Risk Theory Professor Peter Cramton Economics 300Is 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: '( ) 0iUxExpected 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: 1 1 2 21( ) ( ) ( ) ... ( )ni i n niEU pU x pU x p U x p U x • 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 functionClassification "( ) 0, strictly concave U(X) Risk averse"( ) 0, linear U(X) Risk neutral"( ) 0, strictly convex U(X) Risk loverUXUXUXExamples of commonly used Utility functions for risk averse individuals ( ) ln( )()( ) where 0 1( ) 1 e where 0aaxU x xU x xU x x aU x a Example: Alex is considering a job, which is based on commission & pays $3000 with 50% probability & $9000 with 50% probability. Wealth (thousands of dollars) Utility $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. 65 80 9 95 6 3 85 5 CEMeasuring Risk Aversion • The most commonly used risk aversion measure was developed by Pratt "( )( ) '( )UXrXUX• 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 21"( ) 1( ) '( )U X XrXU X X X • Risk aversion decreases as wealth increasesln(x) becomes “more linear” 23450.51.01.5Risk Aversion • If utility is exponential U(X) = -e-aX = -exp (-aX) where a is a positive constant • Pratt’s risk aversion measure is 2"( )( ) '( )aXaXU X a er X aU X ae • Risk aversion is constant as wealth increases CARA = constant absolute risk aversion12340.20.40.60.81.01xe21xeWillingness 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) = 11.45714Willingness 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 $5,000Summary • 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
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