The Wise Guy s Juice: A Reduced Form Skewness Analysis within Australian Fixed-Odds Horse Racing

Transcription

1 The Wise Guy s Juice: A Reduced Form Skewness Analysis within Australian Fixed-Odds Horse Racing By Max Kaftal ABSTRACT I utilize fixed-odds horse race gambling data to estimate a utility function of uncertainty for consumers. Results suggest that bettors accept low return bets due to a preference for skewness, not risk. The fixed odds format enables me to estimate a reduced form model of the bookie s take. Using the Gini coefficient to capture the skewness of a race, I find evidence that bookie takes rise with a race s skewness, a result that further confirms that bettors have a positive preference for skewness. I d like to thank my thesis advisor, Professor Charles Moul, for his countless hours of time, guidance, and support. His advisement has showed me how a combination of hard work and humor can make a difficult project both enjoyable and rewarding.

2 2 In a world of randomness, how people perceive aspects of uncertainty relative to expectation is central to many economic decisions. Gambling is an especially rich area for such research, as negative expected returns imply the existence of other benefits of uncertainty. Exploring higher moments helps to describe these different preferences. The second moment, variance, is equated with risk. Skewness is the third moment and describes the asymmetry of a variable around the vertical axis. Horserace gambling provides a useful setting within this realm of economic literature and has been widely used due to data availability. In this paper, I use Australian fixed odds horseracing data to demonstrate skewness-loving behavior. Bettors relinquish money to a bookie, or wiseguy, who sets the odds. These odds then imply an expected level of bookie profit (the take or juice ). I show the differing bettor behavior between long shots and favorites, which can under certain assumptions, reveal bettor utility over uncertainty. Lastly, I calculate Gini coefficients as a proxy for the skewness of a race and demonstrate how bookie take relates to skewness. If bettors love skewness, bookies should capture this preference with high takes on high skewness races. In my research I find just this: bettors prefer skewness, and bookies adjust take to capture this inclination. Horseracing odds and payouts are generally represented within two frameworks: pari-mutuel and fixed odds. American dog racing and horseracing use the pari-mutuel system, in which track take is fixed (usually between 17%-19%). 1 The odds displayed to the public are preliminary, and final odds are calculated only when the betting windows are closed. Final payouts are calculated as the ratio of the amounts bet on each horse relative to the available winnings (after fixed track take). Only pari-mutuel data has been used in past study, probably because of it is the betting method used within the United States. Within the fixed odds system (used in Las Vegas sports betting and Australian horseracing), though, bookmakers jointly set the odds for all outcomes, and bookie take is therefore variable. 2 Although odds may change over time in fixed odds, the bettor receives the odds given at the moment he accepts the bet. Ali (1977) uses horse racing data on win bets to illustrate what has been termed the long shot anomaly. 3 The long shot anomaly is characterized by favored horses winning more often (the objective probability) than their odds imply (the subjective probability); long-shots necessarily win less often. That is, favorites are under bet and long shots are over bet. Ali explains the long shot anomaly by arguing that bettors place a positive value on variance (i.e., bettors are locally risk-loving). As empirical support for this hypothesis, Ali estimates a log-log utility function. He finds that utility increases with odds at an 1 Golec and Tamarkin, For further details on the comparison between fixed odds and pari-mutuel gambling, see Sauer, A win bet pays out to the bettor if and only if the horse finishes first.

3 3 increasing rate, implying that gamblers are risk-loving. Ali does not explore his utility function s implications for higher moments, but results suggest that bettors also exhibit skewness-aversion. Golec and Tamarkin (1998, hereafter GT) revisit the long shot anomaly, estimating the utility function with a third-order polynomial and a dataset that focuses upon heavy favorites and long shots. After replicating Ali s findings by applying the power specification to their data, they use a cubic specification to show that bettors prefer expectation and skewness but are risk-averse (contrasting Ali). 4 They then provide institutional details in favor of their hypothesis that bettors seek skewness and not variance. Cain, Peel, and Law (2002) dismiss what they consider the simplistic GT approach of constructing generic utility functions for skewness preference. Specifically, more complex utility functions, such as that of Markowitz (1952), allow for preferences to change along the utility curve and with income. Without adjusting for local preferences, they argue that general claims such as risk-loving or skewness-loving are erroneous, as behavior changes along the utility function. Regarding the necessity of skewness-loving for gambling, they demonstrate it is possible for a locally risk-averse bettor to choose a wager with less skewness, if variance and expected return are the same. They further argue the experiment by GT is seriously flawed, as variance and skewness necessarily move together in all wagers considered. The necessary thought-experiment (constant variance and varying skewness) therefore never occurs. Cain, Peel, and Law claim a more specific utility function is needed to prove skewness preference, thereby rejecting the skewness argument put forth by GT s general demand side estimates. Both Ali and GT use pari-mutuel data to explore a utility function for preference of risk or skewness. For their economic applications, though, pari-mutuel is inferior to fixed odds data. Fixed odds data better reflect needed assumptions for utility recovery than pari-mutuel because a bettor knows his payout at the time the bet is taken. Furthermore, the pari-mutuel setting eliminates supply-side factors (i.e., bookie reactions) that may help resolve disputes in the literature; as such supply analysis can sidestep much of the controversy regarding particular utility specifications. After replicating the previous papers, I explore race-level variables with potential demand and cost implications to the bookie. I show that a race s Gini coefficient of odds is an inverse proxy for skewness but unrelated to variance in this setting, resolving the collinearity argument of Cain, Peel, and Law. My reduced form estimates of race take indicate that bookie take decreases as the race s Gini coefficient increases. Thus, bettors prefer skewness and bookies exploit this preference by increasing take. These outcomes bolster the demand side results of skewness preference as found by GT. 4 Strictly speaking, GT s estimates are necessary for their conclusions but not sufficient.

4 4 Data The fixed odds data come from the Australian Racing Board (ARB), the overarching body of the individual states racing agencies. The data used within this study are comprised of 1,463 races, containing 18,912 horses from the 10 largest metro tracks within Sydney, Melbourne, and Brisbane. The data represent every Saturday and holiday race meeting between November 2 nd, 2002, and May 29 th, 2004, that contains a minimum of seven starting horses and a maximum of seventeen starting horses (and uses fixed odds gambling). 5 I use Saturday and holiday races, as they are the most heavily bet and so should provide the most accurate insight. For example, at the Australian Jockey Club in Sydney (one of Australia s largest) in 2003, the attendance averaged 11,797 on the 32 races on Saturdays or public holidays, and only 1,266 on Sundays or midweek. The total bookie turnover averaged $5,359,000 on Saturdays and public holidays and $1,811,000 during midweek and Sundays. 6 Ali uses final odds to group horses. Throughout the race, the horse with the lowest odds represents the favorite and second best is the second favorite. Each group s objective probability is calculated as the percentage of actual race wins compared to races. The odds directly imply subjective probabilities which are then averaged across races for particular favorite positions. I follow Ali and group horses by race, ranking each by increasing official starting odds. 7 The objective probabilities equal the win frequencies of horses within a particular slot (the ratio of wins to total number of races with at least that many horses). The subjective probabilities equal the prospective win frequencies based on odds alone. Let t and X h respectively denote the bookie s take for a race and the final odds of horse h winning. Within this data, X h is gross odds, including the odds of horse h winning plus the amount returned to the bettor in the case of a win. For instance, if a horse has starting odds of 3:1, X h equals 4 (three for the win and one returned to the bettor). Bookie un-take, the percentage of possible winnings for bettors, is 1 t = 1/ subjective probability for horse h winning is the horse s stake in the potential winnings ((1-t)/ X h ). I assume that all money outside of the untake is captured as the bookie take. 8 I then average the subjective probabilities for each rank. # H k = 1 1/ Xh. The 5 Australia also uses pari-mutuel gambling. In the season, fixed odds accounted for $AU 2,937 million of total betting and pari-mutuel accounted for $AU 8,764 million. Australian Racing Board Fact Book ( 6 Australian Jockey Club Annual Report ( 7 GT group by the odds directly, but this approach is unappealing given the range of race sizes (7-17 horses) that I employ. 8 We ignore breakage, the situation where the track obtains additional revenue by rounding payoffs and keeping the remainder.

5 5 Results Confirming the Long Shot Anomaly Known odds visibly change over time under fixed odds gambling, so the market is likely to be more efficient than that found in pari-mutuel. This may negate the long shot anomaly. Within this section, I confirm the long shot anomaly within the framework of fixed odds. Table 1 displays the average objective and subjective probabilities of each grouping. Horses ranked 13th and over were grouped within one category due to their small objective probabilities. Table 1 Rank Wins Objective Subjective Favorites are under bet; four of the top five horses have larger objective probabilities than subjective probabilities. These horses win more than their odds imply. On the contrary, all other horses (long shots), with the curious exception of the 11 th ranked horse, are over bet. The horses with the least likely odds (the 13+ category) are implied to win over three times the number of races that they actually do. 9 Bettors accept much lower expected returns in order to bet on a long shot. Therefore, the long shot anomaly also holds under fixed odds betting. 9 Groups of 14+, 15+, etc. were tested for robustness, mostly yielding similar results.

6 6 Utility Estimates Both Ali and GT use objective probabilities to recover utilities for bettors. Letting the winning horse h return X h, the expected utility of a wager on horse h is p h u(x h ) + (1-p h )u(0). If the utility of a loss is normalized to zero (i.e., u(0)=0) and bettors are indifferent between the wagers on horses at closing odds, then p h u(x h )=p i u(x i ) for all horses in a race. Finally, normalize the utility associated with the longestshot horse H winning to one (i.e., u(x H ) = 1). Then the utility function is defined as u(x h ) = p H /p h First consider the power specification used by Ali: ln(u(x i )) = b 0 + b 1 *ln(x i ) + e i, (n=12) R Table 2 Coefficient S.E. T b b Table 2 replicates Ali s approach, but my results differ from his. As b 1 >1 reveals risk-loving behavior, my results cannot reject that bettors like expectation and are risk-neutral (as well as skewnessneutral). Bettors would more rationally put money in savings accounts if this accurately characterizes their utility. By only recognizing preference for expected return, the specification fails to explain why bettors gamble. Approximating the utility function with a cubic specification allows for a separate measurement of skewness s impact on utility: u(x i ) = b 0 + b 1 X h + b 2 X 2 h+ b 3 X 3 h + ε h GT s conclusion of bettors liking expectation and skewness but disliking variance requires b 1 >0, b 2 <0 and b 3 >0. A result of b 1 >0 implies return loving. Within this regression we are primarily testing b 2 and b 3. The second term reflects risk, as b 2 >0 demonstrates risk-loving and a result of b 2 <0 reflects risk aversion. The third term reflects analogous preferences towards skewness.

7 7 Coefficient (S.E.) b 0 b 1 b 2 b 3 Table 3 Column 1 Column 2 With Third Without Third Term Term (0.0569) ( ) ** (0.0055) ( ) * ** (9.6E-05) (-4.2E-04) 1.47E-05** (4.0E-06) R * Significant at 90% confidence ** Significant at 95% confidence Table 3 shows two results. Column 1 uses the specification of GT but truncates the third term for skewness to possibly demonstrate risk preference of Ali (his form reveals risk neutrality). This specification s estimates seem unrealistic because bettors dislike both expectation and risk. When the cubic term is introduced (Column 2), I find similar results with fixed-odds data as GT found using parimutuel data. All coefficients are significant at the 95% confidence level, and the high R 2 reflects a good fit for this utility function. Although the sign of the coefficients is a necessary condition for GT s conclusions, it is not sufficient to characterize utility. I therefore use the delta method to find the values and standard errors of the derivatives with respect to the odds to describe the utility function more fully. Table 4 q S.E. t du/dx d2u/dx2-2.23e E d3u/dx3 8.81E E The results in Table 4 refute the earlier conclusion regarding risk aversion. The values at the derivative functions of the cubic specification affirm return and skewness-loving, but risk neutrality cannot be rejected at conventional confidence levels.

8 8 The Gini Coefficient as a Measure of Skewness Because I use fixed odds data rather than the previously used pari-mutuel data, the endogenous bookie take creates the possibility that bookie choices could capture bettor preferences. In my data, there are only three observable race characteristics that could fill demand and/or cost roles. On the cost side, the number of horses in a race could imply more work for the bookie. Therefore, costs suggest that races with more horses will have higher takes. On the demand side, both the variance and skewness of races could influence bettor demand. While Table 1 shows that objective and subjective probabilities differ systematically, assuming that the two probabilities are the same may serve as an appropriate first approximation. I consider the analytical definitions under this assumption below. The variance of a bet is defined as Var (b i ) = E(b i E(b i )) 2. Given that the expectation of any bet is E(b i ) = 1 t and denoting the probability of winning as p, this variance expression can be rewritten as Var(b i ) = (1-t) 2 *(1/p i 1 ) Let the variance of a race be the weighted average of the variances of the individual bets: Var(Race) = Σ p i *V(b i )= Σ ((1-t) 2 p i (1/p i -1)) = (N-1)(1-t) 2 The variance of a race therefore increases with the number of horses (N) but is independent of the odds within that race. The skewness coefficient of a bet is defined as Skew(b i ) = (E(b i E(b i )) 3 )/Var(b i ) 3/2. Expanding this definition yields Skew(b i ) = (1/p i 2) / (1/p i 1) ½ Let the skewness of a race be the weighted average of the skewness coefficients of the individual bets: Skew(Race) = Σ p i (1/p i 2) / (1/p i 1) ½ = Σ (1 2p i ) / (1/p i 1) ½ Unlike the variance, the skewness of a race is dependent upon the odds. Imposing the equal-probability assumption within my reduced form regression is unattractive because it ignores the differences between favorites and long-shots. I explore race-specific Gini coefficients as proxies for race-level skewness to alleviate this problem. 10 That is, I calculated the above race-specific skewness levels and separately calculated Gini coefficients based on official starting prices (final odds). For more detailed calculation steps, see the appendix. Long shot bets have higher skewness than bets on favorites, so a positive correlation between race-skewness and Gini may seem intuitive. Bets on favorites, however, will be more heavily weighted because of higher probabilities. Furthermore, the skewness 10 Gini coefficients measure inequality and are often employed for distribution of income. A Gini coefficient of 0 demonstrates perfect equality of all income, while a coefficient of 1 describes a single individual holding all the wealth.

9 9 coefficient being scaled for variance may also undercut the former intuition. A negative correlation is therefore also plausible. I regress the weighted average skewness on the Gini coefficient and the number of horses in the race. I expect the impact of the number of horses in a race (N) to be positive. More horses imply a larger spread of odds; higher skewness is more likely. Table 5 displays results (n=1463). Table 5 R Coefficients S.E T Intercept Gini ** N 0.177** **Significant at 99% confidence The results are highly significant (99% confidence) and depict a very strong fit (R 2 of 0.97), establishing the Gini as a strong inverse proxy for this (admittedly imperfect) measure of race-skewness. As bookie costs and variance are not associated with the Gini, any impact of the Gini on bookie take must arise through consumer preferences on skewness. Bookie Take and Skewness If bettors truly prefer skewness as suggested through the utility function, supply side factors should reflect this behavior. The impact of the number of horses on take is theoretically ambiguous. On the costside, more work is presumably needed to establish optimal odds with more horses in a specific race; bookies should increase take on these races. The positive relationship between the number of horses and skewness is shown in Table 5, and a preference for skewness would then also suggest a higher take. If, however, bettors are risk-averse, the theoretical result that variance increases with the number of horses suggests a lower take. The fact that my prior results cannot reject that bettors are risk-neutral strongly suggests that the relationship between the number of horses and take will be positive. A skewness-loving utility function demonstrates a clear relationship between the Gini and the supply side. Bookies should raise take on low Gini races to capture skewness preference.

10 10 Table 6 Gini N Take Mean Std.Dev Min Max Table 6 displays the summary statistics by race. The average Gini coefficient is 0.4, with a standard deviation of The large spread between extremes is notable for the Ginis. As previously mentioned, there are between 7 and 17 horses in each race, with an average of about 11. The average take is nearly 18%, similar to the common fixed take in pari-mutuel gambling. However, the range between extremes is of interest. It is especially striking that 3 of 1463 races had negative bookie takes. This could possibly be driven by late scratches or unexpected high betting volume on a particular horse, making it difficult for the bookie to adjust quickly. The maximum bookie take is nearly 37%, illustrating the bookie s strong potential to capture profit. I regress the individual race takes on the Gini coefficient and N. It is possible that tracks may differ in their quality and may systematically have different sorts of races (e.g., the nicest track may run the largest races, conflating the impact of unobserved quality and the number of horses). Since each track has its own system or personnel, individuality between bookies might also cause disparity among results. I therefore also consider binary variables for the ten individual metro tracks used. Table 7 displays the important results (n=1463). Table 7 Variable 1 2 Gini (S.E) * (0.013) * (0.011) N (S.E.) * (0.0005) * (0.0004) Track Dummies No Yes R * Significant at 99% confidence The results are consistent with the expectation of skewness-loving bettors. The individual racetracks differ markedly in track take. By introducing track-specific binary variables, the R 2 increases from 0.20

11 11 to More importantly, results after controlling for unobserved track quality are still statistically and economically significant. An additional horse in a race is estimated to increase take 0.8 percentage points. On the margin of greater interest, decreasing a race s Gini coefficient by 0.1 increases take by 0.4 percentage points. Bettors evidently prefer races with evenly matched horses, and these are races that exhibit relatively high skewness. This result confirms GT s finding of skewness-loving among bettors by including supply side factors. Bookies exploit skewness-loving behavior through track take. Conclusion Underlying bettor preference characterizes the rationale behind gambling. Ali and GT have conflicting explanations of the long shot anomaly. Ali argues that agents are risk-loving, while GT instead argue that a preference of skewness drives gambling. Both of these studies, however, are inherently flawed due to the collinearity between variance and skewness. Also, the use of pari-mutuel data only allows exploration of demand side factors with all the consequent utility-specification criticisms. By using fixed odds data, I explore the utility and its relation to bookie take, while accounting for the collinearity. Preference for skewness is first demonstrated by the utility function. By using the Gini coefficient as a proxy for race skewness, I demonstrate that bookies capture this preference by increasing take for races with high skewness. The results of this study have wide implications in consumer preference outside of gambling. Financial returns have been modeled as random variables. Inherent to this method is the idea that investors prefer high returns but dislike risk (unfortunately, risk is the avenue to make returns). But this study suggests investors may value other characteristic of investment aside from expected return or variance. If people value skewness, they may forego relatively high return investments for those with higher skewness (such as an IPO). The results are also associated with the findings in prospect theory, which attempts to explain decision-making preference for when returns are known, but risk varies (the case in fixed odds horse racing). People tend to assume an unlikely event happens more than it actually does. Prospect theory helps explain the long-shot anomaly within this study. Bettors assume long-shots win more frequently than they actually do. Within financial returns, these results may help explain why investors may prefer high risk investments such as in IPO (a highly skewed investment). Investors may perceive the probability of the stock dramatically rising in price as more likely than it realistically is. Although skewness is admittedly difficult to measure, the differentiation between risk, skewness, and other measures of human preference are important to understand behavior under uncertain conditions.

12 12 Appendix 1: Mean, Variance, and Skewness of Individual Ranks Rank Mean Variance Skewness , , , ,943, ,642, ,623, ,638, ,126, ,390, ,079,017, ,519, ,169,066,493 To calculate mean, variance, and skewness of a bet, the GT method was employed, using the following equations (b and p respectively denote subjective and objective probability): μ h = p h ((1-t)/b h ) - 1 σ 2 h = p h (1-p h )((1-t)/b h ) 2 M 3 h = (p h (1-p h )(1-2p h )((1-t) 3 3 )/b h The average returns for each rank accurately reflect the earlier findings of the long shot anomaly. Favorites are underbet, they pay out a less negative return. Longshots are overbet and have far worse average payouts. The variance and skewness are largely consistent with GT s findings. They mostly increase montonically, aside from the anomalous 12 th horse (and the 13 th with regards to variance). This result occurs due to the drastic decease in objective probability within these ranks.

13 13 Appendix 2: Distribution of Numbers of Horses in Races Starting Horses Frequency

14 14 Appendix 3: Calculating Values at Derivatives Using the Delta Method The delta method was used to approximate values and standard errors at the derivatives for the cubic function. I calculated the first three derivatives of utility with respect to the odds (X). U = b 0 + b 1 *X + b 2 *X 2 + b 3 *X 3 + e du/dx = b 1 + 2*b 1 *X + 3*b 3 *X 2 d 2 U/dX 2 = 2*b 2 + 6*b 3 *X d 3 U/dX 3 = 6*b 3 Next, I expressed these derivatives as functions of the parameter vector ([b 0 b 1 b 2 b 3 ]) in matrix form, using expectations for the random variables (e.g., X). [0 1 2*E(X) 3*E(X 2 ) ] [ *E(X) ] [ ] Let the above matrix be denoted as g, and let the original covariance matrix be denoted V. The implications for the derivative point estimates are then q= g*b. The covariance matrix is Q=g*V*g. Standard errors are the square root of the diagonal of Q.

15 15 Appendix 4: Calculating the Gini Coefficient The Gini coefficient is calculated by drawing a 45 degree angle from the (0,0) to (1,1). Letting N denote the number horses in a race, points along the horizontal axis are spaced equidistant apart from each other at 1/N. The subjective odds are sorted in increasing order, added by sequential summation, and plotted along the vertical axis, forming an area (L) under the 45 degree angle. To do this, the lowest subjective probability (p h ) is closest to the origin, and forms a triangle. = ½* 1/N * p h The rest of the areas are the resulting trapezoids for the rest of the horses. = (1/2N) Σ (p h + p k )) The sum of these areas yields a possible area of ½ (1/2*1*1). In order to obtain a coefficient that fully spans 0 and 1, L is multiplied by 2. The Gini coefficient is calculated as Gini = 1-2*L.

16 16 Appendix 5: Skewness and Take R 2 =.21 Coefficient S.E. T Intercept Skewness ** N ** ** Significant at 99% confidence Although both theory and results suggest the Gini coefficient accurately captures skewness in a race, skewness was regressed against take to solidify their interaction. These results show a positive relationship, as expected, between skewness and bookie take. Bookies capture preference by raising take on high skewness races.

17 17 Appendix 6: Data collection The data for this project was provided by Tim Ryan, the CEO of the Australian Racing Board. The original data set contained all Saturday race information from 2/11/2002 through 04/08/2007. All observations involved fixed odds gambling, and all came from the 10 metro racetracks in Melbourne, Brisbane, and Sydney. The data contained all horses, opening odds, all line fluctuations, and closing line (starting price). First, the data were broken down by horse, so that each horse was a single observation. While the data were nominally in spreadsheet format, there were sufficient needs for cleaning that preparing the entire dataset was infeasible. I began with 10,000 observations, and later expanded this to 20,000 observations. Scratched horses were originally left as observations, in case their presence was later necessary. Next, it was necessary to limit races to those with seven to seventeen starting horses. Races with too few horses gave little insight there was little chance for variability and skewness, and the objective probability for an individual horse was high. There were few races with over seventeen horses, and the individual objective probabilities for those long shots were considered too small. After scratched horses were removed from the data set, the usable 1,463 races and 18,912 starting horses were left.

18 18 References Ali, Mukhtar M. Probability and Utility Estimates for Racetrack Bettors. The Journal of Political Economy. Vol. 85, No.4 (Aug., 1977) Cain, M, D. Peel and D. Law. Skewness as an explanation of gambling by locally risk-averse agents. Applied Economics Letters. Vol. 9, No. 15 (Dec. 2002) Golec, Joseph and Maurry Tamarkin. Bettors Love Skewness, Not Risk, at the Horse Track. The Journal of Political Economy. Vol. 106, No. 1 (Feb., 1998) Markowitz, H. The utility of wealth. Journal of Political Economy. Vol. 60, No. 2 (Apr. 1952) Sauer, Raymond D. The Economics of Wagering Markets. Journal of Economic Literature. Vol. 36 No. 4. (Dec. 1998)