Models & Decision with Financial Applications Unit 3: Utility Function and Risk Attitude Duan LI Department of Systems Engineering & Engineering Management The Chinese University of Hong Kong http://www.se.cuhk.edu.hk/ seem5820
2 Model Decision Maker s Preference: How to model preferences: Utility function approach. Every decision involves some kind of trade-off. In decision making under uncertainty, the fundamental trade-off question often is: How much risk is a decision maker willing to assume? Basing decisions on expected monetary values is convenient, but it can lead to decisions that may not seem intuitively appealing, since the EMV does not capture different risk attitudes of different decisionmaker(s). Consider two lotteries, {0.5, 100; 0.5, 1} and {0.5, 200; 0.5, 100}. Although the second lottery has a higher expected value, most people prefer the first one. The phenomenon that people act differently indicates that the expected value does not capture the essence in modeling decision makers risk attitudes.
3 Probability-equivalent Assessment Technique for Assessing Utilities: Let x 1, x 2..., x n be possible consequences from an uncertain event. Let x and x be the most preferred consequence and the least preferred consequence, respectively. An individual s preference on a lottery- that gives a chance p at x and a complementary chance at x is increasing with respect to p (Monotonicity). For any consequence x, there exists some number p(x) such that the decision maker would be indifferent between (1) x for certain and (2) a lottery giving a chance p(x) at x and a complementary chance at x. = p(x) is termed utility of the consequence x. Clearly, we have p(x ) = 0 and p(x ) = 1.
Non uniqueness: If u 1 is a utility function, then for any constant a and any positive constant b, the function u 2 = a + bu 1 is also a utility function. The utility function provides (i) a ranking of the consequences or uncertain events and (ii) the degree of preference of one against the other. Let A and B be consequence or uncertainty event. 1. A B (DM prefers A to B) if and only if u(a) > u(b). 2. A B (DM prefers B to A) if and only if u(a) < u(b). 3. A B (DM is indifferent between A to B) if and only if u(a) = u(b). x and x are not necessary in the candidate list. The consequences may not be in dollar unit, or even be evaluated with respect to multiple attributes. 4
5 Utility Axioms: The behavioral assumptions that form the basis of expected utility are called axioms. These axioms relate to the consistency with which an individual expresses preferences from among a series of risky prospects. 1. Ordering and transitivity. A decision maker can order (establish preference or indifference) any two alternatives. Further the ordering is transitive: A 1 A 2 and A 2 A 3 A 1 A 3 ( Money Pump argument). 2. Reduction of compound uncertain events. A decision maker is indifferent between a compound uncertain event and a simple uncertain event as determined by reduction using standard probability manipulations. 3. Continuity. A decision maker is indifferent between a consequence A and some uncertain event involving only two basic consequences A 1 and A 2, where A 1 A A 2 (We can construct a reference
gamble with some probability p, 0 < p < 1, for which the DM will be indifferent between the reference gamble and A.) 4. Substitutability. A decision maker is indifferent between any original uncertain event that includes A and one formed by substituting for A an uncertain event that is judged to be its equivalent. 5. Monotonicity. Given two reference gambles with the same possible outcomes, a decision maker prefers the one with the higher probability of winning the preferred outcome. 6. Invariance. All that is needed to determine a DM s preferences among uncertain events are the payoffs (or consequences) and the associated probabilities. 7. Finiteness. No consequences are considered infinitely bad or infinitely good. Most of us agree that the above assumptions are reasonable under most circumstances. 6
7 Utility of Uncertain Event: If a DM accepts the axioms, (i) it is possible for him/her to find a utility function to evaluate the consequences, and(ii) he/she should be making decisions in a way that is consistent with maximizing expected utility. Consider a selection between two lotteries, A : {0.5,20;0.5,15} and B : {0.3,40;0.7,10}. Applying the above axioms, we can conclude that For a decision maker A B if and only if his/her utility satisfies 0.5u(20) + 0.5u(15) > 0.3u(40) + 0.7u(10).
Assume that there is an uncertain event which pays off x i with probability p i, i = 1,2,..., n. The expected utility of this uncertain event is n i=1 p iu(x i ). Expected monetary value (EMV): Assume that there is an uncertain event which pays off x i with probability p i, i = 1,2,..., n. The EMV of this uncertain event is equal to n i=1 p ix i. Certainty Equivalent: Assume that there is an uncertain event which pays off x i with probability p i, i = 1,2,..., n. q is said to be the Certainty Equivalent of this uncertainty event if 8 u(q) = n p i u(x i ) i=1 In other words, the decision maker is indifferent between obtaining q for certain and entering the uncertain event: Expected Utility of the Uncertain Event = Utility of the Certainty Equivalent
9 Utility assessment using certainty equivalent Utility function is, in general, applied to the whole wealth. = The preference could change when the DM s current wealth is different. Example: A bettor with utility U(x) = lnx, where x is total wealth, has a choice between two alternatives: A: Win $10,000 with probability 0.2 and win $1,000 with probability 0.8. B: Win $3,000 with probability 0.9 and lose $2,000 with probability 0.1. 1. If the bettor currently has $2,500, he should prefer A to B, since u(a) = 0.2u(12,500)+0.8u(3,500) = 8.415 > u(b) = 0.9u(5,500)+0.1u(500) = 8.373 The CE of A: u 1 (8.415) - 2500 = 2014.28 The CE of B: u 1 (8.373) - 2500 = 1828.60
2. If the bettor currently has $5,000, he should prefer B to A, since u(a) = 0.2u(15,000)+0.8u(6,000) = 8.883 < u(b) = 0.9u(8,000)+0.1u(3,000) = 8.889 The CE of A: u 1 (8.883) - 5000 = 2208.38 The CE of B: u 1 (8.889) - 5000 = 2251.76 3. If the bettor currently has $10,000, he should prefer A to B, since u(a) = 0.2u(20,000)+0.8u(11,000) = 9.4254 > u(b) = 0.9u(13,000)+0.1u(8,000) = 9.4244 The CE of A: u 1 (9.4254) - 10,000 = 2399.36 The CE of B: u 1 (9.4244) - 10,000 = 2386.97 What is the reason behind this pattern of choices between A and B? 10
11 Risk Attitudes A decision maker is risk averse if the decision maker considers no uncertain event more desirable than its EMV, i.e., the decision maker s Certainty Equivalent is less than the EMV. A decision maker is risk averse if his/her utility function is concave (the curve opens downward). If the utility function is second-order differentiable, the utility function is concave if its second order derivative is always nonpositive. Example: u(x) = lnx, where x 0.
A decision maker is risk prone if the decision maker considers every uncertain event more desirable than its EMV, i.e., the decision maker s Certainty Equivalent is always larger than the EMV. A decision maker is risk prone if his/her utility function is convex (the curve opens upward). If the utility function is second-order differentiable, the utility function is convex if its second order derivative is always nonnegative. Example: u(x) = expx. A decision maker is risk neutral if the decision maker evaluates every uncertain event by its EMV, i.e., the decision maker s Certainty Equivalent is always equal to the EMV. A decision maker is risk neutral if his/her utility function is linear. When the variation of the consequence of different options is small with respect to the current wealth, a nonlinear utility can be well approximated by a linear utility function. 12
13 Application in Insurance Example: Assume that the utility function of an individual is given by u = w 1/2. His total wealth is $250,000 of which $160,000 is the worth of his house. There is 10% probability that his house may be destroyed by fire. What is the maximum premium (also called insurance premium) he should be willing to pay for insurance against fire? Define Risk Premium = EMV - Certainty Equivalent. What is the risk premium of the above example problem?
14 Decision Making Theory in the Insurance Market Consider two possible states of the nature, s 1 and s 2, which mayoccur at any time t, where s 1 is the normal state while s 2 is the state of accident. Let w i denote the wealth of the DM in state s i, i = 1,2. The utility function of the DM is denoted by u(w) which is state independent. The utility function is an increasing function of the wealth. u > 0. Assume that the DM is risk averse. u < 0. The probability that state s 1 will occur is p 1.
15 The expected utility function which the DM wants to maximize is Eu = p 1 u(w 1 )+(1 p 1 )u(w 2 ) Indifference curves on the w 1 w 2 plane: Eu = p 1 u(w 1 )+(1 p 1 )u(w 2 ) = constant The DM is indifferent between any two points on an indifference curve. The indifference curves have a negative slope: dw 2 dw 1 = p 1 1 p 1 u (w 1 ) u (w 2 ) < 0 The indifference curves are convex when the DM is risk averse, d(w 2 ) 2 d 2 w 1 = p 1 1 p 1 [ u (w 1 ) u (w 2 ) + u (w 1 ) [u (w 2 )] 2u (w 2 ) dw 2 dw 1 ] > 0
The 45 degree line on the w 1 w 2 plane is called the certainty line (No uncertainty in wealth associated with the states of the world). Along the certainty line the indifference curves have the same slope, p 1 1 p 1. Insurance premium: The maximum premium the DM is willing to get himself/herself fully covered against the risk = The money which the DM pays to move from the current (ŵ 1,ŵ 2 ) to the point on the certainty line that has the same utility value. Insurance policy: If the accident occurs, the insured claims the benefit. If the accident does not occur, the insured loses the money paid in premium. Suppose an insurance firm offers $a worth of benefit for each $1 paid in premium. 16
The insurance deal is said to be fair if the expected profit from such a contract is zero, 17 (1 p 1 )a p 1 = 0 i.e., a = p 1 1 p 1 TheDM sbudgetlinepassesthroughhis/hercurrentposition(ŵ 1,ŵ 2 )) with slope p 1 1 p 1, w 2 = ŵ 2 p 1 1 p 1 (w 1 ŵ 1 ) The fair premium is less than the insurance premium which is the maximum premium that the DM is willing to pay.
18 Fitting of Utility Function The degree of risk aversion of any decision maker changes when his/her status changes! The degree of risk aversion can be measured by the following normalized curvature -like measure: r(x) = u (x) u (x) We may classify decision makers by the classes of their degree of risk aversion, thus fitting the right utility functions for them.
19 Constant risk aversion and the exponential utility function An individual displays constant risk aversion if the risk premium for a gamble does not depend on the initial amount of wealth held by the decision maker. For example, if we observe the following behavior, 50-50 Gamble Expected Certainty Risk Between Value Equivalent Premium 10, 40 25 21.88 3.12 20, 50 35 31.88 3.12 30, 60 45 41.88 3.12 40, 70 55 51.88 3.12 we can conclude that the decision maker is of a constant risk aversion. In other words, a constantly risk aversion person would be just anxious about taking a bet regardless of the amount of money available.
Mathematically, if a decision maker is of a constant risk aversion, his/her degree of risk aversion r(x) is a constant with respect to x. For decision makers with a constant risk aversion, their utility can be expressed as the following exponential form: u(x) = 1 e x/r, where parameter R is called the risk tolerance that determines how risk averse the utility function is. Note r(x) = 1 R for such a utility function. Essentiallythe abovetable is generatedby settingr=35in theabove exponential utility function. This kind of experiments can be designed to test whether a decision maker is of a constant risk aversion. 20
21 How to determine parameter R? The risk tolerance parameter R in the exponential utility function u(x) = 1 e x/r determines how risk averse the utility function is. Larger values of R make the exponential utility function flatter, while smaller values make it more concave or more risk averse. How to determine R? Consider the gamble: Win $Y with probability 0.5 and lose $Y/2 with probability 0.5. The risk tolerance R is then approximately equal to the largest value of Y for which you would prefer to take the gamble rather than not to take it. Suppose that the expected value and variance of the payoffs are µ and σ 2, respectively. Then, Certainty Equivalence µ 0.5σ2 R when the utility function is of an exponential form.
22 Decreasing risk aversion and the logarithmic utility function If an individual s preferences show decreasing risk aversion, then the risk premium decreases if a constant amount is added to all payoffs in a gamble. For example, if we observe the following behavior, 50-50 Gamble Expected Certainty Risk Between Value Equivalent Premium 10, 40 25 20.00 5.00 20, 50 35 31.62 3.38 30, 60 45 42.43 2.57 40, 70 55 52.92 2.08 In other words, decreasing risk aversion means the more money you have, the less nervous you are about a particular bet.
Mathematically, if a decision maker is of a decreasing risk aversion, his/her degree of risk aversion r(x) is decreasing with respect to x. For decision makers with a decreasing risk aversion, their utility can be expressed as the following exponential form: 23 u(x) = ln(ax+b), where a > 0 and x is defined in {x ax+b 0}. Note r(x) = a ax+b which is decreasing with respect to x. Essentially the above table is generated by setting a = 1 and b = 0 in the above logarithmic utility function. This kind of experiments can be designed to test whether a decision maker is of a decreasing risk aversion. The problem of determining utility function is now reduced to parameter estimation problem for a and b.