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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 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? (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. 1 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain Question 2 (20072008 Test 2 question) Mr. S. Hussein, a distant relative of Saddam Hussein, owns a very expensive home worth $1.5m 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. 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. 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, 86% of flights from Toronto to NYC are on time. 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. (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 2 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain develop and release the shampoo product? (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 (d) Draw the P&G's decision tree if it makes the decision with the perfect test in part (c). (e) What is the value of a "perfect test"? Show your calculations. 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. 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. (a) Should "HeyJazz Mining Company" 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 3 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain 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 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. (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? (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?) 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 50 50 Minerals No Minerals Total 100 (a) Find the expected return of each investment. (b) Jill has the utility function U = 5I , where I denotes the payoff. Which investment will she choose? 4 University of Toronto, Department of Economics, ECO 204 Summer 2009 S. Ajaz Hussain (c) Ken has the utility function U = 5I . Which investment will he choose? (d) Laura has the utility function U = 5I 2 . Which investment will she choose? 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. 5 ...
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto- Toronto.

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