Point-Spread Wagering Markets' Analogue to Realized Return in Financial Markets

Abstract

Tests for market efficiency and rational behavior in financial markets commonly utilize realized return as the variable of interest. Researchers who study point-spread wagering markets for sporting events generally agree that the point spread is these markets' analogue to asset price in financial markets. An issue that is less clear, to date, is upon exactly which variable researchers should focus when testing for efficiency and rationality in point-spread betting markets. The objective of this article is to verify that change in point spread is an acceptable proxy for realized return.

Keywords

asset pricing realized return sports wagering

Introduction

Studies that test for market efficiency and rational behavior in financial markets commonly utilize realized return as the variable of interest. The objective of this article is to demonstrate that studies of efficiency and behavior in point-spread wagering markets can employ change in point spread as the analogue to realized return in financial markets. To establish that change in spread is a legitimate proxy for realized return is important for any researchers who test financial economic theories using a sports betting market as the setting, yet who may want to position their studies within the body of literature on financial markets.

The aforementioned tests in traditional financial markets employ various methods, including tests of profitability of trading strategies, comparisons of actual outcomes versus expected outcomes (such as those for corporate earnings), and examinations of cross-sectional variation in returns. All three methodologies are equally well accepted in the finance literature; each has its own analogue when testing for efficiency and rationality in sports wagering markets.

Numerous studies have examined the profitability of betting strategies across gambling markets for all major professional sports (Major League Baseball [MLB], National Hockey League [NHL], National Football League [NFL], and National Basketball Association [NBA]). For example, Woodland and Woodland (1994) and Gandar, Zuber, Johnson, and Dare (2002) study MLB odds betting, and Woodland and Woodland (2001) study odds betting for NHL games. Gray and Gray (1997) test for profitability in the NFL point-spread betting market, while Gandar, Zuber, and Lamb (2001), Paul and Weinbach (2005), and Ashman, Bowman, and Lambrinos (2010) perform tests in the NBA point-spread market.

Other studies have focused on forecast errors (i.e., differences between predicted outcomes and actual outcomes) as another means for testing efficiency of sports betting markets. Dare and MacDonald (1996), Gray and Gray (1997), and Dare and Holland (2004) all examine differences between spreads and outcomes for NFL games. Dare and MacDonald (1996) and Borghesi, Paul, and Weinbach (2010) study forecast errors in the wagering markets for college football and college basketball, respectively.

Yet only a handful of studies focus on variation in changes in point spreads. Avery and Chevalier (1999), Durham, Hertzel, and Martin (2005), and Durham and Perry (2008) all use changes in spreads to investigate whether various sources of sentiment affect the paths of point spreads. Our hope is that this current study will provide the groundwork to support any future research that should decide to employ this key variable—change in spread—in tests for market efficiency or bettor rationality in the sports wagering realm.

Advantages and Mechanics of Point-Spread Wagering Markets

The numerous advantages of performing such tests with a sports wagering market as the setting are well documented. Each asset (i.e., wager) has a single payoff that is dependent upon a single event (a game’s outcome), each game has a distinct “settling up” point when the values of all wagers on the game are unambiguously realized, and each asset has a short, finite life. Researchers who study point-spread wagering markets seem to all generally understand that the point spread is these markets' analogue to asset price in financial markets. An issue that has remained less clear, to date, is upon exactly which variable researchers should be focusing when testing hypotheses and theories about efficiency and rationality in point-spread betting markets. This article endeavors to clarify this point, to emphasize that the key variable of interest should be the change in point spread.

Sports betting markets facilitate two types of betting: odds betting and point-spread betting, the latter of which is of interest in this analysis. While equations for realized return can be constructed for odds betting markets (or for European-style spread betting markets, for that matter), our focus in this study is on developing a reasonable proxy specifically for point-spread betting markets. Point-spread markets are our interest here, given (a) the similarities between these markets' price-setting mechanisms and those of financial markets and (b) that all games for which point spreads can be established will have quoted point spreads, while only some of these games will also have money lines (i.e., have quoted odds).

Point-spread wagering for a game works as follows. A bettor can place a wager on either the favored team or the underdog team “against the point spread,” where the point spread can take any value (in 0.5-point increments) greater than or equal to zero.¹ If the favored team wins by an amount greater than the point spread, an $11 bet on the favorite pays a gross amount of $21 and an $11 wager on the underdog pays $0.² If the favorite wins by less than the point spread or loses the game outright, $11 wagered on the underdog pays $21 and $11 bet on the favorite returns $0. If the favorite wins by exactly the point spread, all bettors receive their original amounts back (i.e., all bets “push”).

A bet’s payoff is fixed; it is independent of the magnitude by which the game’s outcome differs from the spread.³ A bet’s expected payoff, then, depends only on probabilities associated with three different general outcomes: the favorite winning by an amount greater than the spread, the favorite winning by an amount less than the spread (or losing the game outright), and the favorite winning by an amount exactly equal to the spread. The probability of winning a wager on the favorite is simply the cumulative probability associated with all possible outcomes greater than the spread, as depicted by the area under the curve to the right of the point spread in either Figure 1A or B.⁴ The probability of winning a wager on the underdog is equal to the cumulative probability associated with all possible outcomes less than the spread, as shown in the same two figures. The probability of both wagers ending in a “push” is equal to the single probability associated with the possible outcome that is equal to the spread.⁵

A. At any instant, the bookmaker maintains a spread that will attract equal dollars of wagering on both teams in the contest. In the absence of sentiment, the bookmaker sets the spread to be equal to the median of the distribution of all possible outcomes of the game. This choice of spread makes the cumulative probability of winning a bet on the favorite and the cumulative probability of winning a bet on the underdog (which itself is also the cumulative probability of losing a bet on the favorite) both equal to 50%. B. At any instant, the bookmaker sets the spread to attract equal dollars of wagering on both teams in the contest. This choice of spread—in the presence of sentiment for the favored team—makes the cumulative probability of winning a bet on the sentimentally popular favorite less than 50%. The cumulative probability of winning a bet on the underdog is thus greater than 50%.

The point spread is the mechanism that the bookmaker uses to balance the amount of dollars wagered on the two teams in a contest.⁶ In order to satisfy its objective of balancing the amount of dollars wagered on each team, and in the absence of any bettor sentiment, the bookmaker sets the spread equal to the median possible outcome (as depicted in Figure 1A), given whatever information set exists at the time. This choice of spread makes the probability of winning a wager on the favored team and the probability of winning a wager on the underdog team both equal to 50%, ignoring the possibility of a push.

In the presence of sentimental betting, if the bookmaker keeps the point spread at the median of all possible outcomes, information-based bettors will still split their wagers equally across the two teams but the sentimental bettors will cause an imbalance in money wagered in the direction of the team for which they have sentiment. Thus, the bookmaker needs to adjust the spread in the direction of the sentimentally attractive team. Figure 1B shows how the spread would be set in the presence of disproportionate sentiment for the favorite. Such a spread, to the right of the median of possible outcomes, will attract more than half of the information-based bettors' money on the underdog and less than half on the favorite. Concurrently, the sentimental money on the favorite will offset the tilt in dollars wagered by information-based bettors, serving to balance the overall amounts bet on the two teams. (A similar explanation would apply to a case in which disproportionate sentiment exists for the underdog. The book-balancing spread would simply lie to the left of the median possible outcome.)

The opening spread for a given game is defined as the first point spread published by the sports book; it is the spread at which betting commences. At the time the market opens for wagering on the game, that game has a distribution of possible outcomes that is centered on some presumed median. With passage of time during the period when the market is open for wagering, imbalances in wagering on the two teams in the contest may emerge. Any imbalance may be due to either new information, the arrival of sentimental bettors, or even randomness. In response to any imbalance, the bookmaker will change the spread in whatever direction is necessary to continue to attract balanced wagering on the two contestants. The closing spread is the spread that exists when the market for wagering on the contest closes, at the instant when the actual contest begins.

The General Model

Before continuing, we need to develop a simplified one-period model and establish two new variable definitions. At the beginning of the period, a representative bettor places an $11 wager on the favored team at the opening spread. Let p _wfo denote the probability of winning this wager; this probability will be a function of the information set that exists at the time that the market for wagering on the game closes. Next, between when the market opens and the market closes, one of the three things can happen: new information favoring one team in the contest arrives, net sentiment emerges in favor of one team in the contest, or neither new information nor sentimental betting arrives during the period. At the end of the period (i.e., just before the betting market closes), the bettor can place another $11 wager at the closing spread, but this wager is on the underdog team. Let p _wuc be the probability of winning this wager on the underdog team at the closing spread. The two probabilities noted here (p _wfo and p _wuc) are instantaneous functions of the information set that exists at the instant when the betting market closes (which is also when the sporting event begins). This information set is either a superset of, or the same as, the information set that exists when the betting market opens.

The representative bettor now holds two bets: an $11 wager on the favorite at the opening spread and an $11 wager on the underdog at the closing spread. The first wager has a p _wfo chance of winning, as per the preceding discussion; the second wager has a p _wuc chance of winning. But, this second probability can be restated in terms of its complement (i.e., as 1 − p _wfc), where p _wfc is the probability of winning a bet on the favorite at the closing spread. The overall payoff on these two wagers together is then equal to the sum of the two wagers' expected payoffs:

[p_{w f o} \cdot $ 21 + (1 - p_{w f o}) \cdot $ 0] + [(1 - p_{w f c}) \cdot $ 21 + p_{w f c} \cdot $ 0],

which simplifies to $21 + (p _wfo − p _wfc) · $21. A noteworthy point here is that this overall payoff does not involve any uncertainty: the two probability variables are each unique, instantaneous, cumulative probabilities, measured at the time that the betting market closes, as noted earlier. The realized return on the total $22 deployed across the two transactions is thus equal to [$21 + (p _wfo − p _wfc) · $21 − $22] ÷ $22. This realized return simplifies to:

(p_{w f o} - p_{w f c}) \cdot $ 21 / $ 22 - $ 1 / $ 22.

The return is a positive, monotonic function of p _wfo and is a negative, monotonic function of p _wfc.^7,8

Intraweek Information, Sentiment, or Neither

We now proceed to apply Equation 1 to three specific settings, the first one being a setting in which new information about the game arrives during the period. The second setting is one in which sentimental betting arrives during the period in the absence of any new information, and in the last setting, neither new information nor sentiment arrives during the period. In all three settings, no sentiment exists at the instant when the market opens.

New Information

When any new information arrives, the distribution of possible outcomes will shift. Besides causing the bookmaker to change the spread (as we will discuss next), this shift in the distribution will also affect p _wfo (the probability of winning the original wager). More specifically, as shown in Figure 2, if good news arrives about the favored team, the distribution of possible outcomes (and its median) will shift such that p _wfo will be greater than 50%.⁹ (Alternately, if bad news arrives about the favorite, the distribution will shift such that p _wfo will be less than 50%.)

In response to the good news about the favored team, the bookmaker picks the new (closing) spread to be equal to the median of the updated distribution of all possible outcomes of the game. The closing spread will be greater than the opening spread; the change in spread will be positive. In light of the new good information about the favorite, the probability of winning a bet on the favorite at the opening spread is greater than 50% and the probability of winning a bet on the favorite at the new (closing) spread is equal to 50%.

As discussed earlier, the bookmaker will change the spread in order to maintain a balance in the dollars bet on the two teams in the presence of the new information. The closing spread will be equal to the median of the distribution of outcomes that reflects the new information, making the probabilities of winning a bet on the favorite at this new spread (p _wfc) and of winning a bet on underdog at this same spread (1 − p _wfc) both equal to 50%. Equation 1 then simplifies to:

(p_{w f o} - 50 %) \cdot $ 21 / $ 22 - $ 1 / $ 22.

Most importantly, p _wfo is positively, monotonically related to the intraperiod change in point spread. When intraweek news about the favored team is good [bad], the change in spread is positive [negative], and p _wfo increases [decreases] from 50%. The magnitude of the news will determine the size of the change in spread; the size of the change in spread will determine how different p _wfo will be from 50%.

Sentimental Betting

The next setting is one in which sentimental betting arrives during the period in the absence of any new information. Without the arrival of any new information, the distribution of possible outcomes will not shift and, hence, p _wfo will remain at 50%. However, as depicted in Figure 3, if sentimental betting emerges on the favored team, the closing spread will be greater than the opening spread, and p _wfc will be less than 50%. (Alternately, if sentiment emerges for the underdog, the spread will decrease and p _wfc will be greater than 50%). Equation 1 then simplifies to:

(50 % - p_{w f c}) \cdot $ 21 / $ 22 - $ 1 / $ 22.

Most noteworthy here is the fact that p _wfc is negatively, monotonically related to the intraperiod change in point spread. When intraweek sentimental betting arrives on the favorite [underdog], the change in spread is positive [negative] and p _wfc decreases [increases] from 50%. The magnitude of the sentiment will determine the size of the change in spread; the size of the change in spread will determine how different from 50% p _wfc will be.

The bookmaker picks the new (closing) spread in response to the sentimental betting that arrives on the favorite, despite no change in the distribution of possible outcomes for the game. The change in spread is positive. In light of the same information about the game, the probability of winning a bet on the favorite at the opening spread is still 50% and the probability of winning a bet on the favorite at the new (closing) spread is less than 50%.

Neither New Information nor Sentiment

A final possibility is that the point spread does not change during the period at all (i.e., that the closing spread equals the opening spread), in which case p _wfo = p _wfc = 50%. The realized return when two offsetting wagers are made at the same point spread is thus $(50 % - 50 %) \cdot $ 21 / $ 22 - $ 1 / $ 22 = - $ 1 / $ 22$ .

Adapting the Model to Other Settings

More complex settings can exist, beyond the ones that we have examined to this point, namely, one-period settings where wagers are placed only when the market opens and when it closes, and the intraperiod event is either the arrival of new information, the arrival of sentimental betting, or the absence of any new information or sentiment during the period when the market is open for wagering.

All three settings stipulate that no up-front sentiment exists for either team in the contest, so that the opening point spread for a game is based solely on the information set available at the time the wagering market opens. If we were to repeat the preceding analyses and include nonzero sentiment in favor of one team in the contest at the time the market opens, the same general equation for realized return (Equation 1) will still apply for two offsetting $11 wagers: (p _wfo − p _wfc) · $21/$22 − $1/$22. However, the value for p _wfo must be adjusted to reflect the fact that the bookmaker will set the opening spread to a value different from what it would be in the absence of any up-front sentiment in favor of either team in the contest.

Also, a bettor can place a pair of offsetting wagers at any two points in time during which wagering on a game occurs, and not simply at the opening spread and closing spread, as specified in our model. Again, however, the general equation for realized return (Equation 1) still emerges with p _wfo now being the probability of winning a wager on the favored team at the spread that exists at the time when this first bet is placed and p _wfc be the probability of winning a wager on the favored team at the spread that exists when the second, offsetting wager is placed. Any time a bettor offsets an initial wager on one team with a second wager on the opposing team, he or she locks in some sort of realized return, be it positive or negative (or perhaps even zero).

Connecting Realized Return to Change in Spread

In the General Model section, we demonstrated that realized return is a function of the intraperiod difference in probability of winning a wager on the favored team at the opening spread and the probability of winning a wager on the favored team at the closing spread as per Equation 1: (p _wfo − p _wfc) · $21/$22 − $1/$22. The difference term inside the parentheses takes different values, depending on what type of event occurs during the trading period.

If new information about the wagered-upon game arrives during the period, the appropriate realized-return equation is Equation 2: (p _wfo − 50%) · $21/$22 − $1/$22. In the section on New Information, we demonstrated that p _wfo is a positive, monotonic function of the intraperiod change in spread that accompanies the arrival of the new information. Thus, realized return is necessarily a positive, monotonic function of the intraperiod change in spread.

If sentiment about the game arrives during the betting period, the realized return is reflected by Equation 3: (50% − p _wfc) · $21/$22 − $1/$22. In the section on Sentimental Betting, we showed that p _wfc is a negative, monotonic function of the change in spread that accompanies the arrival of sentimental bettors. Again, realized return is a positive, monotonic function of change in spread.

In the case where neither new information nor sentimental betting emerges for the wagered-upon game, both p _wfo and p _wfc equal 50% because the opening and closing spreads are equal and both spreads represent the median outcome. The realized return is fixed: −$1/$22 (or −4.55%). This realized return of −4.55% constitutes a single point in the monotonic relationship between realized return and change in spread.

Conclusion

We have thus established a positive, monotonic relationship between realized return and change in spread in our market for point-spread wagering on sports. Therefore, we assert that to employ change in spread as a proxy for realized return is appropriate. The point-spread wagering market is certainly one in which questions of market efficiency and rational behavior exist; having now established a legitimate proxy variable for realized return makes such tests all the more accessible.

A worthwhile area for future research will be to estimate the distributions of actual outcomes around point spreads (also known as forecast errors) for different batches of point spreads, and perhaps even for different sports. Knowing the values for standard deviation and mean for a distribution of forecast errors for a given opening point spread, we could then use the cumulative normal probability density function to identify the two key probabilities in the general model, namely, the probability of winning a wager on the favored team at the opening spread and the probability of winning a wager on the favored team at the closing spread. In turn, we could then plug these cumulative probabilities into the realized-return equations developed in this article to translate those values into a numerical value for realized return associated with a change in spread from the original spread.

Other future research exercises will be to develop the realized-return equations for other types of wagers besides point-spread wagers. For example, for odds betting, the realized return equation (as noted in an earlier endnote) is (1 − p _wfc) · {[1 + C − c]/[1 + O + c] − 1}, where p _wfc is the probability of the favored team winning the game (given the information set that exists when the betting market closes), C is the closing odds ratio, O is the opening odds ratio, and c is the bookmaker’s commission. For European-style spread betting, the expected payoffs would be more complex. Whereas our expected payoff for a point-spread bet involves cumulative probabilities and fixed gross payoffs of $21 or $0, the expected payoff for a spread bet will involve the summation of products of probabilities and payoffs, where the probabilities are of different possible game outcomes and the payoffs are a function of the game outcomes.

To establish the equations for realized return in these other types of betting markets will be useful. However, to have established a positive, monotonic relationship between realized return and change in spread in the point-spread wagering market is most important, given this market’s numerous similarities to the more traditional financial markets.

Footnotes

Notes

Acknowledgments

The authors thank John Settle, an anonymous Journal of Sports Eocnomics referee, and seminar participants at Portland State University, at the 2011 International Conference on Gambling Studies at Nottingham Business School, and at the 2011 Western Economics Association International meetings in San Diego.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

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