the round the clock priority rebooking service and maybe a hotel at $179. Sounds like a scam. Question 5 (20082009 Final Exam Question) Proctor and Grumble (P&G), a risk neutral decision maker, must decide whether to develop and release a new shampoo "Shine on you" targeted at bald men. If P&G does not develop the shampoo, the outcome is $0m. If P&G develops the shampoo, R&D costs will be $100m and when released into the market, it will be a success (S) with probability 0.6 or a failure (F) with probability 0.4. If the shampoo is a success, gross revenues (before R&D costs) are estimated to be $500m. On the other hand, if the shampoo is a failure, gross revenues (before R&D costs) are estimate to be $0m. (a) Draw P&G's decision tree and indicate whether the shampoo should be developed. Answer See decision tree below, where the EV of the decision are shown: EV(develop and release) = $200m > $0m = Value of no development. Hence P&G should develop and release the shampoo. 4 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain (b) P&G's statistics consultant thinks the probability of success may not be accurate (i.e. there is some margin of error). What is the lowest probability of success for which P&G will choose to develop and release the shampoo product? Answer Let us calculate the "threshold" probability of success needed to make the decision to develop and release the shampoo. Suppose we denote the probability of success as P(S) = p and the probability of failure as: P(F) = 1 p. What is the lowest p for P&G to develop and release the shampoo? It is: EV(develop and release) > 0 400 p + (1 p)(100) > 0 400 p (1 p)(100) > 0 400 p 100 + 100p > 0 500 p > 100 p > 100/500 p > 0.2 Thus, as long as the probability of success is at least 20%, P&G's decision will be to develop and release the shampoo. At the same time, if P(S) is such that 0.6 < P(S) 1, P&G will decide to develop and release the shampoo. Thus the decision to develop and release the shampoo is "robust" to p = 0.6 0.4 or a 40% margin of error in the probability of success. P&G would have to be really off in their estimate of P(S) to make the wrong decision. (c) [This part can be answered without answering parts (a) (b)]. Suppose a marketing company has developed a perfect (100% accurate) test for new products. The test results can be positive (+) or negative (). Fill in the entries in the two way classification table below: S F Total + Total 100 5 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Answer First we know from the beginning of the question that: P(S) = 0.6 or 60% P(F) = 0.4 or 40% Since the test is 100% accurate, it must be "+" 60% of the time and "" 40% of the time. Thus, the entries should be: S F Total + 60 0 60 0 40 40 Total 60 40 100 (d) Draw the P&G's decision tree if it makes the decision with the perfect test in part (c). Answer Here is the decision tree with the perfect test. (e) What is the value of a "perfect test"? Show your calculations. Answer 6 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain The EV of making a decision with a perfect test is $240m. From part (a), the EV of the decision without a test was $200m. Hence: Value of Information = EV(Decision with Information) EV(Decision without information) Value of perfect test = $240m $200m Value of perfect test = $40m Question 6 (Summer 2008 Final Exam Question) For the last five years, Blue Hat Software has successfully marketed a software package. Recently, sales have begun to slip because the software is incompatible with a number of popular application programs. Thus, Blue Hat Software's future profits are uncertain. In the software's current form, managers forecast three different scenarios: maintain current profits of $2m with probability 0.2; a slip in profits to $0.5m with probability 0.5; or losses of $1m. Alternatively, Blue Hat Software can develop a compatible version of its software. Depending on development cost, this strategy is predicted to yield either $1.5m, $1.1m, $0.8m or $0.6m with equal probability. What is Blue Hat Software's optimal strategy? Show all steps and calculations using a decision tree. State all assumptions. Answer: I won't draw the tree, resorting instead to a numerical calculation. We have to compare: EV(No Redesign) with EV(Redesign). Now: EV(No Redesign) = 0.2(2) + 0.5(0.5) + 0.3(1) = $0.35m EV(Redesign) = 0.25(1.5 + 1.1 + 0.8 + 0.6) = $1m Since EV(Redesign) > EV(No Redesign) Blue Hat should forge ahead with the redesign, so long as the cost of redesign is less than EV(Redesign) EV(No Redesign). Question 7 (20072008 Test Question) "HeyJazz Mining Company" has the option to purchase land. The seller's best and final price is $3 million. If the land has commercial mineral deposits, "HeyJazz Mining Company" estimates its value at $5 million. If there are no deposits, the estimated value is $2 million. "HeyJazz Mining Company" believes that the chance of mineral deposits is 5050. 7 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain (a) Should "HeyJazz Mining Company" purchase the land? Answer: You can do this question by drawing a tree if you want. You have to compare the EV(not purchasing land) against the EV(purchasing land). EV(Not purchasing land) = $0 EV(purchasing land) = 0.5($5m) + 0.5($2m) $3m = $0.5m Since EV(purchasing land) > EV(Not purchasing land "HeyJazz Mining Company" should purchase the land. (b) The seller has agreed to let "HeyJazz Mining Company" take samples from the land. Based on past experience, if there are minerals present, the samples will be "positive" 80% of the time. If no minerals are present, the samples will (falsely) give a favorable reading 40% of the time. Fill the table below and determine whether "HeyJazz Mining Company" should purchase the land and how much it should pay for the test. Positive (+) Negative () Total 50 50 Minerals No Minerals Total 100 Answer: First, fill in the table. We know that if there are minerals, the test is "+" 80% of the time. Thus: P(+M) = 0.8. This implies: Positive (+) Negative () Total 8 Minerals 40 No Minerals Total 50 50 100 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain In turn, this implies: Positive (+) Negative () Total Next, we know that when there are no minerals the test gives a false "+" reading 40% of the time. Thus: Positive (+) Negative () Total From which: Positive (+) Negative () Total Minerals 40 10 50 No Minerals 20 30 50 Total 60 40 100 Minerals 40 10 50 50 No Minerals 20 Total 60 100 Minerals 40 10 50 50 No Minerals Total 100 Now, we need to compare the EV of purchasing land with information versus the EV of purchasing land without information (part (a)). The tree and its solution is: 9 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Please note cost of purchasing land has been subtracted from expected payoffs. The company should purchase the land as long as the test does not cost more than $0.6m $0.5m = $0.1m. Question 8 Suppose that Natasha's utility function is given by u(I) = 10I , where I represents annual income in thousands of dollars. (a) Is Natasha risk loving, risk neutral, or risk averse? Explain. Answer: Natasha is risk averse. This is because she has a concave utility function (first derivative is positive and the second derivative is negative). (b) Suppose that Natasha is currently earning an income of $40,000 (I = 40) and can earn that income next year with certainty. She is offered a chance to take a new job that offers a 0.6 probability of earning $44,000, and a 0.4 probability of earning $33,000. Should she take the new job? Answer: The utility of her current salary is 4000.5, which is 20. The expected utility of the new job is EU = (0.6)(4400.5 ) + (0.4)(3300.5 ) = 19.85 which is less than 20. Therefore, she should not take the job. (c) In part (b), would Natasha be willing to buy insurance to protect against the variable income associated with the new job? If so, how much would she be willing to pay for that insurance? (Hint: What is the risk premium?) 10 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Answer: Assuming that she takes the new job, Natasha would be willing to pay a risk premium equal to the difference between $40,000 and the utility of the gamble so as to ensure that she obtains a level of utility equal to 20. We know the utility of the gamble is equal to 19.85. Substituting into her utility function we have, 19.85 = (10I)0.5 and solving for I we find the income associated with the gamble to be $39,410. Thus, Natasha would be willing to pay for insurance equal to the risk premium, $40,000 $39,410 = $590. Question 9 Suppose that two investments have the same three payoffs, but the probability associated with each payoff differs, as illustrated in the table below: Payoff $300 $250 $200 Probabilities for Investment A 0.10 0.80 0.10 Probabilities for Investment B 0.30 0.40 0.30 (a) Find the expected return of each investment. Answer: The expected value of the return on investment A is EV = (0.1)(300) + (0.8)(250) + (0.1)(200) = $250. The expected value of the return on investment B is EV = (0.3)(300) + (0.4)(250) + (0.3)(200) = $250. (b) Jill has the utility function U = 5I , where I denotes the payoff. Which investment will she choose? Answer: Jill's expected utility from investment A is EU = 0.1(5*300) + 0.8(5*250) + 0.1(5*200) = 1,250. Jill's expected utility from investment B is EU = 0.3(5*300) + 0.4(5*250) + 0.3(5*200) =1,250. Since both investments give Jill the same expected utility, she will be indifferent between the two. (c) Ken has the utility function U = 5I . Which investment will he choose? 11 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Answer: Ken's expected utility from investment A is: EU = 0.1(5*300)0.5 + 0.8(5*250)0.5 + 0.1(5*200)0.5 = 35.32. Ken's expected utility from investment B is: EU=0.3(5*300)0.5 + 0.4(5*250)0.5 + 0.3(5*200)0.5 = 35.25. Ken will choose investment A since it has a higher expected utility. (d) Laura has the utility function U = 5I 2 . Which investment will she choose? Answer: Laura's expected utility from investment A is: EU = 0.1(5*300*300) + 0.8(5*250*250) + 0.1(5*200*200) = 315,000. Laura's expected utility from investment B is: EU = 0.3(5*300*300) + 0.4(5*250*250) + 0.3(5*200*200) = 320,000 Laura will choose investment B since it has a higher expected utility. Question 10 As the owner of a family farm whose wealth is $250,000, you must choose between sitting this season out and investing last year's earnings ($200,000) in a safe money market fund paying 5.0% or planting summer corn. Planting costs $200,000, with a sixmonth time to harvest. If there is rain, planting summer corn will yield $500,000 in revenues at harvest. If there is a drought, planting will yield $50,000 in revenues at harvest. As a third choice, you can purchase AgriCorp droughtresistant summer corn at a cost of $250,000 that will yield $500,000 in revenues at harvest if there is rain, and $350,000 in revenues at harvest if there is a drought. You are risk averse and your preferences for family wealth (W) are specified by the relationship U(W ) = W . The probability of a summer drought is 0.30 and the probability of summer rain is 0.70. Which of the three options should you choose? Explain. Answer: You need to calculate expected utility of wealth under the three options. Wealth is equal to the initial $250,000 plus whatever is earned on growing corn, or investing in the 12 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain safe financial asset. Expected utility under the safe option allowing for the fact that your initial wealth is $250,000 is: EU = (250,000 + 200,000(1 + .05)).5 = 678.23 Expected utility with regular corn, again including your initial wealth: EU = 0.7(250,000 + (500,000 200,000)).5 + 0.3(250,000 + (50,000 200,000)).5 EU = 519.13 + 94.87 = 614. Expected utility with droughtresistant corn, again including your initial wealth: E(U) = 0.7(250,000 + (500,000 250,000)).5 + 0.3(250,000 + (350,000 250,000)).5 EU = 494.975 + 177.482 = 672.46 You should choose the option with the highest expected utility, which is the safe option of not planting corn. 13 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain 14 ...
View
Full Document
 Fall '08
 HUSSEIN
 Economics, Microeconomics, S. Ajaz Hussain

Click to edit the document details