utility and the skewness of return in gambling

The Geneva Papers on Risk and Insurance Theory, 29: 145–163, 2004c© 2004 The Geneva Association

Utility and the Skewness of Return in Gambling

MICHAEL CAIN [email protected] of Business, University of Wales, Hen Goleg, College Road, Bangor, LL57 2DG

DAVID PEEL [email protected] School, University of Lancaster, Lancaster, LA1 4YK

Received March 12, 2003; Revised March 12, 2003

Abstract

This paper demonstrates that the intuitively appealing argument based on the postulated trade-off between expectedreturn, variance and skewness of return of a risk-averse gambler does not provide an explanation of observedbetting behaviour. It is shown how the expected utility of a representative gambler faced with a single-prizedoutcome event can be expressed in terms of the mean and variance of return, the mean and skewness of returnor, generally, of the mean and any other single moment of return; and the standard practice of taking a Taylorseries expansion/approximation of the expected utility involving moments of return is usually incorrect. Previousanalyses have suggested that a punter will accept a lower mean return for higher skewness and this work seemsto have involved invalid expansions of the utility function. The upshot is that with certain utility functions whichhave been used in a number of studies, any analysis based on expansion and estimation of the derivatives of theutility 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 expansion

JEL Classification No.: C44, D80, G10

1. Introduction

The traditional rationale for gambling behaviour is that bettors are risk-loving, and thisprovides an explanation of the favourite-longshot bias observed in numerous empiricalstudies of racetrack betting where bets on longshots, low probability bets, have low meanreturns relative to bets on favourites, high probability bets; see for example Weitzman[1965], Dowie [1976], Ali [1977] and Quandt [1986] and, for comprehensive reviews of thesalient literature, Sauer [1998], Thaler and Ziemba [1988] and Vaughan Williams [1999].The assumption of risk-loving behaviour implies that optimal bet size would be unboundedand that punters only bet one horse in a race, but given that punters typically bet small stakesand that some punters bet on more than one horse in a race, recently a number of authorshave suggested that gambling can be consistent with risk-aversion. However, because betstypically offer negative expected returns, agents who are globally risk-averse would notbet. One consistent explanation of observed gambling behaviour is to assume that agentsare everywhere risk-averse but obtain direct utility from gambling; this is the approach set

146 CAIN AND PEEL

out by Conlisk [1993]. Alternatively, an explanation preferred by the authors, it might beassumed that the representative agent’s utility function exhibits regions of risk-loving as wellas risk-averse behaviour as set out by Friedman and Savage [1948] and Markowitz [1952].Some authors have suggested that gambling can be consistent with risk-aversion and thisexplanation incorporates the third moment into the analysis, recognising a preference forskewness 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 [2001]. As recently stated by Golec and Tamarkin, “horsebettors accept low-return, high-variance bets because they enjoy the high skewness offeredby these bets.”

The purpose of this paper is to demonstrate that the intuitively appealing argument basedon the postulated trade-off between expected return, variance and skewness of return ofa risk-averse gambler does not provide an explanation of the observed betting behaviourthat produces the favourite-longshot bias. It is shown how the expected utility of a repre-sentative gambler faced with a single-prized outcome event can be expressed in terms ofthe mean and variance of return, or the mean and skewness of return or, generally, of themean and any other single moment of return; and the standard practice of taking a Taylorseries expansion/approximation of the expected utility involving moments of return, whichprovides a basis for the three moment “explanation” of gambling, is usually incorrect.1

2. Utility model for betting

To illustrate the argument, a standard approach is employed. Following Ali [1977] andGolec and Tamarkin [1998], it is assumed that the representative bettor has utility functionU (·) and bets total wealth, M, which is the unit of measurement of all payouts and of theargument of the utility function. A winning one unit staked bet pays out X = 1 + a, wherea represents the odds quoted against the particular horse (or team) winning, and a losingbet returns nothing. The mean return is µ = pX , where ‘p’ is the win probability, andhence the winning payout or return is X = µ

p . The actual payout will be 0 or X and the(representative) bettor’s expected utility of payout, E = (1 − p)U (0) + pU (X ), can beexpressed as a function of p and µ, E1(p, µ), as

E = E1(p, µ) = (1 − p)U (0) + pU

(µ

p

). (1)

A rational bettor who does not derive utility from the act of gambling per se, will make thebet if E1(p, µ) ≥ U (1) and hence, from (1), it follows that :

p

[U

(µ

p

)− U (0)

]≥ [U (1) − U (0)]. (2)

For (2) to hold, the bettor cannot be globally risk-averse and U (·) must exhibit some risk-loving characteristics as assumed by Friedman and Savage [1948] and Markowitz [1952];perhaps with the bettor risk-loving over favourites and risk-averse over longshots so that

UTILITY AND THE SKEWNESS OF RETURN IN GAMBLING 147

U ′′(0+) > 0 and U ′′(∞) < 0. Equality in (2) establishes the border of the sub-region ofthe (p, µ) plane corresponding to feasible rational betting.

If all X, or equivalently µ = µ(p), are set at an equilibrium value across all competitors,then d E

dp = 0. It is thus possible to differentiate (1) with respect to p and equate to zero inorder to find the combinations of expected return, µ, and probability, p, between which thebettor is indifferent. This produces d E

dp = −U (0) + U (µ

p ) + U ′(µ

p )[ dµ

dp − µ

p ] = 0 and hence

dµ

dp= µ

p+

[U (0) − U

(µ

p

)]U ′(µ

p

) . (3)

If U (0) = 0, then (3) reduces to

dµ

dp= µ

p

[1 − 1

e

], (4)

where e = e(X ) = e(µ

p ) is the elasticity of U (·) at X = µ

p . Observe from (4) that the slopeof the equilibrium expected return-win probability frontier will be positive (or negative)depending on whether the elasticity is greater than (or less than) one.

Differentiating (3) with respect to p yields

d2µ

dp2= − 1

p

[dµ

dp− µ

p

]2 U ′′(µ

p

)U ′(µ

p

) = −U ′′(µ

p

)pU ′(µ

p

)[

U(

µ

p

) − U (0)

U ′(µ

p

)]2

,

which is positive (negative) if U ′′(·) is negative (positive) at X = µ

p .

The favourite-longshot bias is that dµ

dp > 0 and from (4) in the case U (0) = 0, a neces-sary and sufficient condition is that the elasticity is greater than one. From (3) and (2) itfollows that d X

dp = 1p [ dµ

dp − µ

p ] = [U (0)−U (X )]pU ′(X ) ≤ [U (0)−U (1)]

p2U ′(X ) and, whilst d Xdp will be naturally

negative so that dµ

dp <µ

p and U (0) < U (1) < U (X ), it does not follow that dµ

dp > 0 in allcases. However, if the bettor is risk-loving with U (0) = 0, U ′(X ) > 0, U ′′(X ) > 0, thenXU ′(X ) > U (X ), e(X ) > 1 and, from (4) , dµ

dp > 0 in this particular case. These pointsare illustrated with a utility function which captures the form envisaged by Friedman andSavage [1948], who hypothesise that agents are initially (at low levels of wealth) risk-aversethen (at higher levels of wealth) risk-loving and then again (at even higher levels of wealth)risk-averse. A function capturing these properties is:

U (x) = xα

1 + e−βx, x ≥ 0 (0 < α < 1, β > 0).

In this case U ′(x) = αxα−1

(1+e−βx ) + βxαe−βx

(1+e−βx )2 > 0, (which is positive for x > 0), and

U ′′(x) = xα

x2(1 + e−βx )3[(α2 − α) + e−2βx (−α + α2 + β2x2 + 2αβx)

+ e−βx (2α2 − 2α + 2αβx − β2x2)].

148 CAIN AND PEEL

Figure 1. A Friedman–Savage utility function and the (µ, p) frontier. [U (x) = xα

1+e−βx where α = 0.975,β = 0.035; expected 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 thepunter is risk-averse, but for a variety of values of β and α > 0 the second derivative ispositive in the middle of its domain and the agent is risk-loving. This is illustratedwith values of α = 0.975, β = 0.035; the elasticity is plotted for these values in figure 1(a).

In Figure 1(b)–(d) the expected return–win probability frontier is plotted. Whilst afavourite-longshot bias is apparent, in that extreme longshots have lower rates of returnthan favourites, the interesting feature of the plots is that the frontier exhibits two turn-ing points. Consequently, it is demonstrated that with a specification of a utility functionwhich admits both risk-aversion and risk-loving behaviour over its range, as in Friedmanand Savage [1948] and Markowitz [1952], there is a hitherto neglected implication that theequilibrium mean return-win probability frontier may exhibit turning points.2 Such a spec-ification can provide a consistent rationale for the “anomalous” reverse favourite-longshotbias found in the Hong Kong betting market by Busche and Hall [1988], and in US baseballbetting by Woodland and Woodland [1994, 2001] as well as the more universal findings.Essentially, it seems that different empirical studies have been exploring different segmentsof the equilibrium mean return-win probability frontier; see Cain and Peel [2002]. The


existence of a minimum also predicts that Friedman-Savage utility functions imply thatextreme longshot bets, such as football pools or most lotteries, may exhibit higher expectedrates of return than the most extreme longshots observed in horseracing.

3. Moments of return

If the return or payout, R, to a one unit staked bet is X = 1 + a with probability p and 0with probability 1 − p, the mean return is µ = E(R) = pX and hence the winning payoutis X = µ

p . The (higher) central moments of return are thus as follows:

variance: σ 2 = V (R) = E(R − µ)2 = µ2(1 − p)

p

skewness: µ3 = S(R) = E(R − µ)3 = µ3(1 − p)(1 − 2p)

p2

kurtosis: µ4 = K (R) = E(R − µ)4 = µ4(1 − p)[(1 − p)3 + p3]

p3

fifth moment: µ5 = E(R − µ)5 = µ5(1 − p)[(1 − p)4 − p4]

p4.

In general, for n ≥ 1,

odd moments: µ2n+1 = E(R − µ)2n+1 = µ2n+1 p(1 − p)[(1 − p)2n − p2n]

p2n+1

even moments: µ2n = E(R − µ)2n = µ2n p(1 − p)[(1 − p)2n−1 + p2n−1]

p2n.

Note that for 0 < p < 1, all even moments are positive, and odd moments are positiveif 0 < p < 1

2 but negative if 12 < p < 1; and when p = 1

2 all odd moments are zero.

Writing An ≡ [(1−p)2n−p2n ]

[p(1−p)]n− 12

= µ2n+1

σ 2n+1 and Bn ≡ [(1−p)2n+1+p2n+1][p(1−p)]n = µ2n+2

σ 2n+2 , it can be shown that

An = An(p) is decreasing in p over 0 < p < 1, whilst Bn = Bn(p) is a convex function ofp over 0 < p < 1 with a minimal value of 1 when p = 1

2 ; and hence for n ≥ 1, µ2n ≥ σ 2n.

These results seem to have implications for the estimation of the effect of moments ongambling behaviour. Higher central moments, both odd and even, are more important forsmall p (longshots) in the sense that µn

σ n decreases with p(< 12 ), although for p > 1

2 theeven moments again become potentially more influential compared with the variance, andthe odd ones increasingly more negative. With p = 1

2 , and perhaps approximately so nearp = 1

2 , the use of higher central moments in any regression analysis is equivalent to usingpowers of the standard deviation, σ .

Since σ 2 = µ2(1−p)p , it follows that p = µ2

µ2+σ 2 and hence all moments can be expressedas functions of µ and σ 2. In particular,

µ3 = σ 2(σ − µ)(σ + µ)

µ, (5)

150 CAIN AND PEEL

but more generally, for n ≥ 1, µ2n+1 = (σ 4n−µ4n )σ 2

µ2n−1(µ2+σ 2) and µ2n+2 = (σ 4n+2+µ4n+2)σ 2

µ2n (µ2+σ 2) . Likewise,

the expected utility of return can be expressed as a function of µ and σ 2. From (1),

E = E2(µ, σ 2) ≡ σ 2

(σ 2 + µ2)U (0) + µ2

(σ 2 + µ2)U

(σ 2 + µ2

µ

).

Similarly, writing s for the skewness of return, µ3, E may be considered as a function ofµ and s: E = E3(µ, s) = E2(µ, σ 2(µ, s)), by noting from (5) that σ 4 − µ2σ 2 − µs = 0.

Observe that s > 0 if p < 12 (i.e. µ2 < σ 2) and s < 0 if p > 1

2 (µ2 > σ 2); and, furthermore,

σ 2(µ, s) =

1

2[µ2 +

√µ4 + 4µs] when µ2 < 2σ 2

(p < 2

3

)1

2[µ2 −

√µ4 + 4µs] when µ2 > 2σ 2

(p > 2

3

),

and µ > 0, s ≥ −µ3

4.

In a similar manner, the expected utility may be considered implicitly as a function of µ andany one other higher central moment or, in fact, as a function of any two central moments.However, in practice it may be difficult to obtain an explicit expression. For instance, Emay be considered a function of µ and kurtosis, κ , as

E = E4(µ, κ) = E2(µ, σ 2(µ, κ)),

noting that κ ≥ 0 and σ 8 + σ 2(µ6 − κµ2) − κµ4 = 0;and a function of σ 2 and skewness, s, as

E = E5(σ 2, s) = E2(µ(σ 2, s), σ 2),

noting that σ 4 = µs + µ2σ 2 or µ = µ(σ 2, s) = 12σ 2 {−s + √

s2 + 4σ 6} > 0.It follows from the above that remarks about a gambler’s preferred trade-offs between

expected return and skewness, which implicitly hold variance fixed, will generally be flawed.For instance, Golec and Tamarkin, on p. 224, state “We claim that bettors could be risk-averse and favor positive skewness, and primarily trade off negative expected return andvariance for positive skewness.” Even the highly regarded Hirshleifer and Riley [1992], onp. 73, state that “individuals tend to prefer positive skewness” and suggest that this leads toportfolios that are not so well-diversified.

Mean-variance frontier

The above observations provide a framework for the exploration of (moment) frontiersinvolving trade-offs between pairs of moments of return. For instance, the mean-variance,(µ, σ 2), frontier is defined by

E2(µ, σ 2) ≡ σ 2

(σ 2 + µ2)U (0) + µ2

(σ 2 + µ2)U

(σ 2 + µ2

µ

)= constant ≥ U (1), (6)


for which

∂ E2

∂σ 2= µ2(

σ 2 + µ2)2

{U (0) − U

(σ 2 + µ2

µ

)+ (σ 2 + µ2)

µU ′

(σ 2 + µ2

µ

)}

= 1

X2{U (0) − U (X ) + XU ′(X )}

is > 0 or < 0 according as the bettor is apparently risk-loving or risk-averse at X ;

∂ E2

∂µ= −2µσ 2

(σ 2 + µ2)2

{U (0) − U

(σ 2 + µ2

µ

)+ (σ 2 + µ2)

µU ′

(σ 2 + µ2

µ

)}

+ U ′(

σ 2 + µ2

µ

)

= −2(1 − p)

X{U (0) − U (X ) + XU ′(X )} + U ′(X )

and hence, given σ 2, the bettor will prefer larger µ if apparently risk-averse but smaller µ

if sufficiently strongly risk-loving, with p < 12 .

The slope of the (µ, σ 2) frontier, dµ

dσ 2 , can be obtained by differentiating Eq. (6) withrespect to σ 2. This yields

{ −2µσ 2

(σ 2 + µ2)2U (0) + 2µσ 2

(σ 2 + µ2)2U

(σ 2 + µ2

µ

)+ (µ2 − σ 2)

σ 2 + µ2U ′

(σ 2 + µ2

µ

)}dµ

dσ 2

= µ2

(σ 2 + µ2)2

{−U (0) + U

(σ 2 + µ2

µ

)− (σ 2 + µ2)

µU ′

(σ 2 + µ2

µ

)}

and hence

dµ

dσ 2= −

∂ E2∂σ 2

∂ E2∂µ

= {−U (0) + U (X ) − XU ′(X )}X [2(1 − p){−U (0) + U (X ) − XU ′(X )} + XU ′(X )]

.

Observe that

dµ

dσ 2< 0 if

(1 − 2p)XU ′(X )

2(1 − p)< U (X ) − U (0) < XU ′(X ) (risk-loving)

and

dµ

dσ 2> 0 if

U (X ) − U (0)

XU ′(X )> 1 or

U (X ) − U (0)

XU ′(X )<

(1 − 2p)

2(1 − p),

(risk-averse) (strongly risk-loving)

and this determines the direction of trading between µ and σ 2.

152 CAIN AND PEEL

Mean-skewness frontier

The (µ, s) frontier is defined by E3(µ, s) = E2(µ, σ 2(µ, s)) = constant ≥ U (1), whereσ 2(µ, s) is a solution σ 2 = σ 2(µ, s) of σ 4 − µ2σ 2 − µs = 0.

Now, ∂ E3∂s = ∂ E2

∂σ 2 · ∂σ 2

∂s = µ

(2σ 2−µ2)∂ E2∂σ 2 = 1

(2−3p)X · ∂ E2∂σ 2 and, given µ, larger skewness is

preferred if and only if variance is; unless p > 23 . Note that s = s(p, µ) = µ3(1−p)(1−2p)

p2

and ∂s∂p = (3p−2)µ3

p3 {>0 ifp>2/3<0 ifp<2/3,

so that s (<0) is a minimum when p = 2/3.

The slope, dµ

ds , of the mean-skewness frontier is

dµ

ds= − ∂ E2

∂σ 2 · ∂σ 2(µ,s)∂s

∂ E2∂µ

+ ∂ E2∂σ 2 · ∂σ 2(µ,s)

∂µ

=dµ

dσ 2 · ∂σ 2

∂s

1 − dµ

dσ 2 · ∂σ 2

∂µ

= {U (0) − U (X ) + XU ′(X )}X2[3(1 − p)(1 − 2p) {U (0) − U (X )} + (1 − 6p + 6p2)XU ′(X )]

which can be > 0, = 0 or < 0 depending on the values of 3(1 − p)(1 − 2p), (1 − 6p + 6p2)and XU ′(X )/[U (X ) − U (0)]. Note that 3(1 − p)(1 − 2p) > (1 − 6p + 6p2) if and only if∂s∂p < 0.

Other moment frontiers are explored in the Appendix. It does not appear that any ofthe relationships is monotonic and hence there are no one-sided trade-offs throughout thewhole range of bets. To illustrate the possibilities, in figure 2(a) to (d) some moment frontiersare plotted for the power function U (x) = xα , α = 0.95. Note that for this globally risk-averse utility function, the expected return-skewness trade-off is positive for longshots.Only for extreme favourites, in figure 2(d), is the trade-off negative, as conjectured bynumerous authors. In figure 3(a) to (d), moment frontiers are plotted for the Friedman-Savage utility function of Section 2. Plots of the expected return-variance frontier are givenin figure 3(a) and (b), and of the expected return-skewness frontier in figure 3(c) and (d).Observe that each of these frontiers exhibits a minimum and that for extreme longshots, onthe risk-averse segment of the utility function, the trade-off is positive between expectedreturn and skewness; and not negative as implied by the intuitive argument mentioned aboveconcerning risk aversion and skewness. In figure 4(a)–(c) relationships between expectedutility and skewness or variance are plotted; with expected return fixed at 0.90. Observe thatexpected utility exhibits a maximum and ultimately reduces as skewness increases; and,given expected return, there is an optimal level of skewness and likewise a correspondingoptimal level of variance.

4. Expansion and truncation of the utility function

From Section 3 it follows that, for given µ, the variance and all higher central momentsof return will be unbounded as p → 0 (extreme longshots). An immediate implicationof this is that the common practice of employing a truncated Taylor series expansion toapproximate the expected utility will in general be invalid for small p. However, a salient


Figure 2. Some moment trade-offs for the power utility function. [U (x) = xα where α = 0.95; expected utility,EU = constant > U (1) = 1].

feature of the gambling literature is the analysis of behaviour when p is small, for instancebetting on lotteries (see, for example, Walker and Young [2001]), and hence such studiesinvolving expansions are very questionable. This point is an application of the analysis ofLoistl [1976] to that of gambling but the inappropriateness of expanding utility functionswilly-nilly must be called into question here as it should be (a fortiori) more generally inFinance, where utility is still even taken to be a function of moments of return; see, forexample, Hwang and Satchell [1999].

Previous analyses have suggested that a punter will accept a lower mean return for higherskewness and this work seems to have involved invalid expansions of the utility function.Before demonstrating that such expansions may be invalid, recall from the moment frontiersderived in Section 3 that: (i) given σ 2, the bettor will prefer larger µ unless sufficientlystrongly risk-loving and the contingency necessarily odds against, (ii) given µ, skewnessis preferred if and only if σ 2 is, except for strong favourites with fair odds shorter than 2-1on, and (iii) given σ 2, skewness is preferred if and only if µ is not. It thus follows thatthe previous suggestions that punters will simply trade mean return for higher skewness ofreturn is incorrect in general.

154 CAIN AND PEEL

Figure 3. Some moment trade-offs for the Friedman-Savage utility function. [U (x) = xα

1+e−βx where α =0.975, β = 0.035; EU = constant ≥ U (1)].

A utility function U (x) can be expanded around the point x0 by a Taylor series involvingthe derivatives of U (·) at x0, U i (x0) = [ di U (x)

dxi ]x=x0 , if limi→∞ |U i+1(x0)i!(x−x0)(i+1)!U i (x0) | < 1; and

hence only for x such that |x − x0| < limi→∞ | (i+1)U i (x0)U i+1(x0) |. If limi→∞ | (i+1)U i (x0)

U i+1(x0) | = ∞,

then for all x > 0, U (x) = ∑∞i=0

U i (x0)(x−x0)i

i! , and with x0 = µ and any random variable

X with mean µ and finite moments of all orders, E [U (X)] = ∑∞i=0

U i (µ)E(X−µ)i

i! , if finite.

However, this result does not necessarily hold if limi→∞ | (i+1)U i (µ)U i+1(µ) | < ∞.

Consider the following examples:

1. Constant risk aversion

U (x) = 1 − e−αx , 0 ≤ x < ∞, α > 0,

r (x) = −U ′′(x)

U ′(x)= α > 0, and

limi→∞

∣∣∣∣ (i + 1)U i (x0)

U i+1(x0)

∣∣∣∣ = limi→∞

∣∣∣∣ (i + 1)

α

∣∣∣∣ = ∞.


Figure 4. Expected 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 µ = 0.90 fixed; expected utility − U (1) plotted ≥ 0].

Thus, U (x) can be expanded around x0 for all x and x0 in (0, ∞). In addition,

∞∑i=0

U i (x0)(−x0)i

i!= [1 − e−αx0 ] +

∞∑i=1

(−1)(αx0)i e−αx0

i!

= [1 − e−αx0 ] − e−αx0 [eαx0 − 1] = 0 = U (0),

and hence the expansion holds at x = 0.

2. Power function

U (x) = xα , 0 ≤ x < ∞, α > 0.

U ′(x) = αxα−1 > 0, U ′′(x) = α(α − 1)xα−2

{>0 for α > 1

<0 for 0 < α < 1

and limi→∞

∣∣∣∣ (i + 1)U i (x0)

U i+1(x0)

∣∣∣∣ = limi→∞

∣∣∣∣ (i + 1)x0

(α − i)

∣∣∣∣ = x0 > 0.

U (x) can be expanded around x0 only for |x − x0| < x0 i.e. for 0 < x < 2x0. In addition,

at the end points,∑∞

i=0 U i (x0) (−x0)i

i! = 0 = U (0) and∑∞

i=0 U i (x0) xi0

i! = U (x0).2α =U (2x0); and hence the expansion is valid for 0 ≤ x ≤ 2x0.

3. Markowitz

U (x) = 1 − e−αx − αxe−αx , x ≥ 0, α > 0.

U ′(x) = α2xe−αx > 0 (for x > 0),

156 CAIN AND PEEL

U ′′(x) = α2(1 − αx)e−αx

>0 if x <1

α

<0 if x >1

α

and

limi→∞

∣∣∣∣ (i + 1)U i (x0)

U i+1(x0)

∣∣∣∣ = limi→∞

∣∣∣∣ (i + 1)(i − 1 − αx0)

α(αx0 − i)

∣∣∣∣ = ∞.

In this case, U (x) can be expanded around x0 for all x and x0 in (0, ∞). Also, whenx = 0,

∞∑i=0

U i (x0)(−x0)i

i!= [1 − e−αx0 − αx0e−αx0 ] − α2x2

0 e−αx0

+[ ∞∑

i=2

(i − 1 − αx0)(αx0)i

i!

]e−αx0

= [1 − e−αx0 − αx0e−αx0 − α2x2

0 e−αx0]

+ e−αx0[1 + αx0 + α2x2

0 − eαx0] = 0 = U (0),

and hence the expansion is valid for any x ≥ 0.

Discussion

With the utility model of Section 2, the mean return is µ but the actual return, R, is either0 (with probability 1 − p) orX = µ

p (with probability p); and the expected utility of returnis given by (1). The question is, can the utility function be expanded around µ at the pointx = 0 and also at the point x = X? There are no problems with utility functions 1 and 3,and there is no problem at x = 0 with utility function 2, but the latter power function canbe expanded at x = X ≡ µ

p only if X ≤ 2µ i.e. if p ≥ 12 (an odds-on favourite). It thus

follows that expansion of the power utility function is not valid if p < 12 and the upshot is

that with this utility function (which has been used in a number of studies), any analysisbased on expansion and estimation of the derivatives of U (x) at x = µ will be valid only fordata based on odds-on favourites and not for longshots. If the data also includes longshotsthen the way to proceed is to estimate directly the parameters of the chosen utility model. Itcan then be inferred whether or not the (representative) bettor is risk-averse or risk-lovingat a particular point. From the estimated parameters for the chosen model the signs ofU ′′(µ), U ′′′(µ), U ′v(µ) etc. could then be deduced, but since the expansion is not valid itis not possible to interpret these as global unqualified preferences for certain moments ofreturn (variance, skewness, kurtosis etc.) other than with reference to distributions havingsmall ranges around the mean.

Plotted in figure 5(a) to (f) are the expected return—win probability frontiers impliedby second and third order Taylor expansions of the power utility function as well as theexact relationship given by the utility function itself. Observe that the figures are consistentwith the theoretical analysis. In particular note that the third order expansion in figure 5(e)


Figure 5. Exact (µ, p) frontiers for the power utility function, U (x) = x0.95, and frontiers implied by secondand third order truncated Taylor series expansions (with EU = 1).

158 CAIN AND PEEL

implies a positive relationship for small values of p, even though the exact relationship iseverywhere negative. The approximation for p ≥ 1

2 is quite good as might be suggested bythe theoretical analysis.

Now, if expansion is valid,

U (X ) = U (µ) + (X − µ)U′(µ) + (X − µ)2

2!U

′′(µ) + (X − µ)3

3!U

′′′(µ) + · · · ·

U (0) = U (µ) + (0 − µ)U′(µ) + (0 − µ)2

2!U

′′(µ) + (0 − µ)3

3!U

′′′(µ) + · · · ·

and so

E[U (R)] = pU (X ) + (1 − p)U (0) = U (µ) + V (R)

2!U

′′(µ)

+ S(R)

3!U

′′′(µ) + K (R)

4!U

′′′′(µ) + · · · ·

With the power function, U (µ) = µα , dividing throughout by µαwe may regressE[U (R)]/U (µ) on the standardised moments V (R)/µ2, S(R)/µ3, K (R)/µ4, . . . ; for whichthe intercept should be 1. From the earlier comments, there ought to be stability for p > 0.5but not for the unrestricted case or for p < 0.5. In Table 1, this regression is estimated forvarious ranges of p and for progressive truncations of the expected utility; and the stabilityissues are clearly demonstrated. Data is obtained for values of p from 0 to 1 in steps of 0.0001

Table 1. Regression of expected utility truncated to 2, 3 and 4 moment terms (power utility, U (x) = x0.95).

Intercept Variance/µ2 Skewness/µ3 Kurtosis/µ4 R2

(a)

(n = 9999)

0 < p < 1 0.953246 −0.0000966 – – 0.074213

0 < p < 1 0.954830 −0.000338 0.0000000329 – 0.180313

0 < p < 1 0.956850 −0.000717 0.000000235 −1.67 ∗ 10−11 0.291512

(b)

(n = 4999)

0 < p < 0.5 0.921347 −0.0000802 – – 0.109469

0 < p < 0.5 0.923785 −0.000272 0.0000000260 – 0.250167

0 < p < 0.5 0.926783 −0.000562 0.000000180 −1.27 ∗ 10−11 0.385677

(c)

(n = 4999)

0.5 < p < 1 0.998197 −0.034624 – – 0.991151

0.5 < p < 1 0.999748 −0.033474 0.012554 – 0.999874

0.5 < p < 1 0.999962 −0.028171 0.014985 −0.005906 0.999998

Note: In each case the expected utility, EU, is fixed at 1 and the dependent variable is EU/U (µ).


and the equilibrium mean return, µ, simulated using the power function utility : U (x) = x0.95

and for an expected utility value of 1.0. Table 1(a) depicts results for all n = 9999 observa-tions, 1(b) for the 4999 observations with p < 0.5 and 1(c) for the 4999 observations withp > 0.5. Observe the similarity between cases 1(a) and 1(b) but the quite different results ofcase 1(c). In particular, note the very large values of R2 in case 1(c) compared with the verymuch smaller ones in the other two cases, and the proportionately very much smaller changesin the estimated regression coefficients in case 1(c), compared with the other two cases,as progressively more moment terms are included in the regression. Note also that, whenexpansion is valid, the corresponding theoretical combinatorial coefficients, with exponentα = 0.95, are: 1.0, ( α

2 ) = −0.02375, ( α3 ) = +0.0083125, ( α

4 ) = −0.00426015625, etc;and these are well-estimated in case 1(c) but not in either case 1(a) or 1(b). The resultsthus seem to be consistent with the theoretical analysis concerning stability and truncation;that unqualified expansion without reference to the form of the underlying utility functionis very questionable.

5. Conclusion

It has become increasingly common for authors to suggest that gambling could coexist withrisk-aversion. This explanation often incorporates the fact that the third term of a Taylorseries expansion, around the mean return, of an everywhere risk-averse utility functionis positive; and the agent is then alleged to exhibit a preference for positive skewness.Such considerations have led numerous authors to mistakenly suggest that gamblers acceptlow-return, high-variance bets because they enjoy the high skewness offered by these bets.This paper demonstrates that these intuitively appealing arguments based on the postulatedtrade-off between expected return, variance of return and skewness of return, of a risk-averse gambler, are incorrect at least in the case of a representative gambler faced with asingle-prized outcome event. In this latter case, expected utility can be described in termsof expected return and any other single moment of return. It is shown, and demonstratedwith examples, that for an agent who is everywhere risk-averse, the equilibrium relationshipbetween expected return and skewness of return can be positive (for longshots) not negativeas often conjectured. It is also shown that the widespread, almost standard, practice of takingan unqualified Taylor series expansion/approximation of the expected utility, involvingvarious moments of return, can often be in error.

Appendix

The slope of the variance-skewness or mean-kurtosis frontier can be derived in a similarmanner.

Variance-skewness frontier

A (σ 2, s) frontier may be defined by: E5(σ 2, s) = E2(µ(σ 2, s), σ 2) = constant ≥ U (1),whereµ(σ 2, s) is a solutionµ = µ(σ 2, s) = 1

2σ 2 {−s + √s2 + 4σ 6} > 0 ofσ 4 = µs + µ2σ 2.

160 CAIN AND PEEL

Now, ∂ E5∂s = ∂ E2

∂µ· ∂µ

∂s = −µ

(s+2µσ 2) · ∂ E2∂µ

= −µ2

σ 2(σ 2+µ2) · ∂ E2∂µ

and, given σ 2, larger s is preferred

if and only if larger µ is not. The slope, dsdσ 2 , of the variance-skewness frontier is:

ds

dσ 2= − ∂ E5

∂σ 2

∂ E5∂s

= σ 2(σ 2 + µ2)

µ2·

∂ E2∂σ 2

∂ E2∂µ

+ (2σ 2 − µ2)

µ

= (2σ 2 − µ2)

µ− σ 2(σ 2 + µ2)

µ2· dµ

dσ 2

= X [3(1 − p)(1 − 2p){U (X ) − U (0)} − (1 − 6p + 6p2)XU ′(X )]

[2(1 − p){U (X ) − U (0)} − (1 − 2p)XU ′(X )].

Note that dsdσ 2 = dµ

dσ 2 /dµ

ds , as expected.

Mean-Kurtosis frontier

The (µ, κ) frontier is defined by E4(µ, κ) = E2(µ, σ 2(µ, κ)) = constant ≥ U (1), whereσ 2(µ, κ) is a solution σ 2 = σ 2(µ, κ) of σ 8 + σ 2(µ6 − κµ2) − κµ4 = 0.

In this case, differentiating with respect to κ, (4σ 6 + µ6 − κµ2) ∂σ 2

∂κ= µ2(σ 2 + µ2),

∂ E4

∂κ= ∂ E2

∂σ 2· ∂σ 2

∂κ= µ2(σ 2 + µ2)

(4σ 6 + µ6 − κµ2)· ∂ E2

∂σ 2= µ2(σ 2 + µ2)2

(3σ 8 + 4σ 6µ2 + µ8)· ∂ E2

∂σ 2

and, given µ, larger κ is preferred if and only if larger σ 2 is preferred.The slope, dµ

dκ, of the mean-kurtosis frontier is

dµ

dκ= −

∂ E4∂κ

∂ E4∂µ

= −∂ E2∂σ 2 · ∂σ 2(µ,κ)

∂κ

∂ E2∂µ

+ ∂ E2∂σ 2 · ∂σ 2(µ,κ)

∂µ

=dµ

dσ 2 · ∂σ 2(µ,κ)∂κ

1 − dµ

dσ 2 · ∂σ 2(µ,κ)∂µ

.

Noting that

∂σ 2(µ, κ)

∂κ= µ2(σ 2 + µ2)2

[3σ 8 + 4σ 6µ2 + µ8]= 1

(3 − 8p + 6p2)X2> 0,

and

∂σ 2(µ, κ)

∂µ= 2σ 2[σ 8 + 2σ 6µ2 − 2σ 2µ6 − µ8]

µ[3σ 8 + 4σ 6µ2 + µ8]= 2(1 − p)(1 − 2p)X

(3 − 8p + 6p2),

dµ

dκ= [−U (0) + U (X ) − XU ′(X )]

X 3[2(1 − p)(2 − 6p + 6p2){−U (0) + U (X ) − XU ′(X )} + (3 − 8p + 6p2)XU ′(X )]

and dσ 2

dκ= ∂σ 2(µ,κ)

∂µ· dµ

dκ+ ∂σ 2(µ,κ)

∂κgives the slope of the variance-kurtosis frontier.


In general, the slope of the (µ, µ2n+1) frontier, for n ≥ 1, is

dµ

dµ2n+1=

dµ

dσ 2 · ∂σ 2(µ,µ2n+1)∂µ2n+1

1 − dµ

dσ 2 · ∂σ 2(µ,µ2n+1)∂µ

where

∂σ 2(µ, µ2n+1)

∂µ2n+1= µ2n−1(µ2 + σ 2)

[(2n + 1)σ 4n − µ4n − µ2n+1µ2n−1]

and

∂σ 2(µ, µ2n+1)

∂µ= µ2n−2[4nµ2n+1σ 2 + µ2n+1{(2n + 1)µ2 + (2n − 1)σ 2}]

[(2n + 1)σ 4n − µ4n − µ2n+1µ2n−1];

and the slope of the (σ 2, µ2n+1) frontier isdσ 2

dµ2n+1= ∂σ 2(µ, µ2n+1)

∂µ2n+1+ ∂σ 2(µ, µ2n+1)

∂µ· dµ

dµ2n+1

=dµ

dµ2n+1

dµ

dσ 2

=[

∂σ 2(µ,µ2n+1)∂µ2n+1

]1 − dµ

dσ 2∂σ 2(µ,µ2n+1)

∂µ

·

Similarly, the slope of the (µ, µ2n+2) frontier, for n ≥ 1, is

dµ

dµ2n+2=

dµ

dσ 2 · ∂σ 2(µ,µ2n+2)∂µ2n+2

1 − dµ

dσ 2 · ∂σ 2(µ,µ2n+2)∂µ

where

∂σ 2(µ, µ2n+2)

∂µ2n+2= µ2n(µ2 + σ 2)

[(2n + 2)σ 4n+2 + µ4n+2 − µ2n+2µ2n]

and

∂σ 2(µ, µ2n+2)

∂µ= [µ2n+2{(2n + 2)µ2n+1 + 2nµ2n−1σ 2} − (4n + 2)µ4n+1σ 2]

[(2n + 2)σ 4n+2 + µ4n+2 − µ2n+2µ2n];

and the slope of the (σ 2, µ2n+2) frontier is obtainable as

dσ 2

dµ2n+2= ∂σ 2(µ, µ2n+2)

∂µ2n+2+ ∂σ 2(µ, µ2n+2)

∂µ· dµ

dµ2n+2=

[∂σ 2(µ,µ2n+2)

∂µ2n+2

]1 − dµ

dσ 2∂σ 2(µ,µ2n+2)

∂µ

·

162 CAIN AND PEEL

Notes

1. Presented here is a particular application of a more general result that has been recognised in the literature,but which has been repeatedly ignored. See, for example, Brockett and Garven [1998] and Rothschild andStiglitz [1970], who show that moment preference does not match up with a sequence of utility derivatives.In the gambling literature there seems to be widespread ignorance or disregard for such correct analysis. Ourdemonstration and discussion that such disregard is, in general, in error is the first, as far as we are aware, inthe gambling context.

2. If the Utility function has the form U (x) = 1 − e−αx − αxe−αx , x ≥ 0 (α > 0), so that the agent is risk-lovingover favourites and risk-averse over longshots, as suggested by Markowitz [1952], the expected return—winprobability frontier will only exhibit a minimum. To exhibit a maximum value it is necessary that the curve hasa risk-averse segment followed by a risk-loving one.

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utility and the skewness of return in gambling

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