2. Choice under uncertainty PDF

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2. Choice under Uncertainty

Theoretical questions 1. (2003 ZA Q 13) (a) Explain the benefits a risk-averse person could get by insuring against the risk of a loss. (b) A risk-averse person faces two, independent, risks of loss. In one prospect the potential loss is large, while in the other the potential loss is small, but the probability of the loss, p, is the same for both. The premium per dollar of insurance cover, r, against the losses is also the same in both cases. r is greater than p. Is the person acting irrationally if she buys some insurance cover against the large loss but none against the small loss? 2. (2003 ZB Q 7) (a) Will a risk-averse person gamble if the odds are unfair? (b) Will a risk-averse person insure against a risk if the insurance premium is actuarially fair? (c) How would your answers to (a) and (b) change if the person is risk loving? 3. (2004 ZA Q 9) Is it inconsistent for a risk-averse person both to accept some risks and to reduce others? Is it inconsistent for a risk lover both to accept some risks and to reduce others? Explain your answers. 4. (2004 ZB Q 7) Individual A is risk averse. Individual B is risk loving. Defining all relevant terms, explain the various circumstances in which each individual would accept a gamble. When would each individual reject a gamble? 5. (2006 ZA Q 11) (a) Assuming each is offered the same gamble, analyse the conditions in which both A and B take the gamble. Assuming, again, that each is offered the same gamble, carefully analyse how the terms of the gamble should be set to ensure that one individual accepts the gamble and the other rejects it. 1

(b) Now assume that each individual is offered insurance on the same terms against the risk of a loss. The individuals are free to choose their levels of insurance cover except that they are not allowed to over-insure. Suppose, also, that A and B are offered insurance terms whereby the premium rate is less than the probability of loss. Will both A and B necessarily buy full insurance? If insurance terms are changed so that the premium rate exceeds the probability of loss, will both A and B necessarily choose to be uninsured? 6. (2008 ZA Q 13(b)) Compare and explain the nature of indifference curves of a person who is risk-averse with a person who is risk-loving. 7. (2008 ZA Q 13(c)) Evaluate the following statements: i. If offered the opportunity to place a sufficiently small bet at favourable odds a risk-averse person always accepts; but if a risk-averse person is only offered the opportunity to place a very large bet at favourable odds he always declines. ii. A risk-loving person will always bet, no matter how much the odds are against him. 8. (2008 ZB Q 12(a)) Explain and compare the responses of a risk-averse person and a risk-neutral person to the offer of: i. a gamble with favourable odds, ii. insurance with a premium that is higher than an actuarially fair one. 9. (2009 ZB Q 8) (a) Using the contingent commodities approach to modelling choice under conditions of uncertainty, explain fully the benefits a risk-averse person would get by insuring against the risk of a loss if insurance could be bought on actuarially fair terms. (b) Explain how the insurance decision of a risk-averse person will differ if insurance is not available on actuarially fair terms. (c) Explain the (special) conditions in which actuarially fair insurance could be supplied by private insurance providers. 10. (2009 ZA Q 10(a)) Define and explain the following concepts: i. a von Neumann-Morgenstern utility function, ii. actuarially fair insurance, iii. the value of perfect (or complete) information. 11. (2010 ZB Q 5) Two individuals with different degrees of risk aversion work in a steel plant, which faces closure with positive probability. Both are considering buying unemployment insurance. They would buy the same amount of insurance coverage whether the insurance policy is actuarially fair or actuarially unfair. Is this true or false? Explain with suitable diagrams. 2

12. (2012 ZA Q 12(a)) Using either the contingent commodities model or the expected utility model, explain why a risk averse individual would choose full insurance under fair odds.

Gambling 1

13. (2003 ZA Q 7) Susan’s utility function is U = w2 , and her initial wealth w, is 36. Will she accept a gamble in which she wins 13 with probability 32 or loses 11 with probability 31 ? 14. (2007 ZA Q 10(a)) Mary is offered a choice between two gambles. In gamble one, she wins 200 if a coin comes up heads and loses 100 if the coin comes up tails. In gamble two, she wins 20, 000 if a coin comes up heads and loses 10, 000 if the coin comes up tails. Assume that Mary’s initial wealth, W, is 10, 000. i. Would Mary accept either, both, or neither of the gambles if she is risk neutral? ii. Would Mary accept √ either, both, or neither of the gambles if her utility function is given by U = W? iii. Compare and explain your results in parts i. and ii. 15. (2008 ZA Q 13(a)) Describe and explain the position and slope of a person’s budget line if his wealth is initially £100 and he is offered a gamble in which an unbiased die is rolled and the odds on a bet that it will turn up 2 or less are: i. fair, ii. unfavourable, iii. favourable.

Insurance 16. (2007 ZA Q 10(c)) Suppose Ben has a utility function given by  1 W 2 U (W ) = , 500 where U is utility and W is wealth. Ordinarily, Ben expects his wealth to be 40, 500. However, he faces the possibility that his house will burn down, reducing his wealth to 4, 500. This unfortunate event occurs with probability 31. i. What is Ben’s expected utility, if uninsured? ii. Suppose Ben can pool risks with Anne who faces an identical but independent risk that her house will burn down. Will Ben agree to pool risks with Anne? Explain. 3

iii. What is the difference between the amount Ben would be willing to pay for full insurance and the amount he would actually pay if he was able to buy full insurance on actuarially fair terms? 17. (2008 ZB Q 12(b)) John has the utility function 1

U (W ) = W 2 and initial wealth W = $2, 500. He faces the risk of a loss of $1, 600 with probability 21. i. ii. iii. iv.

What is John’s expected utility? What is the actuarially fair price for full insurance? Will John buy full insurance if it is actuarially fair? What is the maximum John is prepared to pay for full insurance?

18. (2009 ZA Q 10(b)) John expects his future earnings to be worth £100. If he falls seriously ill, however, his expected future earnings are only £25. John believes that his chance of falling ill is two-thirds while the chance of remaining in good health is one third. John’s utility function as a function of the value of his earnings is: 1

U (Y ) = Y 2 . Suppose that an insurance company offers to fully insure John against loss of earnings caused by illness and that the insurance premium is actuarially fair. i. Will John accept the insurance? Explain. ii. What is the maximum amount that John would pay for the insurance? Explain. 19. (2010 ZA Q 13) Kate’s utility depends on her income I. Her utility function is given by √ u( I ) = I. Kate’s income is uncertain. With probability 0.8 her income is 100 and with probability 0.2 her income is 64. (a) What is Kate’s expected utility? (b) Show geometrically that Kate would be prepared to pay a positive premium if her income is insured fully. (c) Calculate the maximum premium that Kate would be willing to pay to insure her income fully. (d) What is the smallest insurance premium that the insurance company will accept in order to fully insure Kate’s income? 4

20. (2011 ZA Q 2) Kim owns a house worth 100, 000. There is a probability of 0.1 that the house could be destroyed by fire during the course of a year. Kim’s utility function is √ u(W ) = W, where W denotes wealth. Suppose Kim is offered full insurance by an insurance company. Assuming that Kim has no other wealth, what is the maximum he would be willing to pay for such an insurance policy? 21. (2011 ZB Q 11(a)) Oscar’s utility depends upon his wealth W . His utility function is √ u(W ) = W. Oscar’s current wealth is 100, but he faces the risk of losing 60 with probability3 1. An insurance company makes him the following offer: for every dollar that Oscar pays as premium, the insurance company will pay him 3 if the loss occurs. i. How much insurance will Oscar buy? ii. Oscar’s friend John has the same current wealth as Oscar and faces exactly the same risk. However, his utility function is v (W ) = ln W. If the insurance company makes the same offer to John, how much insurance will he buy? 22. (2012 ZA Q 12(b)) Rai has wealth 81, but if she becomes ill, she will have to pay medical bills, reducing her wealth to 36. The probability that Rai will fall ill is 13. Rai’s utility function is given by √ u(W ) = W, where W denotes wealth. Rai has the option of buying insurance coverage for illness from an insurance company. i. What is the maximum premium that Rai is willing to pay for full insurance? ii. What is the minimum premium that the insurance company would accept to provide full insurance? iii. Suppose Rai can buy X units of cover by paying a premium of 3X (i.e. the insurance company pays X to Rai if she falls ill, and Rai pays the insurance company X3 whether she is ill or not ill). What is the optimal choice of X by Rai? 23. (2013 ZA Q 14(b)) Jo has a wealth of 10, 000, but faces the risk of losing 3600 with probability 0.2. An insurance company offers Jo the following scheme: in exchange for a premium of X, the insurance company would pay out 5X in the event of a loss. 5

i. Suppose Jo’s utility function is given by u(w) = ln (w), where w denotes wealth. What is the optimal choice of X for Jo? ii. Now suppose Jo’s utility function is given by u(w) = 1 − w1 , where w denotes wealth. What is the optimal choice of X for Jo in this case? 24. (2015 ZA Q 14(a)) Lee does not have insurance against car theft. His car is worth 45. He can park his car on the street or pay to park in a garage. If parked on the street, the car is stolen with probability31. If parked in a garage, the car is safe from theft. Including the value of his car, Lee has a wealth of 81. His utility from wealth W is √ u(W ) = W. i. Calculate the maximum amount that Lee is willing to pay to park in a garage. ii. Now suppose Lee’s risk preference changes so that he becomes risk neutral, The utility function representing his preference over wealth levels is given by u(W ) = W. In this case, what is the maximum amount that Lee is willing to pay to park in a garage?

Investment and diversification 1

25. (2005 ZA Q 2) Suppose that Peter’s utility function is given by U (Y ) = Y2 , where Y represents annual income in thousands of pounds. Suppose that Peter is currently earning an income of £25, 000 (Y = 25) and can earn that income next year with certainty. He is offered a chance to invest £10, 000 in a venture that offers a 0.6 probability of earning £21, 000 and a 0.4 probability of earning £1, 000. (a) What is the expected value of the investment? (b) Will Peter undertake the investment? (c) Is Peter risk loving, risk neutral, or risk averse? 26. (2007 ZA Q 10(b)) David has an investment opportunity that pays 33 with probability 21 and loses 30 with probability 21. √ i. If his current wealth is 111, and his utility function is U = W, will he make this investment? 6

ii. Will he make it if he can take a partner who will share equally in the net return from the investment (positive or negative)? iii. Compare and explain your results in parts i. and ii. 27. (2007 ZA Q 10(d)) Suppose that Sally can invest in shares of the Alpha Heater Company and the Beta Air Conditioner Company. If she invests only in Alpha shares her wealth, W , is 16 when the weather is cold and 9 when the weather is hot. If she invests only in Beta shares her wealth is 9 when the weather is cold and 16 when the weather is hot. The probability that the weather is hot√is 21 and the probability that the weather is cold is 21 . Sally’s utility function is U = W . i. Will Sally invest all her funds in either of the shares or will she invest in both? Explain. ii. Explain the principle behind your answer in part i. 28. (2008 ZB Q 12(c)) Three possible investment projects A, B, and C yield the following payoffs in ‘bad times’, ‘normal times’, and ‘good times’. The probabilities of each of these states of the world are given as well: State of the world Probability A Project B C

Bad times 0.6 0 4 0

Normal times 0.2 0 4 9

Good times 0.2 20 4 16

As a result of making an investment, an investor obtains utility that equals the payoff. i. Which investment does the investor prefer? ii. Assume that, by waiting, before making the decision, the investor can obtain full information about the state of the world. Using a decision tree, calculate and explain the value of perfect (complete) information for the investor. Would the investor do better by waiting than investing immediately if the cost of waiting is zero? 29. (2009 ZA Q 10(c)) Consider the choices of two investors, Anne and Barbara. Anne has the utility function U (Y ) = Y, where Y is income. Barbara has the utility function John’s utility function as a function of the value 1 of his earnings is: U (Y ) = Y2 . Each has a total income of £100. Operating independently of one another, they face the following investment opportunity: invest the £100 and receive £144 in good times, but receive only £64 if there are bad times. Each investor estimates that good times happen with a 0.5 probability. 7

i. Will Anne invest? Will Barbara invest? ii. Now, suppose that, before deciding to invest or not each investor can buy an investor newsletter at a price of £15 that tells her whether good times or bad times will occur. Will Anne buy the newsletter? Will Barbara buy the newsletter? 30. (2015 ZA Q 14(b)) Rachel has 100 to invest. Two assets, 1 and 2, are available for investment. An amount y invested in asset 1 yields a total return of 1.1y. An amount x invested in asset 2 yields a risky total return of x with probability 0.5 and 1.21x with probability 0.5. Rachel’s utility function is given by U (w) = ln(w), where w is wealth after investing. Let any portfolio be denoted by ( x, y) where x is the amount invested in the risky asset (asset 2) and y = 100 − x is the amount invested in the safe asset (asset 1). How much should Rachel invest in the risky asset?

Combined problem 31. (2006 ZB Q 11) (a) Define and explain the following concepts: i. expected value, ii. expected utility, iii. risk preferences, iv. certainty equivalent. 1

(b) Given the utility function U = W2 and initial wealth W = 34, would Susan accept a gamble in which she wins 15 with probability 21 or loses 9 with probability 12 ? What initial value of wealth would make Susan indifferent between accepting the gamble and not accepting the gamble? 1

(c) Peter has the utility function U = W 2 and initial wealth W = 100. He faces the risk of a loss of 36 with probability 21. If Peter is offered actuarially fair insurance, will he buy fun insurance? What is the maximum he will be prepared to pay for fun insurance? (d) With the same utility function and initial wealth as described in (c), suppose that, for 100, Peter can buy an asset that will yield 10, 000 with probability201 19. and 0 with probability 20 Show that Peter will not buy this asset on his own but will join a syndicate with 10 equal partners.

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