Choice under Uncertainty

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1 Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory Discuss different attitudes towards risk Analyze the protfolio choice and demand for insurance of risk averse agents Slide 2 Page 1

2 Description of Risky Alternatives Denote Y the set of all possible outcomes. Y could be the decision maker's feasible consumption bundles or the monetary payoffs (a case we will study extensively later). We assume that the probabilities with which certain outcomes occur are objectively known, e.g. the probabilities with which a red number on the spin of an unbiased roulette wheel may occur. Slide 3 The Model Consider s states of the world Restrict y s R y s represents money in a given state of the world Characterize attitudes toward risk with properties of v(y s ) Assume that v is increasing and continuous (Last point implies that expected utility is increasing in probability on highest money prize) Slide 4 Page 2

3 A Standard Prospect Consider s states of the world A standard prospect P is given by with where s is interpreted as the probability of outcome y s occurring. A standard prospect (also called lottery) gives us the probability distribution over outcomes (most often interpreted as different levels of income). Slide 5 Examples of Standard Prospects P 1 1/2 1/ P 2 1/2 1/4 1/ Positive weight on a finite number of outcomes P 1 = (½, ½, 100, 20) and P 2 = (½, ¼, ¼, 100, 40,0) Slide 6 Page 3

4 Compound Prospects It can be the case that outcomes themselves are uncertain: outcomes of a prospect themselves are prospects. For example, there is a probability that you get a certain job and if you get a specific job there is a probability that you will do well and get promoted. This would then be a compound prospect. Slide 7 Compound Prospects Example Job A 1/4 h 3/4 Job B 1/6 No promo promo 5/6 1/3 No promo 2/3 promo Slide 8 Page 4

5 Reduction to Standard Prospect We can also find a reduced probability distribution, e.g. the distribution over getting promoted or not. In order to get the probabilities of these two outcomes we need to multiply the probability of getting a promo in a job with the probability of getting this job in the first place. Then add all the probabilities of getting a promotion in all jobs together to find out what the probability of getting a promotion is. Put more generally, a reduced prospect is a prospect that yields the same ultimate distribution over outcomes as the compound prospect. Slide 9 The Consequentialist Premise Rational Equivalence Assume that the consumer s preferences are defined on compound prospects, but the consumer is indifferent between any two compound prospects that reduce to the same standard prospect. Slide 10 Page 5

6 Rationality and Continuity It is clear that we need to assume that preferences, again denoted by, are rational (complete and transitive) or no utility representation is possible. Again we need more to ensure a utility representation exists, namely Continuity. Continuity guarantees that we can draw indifference curves, that is we will be able to find small enough changes in the probability distribution of a prospect that will make the person indifferent to the original prospect. Slide 11 Independence Axiom The major new axiom The preference on the space of standard prospects P satisfies the independence axiom, if for every and every Slide 12 Page 6

7 Independence Axiom-Intuition Mix each of the two prospects with a third one, then the preference ordering of the two resulting mixtures does not depend on the particular third prospect used. You can think of it as follows. Consider two compound prospects ((1/2), (1/2), P, P ) and ((1/2), (1/2), P, P ) : When head comes up there will either be P or P, depending which compound prospect you face, but if tail comes up there will always be P. So the individual should choose between the two prospects based on her preference for P and P only. Slide 13 Independence Axiom The independence axiom is at the heart of the theory of choice under uncertainty. It is different from any axiom we have encountered in utility theory so far: it exploits in a fundamental manner the structure of uncertainty present in the model. In experiments, this is the axiom most often violated. Slide 14 Page 7

8 Existence von-neumann Morgenstern proved an early representation theorem With rationality, continuity and independence, there exists a function that assigns a number v to each outcome y s s = 1,,S, such that, for and if and only if if and only if Slide 15 Expected Utility Function This function capturing the relationship between the distribution of utility in any given state and the probability distribution is called Expected Utility Function. This is an appropriate name given that we sum up the products of probability of each state occurring and the utility in each state. Similarity to expected value, but obviously not the same!!! Slide 16 Page 8

9 Domains The utility function v(y s ) is defined on the domain of possible outcomes Y. It tells us how a person feels once an outcome has occurred. E.g. this is the utility when you have $100 for sure. The expected utility function is defined on the space of probability distributions. Slide 17 Ordinal v.s. Cardinal The key result is that v is less flexible than it was in ordinal consumer theory: in fact, it implies that expected utility is cardinal. Cardinal has notion of degree of difference: twice as fast, half as warm. Temperature is cardinal: there is a real sense in which something can be twice as hot. There is also a real sense in which an event is twice as likely as another event! Slide 18 Page 9

10 Example to show that cardinality matters Let Y = {0,5,9} and v(y) = y and compare the prospects P = ( 1, 1 ;0,9) 2 2 and P' = (1;5) then, since is less than, for this consumer Slide 19 An Example Now transform v into This function says that v(y) = [v(y)] 3 So: Slide 20 Page 10

11 Since Affine Transformation S s=1 S ( ) π s v ' y s = π s (αv y s s=1 S ( ) + β) = α π s v(y s ) + π s β s=1 S s=1 Therefore if so does S π s=1 s ( αv ( y s ) + β) represents for all Slide 21 Properties of EU: Risk Aversion Definition: The consumer is said to be strictly risk averse if the consumer always prefers the amount for sure to the same amount in expectation of a prospect. That is, let expected value consumer And strictly risk averse consumer be a prospect with, then for a risk averse for all P with π s <1. Risk aversion is equivalent to the concavity of v(y s ). Slide 22 Page 11

12 Other attitudes towards Risk The consumer is said to be: risk neutral if the consumer is always indifferent between the expected value of a prospect to the prospect itself. risk loving if the consumer always prefers a prospect to its expected value. Slide 23 Certainty Equivalent For given preferences it is convenient to define: The certainty equivalent of P,, as the amount for sure at which the agent is indifferent between this amount and the expected value of the lottery: Slide 24 Page 12

13 Risk Aversion and Certainty Equivalent r Slide 25 Some equivalent statements The agent is risk averse v(y s ) is concave for all prospects Slide 26 Page 13

14 An Example: Insurance Consider a risk averse consumer facing a possible loss of wealth. Let her preferences be represented by v(y s ), a strictly concave and twice differentiable function in y s. Denote her initial total wealth w, the potential loss L, and the probability of loss no loss occurs with probability Slide 27 Insurance Assume that the market for insurance is competitive, and that insurance is priced linearly at $p per unit. That is, if the consumer buys $20 worth of coverage, it will cost $p *20. Denote the amount of coverage the consumer buys q. The consumer must decide on the amount of coverage to buy. That is, she must solve max π q 1 v(w pq) + π 2 v(w L + q pq) Slide 28 Page 14

15 Insurance The first order necessary and sufficient condition for an interior solution ( q (0, w p ) v '(w L + q pq)π 2 (1 p) v '(w pq)π 1 p = 0 ) is so π 2 (1 p) π 1 p = v '(w pq) v '(w L + q pq) Slide 29 Zero profit insurance In a competitive market, insurers can enter whenever there is positive expected profit, so in equilibrium which implies that Slide 30 Page 15

16 Fair insurance Zero profit implies that insurance is actuarially fair, The solution for the consumer is thus 1 = v '(w pq) v '(w L + q pq) Slide 31 Full Coverage Case If v(y s ) is strictly concave and increasing, then v (y s ) takes on lower values the higher y. This implies that v '(w L + q pq) = v '(w pq) iff w L + q pq = w pq so q* = L. strictly concave utility function --- remember that when the second derivative of a function is strictly negative, the value of the first derivative changes everywhere in the domain: thus if the first derivative is equal at two points, they must be the same point. Slide 32 Page 16

17 Insurance Market is imperfectly competitive Insurance companies make profit, that is Consumer pays more than the actuarially fair premium. Consumer demands less than full coverage. Slide 33 Less Than Full Coverage With FOC becomes π 2 (1 p) = π 1 p v '(w pq) v '(w L + q pq) π 2 (1 p) < 1 π 1 p v '(w pq) < v '(w L + q pq) w pq > w L + q pq q * < L. Slide 34 Page 17

18 Risk Aversion and Indifference Curves Slide 35 Risk Aversion and Indifference Curves Suppose only one bad thing can happen with probability 2, but depending on the actions of the individual the amount of money in the two states will vary. Can draw an indifference curve diagram in state-contingent space, that is money if bad thing happens on y-axis and money if good thing happens on x-axis. Slide 36 Page 18

19 State-Contingent income space y 2 y 1 = y 2 45 degree line is the certainty line: no matter what happens, income is the same y 1 Slide 37 Indifference curves in statecontingent space We find indifference curves the same way as always. Hold expected utility fixed, see what combinations of (y 1, y 2 ) give us same expected utility. Slide 38 Page 19

20 Slope of indifference Curves Slide 39 Slope of indifference Curves The slope of the indifference curves at the 45-degree line is Slide 40 Page 20

21 Shape of Indifference curves Note that V is an increasing function of y 1 and y 2, because v(y s ) is increasing. This means the better sets must lie to the right of an indifference curve. Are the better sets convex? The better sets are convex iff v(y s ) is concave. Slide 41 Convex Better sets ( ) = π 1 v ( y 1 ) + π 2 v ( y 2 ) ( ) V y 1, y 2 v y 1 π dy 1 2 y = 1 dy 1 v y 2 π 2 y 2 d 2 y 2 ( ) 2 = d y 1 Need v2 y 1 ( ) ( ) ( ) 2 ( ) v 2 y 1 π 1 y 1 v y 2 π 2 y 2? 0 ( ) 0 v ( y ( ) 2 s ) concave. y 1 Slide 42 Page 21

22 Indifference curves and attitudes towards risk For strictly risk averse person, indifference curves in state-contingent space are strictly convex (better sets are strictly convex). For risk neutral person, indifference curves are convex (straight lines, more specifically) For risk loving person, indifference curves are strictly concave (worse sets are strictly convex). No matter what the attitude towards risk, the slope of an indifference curve where it intersects with the 45 degree (certainty line) is equal to the risk ratio. Slide 43 State-Contingent income space y 2 y 1 = y 2 Better set Indifference curve and better set for a strictly risk averse person Slope= - π 1 / π 2 y 1 Slide 44 Page 22

23 Example: Insurance Demand in Graph y in bad state w - pl w - L Optimal bundle with actuarially fair premium Income bundle w/o insurance w - pl w y in good state Slide 45 Insurance Demand in Graph y in bad state Optimal bundle with not actuarially fair premium w-l+q-pq (1-p)q w - L Income bundle w/o insurance pq w-pq w y in good state Slide 46 Page 23

24 Demand for Insurance Conclusion Strictly risk averse people are willing to take on risks. If the premium is not actuarially fair, strictly risk averse people will not be fully covered and therefore they accept some risk. Slide 47 Conclusion We have investigated choice under uncertainty. We made the assumption that individuals are expected utility maximizers (crucial assumption is Independence axiom). Cardinal properties on the utility function give us attitudes toward risk. Most often individuals are assumed to be risk averse, i.e. they have a decreasing marginal utility of money. We found out that risk averse people are willing to take on risk. Slide 48 Page 24

25 Risk Aversion and Demand for Risky Assets Degrees of absolute/relative risk aversion and their impact on portfolio choice Slide 49 Demand for Risky Asset Will allow for continuous probability distribution Compare investment decisions of people with different wealth levels/degrees of risk aversion Slide 50 Page 25

26 Risk Aversion The definition of risk aversion can be rewritten. An agent is risk averse if for all F ( ) v(y) df(y) v y df(y) This inequality is also known as Jensen s inequality and is the defining property of a concave function. Risk aversion is equivalent to the concavity of v( ). Slide 51 Demand for Risky Asset An asset is a divisible claim to a financial return in the future. Suppose that there are two assets, a safe asset with a return of 1 dollar per dollar invested and a risky asset with a random return of z dollar per dollar invested. The random return z has a distribution function F(z) that we assume satisfies zdf(z)>1; that is, its mean return exceeds that of the safe asset. Slide 52 Page 26

27 Demand for Risky Asset An individual has initial income M to invest, which can be divided in any way between the two assets. Let and denote the amounts of wealth invested in the risky and the safe asset, respectively. Thus, for any realization z of the random return, the individual's portfolio (, ) pays z+. Of course, we must also have + =M. Assume, i.e. the mean return of the risky assets exceeds the return of the safe asset. Slide 53 Demand for Risky Asset How does the individual choose and? max α,β 0 ( ) df ( z) v α z + β s.t. α + β = M. Or equivalently max 0 α M v ( α z + M α ) df ( z) Slide 54 Page 27

28 Demand for Risky Asset FOC is necessary and sufficient. Note that at α=0 we have We conclude α*>0. Slide 55 Demand for Risky Asset Strictly risk averse people are willing to take on risks: They will invest a fraction of their wealth in risky assets if their expected return exceeds the return of the safe asset. We also saw that with the demand for insurance: if the premium is not actuarially fair, strictly risk averse people will not be fully covered and therefore they accept some risk. Slide 56 Page 28

29 Degree of Risk Aversion How can we quantify risk aversion? One idea is size of, but this is changed by affine transformations, so is clearly wrong The Arrow - Pratt measure of (absolute) risk aversion Note that the Arrow-Pratt measure is independent of affine transformations of v(y). Slide 57 Logarithmic Utility Example of decreasing absolute risk aversion Slide 58 Page 29

30 Comparison across Individuals Given two utility functions v 1 (y) and v 2 ( y), the following statements are equivalent: 1. Person 2 is more risk averse than person A(y, v 2 ) A(y, v 1 ) for every y. 3. There exists an increasing concave function ( ) such that v 2 (y)= (v 1 (y)) at all y; that is, v 2 ( ) is a concave transformation of v 1 ( ). 4. y c (F, v 2 ) y c (F, v 1 ) for any F( ). 5. Whenever person 2 finds a prospect at least as good as a riskless outcome y, then person 1 finds that same prospect at least as good as y Slide 59 Demand for Risky Asset Now we have two individuals and one is more risk averse than the other. If person 2 is more risk averse than person 1, person 2 will invest less in the risky asset than person 1. Slide 60 Page 30

31 Demand for Risky Asset Recall FOC for asset demand problem and assume M> 1 *>0 : For person 2 we can write the FOC for the asset demand problem as : Slide 61 Demand for Risky Asset Next we show that at 1 *, FOC for person 2 is already negative. This means by decreasing we can make person 2 better off. This is true because and Slide 62 Page 31

32 Comparison across Wealth Levels Suppose M goes up for individual 1. How does this change person 1 s demand for the risky asset? We have Since 1 * is a function of M, and FOC must also hold at new wealth level, we have ( ( )) ( z 1) df ( z) = 0 v 1 ' α * 1 ( M) z + M α * 1 M ( ( )) ( z 1) z 1 v 1 '' α * 1 ( M) z + M α * 1 M ( ) α * M 1 ( ) M + 1 df z ( ) = 0 Slide 63 Comparison across Wealth Levels A( y)v' y ( )( z 1) ( z 1) α * 1 ( M ) A( y)v' ( y) ( z 1) 2 α * 1 ( M ) M + A( y)v' ( y) ( z 1)dF z ( ) = 0 M +1 df z df ( z) ( ) = 0 Slide 64 Page 32

33 A( y)v' ( y) ( z 1) 2 α * 1 ( M ) where A( y)v' y Note the following M * M df ( z) = α 1 M ( )( z 1) 2 df ( z) > 0 On the other hand, the sign of A ( α * 1 ( M) ( z 1) + M)v ' α * 1 M depends on A(y). ( ) A( y)v' ( y) ( z 1) 2 df ( ( ) ( z 1) + M) ( z 1) df ( z) ( z) Slide 65 Note that if A(y) =c, where c is a constant, i.e. A(y) is independent of y and therefore M, we can write ( ( ) + M) ( z 1) df c v ' α * 1 ( M) z 1 Since by the FOC for optimal investment v ' α * 1 ( M) ( z 1) + M ( ) z 1 ( ) df ( z) = 0 ( z) = 0 This implies ( ) c α * 1 M M * M ( ) α 1 M = 0 v' ( y) ( z 1) 2 df ( z) + 0 = 0 Slide 66 Page 33

34 Note that if A (y)<0, i.e. decreasing absolute risk aversion A( ( α * 1 ( M )( z 1) + M ))v' ( α * 1 ( M )( z 1) + M )( z 1)dF z ( ) < 0 Since by the FOC for optimal investment v ' α * 1 ( M) ( z 1) + M ( ) z 1 ( ) df ( z) = 0 This implies α * 1 ( M ) M ( ) * M α 1 M > 0 A( y)v' ( y) ( z 1) 2 df ( z) > 0 Slide 67 Note that if A (y)>0, increasing absolute risk aversion A( ( α * 1 ( M )( z 1) + M ))v' ( α * 1 ( M )( z 1) + M )( z 1)dF z ( ) > 0 Since by the FOC for optimal investment v ' α * 1 ( M) ( z 1) + M ( ) z 1 ( ) df ( z) = 0 This implies α * 1 ( M ) M ( ) * M α 1 M < 0 A( y)v' ( y) ( z 1) 2 df ( z) < 0 Slide 68 Page 34

35 Comparison across Wealth Levels If we have decreasing risk aversion, then the individual will demand more of the risky asset as her wealth goes up. Again we can capture the idea of decreasing risk aversion in different ways. The basic idea is that you can look at a given wealth level and add an increment z. Then the individual behaves with respect to z differently given the initial wealth. Slide 69 The Coefficient of Relative Risk Aversion Log utility is example of Constant Relative Risk Aversion preferences (CRRA) Slide 70 Page 35

36 Example: Asset demand as proportion of wealth with CRRA Let v(y) = lny. Show that as the wealth of this person goes up, the same proportion of wealth is invested in the risky asset. Slide 71 CRRA and constant proportion of wealth Let /M =, that is is the fraction of income invested in the risky asset. Slide 72 Page 36

37 CRRA and constant proportion of wealth Note that ( z 1) ( γz +1 γ ) γmz + M γm ( ) 2 therefore ( ) ( z 1) 2 M γ M M γmz + M γm ( ) 2 ( ) γ M M = 0. df ( z) = 1 M df ( z) = 0 z 1 γmz + M γm ( ) df z ( ) = 0 With CRRA, a constant proportion of wealth will be invested in the risky asset. Slide 73 CRRA and constant proportion of wealth CRRA is often used in finance theory: It has the implication that no matter how wealth is distributed between individuals, the portfolio decisions of individuals in terms of budget shares do not vary. Slide 74 Page 37

38 Conclusion We have investigated choice under uncertainty. We made the assumption that individuals are expected utility maximizers (crucial assumption is Independence axiom). Cardinal properties on the utility function give us attitudes toward risk. Most often individuals are assumed to be risk averse, i.e. they have a decreasing marginal utility of money. We found out that risk averse people are willing to take on risks by deriving the demand for insurance and risky assets. Slide 75 Conclusion They will invest in risky assets if the expected return is higher than the return of the safe asset. They will prefer to be only partially insured if the insurance premium is not actuarially fair. We have developed a measure of risk aversion. People who are more risk averse will invest less in risky assets and take out more insurance than a person who is less risk averse. People s attitude towards risk may change as they have more wealth. Decreasing relative risk aversion yields the result that the wealthy invest proportionally more in risky assets than the less wealthy. Slide 76 Page 38

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