Expected Utility and Risk Aversion

Expected Utility and Risk Aversion Expected utility and risk aversion 1/ 58

Introduction Expected utility is the standard framework for modeling investor choices. The following topics will be covered: 1 Analyze conditions on individual preferences that lead to an expected utility function. 2 Consider the link between utility, risk aversion, and risk premia for particular assets. 3 Examine how risk aversion a ects an individual s portfolio choice between a risky and riskfree asset. Expected utility and risk aversion 2/ 58

Preferences when Returns are Uncertain Economists typically analyze the price of a good using supply and demand. We can do the same for assets. The main distinction between assets is their future payo s: Risky assets have uncertain payo s, so a theory of asset demands must specify investor preferences over di erent, uncertain payo s. Consider relevant criteria for ranking preferences. One possible measure is the asset s average payo. Expected utility and risk aversion 3/ 58

Criterion: Expected Payo Suppose an asset o ers a single random payo at a particular future date, and this payo has a discrete distribution with n possible outcomes (x 1 ; :::; x n ) and corresponding probabilities (p 1 ; :::; p n ), where P n i=1 p i = 1 and p i 0. Then the expected value of the payo (or, more simply, the expected payo ) is x E [ex] = P n i=1 p i x i. Is an asset s expected value a suitable criterion for determining an individual s demand for the asset? Consider how much Paul would pay Peter to play the following coin ipping game. Expected utility and risk aversion 4/ 58

St. Petersburg Paradox, Nicholas Bernoulli, 1713 Peter continues to toss a coin until it lands heads. He agrees to give Paul one ducat if he gets heads on the very rst throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on. If the number of coin ips taken to rst obtain heads is i, then p i = 1 2 i and x i = 2 i 1 : Thus, Paul s expected payo equals x = P 1 i=1 p i x i = 1 2 1 + 1 4 2 + 1 8 4 + 1 16 8 + ::: (1) = 1 2 (1 + 1 2 2 + 1 4 4 + 1 8 8 + ::: = 1 2 (1 + 1 + 1 + 1 + ::: = 1 Expected utility and risk aversion 5/ 58

St. Petersburg Paradox What is the paradox? Daniel Bernoulli (1738) explained it using expected utility. His insight was that an individual s utility from receiving a payo di ered from the size of the payo. Instead of valuing an asset as x = P n i=1 p i x i, its value, V, would be V E [U (ex)] = X n i=1 p i U i where U i is the utility associated with payo x i. He hypothesized that U i is diminishingly increasing in wealth. Expected utility and risk aversion 6/ 58

Criterion: Expected Utility Von Neumann and Morgenstern (1944) derived conditions on an individual s preferences that, if satis ed, would make them consistent with an expected utility function. De ne a lottery as an asset that has a risky payo and consider an individual s optimal choice of a lottery from a given set of di erent lotteries. The possible payo s of all lotteries are contained in the set fx 1 ; :::; x n g. A lottery is characterized by an ordered set of probabilities np P = fp 1 ; :::; p n g, where of course, p i = 1 and p i 0. Let a i=1 di erent lottery be P = fp 1 ; :::; p ng. Let,, and denote preference and indi erence between lotteries. Expected utility and risk aversion 7/ 58

Preferences Over Di erent Random Payo s Speci cally, if an individual prefers lottery P to lottery P, this can be denoted as P P or P P. When the individual is indi erent between the two lotteries, this is written as P P. If an individual prefers lottery P to lottery P or she is indi erent between lotteries P and P, this is written as P P or P P. N.B.: all lotteries have the same payo set fx 1 ; :::; x n g, so we focus on the (di erent) probability sets P and P. Expected utility and risk aversion 8/ 58

Expected Utility Axioms 1-3 Theorem: There exists an expected utility function V (p 1 ; :::; p n ) if the following axioms hold: Axioms: 1) Completeness For any two lotteries P and P, either P P, or P P, or P P. 2) Transitivity If P P and P P, then P P. 3) Continuity If P P P, there exists some 2 [0; 1] such that P P + (1 )P, where P + (1 )P denotes a compound lottery ; namely, with probability one receives the lottery P and with probability (1 ) one receives the lottery P. Expected utility and risk aversion 9/ 58

Expected Utility Axioms 4-5 4) Independence For any two lotteries P and P, P P if and only if for all 2 (0,1] and all P : P + (1 )P P + (1 )P Moreover, for any two lotteries P and P y, P P y if and only if for all 2(0,1] and all P : P + (1 )P P y + (1 )P 5) Dominance Let P 1 be the compound lottery 1 P z + (1 1 )P y and P 2 be the compound lottery 2 P z + (1 2 )P y. If P z P y, then P 1 P 2 if and only if 1 > 2. Expected utility and risk aversion 10/ 58

Discussion: Machina (1987) The rst three axioms are analogous to those used to establish a real-valued utility function in consumer choice theory. Axiom 4 (Independence) is novel, but its linearity property is critical for preferences to be consistent with expected utility. To understand its meaning, suppose an individual chooses P P. By Axiom 4, the choice between P + (1 )P and P + (1 )P is equivalent to tossing a coin that with probability (1 ) lands tails, in which both lotteries pay P, and with probability lands heads, in which case the individual should prefer P to P. Expected utility and risk aversion 11/ 58

Allais Paradox But, there is some experimental evidence counter to this axiom. Consider lotteries over fx 1 ; x 2 ; x 3 g = f$0; $1m; $5mg and two lottery choices: C1: P 1 = f0; 1; 0g vs P 2 = f:01; :89; :1g C2: P 3 = f:9; 0; :1g vs P 4 = f:89; :11; 0g Which do you choose in C1? In C2? Expected utility and risk aversion 12/ 58

Allais Paradox Experimental evidence suggests most people prefer P 1 P 2 and P 3 P 4. But this violates Axiom 4. Why? De ne P 5 = f1=11; 0; 10=11g and let = 0:11. Note that P 2 is equivalent to the compound lottery: P 2 P 5 + (1 ) P 1 0:11f1=11; 0; 10=11g + 0:89f0; 1; 0g f:01; :89; :1g Expected utility and risk aversion 13/ 58

Allais Paradox Note also that P 1 is trivially the compound lottery P 1 + (1 ) P 1. Hence, if P 1 P 2, the independence axiom implies P 1 P 5. Now also de ne P 6 = f1; 0; 0g, and note that P 3 equals the following compound lottery: while P 4 P 3 P 5 + (1 ) P 6 0:11f1=11; 0; 10=11g + 0:89f1; 0; 0g f:9; 0; :1g is equivalent to the compound lottery P 4 P 1 + (1 ) P 6 0:11f0; 1; 0g + 0:89f1; 0; 0g f:89; 0:11; 0g Expected utility and risk aversion 14/ 58

Allais Paradox But if P 3 P 4, the independence axiom implies P 5 P 1, which contradicts the choice of P 1 P 2 that implies P 1 P 5. Despite the sometimes contradictory experimental evidence, expected utility is still the dominant paradigm. However, we will consider di erent models of utility at a later date, including those that re ect psychological biases. Expected utility and risk aversion 15/ 58

Deriving Expected Utility: Axiom 1 We now prove the theorem by showing that if an individual s preferences over lotteries satisfy the preceding axioms, these preferences can be ranked by the individual s expected utility of the lotteries. De ne an elementary or primitive lottery, e i, which returns outcome x i with probability 1 and all other outcomes with probability zero, that is, e i = fp 1 ; :::p i 1 ;p i ;p i+1 :::;p n g = f0; :::0; 1; 0; :::0g where p i = 1 and p j = 0 8j 6= i. Without loss of generality, assume that the outcomes are ordered such that e n e n 1 ::: e 1. This follows from the completeness axiom for this case of n elementary lotteries Expected utility and risk aversion 16/ 58

Deriving Expected Utility: Axiom 3, Axiom 4 From the continuity axiom, for each e i, there exists a U i 2 [0; 1] such that e i U i e n + (1 U i )e 1 (2) and for i = 1, this implies U 1 = 0 and for i = n, this implies U n = 1. Now a given arbitrary lottery, P = fp 1 ; :::; p n g, can be viewed as a compound lottery over the n elementary lotteries, where elementary lottery e i is obtained with probability p i. P p 1 e 1 + ::: + p n e n Expected utility and risk aversion 17/ 58

Deriving Expected Utility: Axiom 4 By the independence axiom, and equation (2), the individual is indi erent between lottery, P, and the following lottery: p 1 e 1 + ::: + p n e n p 1 e 1 + ::: + p i 1 e i 1 + p i [U i e n + (1 U i )e 1 ] +p i+1 e i+1 + ::: + p n e n (3) where the indi erence relation in equation (2) substitutes for e i on the right-hand side of (3). By repeating this substitution for all i, i = 1; :::; n, the individual will be indi erent between P and p 1 e 1 + ::: + p n e n! nx p i U i e n + 1 i=1! nx p i U i e 1 (4) i=1 Expected utility and risk aversion 18/ 58

Deriving Expected Utility: Axiom 5 Now de ne n P i=1 p i U i. Thus, P e n + (1 )e 1 Similarly, we can show that any other arbitrary lottery P = fp1 ; :::; p ng e n + (1 )e 1, where P n pi U i. We know from the dominance axiom that P P i >, P implying n pi U P i > n p i U i. i=1 i=1 So we can de ne the function i=1 nx V (p 1 ; :::; p n ) = p i U i (5) which implies that P P i V (p 1 ; :::; p n) > V (p 1 ; :::; p n ). i=1 Expected utility and risk aversion 19/ 58

Deriving Expected Utility: The End The function in (5) is known as von Neumann-Morgenstern expected utility. It is linear in the probabilities and is unique up to a linear monotonic transformation. The intuition for why expected utility is unique up to a linear transformation comes from equation (2). Here we express elementary lottery i in terms of the least and most preferred elementary lotteries. However, other bases for ranking a given lottery are possible. For U i = U(x i ), an individual s choice over lotteries is the same under the transformation au(x i ) + b, but not a nonlinear transformation that changes the shape of U(x i ). Expected utility and risk aversion 20/ 58

St. Petersburg Paradox Revisited Suppose U i = U(x i ) = p x i. Then the expected utility of the St. Petersburg payo is V = nx p i U i = i=1 1X 1 p 1X 2 2 i i 1 = 2 1 2 (i+1) = i=1 = 2 2 3 2 + 2 2 + ::: 1X 1 i = p2 1 = i=0 2 1 p = 1:707 2 1 p 2 = i=1 1 1 1 p2 1 1X i=2 1 p 2 A certain payment of 1:707 2 = 2:914 ducats has the same expected utility as playing the St. Petersburg game. 2 i 2 Expected utility and risk aversion 21/ 58

Super St. Petersburg The St. Petersburg game has in nite expected payo because the probability of winning declines at rate 2 i, while the winning payo increases at rate 2 i. In a super St. Petersburg paradox, we can make the winning payo increase at a rate x i = U 1 (2 i 1 ) to cause expected utility to increase at 2 i. For square-root utility, x i = (2 i 2) 2 = 2 2i 2 ; that is, x 1 = 1, x 2 = 4, x 3 = 16, and so on. The expected utility of super St. Petersburg is 1X V = nx p i U i = i=1 1X i=1 1 2 i p 2 2i 2 = i=1 1 2 i 2i 1 = 1 (6) Should we be concerned that if prizes grow quickly enough, we can get in nite expected utility (and valuations) for any chosen form of expected utility function? Expected utility and risk aversion 22/ 58

Von Neumann-Morgenstern Utility The von Neumann-Morgenstern expected utility can be generalized to a continuum of outcomes and lotteries with continuous probability distributions. Analogous to equation (5) is Z Z V (F ) = E [U (ex)] = U (x) df (x) = U (x) f (x) dx (7) where F (x) is the lottery s cumulative distribution function over the payo s, x. V can be written in terms of the probability density, f (x), when F (x) is absolutely continuous. This is analogous to our previous lottery represented by the discrete probabilities P = fp 1 ; :::; p n g. Expected utility and risk aversion 23/ 58

Risk Aversion Diminishing marginal utility results in risk aversion: being unwilling to accept a fair lottery. Why? Let there be a lottery that has a random payo, e", where e" = "1 with probability p " 2 with probability 1 p (8) The requirement that it be a fair lottery restricts its expected value to equal zero: E [e"] = p" 1 + (1 p)" 2 = 0 (9) which implies " 1 =" 2 = (1 p) =p, or solving for p, p = " 2 = (" 1 " 2 ). Since 0 < p < 1, " 1 and " 2 are of opposite signs. Expected utility and risk aversion 24/ 58

Risk Aversion and Concave Utility Suppose a vn-m maximizer with current wealth W is o ered a fair lottery. Would he accept it? With the lottery, expected utility is E [U (W + e")]. Without it, expected utility is E [U (W )] = U (W ). Rejecting it implies U (W ) > E [U (W + e")] = pu (W + " 1 ) + (1 p)u (W + " 2 ) (10) U (W ) can be written as U(W ) = U (W + p" 1 + (1 p)" 2 ) (11) Substituting into (10), we have U (W + p" 1 + (1 p)" 2 ) > pu (W + " 1 )+(1 p)u (W + " 2 ) (12) which is the de nition of U being a concave function. Expected utility and risk aversion 25/ 58

Risk Aversion, Concavity A function is concave if a line joining any two points lies entirely below the function. When U(W ) is a continuous, second di erentiable function, concavity implies U 00 (W ) < 0. Expected utility and risk aversion 26/ 58

Risk Aversion, Concavity To show that concave utility implies rejecting a fair lottery, we can use Jensen s inequality which says that for concave U() E[U(~x)] < U(E[~x]) (13) Therefore, substituting ~x = W + e" with E[e"] = 0, we have E [U(W + e")] < U (E [W + e"]) = U(W ) (14) which is the desired result. Expected utility and risk aversion 27/ 58

Risk Aversion and Risk Premium How might aversion to risk be quanti ed? One way is to de ne a risk premium as the amount that an individual is willing to pay to avoid a risk. Let denote the individual s risk premium for a lottery, e". is the maximum insurance payment an individual would pay to avoid the lottery risk: U(W ) = E [U(W + e")] (15) W is de ned as the certainty equivalent level of wealth associated with the lottery, e". For concave utility, Jensen s inequality implies > 0 when e" is fair: the individual would accept wealth lower than her expected wealth following the lottery, E [W + e"], to avoid the lottery. Expected utility and risk aversion 28/ 58

Risk Premium For small e" we can take a Taylor approximation of equation (15) around e" = 0 and = 0. Expanding the left-hand side about = 0 gives U(W ) = U(W ) U 0 (W ) (16) and expanding the right-hand side about e" gives E [U(W + e")] = E U(W ) + e"u 0 (W ) + 1 2 e"2 U 00 (W ) (17) = U(W ) + 0 + 1 2 2 U 00 (W ) where 2 E e" 2 is the lottery s variance. Expected utility and risk aversion 29/ 58

Risk Premium cont d Equating the results in (16) and (17) gives = 1 2 2 U00 (W ) U 0 (W ) 1 2 2 R(W ) (18) where R(W ) U 00 (W )=U 0 (W ) is the Pratt (1964)-Arrow (1971) measure of absolute risk aversion. Since 2 > 0, U 0 (W ) > 0, and U 00 (W ) < 0, concavity of the utility function ensures that must be positive An individual may be very risk averse ( U 00 (W ) is large), but may be unwilling to pay a large risk premium if he is poor since his marginal utility U 0 (W ) is high. Expected utility and risk aversion 30/ 58

U 00 (W ) and U 0 (W ) Consider the following negative exponential utility function: U(W ) = e bw ; b > 0 (19) Note that U 0 (W ) = be bw > 0 and U 00 (W ) = b 2 e bw < 0. Consider the behavior of a very wealthy individual whose wealth approaches in nity lim W!1 U0 (W ) = lim W!1 U00 (W ) = 0 (20) There s no concavity, so is there no risk aversion? R(W ) = b2 e bw = b (21) be bw Expected utility and risk aversion 31/ 58

Absolute Risk Aversion: Dollar Payment for Risk We see that negative exponential utility, U(W ) = e bw, has constant absolute risk aversion. If, instead, we want absolute risk aversion to decline in wealth, a necessary condition is that the utility function must have a positive third derivative: @R(W ) @W = @ U 00 (W ) U 0 (W ) @W = U000 (W )U 0 (W ) [U 00 (W )] 2 [U 0 (W )] 2 (22) Expected utility and risk aversion 32/ 58

R(W ) ) U(W ) The coe cient of risk aversion contains all relevant information about the individual s risk preferences. Note that R(W ) = U00 (W ) U 0 (W ) = @ (ln [U0 (W )]) @W (23) Integrating both sides of (23), we have Z R(W )dw = ln[u 0 (W )] + c 1 (24) where c 1 is an arbitrary constant. Taking the exponential function of (24) gives e R R(W )dw = U 0 (W )e c 1 (25) Expected utility and risk aversion 33/ 58

R(W ) ) U(W ) cont d Integrating once again, we obtain Z R e R(W )dw dw = e c 1 U(W ) + c 2 (26) where c 2 is another arbitrary constant. Because vn-m expected utility functions are unique up to a linear transformation, e c 1 U(W ) + c 2 re ects the same risk preferences as U(W ). Expected utility and risk aversion 34/ 58

Relative Risk Aversion Relative risk aversion is another frequently used measure de ned as R r (W ) = WR(W ) (27) Consider risk aversion for some utility functions often used in models of portfolio choice and asset pricing. Power utility can be written as U(W ) = 1 W ; < 1 (28) implying that R(W ) = R r (W ) = 1. ( 1)W 2 (1 ) = W 1 W Hence, it displays constant relative risk aversion. and, therefore, Expected utility and risk aversion 35/ 58

Logarithmic Utility: Constant Relative Risk Aversion Logarithmic utility is a limiting case of power utility. Since utility functions are unique up to a linear transformation, write the power utility function as 1 W 1 = W 1 Next take its limit as! 0. Do so by rewriting the numerator and applying L Hôpital s rule: W 1 e ln(w ) 1 ln(w )W lim = lim = lim!0!0!0 1 = ln(w ) (29) Thus, logarithmic utility is power utility with coe cient of relative risk aversion (1 ) = 1 since R(W ) = W 2 W 1 = 1 W and R r (W ) = 1. Expected utility and risk aversion 36/ 58

HARA: Power, Log, Quadratic Hyperbolic absolute-risk-aversion (HARA) utility generalizes all of the previous utility functions: U(W ) = 1 W 1 + (30) s:t: 6= 1, > 0, W 1 + > 0, and = 1 if = 1. Thus, R(W ) = W 1. 1 + Since R(W ) must be > 0, it implies > 0 when > 1. R r (W ) = W W 1. 1 + HARA utility nests constant absolute risk aversion ( = 1, = 1), constant relative risk aversion ( < 1, = 0), and quadratic ( = 2) utility functions. Expected utility and risk aversion 37/ 58

Another Look at the Risk Premium A premium to avoid risk is ne for insurance, but we may also be interested in a premium to bear risk. This alternative concept of a risk premium was used by Arrow (1971), identical to the earlier one by Pratt (1964). Suppose that a fair lottery e", has the following payo s and probabilities: e" = + with probability 1 2 with probability 1 2 (31) How much do we need to deviate from fairness to make a risk-averse individual indi erent to this lottery? Expected utility and risk aversion 38/ 58

Risk Premium v2 Let s de ne a risk premium,, in terms of probability of winning p: = Prob(win) Prob(lose) = p (1 p) = 2p 1 (32) Therefore, from (32) we have Prob(win) p = 1 2 (1 + ) Prob(lose) = 1 p = 1 2 (1 ) We want that equalizes the utilities of taking and not taking the lottery: U(W ) = 1 2 (1 + )U(W + ) + 1 (1 )U(W ) (33) 2 Expected utility and risk aversion 39/ 58

Risk Aversion (again) Let s again take a Taylor approximation of the right side, around = 0 U(W ) = 1 2 (1 + ) U(W ) + U 0 (W ) + 1 2 2 U 00 (W ) (34) + 1 2 (1 ) U(W ) U 0 (W ) + 1 2 2 U 00 (W ) = U(W ) + U 0 (W ) + 1 2 2 U 00 (W ) Rearranging (34) implies = 1 2 R(W ) (35) which, as before, is a function of the coe cient of absolute risk aversion. Expected utility and risk aversion 40/ 58

Risk Aversion (again) Note that the Arrow premium,, is in terms of a probability, while the Pratt measure,, is in units of a monetary payment. If we multiply by the monetary payment received,, then equation (35) becomes = 1 2 2 R(W ) (36) Since 2 is the variance of the random payo, e", equation (36) shows that the Pratt and Arrow risk premia are equivalent. Both were obtained as a linearization of the true function around e" = 0. Expected utility and risk aversion 41/ 58

A Simple Portfolio Choice Problem Let s consider the relation between risk aversion and an individual s portfolio choice in a single period context. Assume there is a riskless security that pays a rate of return equal to r f and just one risky security that pays a stochastic rate of return equal to er. Also, let W 0 be the individual s initial wealth, and let A be the dollar amount that the individual invests in the risky asset at the beginning of the period. Thus, W 0 A is the initial investment in the riskless security. Denote the individual s end-of-period wealth as ~W : ~W = (W 0 A)(1 + r f ) + A(1 + ~r) (37) = W 0 (1 + r f ) + A(~r r f ) Expected utility and risk aversion 42/ 58

Single Period Utility Maximization A vn-m expected utility maximizer chooses her portfolio by maximizing the expected utility of end-of-period wealth: max E[U( ~W )] = max E [U (W 0(1 + r f ) + A(~r r f ))] (38) A A Maximization satis es the rst-order condition wrt. A: h i E U 0 ~W (~r r f ) = 0 (39) Note that the second order condition h E U 00 ~W (~r r f ) 2i 0 (40) is satis ed because U 00 W ~ 0 from concavity. Expected utility and risk aversion 43/ 58

Obtaining A from FOC If E[~r r f ] = 0, i.e., E [~r] = r f, then we can show A=0 is the solution. When A=0, ~W = W 0 (1 + r f ) and, therefore, U 0 W ~ = U 0 (W 0 (1 + r f )) is nonstochastic. Hence, h i E U 0 W ~ (~r r f ) = U 0 (W 0 (1 + r f )) E[~r r f ] = 0. Next, suppose E[~r r f ] > 0. A h= 0 is not a solutioni because E U 0 W ~ (~r r f ) = U 0 (W 0 (1 + r f )) E[~r r f ] > 0 when A = 0. Thus, when E[~r] r f > 0, let s show that A > 0. Expected utility and risk aversion 44/ 58

Why must A > 0? Let r h denote a realization of ~r > r f, and let W h be the corresponding level of ~W Also, let r l denote a realization of ~r < r f, and let W l be the corresponding level of ~W. Then U 0 (W h )(r h r f ) > 0 and U 0 (W l )(r l r f ) < 0. For U 0 W ~ (~r r f ) to average to zero for all realizations of ~r, it must be that W h > W l so that U 0 W h < U 0 W l due to the concavity of the utility function. Why? Since E[~r] r f > 0, the average r h is farther above r f than the average r l is below r f. To preserve (39), the multipliers must satisfy U 0 W h < U 0 W l to compensate, which occurs when W h > W l and which requires that A > 0. Expected utility and risk aversion 45/ 58

How does A change wrt W 0? We ll use implicit di erentiation to obtain da(w 0) dw 0 : De ne f (A; W 0 ) E h i U fw and let v (W 0 ) = max f (A; W 0) be the maximized value of expected A utility when A, is optimally chosen. Also de ne A (W 0 ) as the value of A that maximizes f for a given value of the initial wealth parameter W 0. Now take the total derivative of v (W 0 ) with respect to W 0 by applying the chain rule: dv (W 0 ) dw 0 = @f (A;W 0) da(w 0 ) @A dw 0 + @f (A(W 0);W 0 ) @W 0. However, @f (A;W 0) @A maximum. = 0 since it is the rst-order condition for a Expected utility and risk aversion 46/ 58

How does A change wrt W 0 cont d The total derivative simpli es to dv (W 0) dw 0 = @f (A(W 0);W 0 ) @W 0 : Thus, the derivative of the maximized value of the objective function with respect to a parameter is just the partial derivative with respect to that parameter. Second, consider how the optimal value of the control variable, A (W 0 ), changes when the parameter W 0 changes. Derive this relationship by taking the total derivative of the F.O.C. (39), @f (A (W 0 ) ; W 0 ) =@A = 0, with respect to W 0 : @(@f (A(W 0 );W 0 )=@A) @W 0 = 0 = @2 f (A(W 0 );W 0 ) @A 2 da(w 0 ) dw 0 + @2 f (A(W 0 );W 0 ) @A@W 0 Expected utility and risk aversion 47/ 58

How does A change wrt W 0 cont d Rearranging the above gives us da (W 0 ) dw 0 = @2 f (A (W 0 ) ; W 0 ) @A@W 0 = @2 f (A (W 0 ) ; W 0 ) @A 2 (41) We can then evaluate it to obtain h i da (1 + r f )E U 00 ( ~W )(~r r f ) = i (42) dw 0 E hu 00 ( ~W )(~r r f ) 2 The denominator of (42) is positive because of concavity. da Therefore, the sign of dw 0 depends on the numerator. Expected utility and risk aversion 48/ 58

Implications for da dw 0 with DARA Consider an individual with absolute risk aversion that is decreasing in wealth. Assuming E [~r] > r f so that A > 0: R W h < R (W 0 (1 + r f )) (43) where, as before, R(W ) = U 00 (W )=U 0 (W ). Multiplying both terms of (43) by U 0 (W h )(r h r f ), which is a negative quantity, the inequality sign changes: U 00 (W h )(r h r f ) > U 0 (W h )(r h r f )R (W 0 (1 + r f )) (44) Then for A > 0, we have W l < W 0 (1 + r f ). If absolute risk aversion is decreasing in wealth, this implies R(W l ) > R (W 0 (1 + r f )) (45) Expected utility and risk aversion 49/ 58

Implications for da dw 0 with DARA Multiplying (45) by U 0 (W l )(r l r f ), which is positive, so that the sign of (45) remains the same, we obtain U 00 (W l )(r l r f ) > U 0 (W l )(r l r f )R (W 0 (1 + r f )) (46) Inequalities (44) and (46) are the same whether the realization is ~r = r h or ~r = r l. Therefore, if we take expectations over all realizations of ~r, we obtain h i h i E U 00 ( ~W )(~r r f ) > E U 0 ( ~W )(~r r f ) R (W 0 (1 + r f )) The rst term on the right-hand side is just the FOC. (47) Expected utility and risk aversion 50/ 58

Implications for risk-taking with ARA/RRA Inequality (47) reduces to h i E U 00 ( ~W )(~r r f ) > 0 (48) Thus, DARA ) da=dw 0 > 0: amount invested A increases in initial wealth. What about the proportion of initial wealth? To analyze this, de ne da dw 0 A W 0 = da W 0 dw 0 A which is the elasticity measuring the proportional increase in the risky asset for an increase in initial wealth. (49) Expected utility and risk aversion 51/ 58

Implications for risk-taking with RRA Adding 1 A A to the right-hand side of (49) gives = 1 + (da=dw 0)W 0 A A (50) Substituting da=dw 0 from equation (42), we have h i i W 0 (1 + r f )E U 00 ( ~W )(~r r f ) + AE hu 00 ( ~W )(~r r f ) 2 = 1+ i AE hu 00 ( ~W )(~r r f ) 2 Collecting terms in U 00 ( ~W )(~r r f ), this can be rewritten as (51) Expected utility and risk aversion 52/ 58

Implications for risk-taking with RRA h i E U 00 ( ~W )(~r r f )fw 0 (1 + r f ) + A(~r r f )g = 1 + i (52) AE hu 00 ( ~W )(~r r f ) 2 h i E U 00 ( ~W )(~r r f ) ~W = 1 + i (53) AE hu 00 ( ~W )(~r r f ) 2 The denominator in (53) is positive for A > 0 by concavity. Therefore, > 1, so that the individual invests proportionally more h in the risky asset i with an increase in wealth, if E U 00 ( ~W )(~r r f ) ~W > 0. Can we relate this to the individual s risk aversion? Expected utility and risk aversion 53/ 58

Implications for risk-taking with DRRA Consider an individual whose relative risk aversion is decreasing in wealth. Then for A > 0, we again have W h > W 0 (1 + r f ). When R r (W ) WR(W ) is decreasing in wealth, this implies W h R(W h ) < W 0 (1 + r f )R (W 0 (1 + r f )) (54) Multiplying both terms of (54) by U 0 (W h )(r h r f ), which is a negative quantity, the inequality sign changes: W h U 00 (W h )(r h r f ) > U 0 (W h )(r h r f )W 0 (1+r f )R (W 0 (1 + r f )) (55) Expected utility and risk aversion 54/ 58

Implications for risk-taking with DRRA For A > 0, we have W l < W 0 (1 + r f ). If relative risk aversion is decreasing in wealth, this implies W l R(W l ) > W 0 (1 + r f )R (W 0 (1 + r f )) (56) Multiplying (56) by U 0 (W l )(r l r f ), which is positive, so that the sign of (56) remains the same, we obtain W l U 00 (W l )(r l r f ) > U 0 (W l )(r l r f )W 0 (1+r f )R (W 0 (1 + r f )) (57) Inequalities (55) and (57) are the same whether the realization is ~r = r h or ~r = r l. Therefore, taking expectations over all realizations of ~r yields Expected utility and risk aversion 55/ 58

Implications for risk-taking with DRRA h E W ~ U 00 ( ~W )(~r r f )i > E h i U 0 ( ~W )(~r r f ) W 0 (1+r f )R(W 0 (1+r f )) (58) The rst term on the right-hand side is just the FOC, so inequality (58) reduces to h E W ~ U 00 ( ~W )(~r r f )i > 0 (59) Hence, decreasing relative risk aversion implies > 1 so an individual invests proportionally more in the risky asset as wealth increases. The opposite is true for increasing relative risk aversion: < 1 so that this individual invests proportionally less in the risky asset as wealth increases. Expected utility and risk aversion 56/ 58

Risk-taking with ARA/RRA The main results of this section can be summarized as: Risk Aversion Decreasing Absolute Constant Absolute Increasing Absolute Decreasing Relative Constant Relative Increasing Relative Investment Behavior @A @W 0 > 0 @A @W 0 = 0 @A @W 0 < 0 @A @W 0 > A W 0 @A @W 0 = A W 0 @A @W 0 < A W 0 Expected utility and risk aversion 57/ 58

Conclusions We have shown: Why expected utility, rather than expected value, is a better criterion for choosing and valuing assets. What conditions preferences can satisfy to be represented by an expected utility function. The relationship between a utility function, U(W ), and risk aversion. How ARA/RRA a ects the choice between risky and risk-free assets. Expected utility and risk aversion 58/ 58