William Neilson Texas AMUniversity. Abstract

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1 Caibration resuts for rank dependent expected utiity Wiiam Neison Texas AMUniversity Abstract If its utiity function is everywhere increasing and concave, rank dependent expected utiity shares a troubing property with expected utiity aversion to the same moderate stakes risk at every weath eve impies an extreme aversion to arge stakes risks. In fact, the probem may be even worse for rank dependent expected utiity, since the moderate stakes risk need not be actuariay fair. I am gratefu to Rajiv Sarin for hepfu advice, and to the Private Enterprise Research Center and the Program in the Economics of Pubic Poicy for financia support. Citation: Neison, Wiiam, (001) "Caibration resuts for rank dependent expected utiity." Economics Buetin, Vo. 4, No. 10 pp. 1 5 Submitted: September 7, 001. Accepted: September 1, 001. URL: 01D8000A.pdf

2 1. Introduction In addition to the ong ist of experimenta studies questioning its vaidity, there are now theoretica reasons why the expected utiity mode might be inappropriate. A property of expected utiity for sma gambes has ong been known for any gambe ~x and for su cienty smat>0, an expected utiity maximizer with a smooth utiity function prefers the gambe t~x to its expected vaue if ~x has positive expected vaue. So, expected utiity maximizers are risk neutra for otteries with su cienty sma payo s. Rabin (000) discusses the other end of the spectrum. He shows that for any increasing, concave utiity function, an expected utiity maximizer who is averse to the same moderate risk at every weath eve must necessariy be extremey averse to arge risks. In perhaps his most striking exampe, somebody who turns down a 50:50 ose $100/gain $15 gambe at every weath eve must turn down any gambe with a 50% chance of osing $600, no matter how arge the gain is. Putting his resut another way, an expected utiity maximizer who is averse to any singe actuariay favorabe gambe at every weath eve must be extremey (even ridicuousy) risk averse over arge gambes. This argument provides a severe criticism of standard expected utiity modes. The purpose of this note is to expore whether Rabin s resut hods when expected utiity is weakened. In particuar, I examine the resut in the context of rank-dependent expected utiity (Quiggin, 198, 1993). There are severa good reasons for doing so. First, among nonexpected utiity modes, rank-dependent expected utiity has received the most recent attention and has been used in the most appications. Second, rank-dependent expected utiity exhibits rst-order risk aversion, i.e. the risk premium for the noise variabet~" is proportiona to t at t = 0, whereas expected utiity exhibits second-order risk aversion, where the risk premium is proportiona tot att=0(see Sega and Spivak, 1990). First-order risk aversionhas beenused to expaina number of empirica anomaies, such as the purchase of fu insurance with a oading factor (Sega and Spivak, 1990), the equity-premium puzze (Epstein and Zin, 1990), and tax compiance (Bernasconi, 1998). Perhaps second-order risk aversion is the driving force behind Rabin s resut. The resuts of the rank-dependent expected utiity anaysis are fairy simpe Rabin s negative resut sti hods. Furthermore, experimenta evidence andtheoretica workonthe shape ofthe probabiityweighting function suggest that the rank-dependent resut is even worse than the expected utiity one. With rank-dependent expected utiity and the most common current weighting functions, aversion to a singe, appropriatey chosen, actuariay unfavorabe gambe at every weath eve can ead to extreme eves of risk aversion over arge gambes. The resuts have two impications for Rabin s criticism of expected utiity. First, it is unreated to the properties of approximate risk neutraity and second-order risk aversion, since for rank-dependent expected utiity approximate risk neutraity fais but Rabin s criticism is maintained. Second, his criticism extends to an important generaization of expected utiity theory. Section brie y reviews Rabin s resut for expected utiity preferences. Section3 expores the same resut for rank-dependent expected utiity preferences.. Expected Utiity An expected utiity maximizer has a preference functionv(f) over probabiity distributions taking the formv(f) = R U(w+x)dF(x), whereu is the utiity function de ned over 1

3 weath eves. Rabin s caibration resut uses the foowing construction: 8 >< m(k) = >: n 1 (1 g) n g kp i ( g ) 1 1 ³ 1 g 1 1 ³ 1 g if kp kp ³ g ³ g i>0 i 0 (1) Rabin s Expected Utiity Caibration Theorem. Let U be twice di erentiabe with U 0 (w)>0,u 00 (w)<0 for aw. If there existsg>>0 such that, for aw, an expected utiity maximizer with utiity functionu prefers not to take a 50:50 chance of osing and winningg, then she aso prefers not to take a 50:50 chance of osingk and winningmg for any positive integerk and anym<m(k) as de ned in expression (1). Rabin s resut takes the foowing form: If an individua turns down 50:50 ose/gaing bets at every weath eve, then the same individua aso turns down a 50:50 osek/gainmg bet wherekis a positive integer andmis bounded bym(k) de ned above. Some vaues of,g, andk yied an in nite vaue form(k), so that the individua wi turn down any bet invoving a 50% chance of osingk, independent of the amount of the potentia gain. Furthermore, for anyg>>0, there exists an integerk 1 < 0 such that fork K 1, m(k) = 1: This means that if the individua turns down any 50:50 ose/gaing gambe, there is a arge enough integerk 1 such that she turns down any gambe with a 50% chance of osingk 1 or more. The vauek 1 is the smaest integerkfor which à 1 1! kx µ g i 0, g which depends ony on the ratiog=. Tabe 1 gives the vaues ofk 1 for di erent vaues of g=. As shown by the tabe,k 1 can take on some sma vaues, impying that the individua is extremey risk averse. So, for exampe, if an individua is averse to a 50:50 chance of osing $100 or gaining $105 at every weath eve, then she wi not take any gambe with a 50% chance of osing $900, no matter how arge the gain. Even more strikingy, being averse to a 50:50 chance of osing $ or gaining $1:5 at every weath eve means that she wi not takeany gambes with a 50% chance of osing. This is a serious criticism of the expected utiity hypothesis.. Tabe 1. g= K 1 g= K

4 3. Rank-dependent expected utiity Rank-dependent expected utiity preferences di er from expected utiity preferences in that the probabiity distribution is transformed before expected utiity is cacuated. The function ¼ is de ned to be a probabiity transformation function if it is stricty increasing with ¼(0) = 0 and¼(1) = 1. The preference function takes the formv(f) = R U(w)d¼(F(w)) whereu is a utiity function, as before, and¼ is a probabiity transformation function. Risk attitudes are governed by bothu and¼ if both are concave the individua is risk averse. When there are ony two outcomes, as there are in Rabin s setting, the rank-dependent expected utiity preference function takes a very simpe form. Lettingp be the probabiity of the oss, the preference function takes the form¼(p )U(w ) + [1 ¼(p )]U(w +g). So, the preference function di ers from the expected utiity preference function in that the expectation is taken using¼(p ) as the probabiity of the oss instead ofp. Proposition 1. LetUbe twice di erentiabe withu 0 (w)>0,u 00 (w)<0 for aw. Let ¼ be a probabiity transformation function, and et ¹p =¼ ³ 1 1. If there existsg>>0 such that, for aw, a rank-dependent expected utiity maximizer with utiity functionu and transformation function ¼ prefers not to take a ¹p : (1 ¹p) chance of osing and winning g, then she aso prefers not to take a ¹p : (1 ¹p) chance of osingk and winningmg for any positive integerk and anym<m(k) as de ned in expression (1). Proof. If the individua prefers not to take the gambe, it must be the case that ¼(¹p)U(w )+[1 ¼(¹p)]U(w +g)<u(w): Since¼(¹p) = 1, this impies that 1 U(w )+ 1 U(w+g)<U(w). By Rabin s EU caibration theorem, 1U(w k)+ 1 U(w +mg)<u(w) for any positive integerk and anym<m(k) given by (1). Thus, ¼(¹p)U(w k)+[1 ¼(¹p)]U(w +mg)<u(w) for any positive integerk and anym<m(k). The ony di erence between Rabin s expected utiity caibration theorem and its rankdependent counterpart is the probabiity of the oss. The conditions on the utiity function are identica. So, given an increasing concave utiity function U and a probabiity transformation function¼, if g,, and k satisfy the conditions of Rabin s EU theorem, they aso satisfy the conditions of Proposition 1, but using ¹p as the probabiity of the oss instead of 1. This is unsurprising because ¹p is simpy the soution to¼(p) = 1, so that the transformed probabiity distribution coincides with the 50:50 one used in Rabin s EU caibration theorem. Consequenty, the probem posed for expected utiity by Rabin s anaysis aso pertains to rank-dependent expected utiity. Aversion to an appropriate moderate-stakes risk at every weath eve impies extreme aversion to risks with arge stakes. Even though the expected utiity caibration resut and its rank-dependent counterpart are very simiar, their interpretations hod signi cant di erences. Most experimenta studies of the probabiity transformation function¼, incuding Tversky and Kahneman (199), Camerer and Ho (1994), Wu and Gonzaes (1996), Fox, Rogers, and Tversky (1996), and Gonzaes andwu (1999), aong withpreec s (1998) axiomatic characterizationof the transformation function, nd that¼ 1 ( 1)> 1. For Proposition 1, this means that the ¹p : (1 ¹p) chance 3

5 of osing or winningg may be actuariay unfavorabe. So, for exampe, if¼ 1 ( 1 ) = 0:6, if the individua disikes 60:40 ose $100/win $110 gambes at every weath eve, by Proposition 1 and Tabe 1, that individua wi not take any gambe with a 60% chance of osing $500. Disiking the 60:40 ose $100/win $110 gambe does not require risk aversion, but instead requires a su cienty sma amount of risk preference. Disiking any gambe with a 60% chance of osing $500, though, requires an extreme amount of risk aversion. Consequenty, the negative caibration resut for rank-dependent expected utiity using the now-standard speci cations of the probabiity transformation function is even more striking than its expected utiity counterpart. References Bernasconi, M. (1998) Tax Evasion and Orders of Risk Aversion Journa of Pubic Economics 67, Camerer, C.F., and T.H. Ho (1994) Vioations of the Betweenness Axiom and Noninearity in Probabiity Journa of Risk and Uncertainty 8, Epstein, L.G., and S.E. Zin (1990) First-Order Risk Aversion and the Equity Premium Puzze Journa of Monetary Economics 6, Fox, C.R., B.A. Rogers, and A. Tversky (1996) Options Traders Exhibit Subadditive Decision Weights Journa of Risk and Uncertainty 13, Gonzaez, R., and G. Wu (1999) On the Shape of the Probabiity Weighting Function Cognitive Psychoogy Preec, D. (1998) The Probabiity Weighting Function Econometrica 66, Quiggin, J. (198) A Theory of Anticipated Utiity Journa of Economic Behavior and Organization 3, Quiggin, J. (1993) Generaized Expected Utiity Theory: The Rank-Dependent Mode, Kuwer Academic Pubishers: Dordrecht. Rabin, M. (000) Risk Aversion and Expected-Utiity Theory: Econometrica 68, A Caibration Theorem Sega, U., and A. Spivak (199) First Order versus Second Order Risk Aversion Journa of Economic Theory 51, Tversky, A., and D. Kahneman (199) Advances in Prospect Theory: Cumuative Representation of Uncertainty Journa of Risk and Uncertainty 5, Wu, G., and R. Gonzaez (1996) Curvature of the Probabiity Weighting Function Management Science 4,