Utility and the Skewness of Return in Gambling

Transcription

1 The Geneva Paers on Risk and Insurance Theory, 9: , 004 c 004 The Geneva Association Utility and the Skewness of Return in Gambling MICHAEL CAIN School of Business, University of Wales, Hen Goleg, College Road, Bangor, LL57 DG m.cain@bangor.ac.uk DAVID PEEL Management School, University of Lancaster, Lancaster, LA1 4YK d.eel@lancaster.ac.uk Received March 1, 003; Revised March 1, 003 Abstract This aer demonstrates that the intuitively aealing argument based on the ostulated trade-off between exected return, variance and skewness of return of a risk-averse gambler does not rovide an exlanation of observed betting behaviour. It is shown how the exected utility of a reresentative gambler faced with a single-rized outcome event can be exressed in terms of the mean and variance of return, the mean and skewness of return or, generally, of the mean and any other single moment of return; and the standard ractice of taking a Taylor series exansion/aroximation of the exected utility involving moments of return is usually incorrect. Previous analyses have suggested that a unter will accet a lower mean return for higher skewness and this work seems to have involved invalid exansions of the utility function. The ushot is that with certain utility functions which have been used in a number of studies, any analysis based on exansion and estimation of the derivatives of the utility function may be valid only for data based on odds-on favourites and not for longshots. Key words: mean-variance frontier, Kurtosis, favourite-longshot bias, Taylor series exansion JEL Classification No.: C44, D80, G10 1. Introduction The traditional rationale for gambling behaviour is that bettors are risk-loving, and this rovides an exlanation of the favourite-longshot bias observed in numerous emirical studies of racetrack betting where bets on longshots, low robability bets, have low mean returns relative to bets on favourites, high robability bets; see for examle Weitzman [1965], Dowie [1976], Ali [1977] and Quandt [1986] and, for comrehensive reviews of the salient literature, Sauer [1998], Thaler and Ziemba [1988] and Vaughan Williams [1999]. The assumtion of risk-loving behaviour imlies that otimal bet size would be unbounded and that unters only bet one horse in a race, but given that unters tyically bet small stakes and that some unters bet on more than one horse in a race, recently a number of authors have suggested that gambling can be consistent with risk-aversion. However, because bets tyically offer negative exected returns, agents who are globally risk-averse would not bet. One consistent exlanation of observed gambling behaviour is to assume that agents are everywhere risk-averse but obtain direct utility from gambling; this is the aroach set

2 146 CAIN AND PEEL out by Conlisk [1993]. Alternatively, an exlanation referred by the authors, it might be assumed that the reresentative agent s utility function exhibits regions of risk-loving as well as risk-averse behaviour as set out by Friedman and Savage [1948] and Markowitz [195]. Some authors have suggested that gambling can be consistent with risk-aversion and this exlanation incororates the third moment into the analysis, recognising a reference for skewness of risk-averse agents, documented by Scott and Horvath [1980]; see also Arditti [1967], Woodland and Woodland [1999], Garrett and Sobel [1999], Golec and Tamarkin [1998] and Walker and Young [001]. As recently stated by Golec and Tamarkin, horse bettors accet low-return, high-variance bets because they enjoy the high skewness offered by these bets. The urose of this aer is to demonstrate that the intuitively aealing argument based on the ostulated trade-off between exected return, variance and skewness of return of a risk-averse gambler does not rovide an exlanation of the observed betting behaviour that roduces the favourite-longshot bias. It is shown how the exected utility of a reresentative gambler faced with a single-rized outcome event can be exressed in terms of the mean and variance of return, or the mean and skewness of return or, generally, of the mean and any other single moment of return; and the standard ractice of taking a Taylor series exansion/aroximation of the exected utility involving moments of return, which rovides a basis for the three moment exlanation of gambling, is usually incorrect. 1. Utility model for betting To illustrate the argument, a standard aroach is emloyed. Following Ali [1977] and Golec and Tamarkin [1998], it is assumed that the reresentative bettor has utility function U( ) and bets total wealth, M, which is the unit of measurement of all ayouts and of the argument of the utility function. A winning one unit staked bet ays out X 1 + a, where a reresents the odds quoted against the articular horse (or team) winning, and a losing bet returns nothing. The mean return is X, where isthe win robability, and hence the winning ayout or return is X. The actual ayout will be 0 or X and the (reresentative) bettor s exected utility of ayout, E (1 )U(0) + U(X), can be exressed as a function of and, E 1 (,), as E E 1 (,) (1 )U(0) + U ( ). (1) A rational bettor who does not derive utility from the act of gambling er se, will make the bet if E 1 (,) U(1) and hence, from (1), it follows that : [ ( ) ] U U(0) [U(1) U(0)]. () For () to hold, the bettor cannot be globally risk-averse and U( ) must exhibit some riskloving characteristics as assumed by Friedman and Savage [1948] and Markowitz [195]; erhas with the bettor risk-loving over favourites and risk-averse over longshots so that

3 UTILITY AND THE SKEWNESS OF RETURN IN GAMBLING 147 U (0+) > 0 and U ( ) < 0. Equality in () establishes the border of the sub-region of the (,) lane corresonding to feasible rational betting. If all X, or equivalently (), are set at an equilibrium value across all cometitors, then de 0. It is thus ossible to differentiate (1) with resect to and equate to zero in d order to find the combinations of exected return,, and robability,, between which the bettor is indifferent. This roduces de d U(0) + U( ) + U ( )[ d ] 0 and hence [ ( d U(0) U )] + U ( ). (3) If U(0) 0, then (3) reduces to d [ 1 1 ], (4) e where e e(x) e( )isthe elasticity of U( )atx. Observe from (4) that the sloe of the equilibrium exected return-win robability frontier will be ositive (or negative) deending on whether the elasticity is greater than (or less than) one. Differentiating (3) with resect to yields d [ d 1 d ] U ( ) U ( ) U ( ) [ ( U ) ] U(0) U ( ) U ( ), which is ositive (negative) if U ( )isnegative (ositive) at X. The favourite-longshot bias is that > 0 and from (4) in the case U(0) 0, a necessary and sufficient condition is that the elasticity is greater than one. From (3) and () it d follows that dx d 1 [ d ] [U(0) U(X)] [U(0) U(1)] and, whilst dx will be naturally U (X) U (X) d negative so that d < and U(0) < U(1) < U(X), it does not follow that d > 0inall cases. However, if the bettor is risk-loving with U(0) 0, U (X) > 0, U (X) > 0, then XU (X) > U(X), e(x) > 1 and, from (4), > 0inthis articular case. These oints d are illustrated with a utility function which catures the form envisaged by Friedman and Savage [1948], who hyothesise that agents are initially (at low levels of wealth) risk-averse then (at higher levels of wealth) risk-loving and then again (at even higher levels of wealth) risk-averse. A function caturing these roerties is: U(x) In this case U (x) U (x) x α, x 0(0<α<1, β>0). 1 + e βx αxα 1 (1+e βx ) + βxα e βx > 0, (which is ositive for x > 0), and (1+e βx ) x α x (1 + e βx ) 3 [(α α) + e βx ( α + α + β x + αβx) + e βx (α α + αβx β x )].

4 148 CAIN AND PEEL Figure 1. A Friedman Savage utility function and the (, ) frontier. [U(x) xα 1+e βx where α 0.975, β 0.035; exected utility, EU constant U(1)]. It follows that U (0+), U ( ) 0, U (x) > 0 for x > 0, and U (0+), U ( ) 0. For large, and also for small, x the second derivative is negative so that the unter is risk-averse, but for a variety of values of β and α>0 the second derivative is ositive in the middle of its domain and the agent is risk-loving. This is illustrated with values of α 0.975,β 0.035; the elasticity is lotted for these values in figure 1(a). In Figure 1(b) (d) the exected return win robability frontier is lotted. Whilst a favourite-longshot bias is aarent, in that extreme longshots have lower rates of return than favourites, the interesting feature of the lots is that the frontier exhibits two turning oints. Consequently, it is demonstrated that with a secification of a utility function which admits both risk-aversion and risk-loving behaviour over its range, as in Friedman and Savage [1948] and Markowitz [195], there is a hitherto neglected imlication that the equilibrium mean return-win robability frontier may exhibit turning oints. Such a secification can rovide a consistent rationale for the anomalous reverse favourite-longshot bias found in the Hong Kong betting market by Busche and Hall [1988], and in US baseball betting by Woodland and Woodland [1994, 001] as well as the more universal findings. Essentially, it seems that different emirical studies have been exloring different segments of the equilibrium mean return-win robability frontier; see Cain and Peel [00]. The

5 UTILITY AND THE SKEWNESS OF RETURN IN GAMBLING 149 existence of a minimum also redicts that Friedman-Savage utility functions imly that extreme longshot bets, such as football ools or most lotteries, may exhibit higher exected rates of return than the most extreme longshots observed in horseracing. 3. Moments of return If the return or ayout, R, to a one unit staked bet is X 1 + a with robability and 0 with robability 1, the mean return is E(R) X and hence the winning ayout is X. The (higher) central moments of return are thus as follows: variance: σ V (R) E(R ) (1 ) skewness: 3 S(R) E(R ) 3 3 (1 )(1 ) kurtosis: 4 K (R) E(R ) 4 4 (1 )[(1 ) ] 3 fifth moment: 5 E(R ) 5 5 (1 )[(1 ) 4 4 ]. 4 In general, for n 1, odd moments: n+1 E(R ) n+1 n+1 (1 )[(1 ) n n ] n+1 even moments: n E(R ) n n (1 )[(1 ) n 1 + n 1 ]. n Note that for 0 < < 1, all even moments are ositive, and odd moments are ositive if 0 < < 1 but negative if 1 < < 1; and when 1 all odd moments are zero. Writing A n [(1 )n n ] n+1 and B [(1 )] n 1 σ n+1 n [(1 )n+1 + n+1 ] n+,itcan be shown that [(1 )] n σ n+ A n A n ()isdecreasing in over 0 < < 1, whilst B n B n ()isaconvex function of over 0 < < 1 with a minimal value of 1 when 1 ; and hence for n 1, n σ n. These results seem to have imlications for the estimation of the effect of moments on gambling behaviour. Higher central moments, both odd and even, are more imortant for small (longshots) in the sense that n decreases with (< 1 σ n ), although for > 1 the even moments again become otentially more influential comared with the variance, and the odd ones increasingly more negative. With 1, and erhas aroximately so near 1, the use of higher central moments in any regression analysis is equivalent to using owers of the standard deviation, σ. Since σ (1 ),itfollows that and hence all moments can be exressed +σ as functions of and σ.inarticular, 3 σ (σ )(σ + ), (5)

6 150 CAIN AND PEEL but more generally, for n 1, n+1 (σ 4n 4n )σ n 1 ( +σ ) and n+ (σ 4n+ + 4n+ )σ n ( +σ ). Likewise, the exected utility of return can be exressed as a function of and σ. From (1), E E (, σ ) σ (σ + ) U(0) + (σ + ) U ( σ + Similarly, writing s for the skewness of return, 3, E may be considered as a function of and s: E E 3 (, s) E (, σ (, s)), by noting from (5) that σ 4 σ s 0. Observe that s > 0if < 1 (i.e. <σ ) and s < 0if > 1 ( >σ ); and, furthermore, 1 [ + ( ) 4 + 4s] when < σ < 3 σ (, s) 1 [ ( 4 + 4s] when > σ > 3 ), ). and >0, s 3 4. In a similar manner, the exected utility may be considered imlicitly as a function of and any one other higher central moment or, in fact, as a function of any two central moments. However, in ractice it may be difficult to obtain an exlicit exression. For instance, E may be considered a function of and kurtosis, κ,as E E 4 (, κ) E (, σ (, κ)), noting that κ 0 and σ 8 + σ ( 6 κ ) κ 4 0; and a function of σ and skewness, s,as E E 5 (σ, s) E ((σ, s),σ ), noting that σ 4 s + σ or (σ, s) 1 σ { s + s + 4σ 6 } > 0. It follows from the above that remarks about a gambler s referred trade-offs between exected return and skewness, which imlicitly hold variance fixed, will generally be flawed. For instance, Golec and Tamarkin, on. 4, state We claim that bettors could be riskaverse and favor ositive skewness, and rimarily trade off negative exected return and variance for ositive skewness. Even the highly regarded Hirshleifer and Riley [199], on. 73, state that individuals tend to refer ositive skewness and suggest that this leads to ortfolios that are not so well-diversified. Mean-variance frontier The above observations rovide a framework for the exloration of (moment) frontiers involving trade-offs between airs of moments of return. For instance, the mean-variance, (, σ ), frontier is defined by E (, σ σ ) (σ + ) U(0) + ( σ (σ + ) U + ) constant U(1), (6)

7 UTILITY AND THE SKEWNESS OF RETURN IN GAMBLING 151 for which E σ { ( σ + ) ( σ + ) U(0) U 1 X {U(0) U(X) + XU (X)} + (σ + ( ) σ U + )} is > 0or< 0 according as the bettor is aarently risk-loving or risk-averse at X; E σ { ( σ + U(0) U (σ + ) ( σ + U + ) (1 ) {U(0) U(X) + XU (X)}+U (X) X ) + (σ + ( ) σ U + )} and hence, given σ, the bettor will refer larger if aarently risk-averse but smaller if sufficiently strongly risk-loving, with < 1. The sloe of the (, σ ) frontier,, can be obtained by differentiating Eq. (6) with dσ resect to σ. This yields { σ ( σ σ U(0) + (σ + ) (σ + ) U + ) + ( σ ( ) σ σ + U + )} dσ { ( σ + ) U(0) + U (σ + ( ) σ U + )} (σ + ) and hence E dσ σ E { U(0) + U(X) XU (X)} X[(1 ){ U(0) + U(X) XU (X)}+XU (X)]. Observe that dσ < 0 if (1 )XU (X) (1 ) < U(X) U(0) < XU (X) (risk-loving) and dσ > 0 U(X) U(0) if > 1 or XU (X) (risk-averse) U(X) U(0) (1 ) < XU (X) (1 ), (strongly risk-loving) and this determines the direction of trading between and σ.

8 15 CAIN AND PEEL Mean-skewness frontier The (, s) frontier is defined by E 3 (, s) E (, σ (, s)) constant U(1), where σ (, s)isasolution σ σ (, s)ofσ 4 σ s 0. Now, E 3 E σ E 1 s σ s (σ ) σ ( 3)X E and, given, larger skewness is σ referred if and only if variance is; unless > 3. Note that s s(,) 3 (1 )(1 ) and s (3 )3 3 { <0 >0if>/3 if</3, so that s (<0) is a minimum when /3. The sloe,, of the mean-skewness frontier is ds ds E σ (,s) σ s E + E σ σ (,s) σ dσ s 1 dσ σ {U(0) U(X) + XU (X)} X [3(1 )(1 ) {U(0) U(X)} + ( )XU (X)] which can be > 0, 0or< 0 deending on the values of 3(1 )(1 ), ( ) and XU (X)/[U(X) U(0)]. Note that 3(1 )(1 ) > ( )ifand only if s < 0. Other moment frontiers are exlored in the Aendix. It does not aear that any of the relationshis is monotonic and hence there are no one-sided trade-offs throughout the whole range of bets. To illustrate the ossibilities, in figure (a) to (d) some moment frontiers are lotted for the ower function U(x) x α, α Note that for this globally riskaverse utility function, the exected return-skewness trade-off is ositive for longshots. Only for extreme favourites, in figure (d), is the trade-off negative, as conjectured by numerous authors. In figure 3(a) to (d), moment frontiers are lotted for the Friedman- Savage utility function of Section. Plots of the exected return-variance frontier are given in figure 3(a) and (b), and of the exected return-skewness frontier in figure 3(c) and (d). Observe that each of these frontiers exhibits a minimum and that for extreme longshots, on the risk-averse segment of the utility function, the trade-off is ositive between exected return and skewness; and not negative as imlied by the intuitive argument mentioned above concerning risk aversion and skewness. In figure 4(a) (c) relationshis between exected utility and skewness or variance are lotted; with exected return fixed at Observe that exected utility exhibits a maximum and ultimately reduces as skewness increases; and, given exected return, there is an otimal level of skewness and likewise a corresonding otimal level of variance. 4. Exansion and truncation of the utility function From Section 3 it follows that, for given, the variance and all higher central moments of return will be unbounded as 0(extreme longshots). An immediate imlication of this is that the common ractice of emloying a truncated Taylor series exansion to aroximate the exected utility will in general be invalid for small. However, a salient

9 UTILITY AND THE SKEWNESS OF RETURN IN GAMBLING 153 Figure. Some moment trade-offs for the ower utility function. [U(x) x α where α 0.95; exected utility, EU constant > U(1) 1]. feature of the gambling literature is the analysis of behaviour when is small, for instance betting on lotteries (see, for examle, Walker and Young [001]), and hence such studies involving exansions are very questionable. This oint is an alication of the analysis of Loistl [1976] to that of gambling but the inaroriateness of exanding utility functions willy-nilly must be called into question here as it should be (a fortiori) more generally in Finance, where utility is still even taken to be a function of moments of return; see, for examle, Hwang and Satchell [1999]. Previous analyses have suggested that a unter will accet a lower mean return for higher skewness and this work seems to have involved invalid exansions of the utility function. Before demonstrating that such exansions may be invalid, recall from the moment frontiers derived in Section 3 that: (i) given σ, the bettor will refer larger unless sufficiently strongly risk-loving and the contingency necessarily odds against, (ii) given, skewness is referred if and only if σ is, excet for strong favourites with fair odds shorter than -1 on, and (iii) given σ, skewness is referred if and only if is not. It thus follows that the revious suggestions that unters will simly trade mean return for higher skewness of return is incorrect in general.

10 154 CAIN AND PEEL Figure 3. Some moment trade-offs for the Friedman-Savage utility function. [U(x) 0.975,β 0.035; EU constant U(1)]. x α 1+e βx where α A utility function U(x) can be exanded around the oint x 0 by a Taylor series involving the derivatives of U( ) atx 0, U i (x 0 ) [ di U(x) ] dx i xx0, if lim i U i+1 (x 0 )i!(x x 0 ) (i+1)!u i (x 0 < 1; and ) hence only for x such that x x 0 < lim i (i+1)u i (x 0 ) U i+1 (x 0. If lim ) i (i+1)u i (x 0 ) U i+1 (x 0, ) then for all x > 0, U(x) U i (x 0 )(x x 0 ) i i0, and with x i! 0 and any random variable X with mean and finite moments of all orders, E [U(X)] U i ()E(X ) i i0, if finite. i! However, this result does not necessarily hold if lim i (i+1)u i () <. U i+1 () Consider the following examles: 1. Constant risk aversion U(x) 1 e αx, 0 x <, α > 0, r(x) U (x) α>0, and U (x) lim (i + 1)U i (x 0 ) U i+1 (x 0 ) lim (i + 1) i α. i

11 UTILITY AND THE SKEWNESS OF RETURN IN GAMBLING 155 Figure 4. Exected utility and its trade-offs with skewness and variance for the Friedman-Savage utility function. [U(x) xα 1+e βx, α 0.975, β 0.035; mean return 0.90 fixed; exected utility U(1) lotted 0]. Thus, U(x) can be exanded around x 0 for all x and x 0 in (0, ). In addition, U i (x 0 ) ( x 0) i [1 e αx 0 ( 1)(αx 0 ) i e αx 0 ] + i! i! i0 i1 [1 e αx 0 ] e αx 0 [e αx 0 1] 0 U(0), and hence the exansion holds at x 0.. Power function U(x) x α, 0 x <, α>0. { >0 for α>1 U (x) αx α 1 > 0, U (x) α(α 1)x α <0 for 0 <α<1 and lim (i + 1)U i (x 0 ) i U i+1 (x 0 ) lim (i + 1)x 0 i (α i) x 0 > 0. U(x) can be exanded around x 0 only for x x 0 < x 0 i.e. for 0 < x < x 0.Inaddition, at the end oints, i0 U i (x 0 ) ( x 0) i 0 U(0) and i! i0 U i (x 0 ) xi 0 U(x i! 0 ). α U(x 0 ); and hence the exansion is valid for 0 x x Markowitz U(x) 1 e αx αxe αx, x 0, α>0. U (x) α xe αx > 0 (for x > 0),

12 156 CAIN AND PEEL >0 if x < 1 U (x) α (1 αx)e αx α <0 if x > 1 α lim (i + 1)U i (x 0 ) U i+1 (x 0 ) lim (i + 1)(i 1 αx 0 ) i α(αx 0 i). i and In this case, U(x) can be exanded around x 0 for all x and x 0 in (0, ). Also, when x 0, U i (x 0 ) ( x 0) i i! i0 [1 e αx 0 αx 0 e αx 0 ] α x0 e αx 0 [ ] + (i 1 αx 0 ) (αx 0) i e αx 0 i! i [ 1 e αx 0 αx 0 e αx 0 α x0 ] e αx 0 + e αx [ αx 0 + α x0 ] eαx 0 0 U(0), and hence the exansion is valid for any x 0. Discussion With the utility model of Section, the mean return is but the actual return, R, iseither 0 (with robability 1 )orx (with robability ); and the exected utility of return is given by (1). The question is, can the utility function be exanded around at the oint x 0 and also at the oint x X? There are no roblems with utility functions 1 and 3, and there is no roblem at x 0 with utility function, but the latter ower function can be exanded at x X only if X i.e. if 1 (an odds-on favourite). It thus follows that exansion of the ower utility function is not valid if < 1 and the ushot is that with this utility function (which has been used in a number of studies), any analysis based on exansion and estimation of the derivatives of U(x)atx will be valid only for data based on odds-on favourites and not for longshots. If the data also includes longshots then the way to roceed is to estimate directly the arameters of the chosen utility model. It can then be inferred whether or not the (reresentative) bettor is risk-averse or risk-loving at a articular oint. From the estimated arameters for the chosen model the signs of U (), U (), U v () etc. could then be deduced, but since the exansion is not valid it is not ossible to interret these as global unqualified references for certain moments of return (variance, skewness, kurtosis etc.) other than with reference to distributions having small ranges around the mean. Plotted in figure 5(a) to (f) are the exected return win robability frontiers imlied by second and third order Taylor exansions of the ower utility function as well as the exact relationshi given by the utility function itself. Observe that the figures are consistent with the theoretical analysis. In articular note that the third order exansion in figure 5(e)

13 UTILITY AND THE SKEWNESS OF RETURN IN GAMBLING 157 Figure 5. Exact (, ) frontiers for the ower utility function, U(x) x 0.95, and frontiers imlied by second and third order truncated Taylor series exansions (with EU 1).

14 158 CAIN AND PEEL imlies a ositive relationshi for small values of, even though the exact relationshi is everywhere negative. The aroximation for 1 is quite good as might be suggested by the theoretical analysis. Now, if exansion is valid, and so (X U(X) U() + (X )U ) (X () + U )3 () + U () +! 3! (0 U(0) U() + (0 )U ) (0 () + U )3 () + U () +! 3! E[U(R)] U(X) + (1 )U(0) U() + V (R) U ()! + S(R) 3! U () + K (R) U () + 4! With the ower function, U() α,dividing throughout by α we may regress E[U(R)]/U()onthe standardised moments V (R)/, S(R)/ 3, K (R)/ 4,...;for which the intercet should be 1. From the earlier comments, there ought to be stability for > 0.5 but not for the unrestricted case or for < 0.5. In Table 1, this regression is estimated for various ranges of and for rogressive truncations of the exected utility; and the stability issues are clearly demonstrated. Data is obtained for values of from 0 to 1 in stes of Table 1. Regression of exected utility truncated to, 3 and 4 moment terms (ower utility, U(x) x 0.95 ). Intercet Variance/ Skewness/ 3 Kurtosis/ 4 R (a) (n 9999) 0 < < < < < < (b) (n 4999) 0 < < < < < < (c) (n 4999) 0.5 < < < < < < Note: Ineach case the exected utility, EU, isfixedat1and the deendent variable is EU/U().

15 UTILITY AND THE SKEWNESS OF RETURN IN GAMBLING 159 and the equilibrium mean return,, simulated using the ower function utility : U(x) x 0.95 and for an exected utility value of 1.0. Table 1(a) deicts results for all n 9999 observations, 1(b) for the 4999 observations with < 0.5 and 1(c) for the 4999 observations with > 0.5. Observe the similarity between cases 1(a) and 1(b) but the quite different results of case 1(c). In articular, note the very large values of R in case 1(c) comared with the very much smaller ones in the other two cases, and the roortionately very much smaller changes in the estimated regression coefficients in case 1(c), comared with the other two cases, as rogressively more moment terms are included in the regression. Note also that, when exansion is valid, the corresonding theoretical combinatorial coefficients, with exonent α 0.95, are: 1.0, ( α ) , ( α 3 ) , ( α 4 ) , etc; and these are well-estimated in case 1(c) but not in either case 1(a) or 1(b). The results thus seem to be consistent with the theoretical analysis concerning stability and truncation; that unqualified exansion without reference to the form of the underlying utility function is very questionable. 5. Conclusion It has become increasingly common for authors to suggest that gambling could coexist with risk-aversion. This exlanation often incororates the fact that the third term of a Taylor series exansion, around the mean return, of an everywhere risk-averse utility function is ositive; and the agent is then alleged to exhibit a reference for ositive skewness. Such considerations have led numerous authors to mistakenly suggest that gamblers accet low-return, high-variance bets because they enjoy the high skewness offered by these bets. This aer demonstrates that these intuitively aealing arguments based on the ostulated trade-off between exected return, variance of return and skewness of return, of a riskaverse gambler, are incorrect at least in the case of a reresentative gambler faced with a single-rized outcome event. In this latter case, exected utility can be described in terms of exected return and any other single moment of return. It is shown, and demonstrated with examles, that for an agent who is everywhere risk-averse, the equilibrium relationshi between exected return and skewness of return can be ositive (for longshots) not negative as often conjectured. It is also shown that the widesread, almost standard, ractice of taking an unqualified Taylor series exansion/aroximation of the exected utility, involving various moments of return, can often be in error. Aendix The sloe of the variance-skewness or mean-kurtosis frontier can be derived in a similar manner. Variance-skewness frontier A(σ, s) frontier may be defined by: E 5 (σ, s) E ((σ, s),σ ) constant U(1), where (σ, s)isasolution (σ, s) 1 σ { s + s + 4σ 6 } > 0ofσ 4 s + σ.

16 160 CAIN AND PEEL Now, E 5 E s s (s+σ ) E σ (σ + ) E and, given σ, larger s is referred ds if and only if larger is not. The sloe,, of the variance-skewness frontier is: dσ Note that ds dσ E5 σ E 5 s (σ ) σ (σ + ) E σ E σ (σ + ) dσ + (σ ) X[3(1 )(1 ){U(X) U(0)} ( )XU (X)]. [(1 ){U(X) U(0)} (1 )XU (X)] ds / dσ dσ ds Mean-Kurtosis frontier, as exected. The (, κ) frontier is defined by E 4 (, κ) E (, σ (, κ)) constant U(1), where σ (, κ)isasolution σ σ (, κ)ofσ 8 + σ ( 6 κ ) κ 4 0. In this case, differentiating with resect to κ, (4σ κ ) σ κ (σ + ), E 4 κ E σ σ κ (σ + ) (4σ κ ) E σ (σ + ) (3σ 8 + 4σ ) E σ and, given, larger κ is referred if and only if larger σ is referred. The sloe,, of the mean-kurtosis frontier is dκ Noting that E4 dκ κ E 4 E σ (,κ) σ κ E + E σ σ (,κ) σ (,κ) dσ κ 1 dσ σ (,κ). σ (, κ) κ (σ + ) [3σ 8 + 4σ ] 1 ( )X > 0, and σ (, κ) σ [σ 8 + σ 6 σ 6 8 ] (1 )(1 )X, [3σ 8 + 4σ ] ( ) dκ [ U(0) + U(X) XU (X)] X 3 [(1 )( ){ U(0) + U(X) XU (X)}+( )XU (X)] and dσ dκ σ (,κ) dκ + σ (,κ) gives the sloe of the variance-kurtosis frontier. κ

17 UTILITY AND THE SKEWNESS OF RETURN IN GAMBLING 161 In general, the sloe of the (, n+1 ) frontier, for n 1, is where and n+1 σ (, n+1 ) dσ n+1 1 σ (, n+1 ) dσ σ (, n+1 ) n 1 ( + σ ) n+1 [(n + 1)σ 4n 4n n+1 n 1 ] σ (, n+1 ) n [4n n+1 σ + n+1 {(n + 1) + (n 1)σ }] ; [(n + 1)σ 4n 4n n+1 n 1 ] and the sloe of the (σ, n+1 ) frontier is dσ σ (, n+1 ) + σ (, n+1 ) n+1 n+1 n+1 dσ [ σ (, n+1 )] n+1 1 σ (, n+1 ) dσ n+1 Similarly, the sloe of the (, n+ ) frontier, for n 1, is where and n+ σ (, n+ ) dσ n+ 1 σ (, n+ ) dσ σ (, n+ ) n ( + σ ) n+ [(n + )σ 4n+ + 4n+ n+ n ] σ (, n+ ) [ n+{(n + ) n+1 + n n 1 σ } (4n + ) 4n+1 σ ] ; [(n + )σ 4n+ + 4n+ n+ n ] and the sloe of the (σ, n+ ) frontier is obtainable as dσ σ (, n+ ) + σ (, n+ ) n+ n+ n+ [ σ (, n+ )] n+ 1 σ (, n+ ) dσ

18 16 CAIN AND PEEL Notes 1. Presented here is a articular alication of a more general result that has been recognised in the literature, but which has been reeatedly ignored. See, for examle, Brockett and Garven [1998] and Rothschild and Stiglitz [1970], who show that moment reference does not match u with a sequence of utility derivatives. In the gambling literature there seems to be widesread ignorance or disregard for such correct analysis. Our demonstration and discussion that such disregard is, in general, in error is the first, as far as we are aware, in the gambling context.. If the Utility function has the form U(x) 1 e αx αxe αx, x 0(α>0), so that the agent is risk-loving over favourites and risk-averse over longshots, as suggested by Markowitz [195], the exected return win robability frontier will only exhibit a minimum. To exhibit a maximum value it is necessary that the curve has a risk-averse segment followed by a risk-loving one. References ALI, M.M. [1977]: Probability and Utility Estimates for Racetrack Bettors, Journal of Political Economy, 85, ARDITTI, F.D. [1967]: Risk and the Required Return on Equity, Journal of Finance,, BROCKETT, P.L. and GARVEN, J.R. [1998]: A Re-Examination of the Relationshi Between Preferences and Moment Orderings by Rational Risk Averse Investors, Geneva Paers on Risk and Insurance Theory, 3, BUSCHE, K. and HALL, C.D. [1988]: An Excetion to the Risk Preference Anomaly, Journal of Business, 61, CAIN, M. and PEEL, D.A. [00]: The Utility of Gambling and the Favourite-Longshot Bias, Euroean Journal of Finance (forthcoming). CONLISK, J. [1993]: The Utility of Gambling, Journal of Risk and Uncertainty,6, DOWIE, D. [1976]: On the Efficiency and Equity of Betting Markets, Economica, 43, FRIEDMAN, M. and SAVAGE, L.J. [1948]: The Utility Analysis of Choices Involving Risk, Journal of Political Economy, LV1, GARRETT, T.A. and SOBEL, R.S. [1999]: Gambler s Favour Skewness, Not Risk. Further Evidence for United States Loterry Games, Economic Letters, 63, GOLEC, J. and TAMARKIN, M. [1998]: Bettors Love Skewness, Not Risk, at the Horse Track, Journal of Political Economy, 106, HIRSHLEIFER, J. and RILEY, J.G. [199]: The Analytics of Uncertainty and Information. Cambridge Surveys of Economic Literature, Cambridge University Press. HWANG, S. and SATCHELL, S.E. [1999]: Modelling Emerging Market Risk Premia Using Higher Moments, International Journal of Finance and Economics, 4(4), LOISTL, O. [1976]: The Erroneous Aroximation of Exected Utility by Means of a Taylor s Series Exansion: Analytic and Comutational Results, American Economic Review, 66, MARKOWITZ, H. [195]: The Utility of Wealth, Journal of Political Economy, 56, QUANDT, R.E. [1986]: Betting and Equilibrium, Quarterly Journal of Economics, 101, ROTHSCHILD, M. and STIGLITZ, J.E. [1970]: Increasing Risk: I. A Definition, Journal of Economic Theory,, SAUER, R.D. [1998]: The Economics of Wagering Markets, Journal of Economic Literature, XXXV1, SCOTT, R.C. and HORVATH, P.A. [1980]: On The Direction of Preference for Moments of Higher Order Than The Variance, The Journal of Finance, XXXV(4), THALER, R.H. and ZIEMBA, W.T. [1988]: Anomalies: Parimutuel Betting Markets: Racetracks and Lotteries, Journal of Economic Persectives,, VAUGHAN WILLIAMS, L. [1999]: Information Efficiency in Betting Markets: A Survey, Bulletin of Economic Research, 5, 1 30.

19 UTILITY AND THE SKEWNESS OF RETURN IN GAMBLING 163 WALKER, I and YOUNG, J. [001]: An Economist s Guide to Lottery Design, The Economic Journal, 111, F700 F7. WEITZMAN, M. [1965]: Utility Analysis and Grous Behaviour: An Emirical Study, Journal of Political Economy, 73, WOODLAND, B.M. and WOODLAND, L.M. [1994]: Market Efficiency and the Favorite-Longshot Bias; the Baseball Market, Journal of Finance, 49, WOODLAND, B.M. and WOODLAND, L.M. [1999]: Exected Utility, Skewness, and the Baseball Betting Market, Alied Economics, 31, WOODLAND, B.M. and WOODLAND, L.M. [001]: Market Efficiency and Profitable Wagering in the National Hockey League: Can Bettors Score on Longshots? Southern Economic Journal, 67,