Risk preferences and information flows in racetrack betting markets.

I. Introduction

Empirical and theoretical research on racetrack betting markets has blossomed in recent years due in part to the importance of the industry in terms of its dollar volume, number of participants, and similarity to security markets. One important difference from security

Virtually all of the empirical work analyzing risk preferences in horse race betting markets shows the representative bettor to be risk loving; that is, the representative bettor derives more utility from the uncertain gamble than from the certain wealth that is wagered. The one exception is a study by Busche and Hall (1988) who find that the Hong Kong race track betting market is dominated by either risk-neutral or risk-averse participants. In all of these studies, the "representative" bettor's risk preference is revealed by the final odds determined at the close of betting. A question remains, however, as to whether there really is a "representative" bettor.

Even though risk-loving participants may dominate, it is possible that individuals with utility functions exhibiting risk aversion participate in the betting process. Risk-averse individuals may enter into these markets if they attach an entertainment value to a day at the track for which they are willing to pay a price. If entertainment is a commodity providing satisfaction, and thus included with wealth as an argument in the utility function, an investor may participate in an unfair gamble yet be risk averse.

It is also feasible for risk-averse bettors to participate in racetrack betting markets if they believe they have superior analytical ability or special information that will allow them to earn positive returns after adjusting for the track take. These bettors trade based upon a comparison of their predetermined odds and the subjective odds determined by the pari-mutuel betting system. Their predetermined odds, in turn, are most likely based on horse-specific information such as the health of the horse on race day or performance in recent workouts. Given the existence of asymmetric information sets and varying analytical ability, it is possible for risk-averse individuals to earn positive returns despite the relatively large proportion of funds kept by the organizers.

It is even possible for participants with identical utility functions to place bets based simply on a divergence of opinion. Under these conditions, horses may be either under- or overbet based on the heterogeneous expectations of participants. Lusht and Saunders (1989) examine this hypothesis but find no evidence to support it in their sample.

The purpose of this paper is threefold: first, to describe the effects of different risk preferences on the subjective and objective probabilities of winning at racetracks; second, to empirically demonstrate the emergence during the wagering period of bettors with different risk preferences; third, to show that information associated with changing risk preferences can be used in betting strategies that earn positive returns.

II. Risk-Loving Behavior in Racetrack Betting Markets

Models of racetrack betting behavior are commonly cast in terms of the objective and subjective probabilities of a horse winning a given race. Objective probabilities of winning are based on the actual racing performances of the horses involved, while subjective probabilities are reflected in the odds jointly determined by the bettors.

The odds of horse i winning a given race are described in terms of the amount bet on horse i and the proportion of funds that are kept by the track. The odds for horse i, [O.sub.i], are defined as

[O.sub.i] = [(1 - t)/[x.sub.i]] - 1, (1)

where t represents the track take and breakage and [x.sub.i] denotes the amount bet on horse i.(1) Normalizing the total amount bet in a race to unity, Asch, Malkiel, and Quandt (1982) interpret the [x.sub.i]'s as the subjective probabilities that horse i will win.

The net payoff per dollar bet on horse i to win is $[O.sub.i] if horse i wins and -$1 otherwise. Thus, defining betting outcomes for horse i in terms of expected return and variance results in

E([r.sub.i]) = [p.sub.i]([O.sub.i]) + (1 - [p.sub.i])(-1) = [p.sub.i][(1 - t)/[x.sub.i]] - 1 (2)

and

[[[Sigma].sup.2].sub.i] = [p.sub.i]([[O.sup.2].sub.i]) + (1 - [p.sub.i]) - [[E([r.sub.i])].sup.2], (3)

where [p.sub.i] refers to the objective probability of horse i winning. For any pair of horses such that E([r.sub.i]) [greater than] E([r.sub.j]), equation (2) implies that

[p.sub.i]/[x.sub.i] [greater than] [p.sub.j]/[x.sub.j]. (4)

Quandt (1986) shows that equation (4) implies [p.sub.i] [greater than] [p.sub.j] in a market dominated by risk lovers; i.e., if E([r.sub.i]) [greater than] E([r.sub.j]), [[[Sigma].sup.2].sub.i] [less than] [[[Sigma].sup.2].sub.i]. If horses are ordered such that E([r.sub.1]) [greater than] E([r.sub.2]) [greater than] ... [greater than] E([r.sub.n]) and if [[[Sigma].sup.2].sub.1] [less than] [[[Sigma].sup.2].sub.2] [less than] ... [less than] [[[Sigma].sup.2].sub.n], then [p.sub.1] [greater than] [p.sub.2] [greater than] ... [greater than] [p.sub.n]. Quandt proves that for [p.sub.1] [greater than] [p.sub.2] [greater than] ... [greater than] [p.sub.n], there exists a k such that

[p.sub.1]/[x.sub.1] [greater than] ... [greater than] [p.sub.k]/[x.sub.k] [greater than] 1 [greater than or equal to] [p.sub.k+1]/[x.sub.k+1] [greater than] ... [greater than] [p.sub.n]/[x.sub.n]. (5)

If the indifference mapping in risk-return space is negatively sloped, equation (5) indicates that favorites are underbet and longshots are overbet in the sense that [p.sub.i] [greater than] [x.sub.i] for favorites and [p.sub.i] [less than] [x.sub.i] for long shots, and the ratio of objective to subjective odds, [p.sub.i]/[x.sub.i] increases monotonically as the objective probabilities increase.

[TABULAR DATA FOR TABLE 1 OMITTED]

Although the organizers structure the market such that the probability of winning is not commensurate with the payoff received, risk-loving participants are lured to horse racing by the potential for a significant payoff. The risk-loving individuals are willing to accept a lower expected return and higher variance for the opportunity, albeit with low probability, to earn a significant payoff.

Here, we perform an empirical test for the presence of risk-loving behavior using a sample of 8,021 horses involved in 939 races at Philadelphia (PA) and Garden State (NJ) Parks over a recent twelve-year period. The data represent results for many horses racing in various conditions and can be considered a random sample reasonably representing American racetracks.

The horses are grouped into eight odds categories, with midpoints at 0.25, 1, 2, 3.25, 5, 8, 15, and 40 to 1. These groupings provide a fairly detailed breakdown of horses over the interval from favorites to longshots.

Next, we calculate the expected returns, variances, and probability ratios from the final odds available at the end of the wagering period. Given a track take and breakage of 18.5 percent for the sample, subjective probabilities, [x.sub.i], are computed for each horse from equation (1) and then averaged within each odds group. The objective probabilities, [p.sub.i], are estimated as the percentage of winning horses in each odds group. The expected returns and variances are computed from equations (2) and (3).

Table 1 summarizes the results based on the odds at the end of the wagering period. By construction, the average subjective probabilities, [x.sub.i], decrease with increasing odds categories. As suggested by Quandt (1986), the average objective probabilities, [p.sub.i], decrease monotonically with increasing odds categories. Also, the ratio of objective to subjective probabilities, [p.sub.i]/[x.sub.i], is greater than unity for favorites, less than unity for longshots, and increases with the objective probabilities, although not monotonically. This positive relation between the p/x ratios and the objective probabilities measured at the end of the wagering period indicates favorites are underbet and long shots are overbet as predicted by the Quandt model under the assumption of risk-loving behavior. These results are consistent with prior research that finds a persistent bias in favor of low-odds horses at American racetracks (e.g. see Griffith (1949) and Lusht and Saunders (1989)).

The expected returns and variances shown in Table I are inversely related, but not perfectly so. Given the expected convexity of the mean-variance relation, a semilogarithmic regression is performed on the data. The resulting equation (with t-statistics in parentheses) linking expected return and variance is described by

[Mathematical Expression Omitted].

Figure I graphs the mean-variance pairs for each odds group with the calculated regression curve drawn through the points. These results are also consistent with prior research for other American racetracks. Using ending odds, Ali (1977) and Asch, Malkiel, and Quandt (1982), among others, observe utility functions that imply negatively sloped risk-return indifference mappings for their representative bettors.

III. Risk-Averse Behavior in Racetrack Betting Markets

The assertion that bettors are risk lovers implies that the utility function of the representative bettor includes wealth as its only argument and the horse racing market is efficient. However, these assumptions concerning bettors at racetracks may not hold globally.

The inclusion of an entertainment value in the utility function of bettors and/or the existence of asymmetric information sets allows for the possibility of risk-averse behavior to occur in racetrack betting markets. If risk-averse behavior is present, general statements about the utility function of a representative bettor must be modified to allow for a much more complicated market-equilibrating process than is currently assumed.

Given the assumption of risk aversion, a progressive ordering of horses from favorite to longshot such that E([r.sub.1]) [less than] E([r.sub.2]) [less than] ... [less than] E([r.sub.n]) and [[[Sigma].sup.2].sub.1] [less than] [[[Sigma].sup.2].sub.2] [less than] ... [less than] [[[Sigma].sup.2].sub.n] implies that [p.sub.1] [greater than] [p.sub.2] [greater than] ... [greater than] [p.sub.n]. There must then be a q such that

[p.sub.1]/[x.sub.1] [less than] ... [less than] [p.sub.q]/[x.sub.q] [less than] 1 [less than or equal to] [p.sub.q+1]/[x.sub.q+1] [less than] ... [less than] [p.sub.n]/[x.sub.n]. (6)

Equation (6) indicates that by assuming risk aversion, favorites are overbet and longshots are underbet in the sense that [p.sub.i] [less than] [x.sub.i] for favorites and [p.sub.i] [greater than] [x.sub.i] for longshots. Also, the ratio of objective to subjective probabilities, [p.sub.i]/[x.sub.i], will decrease monotonically as the objective probabilities, [p.sub.i], increase.

A person may choose to place a bet at any time during the betting period for any number of reasons. If the bet is placed at or near the end of the wagering period, the individual has the opportunity to observe as much about the horses in the race as possible, including the progression of odds determined by the behavior of other bettors. Or, the individual may have been busy analyzing "expert" commentary on the horses and then placed a bet after that process was completed. The results reported in Table 1, as well as in prior research, however, indicate that individuals with risk-loving behavior dominate the betting at the end of the period.

Since we show risk-loving behavior dominates at the end of the wagering period, risk-averse behavior, if it exists, cannot be revealed in the final odds in our data set of 939 races. In the pari-mutuel wagering system used at most racetracks throughout the world, the participants are allowed to place bets on a horse(s) of choice at any time during a twenty- to twenty-five-minute pre-race wagering period. The odds that result from this betting are listed on a tote board, which is updated about once a minute. If we are to test for the presence of risk-averse betting [TABULAR DATA FOR TABLE 2 OMITTED] behavior, we must therefore analyze the betting at points other than at the end of the period. For our sample of 8,021 horses, we collected the odds as they were posted on the tote board to analyze bettor risk preferences throughout the betting period.

To determine the risk preferences of individuals that place bets earlier in the wagering period, the subjective probabilities, means, and variances are calculated approximately five minutes into the wagering period. These early odds are near the other extreme on the time continuum from the ending odds dominated by risk lovers, but late enough into the period to allow a betting pool of sufficient size to permit meaningful interpretive results.(2)

The testable predictions of the model are defined in terms of the objective and subjective probability ratios. Evidence for racetrack betting dominated by risk-averse participants is indicated by a positively sloped risk-return indifference curve and a falling ratio of objective to subjective odds, [p.sub.i]/[x.sub.i], as [p.sub.i] increases.

Table 2 presents summary results of the betting based on early odds. As before, the average subjective probabilities, [x.sub.i], decrease with increasing odds categories, and the average objective probabilities, [p.sub.i], decrease monotonically with increasing odds categories. But, the ratio of objective to subjective probabilities, [p.sub.i]/[x.sub.i], is less than unity for favorites; greater than unity for longshots; and increases, although not monotonically, with increasing odds categories.

The results in Table 2, which are based on early odds, can be readily compared with those of Table 1, which are based on ending odds. In both tables, subjective and objective probabilities decrease with increasing odds categories. However, the ratio of objective to subjective probabilities decreases with increasing odds categories for ending odds but increases for early odds. This inverse relation between the p/x ratios and the objective probabilities indicates that favorites are overbet and longshots are underbet early in the wagering period. These results are consistent with the predictions of the model under the assumption of risk aversion. Contrasting the results reported in Table 1 with those in Table 2, we conclude that during the betting period, favorites are initially overbet and ultimately underbet, whereas longshots are initially underbet and ultimately overbet. In other words, the results indicate that the risk preferences of bettors adding to the betting pool vary over the wagering period.

As before, we fit a regression line through the mean-variance points corresponding this time to early betting (t-statistics in parentheses):

[Mathematical Expression Omitted].

Since we observe a significantly positive relation between mean returns and variances, the null hypothesis that the representative bettor in the early wagering period possesses risk-neutral or risk-loving behavior can be rejected. The mean-variance pairs and the regression line are shown in Figure II. In sum, the results are not inconsistent with the presence of risk aversion in American racetrack wagering markets.(3)

The results are, moreover, consistent with those of Busche and Hall (1988). However, the Busche and Hall results are based on ending odds. It appears that in Hong Kong risk-averse bettors dominate, whereas in the United States risk-averse bettors are present early in the wagering period but are ultimately dominated by risk-loving participants. Perhaps cultural differences can explain the divergent results for the two racetrack markets.

IV. Risk Preferences in Subsets of Races

Individuals that bet early in the wagering period may do so for various reasons. For example, they, having no opinion of their own, may choose to act on the predictions of the handicappers in the local newspaper. Since they are working with predetermined selections, they place bets early to then enjoy the surroundings. Or, there may be informed bettors, such as trainers or owners, that have special knowledge and benefit little from betting late in the period. The results summarized in Table 2 and Figure II, however, make clear that, for whatever their reasons, risk-averse individuals dominate early betting over a large number of races.

The results in Table 1 and Figure I make it equally clear that late betting is dominated by risk lovers. Their motivation is to wait as long as possible to find longshots that will provide the greatest opportunity for a high payoff, albeit with very low probability.

The large number of races examined here occurred over several years and were run in various situations involving differing weather conditions, quality of competition, size of purse, and so forth. We wonder, however, whether the odds shift we observe from early to late betting is affected in races where special circumstances exist.

For example, some races are run on cold, rainy days. If the primary motivation of risk-averse bettors is to seek entertainment at the track, they might decide to stay home on such days, so we may observe less overbetting of favorites (or less early betting in general) in the early wagering in these races. Or, in some races, the early odds may be widely distributed across horses. In this case, risk-loving bettors could more easily identify their longshots. This would provide motivation for them to jump to the longshots and avoid favorites, thus causing an even more pronounced overbetting of high-odds horses. The reverse could also be true: if odds are tightly distributed so that risk lovers cannot as easily discern longshots, their tendency to overbet high-risk horses might be less pronounced. Subsets of races, run under varying conditions, may therefore show either more or less of an odds shift over the wagering period than the whole sample reveals.

We are unable to examine the first scenario of less early overbetting of favorites in races where risk-averse bettors are discouraged from attending because of bad weather or other factors. Our data set does not contain the dates on which the races were run and that information is impossible to obtain. We can, however, examine the second case. To do so, we first analyze risk preferences in races where early odds are most likely to show a clear distinction between favorites and longshots.

From the entire sample of 939 races, we first select races where the standard deviations of early odds within a given race are either greater than 10-to-1 or greater than 15-to-1. The choice of these two values for the standard deviations of odds is arbitrary, but should result in subsets of races in which early odds are reasonably widely distributed across the horses running in them.(4) Analyzing races with odds dispersions of more than 15-to-1 is not feasible because of a lack of horses in some odds categories in those races. Our hypothesis is that risk-loving bettors, seeing longshots clearly delineated in these races, will skew the odds more sharply toward an overbetting of longshots. Evidence of this should be revealed in ending odds by a greater difference in the p/x ratios ranked from low to high win probabilities and a steeper negative slope of the risk-return indifference curve.

The subjective and objective probabilities, the p/x ratios, mean returns, and variances are then calculated in the selected odds groupings for races whose early odds have standard deviations that exceed 10-to-1 and 15-to-1. There are 526 races involving 4,937 horses in the first category and 313 races involving 3,063 horses in the second category. The results are summarized in Panel A of Table 3. (For brevity, we omit the p and x probabilities from the table.)

The p/x ratios in Panel A indicate a more pronounced overbetting of longshots at the end of the wagering period for races with a relatively wide [TABULAR DATA FOR TABLE 3 OMITTED] dispersion of early odds compared with the whole sample. In fact, the p/x ratios are more widely separated the greater the dispersion in early odds.

The shift toward more overbetting of longshots at the end of the wagering period is illustrated in the mean-variance trade-off regressions reported in Panel A of Table 4.(5) In the regressions for the end of the wagering period, the slope coefficients get progressively more negative when moving from the whole sample to a greater dispersion of early odds.

The conclusion from these results is that risk-loving behavior at the end of the wagering period is amplified when bettors are faced with a relatively clear distinction between favorites and longshots in the early odds. Results of tests on the effects of such information on risk preferences in the early betting are presented in Panel B of Tables 3 and 4.

[TABULAR DATA FOR TABLE 4 OMITTED]

For early betting, the p/x ratios are more tightly distributed and closer to one than for the whole sample, indicating a lessening of the influence of risk-averse behavior. The mean-variance regressions indicate a flattening of the risk-return indifference curves when moving from the whole sample to a greater dispersion of early odds. In fact, the slope coefficient for races with a standard deviation of greater than 15-to-1 in early odds is not significantly different from zero, indicating risk neutrality, but we cannot, with confidence, jump to this conclusion because of the poor regression fit and the resulting low [R.sup.2]

One explanation for these results is that the greater dispersion in early odds may be sufficient to attract some risk-loving bettors to longshots early in the wagering period, but not enough to dominate the betting. Risk-averse bettors, on the other hand, would choose the early favorites anyway, for reasons indicated earlier.

If early odds are relatively widely dispersed, risk-loving bettors may be able to more clearly identify longshots and, if so, will skew the odds even further toward an overbetting of high-risk horses. If early odds are more tightly distributed, however, we wonder whether risk-loving bettors will continue to place bets on high-odds horses when the possibility of a large payoff on a particular horse is less evident.

We again return to our sample of 939 races and select those where the standard deviations of early odds are either less than 10-to-1 or less than [TABULAR DATA FOR TABLE 5 OMITTED] 7-to-1.(6) Analyzing races with odds dispersions of less than 7-to-1 is not feasible because of a lack of horses in some odds categories in those races. Our hypothesis is that risk-loving bettors, seeing longshots less clearly delineated in these races, will skew the odds less sharply toward an overbetting of longshots. Evidence of this should be revealed in ending odds by higher p/x ratios for high-risk horses and smaller negative slopes of the risk-return indifference curves.

We carry out the same calculations as before for races whose early odds have standard deviations that are less than ten to one and seven to one. There are 413 races involving 3,084 horses in the first category and 264 races involving 1,882 horses in the second category. The results are summarized in Panel A of Table 5.

[TABULAR DATA FOR TABLE 6 OMITTED]

The p/x ratios for the end of the wagering period in Panel A are higher than those for the whole sample for the highest risk horses (odds midpoints of 40-to-1 and 15-to-1) and are greater than one. For lower risk groupings, the patterns of the p/x ratios roughly mimic those for the whole sample, except for the lower values for the 1-to-1 odds groupings. These results indicate that underbetting of the highest risk horses occurs in races where the early odds are relatively less dispersed, and betting patterns on lower risk horses remains roughly the same as it is for all races. We can conjecture, but not conclude from these results, that risk-loving bettors may avoid placing bets in races where they cannot clearly distinguish the true longshots from other horses.

The mean-variance regression results are reported in Panel A of Table 6. The regressions indicate a progressive shift from risk-loving to risk-neutral to risk-averse behavior when moving from the whole sample to races with progressively less dispersion in odds, but this is most likely due to the very large expected returns for the highest odds groupings (the 40-to-1 grouping even yields positive expected returns). The lack of a well-defined relation between expected returns and variance is further evidenced by the weaker regression results and lower [R.sup.2]'s than for the whole sample. These results fall short of allowing us to conclude that risk-averse behavior dominates the betting at the end of the wagering period in races where early odds are more tightly distributed because risk lovers are avoiding such races. They do, however, indicate that the behavior of risk-loving bettors is not as systematic in these races as it is in the larger sample.

Bettor behavior early in the wagering period in races where odds are relatively closely distributed is summarized in Panel B of Tables 5 and 6. Here, the wagering behavior is weighted more heavily toward risk-averse bettors than for the whole sample. The p/x ratios in Table 5 are higher for high-risk horses and lower for low-risk horses, indicating a more pronounced overbetting of favorites and underbetting of longshots. This is also shown in the mean-variance regressions where the slope coefficients are greater than for the whole sample.(7)

V. The Information Content of Changing Odds Patterns

Given the results of the analyses in the previous sections, we conclude that the risk preferences of the bettors adding to the betting pool and, therefore, the relation between objective and subjective winning probabilities, vary over the wagering period. In a large sample containing races run over wide-ranging conditions and circumstances, longshots are underbet and favorites are overbet in the early wagering, and, as the betting progresses, this pattern reverses. In a subset of races where a relatively clear distinction can be made between longshots and favorites, the overbetting of favorites in the early wagering is lessened and the influence of risk-loving bettors is strengthened at the end of wagering. When odds differences are less clearly delineated, risk-loving behavior is less evident in the late wagering and risk-averse behavior is more evident in the early wagering. In this case, we would expect a strong overbetting of favorites early in the wagering period and less of a shift in odds toward an overbetting of longshots at the end.

What all this means, of course, is that there may be significant information content in the changing odds patterns before a race and an astute bettor observing these changes may be able to gain an advantage from knowing them. Suppose a bettor has knowledge of the early odds in a given race, and the changes that occur in these odds over the betting period are recorded. A wagering model for any horse i to win a race j may be specified as

[W.sub.ij] = F([[O.sup.E].sub.ij], [[Omega].sub.ij]), (7)

where [W.sub.ij] is a continuous random variable indicating the winning ability of horse i in race j, [[[O].sup.E].sub.ij] is the odds posted early in the betting period for horse i in race j, and [[Omega].sub.ij] is the change in the odds over the wagering period for horse i in race j.

Since [W.sub.ij] is not generally measurable, a dichotomous signaling variable, [Y.sub.ij], can be specified to take a value of one for the winning horse in race j and zero for each loser. This allows for the specification of the logistic model,

[Y.sub.ij] = a + [b.sub.1] [[O.sup.E].sub.ij] + [b.sub.2][[Omega].sub.ij] + [u.sub.ij], (8)

where a, [b.sub.1], and [b.sub.2] represent the parameters of the model, and [u.sub.ij] represents a stochastic error term.(8)

Let [P.sub.ij] be the probability of horse i winning race j. Maximum likelihood estimators, [a.sup.*], [[b.sub.1].sup.*], and [[[b.sub.2].sup.*], call be obtained from a sample of N races. The estimated will probability for each horse in each race is

[Mathematical Expression Omitted] (9)

where [h.sub.j] is the number of horses running ill each race j, j = 1, ..., N.

The parameters of the logit model are estimated from the entire sample of 939 races, with [[O.sup.E].sub.ij] measured by the posted odds available early in the betting period. The odds change variable, [[Omega].sub.ij], is measured as the natural logarithm of the odds ratio

ln([[O.sup.F].sub.ij]/[[O.sup.E].sub.ij], where [[O.sup.F].sub.ij] is the final odds posted at the end of the wagering period. The results are presented in Panel A of Table 7.

The chi-square statistics for the coefficients and for the likelihood ratio indicate that both the early odds level and the change in odds during the betting period are highly significant ill determining will probabilities. The negative signs on both coefficients indicate that, for a given odds change, horses with low early odds have higher win probabilities than early longshots, and that, for a given odds level, the smaller the change (or bigger the drop) in odds, the greater the win probability. Moreover, the most important factor in generating high win probabilities in both size and significance in the model is the presence of low early odds.

Such a pattern in odds movements would be present in those races, analyzed in the previous section, where odds differences between longshots and favorites are not clearly delineated. In those races, odds on favorites might be bid down sharply early in the wagering period by risk-averse bettors. These bettors may have superior information (the "smart money") and bet early, comfortable with their horse-specific knowledge on the likely winner. there would be little advantage to [TABULAR DATA FOR TABLE 7 OMITTED] betting later if they knew that, say, horse i had the best chance of winning, unless something were to happen to the horse during the betting period. The early bettors may also include risk-averse but uninformed participants that act oil the predictions of expert handicappers who, through superior analytical ability, may more accurately predict winners. Risk-averse bettors who have come to the track mainly for an entertaining day may enjoy watching the horses and soaking up the pre-race excitement. When they finally bet, being risk-averse, they may choose the horse with the lowest odds (made such by the smart money and handicappers), thus driving the odds down further. Through all this, risk-loving bettors may curtail their betting or even avoid betting because of a lack of information on clear longshots, thus saving their money for races where it can be put more at risk. The result would be low early odds and either little movement in or a further drop in the odds of horses with high win probabilities. This scenario is, of course, but one of many, but it is consistent with our results and the predictions of the wagering model.

By focusing on the early odds and changing odds patterns during the wagering period, abettor can significantly improve the probability of winning. If the betting market does not currently use this information, an opportunity exists for a bettor to earn abnormal returns by selecting horses with the highest win probabilities.

To test whether information on early odds and the changes that occur in these odds can be used to generate abnormal returns, we provide a set of simulations for different betting strategies. Using a methodology similar to that of Asch, Malkiel, and Quandt (1984), two naive strategies are tested: (1) bet on the horse with the highest win probability in each race; (2) bet on the horse with the highest win probability if that probability exceeds a filter coefficient taking values of .20, .25, .30, and .35.

If the estimated win probabilities are obtained by replacing the coefficients from equation (8) with the maximum likelihood estimates derived from the same set of data, the results may be biased. The 939 race data set is therefore divided into two randomly selected subsamples consisting of 469 and 470 races each. The parameters of the logit model are estimated on the first half and the win probabilities for each horse in each race are determined from the data in the second half. The process is then repeated by estimating the coefficients from the second half of the data and deriving simulated probabilities for the first half. The logit results for the two subsamples are reported in Panels B and C of Table 7. Meaningful conclusions can be expected from this procedure since both halves yield results that are similar to the full sample and that are reasonably consistent between the subsamples.

We obtain simulated win probabilities from the logit models estimated on each half of the data set, and we implement our betting strategies. Mean returns are determined from equation (2) and standard errors from equation (3). One-tailed t-tests are performed to determine whether the coefficients are significantly greater than the track take (18.5 percent) and, if positive, whether they are significantly greater than zero. The results of these simulations are reported in Table 8.

With the exception of the .25 filter coefficient, which yields a positive return sufficient to beat the track take on one side and a nonsignificant return on the other, the results are consistent for both subsamples. For six of the remaining eight cases, the null hypothesis that the expected returns are less than or equal to the track [TABULAR DATA FOR TABLE 8 OMITTED] take is rejected at the 5 percent level, and the remaining two cases are rejected at the 10 percent level.(9)

The interpretation of the return results reported in Table 8 is relatively straightforward. Bettors placing wagers on horses in a random fashion over a large number of races can expect a negative return equivalent to the track take of 18.5 percent. That is, for each one dollar bet in these races, approximately 81 cents would be received. What is most striking about the results, however, is that betting strategies using the last two filters generate positive returns significantly greater than the track take. In one case, this positive return is significantly greater than zero. In this case, bettors choosing horses with win probabilities defined by our model that exceed 35 percent, could expect (at a 95 percent confidence level) to receive, on average, $1.36 for each dollar bet.

To our knowledge, these results represent the first empirical finding of positive returns using the information content of changing odds during the wagering period. The use of information on beginning odds levels and changes in the odds during the betting period can allow bettors to earn abnormal returns that, for strategies employing sufficiently high filtering coefficients, are positive. The results of the simulations are also not inconsistent with the existence of risk-averse participants with superior information and analytical ability.

While these results are interesting in the sense that racetrack wagering could, if odds movements are analyzed throughout the betting period, lead to positive returns, the procedure does not, unfortunately, appear to be practical for a typical bettor. To earn positive returns, a bettor would have to, most likely, enter odds data into a laptop computer at or near the end of the betting period and place a wager after the results are computed. Moreover, the use of this procedure would result in bets being placed in only about 10 percent of the races. The conclusion must be, therefore, that if a person wishes to earn positive returns at the racetrack, he or she must pay a significant price in terms of time spent analyzing many races that will ultimately bear no fruit.

VI. Conclusions

Racetrack betting markets represent a unique environment to study risk preferences of participants seeking a return on money put at risk. The general assumption from prior research is that bettors in markets where the a priori rate of return is negative are risk loving. However, risk-averse bettors may participate in these markets if their involvement provides a value, called entertainment, for which they are willing to pay a price. Moreover, risk-averse bettors might participate if they expect to earn a positive return because of their superior information sets or analytical abilities.

The empirical evidence presented in this paper suggests that, over a large number of races representing a wide variety of conditions, risk-loving behavior dominates the betting late in the wagering period, but risk-averse behavior dominates in the early wagering. The presence of risk-averse bettors in racetrack betting markets requires a more complicated market-adjustment process than prior research suggests. We recommend that future research incorporate this diversity of risk preferences.

When a subset of races involving a relatively large dispersion of early odds is analyzed, the shift in odds and the risk preferences they embody are skewed more toward all overbetting of longshots and risk-loving behavior than for the whole sample. When races involving a relatively small dispersion of early odds are examined, betting is more heavily weighted toward all overbetting of favorites and risk-averse behavior.

The actions of both risk-loving and risk-averse participants can be observed by the changing subjective odds posted on a tote board before the race. In this paper, we find that information flows from the tote board concerning the three groups' behavior may be used to earn abnormal and possibly positive returns.

1 The track take is the proportion of funds bet that is retained by the track to cover expenses and profits, while breakage reflects the practice of racetracks of rounding down payoffs to the nearest $.05 or $. 10. In the sample used in this study, the take and breakage average 18.5 percent.

2 Through observation at the sampled racetracks, we determined that approximately 10 percent to 15 percent of the betting pool is in place after five minutes of betting in a typical race. Of course, the amount of money bet varies greatly from race to race depending on the size of the field, importance of the race, etc.

3 Although not presented here, we also computed probabilities, returns, and variances associated with the approximate midpoint of the betting period. The probability ratios deviate little from one over all odds groupings, and the slope of the risk-return indifference curve is not significantly different from zero. The results indicate that risk-loving bettors may enter gradually into the market and, by the end of the wagering period, dominate the betting.

4 The average standard deviation of early odds in the greater than 10-to-1 category is 21.6, and it is 26.1 for the 15-to-1 category. The average standard deviations of ending odds in each category are 20.6 and 23.8, respectively. For the entire sample, the average standard deviations over all races are 18.1 for both beginning and ending odds. In the subsets, then, the greater variability exhibited in the early odds is maintained in the ending odds.

5 Graphs of the mean-variance regressions (similar to those shown in [ILLUSTRATION FOR FIGURE I AND II OMITTED]) are available upon request from the authors.

6 The average standard deviation of early odds in the less than 10-to-1 category is 6.4 and for the 7-to-1 category it is 4.9. Ending odds standard deviations are 8.6 and 6.8, respectively. In the subsets, then, the lower variability in the early odds is maintained in the ending odds.

7 There are no winning horses in the 40-to-1 odds category in the races where the standard deviation of early odds is less than 7-to-1. This category is therefore eliminated from the regression since ln([[Sigma].sup.2]) in this category is undefined.

8 For a detailed discussion of the use of the logit procedure in analyzing returns in racetrack betting markets, see Bolton and Chapman (1986).

9 By definition, the model selects horses with low early odds. The spread in these early odds is also low, ranging, on average, from about 0.5-to-1 to less than 10-to-1 for smaller probability filters, and 0.5-to-1 to 3.5-to-1 for the 35 percent filter.

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We are indebted to an anonymous referee whose comments and suggestions added materially to this paper.