# Risk and Insurance. Vani Borooah University of Ulster

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1 Risk and Insurance Vani Borooah University of Ulster

2 Gambles An action with more than one possible outcome, such that with each outcome there is an associated probability of that outcome occurring. If the outcomes are good (G) and bad (B), denote the associated probabilities by p G and p B

3 Payoffs and Utilities With each outcome is associated a payoff which can be expressed in terms of money: \$c G and \$c B With each payoff is associated a utility, u(c): u(c G ) is the utility in the good situation u(c B ) is the utility in the bad situation. We assume that utility increases with payoff Note: a payoff is different from the utility from the payoff

4 Expected Return and Utility Expected Return: The expected return from the gamble is: ER=p G c G +p B c B Expected Utility: The expected utility from the gamble is: EU=p G u(c G )+p B u(c B ) Note: The return expected from a gamble is different from the utility expected from the gamble

5 Facing a Gamble You are faced with a gamble: If you accept the gamble you will, in exchange for \$W (the amount staked ), receive C G with probability p G and C B with probability p B If you reject the gamble you will keep your \$W You have to decide whether or not to accept the gamble?

6 Expected Utility Rule If you accept the gamble, your expected utility is EU=p G u(c G )+p B u(c B ) If you reject the gamble, your (certain) utility is u(w) The expected utility rule requires you to compare EU and u(w) and: accept if EU>u(W) reject if EU<u(W) indifferent if EU=u(W)

7 Certainty Equivalent How much should the stake be to make you indifferent between accepting and rejecting the gamble? Or what value of W will equate: U(W) = EU=p G u(c G )+p B u(c B ) Suppose W * solves the above equation Then W * is known as the certainty equivalent of the gamble it expresses the worth of the gamble: \$W *

8 Choice Using Certainty Equivalent If the certainty equivalent is W * and W is the stake, you will: 1. Accept the gamble if W < W * 2. Reject the gamble if W > W * 3. Indifferent to the gamble if W = W *

9 Risk Premium The risk premium associated with a gamble is the maximum amount a person is prepared to pay to avoid the gamble RP = ER - CE

10 An Example Suppose you have to pay \$2 to enter a competition. The prize is \$19 and the probability of winning is 1/3. You have a utility function u(x)=log x and your current wealth is \$10. What is the certainty equivalent of this competition? What is the risk premium? Should you enter the competition?

11 Answer:I 1 2 EU log( ) log(10 2) log(27) log(8) 3 3 1/3 2/3 log(27 ) log(8 ) log(3) log(4) log(12) CE 12 The gamble is worth \$12 to him. But, if he rejects the gamble, he has only \$10. So, he will accept the gamble.

12 Answer:II The expected wealth from the lottery is: 1 2 ER ( ) (10 2) So, RP

13 Attitudes to Risk Intuitively, whether someone accepts a gamble or not depends on his attitude to risk Again intuitively, we would accept adventurous persons to accept gambles that more cautious persons would reject To make these concepts more precise we define three broad attitudes to risk

14 Three Attitudes to Risk The Risk Averse Person The Risk Neutral Person The Risk Loving Person To define these attitudes, we use the concept of a fair gamble In essence, a fair gamble allows you receive the same amount of money through two distinct ways: Gambling or not gambling

15 A Fair Gamble A fair gamble is one in which the sum that is bet (W) is equal to the expected return: W = ER = p G c G +p B c B You are offered a gamble in which you bet W=\$500 and receive: \$250 with p B = 0.5 or \$750 with p G = 0.5 ER=\$500=W: fair gamble

16 An Unfair Gamble An unfair gamble is one in which the sum that is bet (W) is different (usually less) from the expected return: W < ER = p G c G +p B c B You are offered a gamble in which you bet W=\$500 and receive: \$250 with p B = 0.6 or \$750 with p G = 0.4 ER=\$450<W: unfair gamble

17 Attitudes to Risk and Fair Gambles A risk averse person will never accept a fair gamble A risk loving person will always accept a fair gamble A risk neutral person will be indifferent towards a fair gamble

18 What Does This Mean? Given the choice between earning the same amount of money through a gamble or through certainty The risk averse person will opt for certainty The risk loving person will opt for the gamble The risk neutral person will be indifferent

19 Diminishing Marginal Utility Why does the risk averse person reject the fair gamble? Answer: because her marginal utility of money diminishes

20 Example Your wealth is \$10. I toss a coin and offer you \$1 if it is heads and take \$1 from you if it is tails This is a fair gamble: =10, but you reject it Because, your gain in utility from another \$1 is less than your loss in utility from losing \$1 Your MU diminishes, you are risk averse Conversely, if you are risk averse, your MU diminishes

21 A risk averse person / with diminishing MU / with a concave utility function will reject a fair gamble u(c) u(750) u(500) EU u(250) c The certainty equivalent of the gamble is \$400; the risk premium is \$100

22 A risk neutral person / with constant MU / with a linear utility function will be indifferent between accepting/rejecting a fair gamble u(c) u(750) u(500) =EU u(250) c The certainty equivalent of the gamble is \$500; the risk premium is \$0

23 A risk loving person / with increasing MU / with a convex utility function will accept a fair gamble u(c) u(250) u(750) EU u(500) c The certainty equivalent of the gamble is \$600; the risk premium is -\$100

24 Contingent Commodities With contingent commodities, the nature of the commodity depends upon the contingency: A house before a storm is a different good after a storm A car before an accident is a different good after an accident A holiday in sunshine is a different good from a holiday during which it rains

25 Trade in Contingent Markets The risk of a gamble is the difference between the payoff in the good state (C G ) and that in the bad state (C B ): Risk = C G - C B When we buy insurance we try to reduce risk by trading between two contingent states: good and bad We do this by buying wealth in the bad state and paying for it from wealth in the good state The rate at which we can make this exchange depends on the premium \$ (per \$ of insurance bought) charged by the insurance company \$(1- ) of additional C B can be bought by giving up \$ of C G So \$1 of additional C B can be bought by giving up ( /1- ) of C G

26 The Insurance Budget Line Z is amount of insurance C G = C G - Z < C G C B = C B - Z + Z = C B + (1- )Z > C B The slope of the budget line is - /(1- ): as insurance gets cheaper, the BL becomes flatter C G No Insurance point: Z=0 C G 45 0 line: C G = C B or full insurance C B C B

27 The Contingent Consumption Indifference Curves On each curve, different combinations of C G and C B give the same level of Expected Utility: p G U(C G ) + p B U(C B ) C G Higher EU on black curve than on red C B

28 Equilibrium in the Insurance Market Given the terms offered by the insurance company, consumer maximises EU at point A C G X: no insurance point C G A: equilibrium point C G * Z = C B* - C B is amount of insurance bought C B C B * C B

29 Different Types of Equilibrium in the Insurance Market Given, the terms offered by the insurance company, consumer maximises EU at point X or at Y or at some point in between X and Y C G C G C G * X: no insurance equilibrium, Z=0, insurance too expensive Y: full insurance equilibrium, insurance cheap C B C G * C B

30 Condition for Equilibrium Indifference Curve should be tangential to budget line This means that the slope of indifference curve equals slope of budget line Slope of indifference curve is marginal rate of substitution: how much of wealth in the good state you are prepared to give up to get another \$ of wealth in the bad state and still be on the same IC Slope of budget line is rate of exchange: how much of wealth in the good state you have to give up to get another \$ of wealth in the bad state

31 Interpreting Equilibrium MRS BG = /(1- ) p u ( C ) B B 1 p u ( C ) 1 B G Note: p G = 1 - p B

32 An Actuarially Fair Premium An actuarially fair premium is one which is equal to the probability of the adverse contingency: = p B When the premium is actuarially fair: u (C B )=u (C G ) So, under diminishing marginal utility: C B = C G Implying full insurance

33 Under what market conditions will an actuarially fair premium be charged? The expected profit of an insurance company is: Z - p B Z 0 When the insurance industry is competitive, free entry of new firms will compete away excess profits: Z - p B Z = 0 Which implies: = p B

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