Risk Theory Professor Peter Cramton Economics 300 Is expected value a good criterion to decide between lotteries One criterion to choose between two lotteries is to choose the one with a higher expected value Does this criterion provide reasonable predictions Let s examine a case Lottery A Get 3125 for sure i e expected value 3125 Lottery B win 4000 with probability 0 75 and win 500 with probability 0 25 i e expected value also 3125 Which do you prefer Is expected value a good criterion to decide between lotteries Probably most people will choose Lottery A because they dislike risk risk averse However according to the expected value criterion both lotteries are equivalent Expected value is not a good criterion for people who dislike risk If someone is indifferent between A and B it is because risk is not important for him risk neutral Expected utility The standard criterion to choose among lotteries Individuals do not care directly about the monetary values of the prizes they care about the utility that the money provides U x denotes the utility function for money We will always assume that individuals prefer more money than less money so U xi 0 Expected utility The standard criterion to choose among lotteries The expected utility is computed in a similar way to the expected value However one does not average prizes money but the utility derived from the prizes The formula of expected utility is n EU pU i xi p1U x1 p2U x2 pnU xn i 1 The individual will choose the lottery with the highest expected utility Utility is invariant to linear transformation V x a bU x for b 0 is an equivalent utility function Classification U X 0 strictly concave U X Risk averse U X 0 linear U X Risk neutral U X 0 strictly convex U X Risk lover Examples of commonly used Utility functions for risk averse individuals U x ln x U x x U x x a U x 1 e where 0 a 1 ax where a 0 Example Alex is considering a job which is based on commission pays 3000 with 50 probability 9000 with 50 probability Utility 95 85 80 65 3 5 CE 6 9 Wealth thousands of dollars 3000 is worth 65 units of utility to Alex and 9000 is worth 95 units of utility The utility of the job s earnings is the average of 65 95 or 80 units of utility We can see from the TU curve that a job paying 6000 with certainty would be worth more to Alex 85 units of utility A job that paid 5000 with certainty would be worth the same level of utility to Alex as the risky job Measuring Risk Aversion The most commonly used risk aversion measure was developed by Pratt U X r X U X For risk averse individuals U X 0 r X will be positive for risk averse individuals r X coefficient of absolute risk aversion r X is same for any equivalent U i e a bU Risk Aversion If utility is logarithmic in consumption U X ln X where X 0 Pratt s risk aversion measure is U X X 2 1 r X 1 U X X X Risk aversion decreases as wealth increases ln x becomes more linear 1 5 1 0 0 5 2 3 4 5 Risk Aversion If utility is exponential U X e aX exp aX where a is a positive constant Pratt s risk aversion measure is 2 aX U X a e r X a aX U X ae Risk aversion is constant as wealth increases CARA constant absolute risk aversion 1 0 1 e 2 x 0 8 0 6 1 e x 0 4 0 2 1 2 3 4 Willingness to Pay for Insurance Consider a person with a current wealth of 100 000 who faces a 25 chance of losing his automobile worth 20 000 Suppose also that the utility function is U X ln x Willingness to Pay for Insurance The person s expected utility will be E U 0 75U 100 000 0 25U 80 000 E U 0 75 ln 100 000 0 25 ln 80 000 E U 11 45714 Willingness to Pay for Insurance The individual will be willing to pay more than 5 000 to avoid the gamble How much will he pay E U U 100 000 y ln 100 000 y 11 45714 100 000 y e11 45714 y 5 426 The maximum premium he is willing to pay is 5 426 426 more than actuarially fair insurance of 5 000 Summary Expected value is an adequate criterion to choose among lotteries if the individual is risk neutral However it is not adequate if the individual dislikes risk risk averse If someone prefers to receive B rather than playing a lottery in which expected value is B then we say that the individual is risk averse If U x is the utility function then we always assume that U x 0 If an individual is risk averse then U x 0 that is the marginal utility is decreasing with money U x is decreasing If an individual is risk averse then his utility function U x is concave We have studied a standard measure of risk aversion and insurance