How To Find Out If Points Spread Is Related To A Bettors' Return On A Bet

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1 Do Investors Categorize? Evidence from the College Football Betting Market Greg Durham* College of Business Montana State University Bozeman, MT Mukunthan Santhanakrishnan* College of Business Idaho State University Pocatello, ID * Corresponding author. Please contact via or via phone:

2 Do Investors Categorize? Evidence from the College Football Betting Market Abstract We examine whether bettors categorize assets in the college football betting market. The singlecash-flow nature of wagers makes this market a powerful setting in which to test theories of investor behavior. We find that while prices of wagers co-move positively with prices of samecategory wagers, the underlying cash flows of these wagers do not co-move positively with underlying cash flows of similar wagers. These findings are consistent with the notion that bettors categorize bets, complementing the evidence from other financial markets in suggesting that investors place their investment alternatives into categories based on simple cues.

3 1 INTRODUCTION In a frictionless market with rational investors, any price changes are due to changes in fundamental values of the assets. On the other hand, if investors group assets into categories based on certain salient characteristics, then a change in preference towards or away from a category may cause changes in prices of assets within that group. Thus, when investors categorize assets, prices within a group of assets may move together even when the fundamental values of these assets demonstrate no comovement among them. Such categorization by investors could lead to an allocation of capital across investments that is inefficient in terms of risk and return. Recent articles provide evidence consistent with the notion that investors categorize assets along various dimensions. Returns on stocks that are added to the Standard & Poor s 500 Index co-move more with returns on other S&P 500 stocks than with returns on stocks that are not part of the index (Barberis, Shleifer, and Wurgler (2005)). 1 Stock returns of companies exhibit a strong degree of comovement with returns of other companies headquartered in the same geographical area (Pirinsky and Wang (2006)); however, these return comovements are not accompanied by positive comovement in the underlying cash flows (measured as contemporaneous quarterly earnings). Investors also appear to categorize investments by level of stock price, as Green and Hwang (2008) show in a study of stock splits. Studies of comovement around index additions and stock splits assume these events to be information-free, thereby also necessarily assuming no changes in underlying cash flows. While fundamental values should not change surrounding either type of event, some literature suggests that market participants expectations of cash flows are influenced by these events (Denis et al. (2003), Kalay and Kronlund (2009)). If such expectations do change, then return comovement may be related to comovement in expected future cash flows, emphasizing the need to examine 1 Other studies (Greenwood (2008) and Boyer (2008)) examine additions to and removals from other key stock-price indices and find similar results.

4 2 not only returns, but underlying cash flows as well. Pirinsky and Wang (2006) do examine both returns and cash flows; however, they do so in a setting where valuations are based on uncertain underlying cash flows over multiple periods. Therefore, while returns may possibly co-move in anticipation of comovement in contemporaneous earnings, return comovements may also reflect investors anticipation of comovement in earnings in any of a number of future periods. Since investors expectations of earnings in future periods are difficult to measure, the hypothesis that the observed return comovements reflect changes in asset fundamentals is difficult to reject with a great degree of confidence. Unique features of the college football betting market make it an ideal laboratory in which to distinguish between comovement in prices (i.e., in point spreads) caused by comovement in fundamentals and comovement caused by categorization. First, the single-cash-flow nature of wagers allows for a direct examination of comovement in underlying cash flows; furthermore, these cash flows are conveniently easy to measure. Second, like their investor counterparts in financial markets, point-spread bettors maintain expected-wealth maximization as a primary component of their value function, despite wagers having negative net present values. And finally, because of its similarities to financial markets in terms of participating clienteles, market makers, information dissemination, etc. the college football betting market is well suited for investigating investor behavior. Exploiting the advantages of this market and its similarities to securities markets, we test whether investors separate wagers into categories or styles, based upon the pair of teams involved in each gambling event. Bettors may be constrained by attention scarcity due to the weekly frequency with which they have to make wealth-allocation decisions, as well as to the number of games played each week. 2 In addition, the bets one-week lives require that bettors 2 Corwin and Coughenour (2008) find specialists on the New York Stock Exchange to be similarly con-

5 3 update their information sets every week, complicating their task of processing information. In order to simplify the task and the choices involved, bettors may categorize wagers based on simple cues such as conference affiliations of the teams involved in each game. One categorization that we specifically consider is wagers on games involving two majorconference teams and wagers on games involving at least one non-major-conference team. 3 We choose this categorization because games involving major-conference teams are likely to garner more pre-game attention among viewers, media outlets, and expert analysts. A second categorization that we consider is wagers on conference games and those on non-conference games. The conference-games category comprises essentially the same set of games each year; even casual bettors understand this common distinction between conference and non-conference games. If investors do categorize along these lines, we expect to see comovement among spreads within categories of wagers. We test for comovement consistent with categorization, following Barberis, Shleifer, and Wurgler s (2005) methodology and using bivariate ordinary-least-squares (OLS) regressions to examine whether changes in spreads are associated with changes in spreads for similar wagers. 4 Specifically, we construct two indices based on changes in spreads, one for similar wagers and another for dissimilar wagers. 5 We explore whether change in point spread for a wager is influenced more by the similar-wagers index or by the dissimilar-wagers index. For all four wager categories, we find that point spreads for a particular category of wagers co-move positively with spreads for similar wagers and are unrelated to spreads of dissimilar wagers. (For example, for wagers on major-conference games, the coefficient on the majorstrained due to the sheer number of stocks for which they make markets. 3 A major-conference team is a team from any conference that, during our sample period, had a contractual agreement with any of the following bowl games: Rose, Orange, Cotton, Sugar, or Fiesta. 4 The variable of interest in this study is change in point spread, which is sports betting s analog to change in price in financial markets. Also, a point spread is commonly referred to simply as a spread. 5 For expositional purposes, a similar wager is defined as a wager from the same category as the wager of interest. A dissimilar wager is any wager from the other category of wagers that does not include the wager of interest.

6 4 conference index is positive and significant, while the coefficient on the index for the complement set of wagers is not significant.) However, to determine whether such comovement is consistent with rationality or with categorization, we must also consider the relationship among underlying cash flows within and/or across different categories of wagers. Thus, we next test for comovement in the underlying cash flows among these same categories of wagers. Because a bet s cash flow is based on whether that bet wins or not, we employ an ordered probit analysis, wherein the dependent variable reflects whether a bet won, pushed, or lost. 6 Our two explanatory variables are constructed using outcomes for similar wagers and for dissimilar wagers during the same week. For all categories of wagers, we find that the likelihood of winning a bet is inversely related to the outcomes of similar wagers: the underlying cash flows among similar wagers do not co-move positively. Theoretical studies show that investors preference for categorizing can partly explain why prices co-move without any accompanying comovement in the underlying cash flows. Our dual findings that spreads co-move in the same direction among similar wagers and that the underlying cash flows do not co-move in the same direction are consistent with the presence of categorization behavior in the college football wagering market. These findings complement the evidence from other financial markets in suggesting that investors sort their investment alternatives into categories based on simple cues. This paper proceeds with a literature review in Section I, followed by an overview of our market in Section II and a discussion of our dataset, variables, and methodology in Sections III and IV. We report and analyze our results in Section V and conclude our paper in Section VI. 6 If the favorite wins by an amount exactly equal to the spread, the game is considered to be a push, so that each bet neither wins nor loses. All bettors simply receive their initially wagered amounts back.

7 5 I. RELATED LITERATURE Because categorizing individual entities into groups is a well-documented form of human thought (Rosch and Lloyd (1978), Wilson and Keil (1999)), researchers have started to examine reasons for, and the effects of, categorizing by investors in capital markets. Investors may categorize assets into groups in order to process vast amounts of information (Kahneman (1973)) or to reduce the number of decision choices (Mullainathan (2002)). The number of choices emerges as a key variable in this literature: the way that the alternatives are grouped together influences individuals consumption and wealth-allocation decisions (Fox, Ratner, and Lieb (2005)), as well as the satisfaction that they derive from such decisions (Mogilner, Rudnick, and Iyengar (2008)). Various researchers have modeled categorization behavior. Peng and Xiong (2006) model the learning process of investors with limited attention in a market with multiple assets and predict that investors develop category learning behavior. Behaving as such, investors allocate attention to market- and sector-level information rather than to firm-level information. Further, Veldkamp (2006) argues that when information acquisition is costly, investors would acquire a common set of information for a subset of assets. Barberis and Shleifer (2003) also point out that, to simplify the wealth allocation process, investors categorize stocks into various styles (such as value, growth, large capitalization, etc.) and allocate resources at the category level rather than at the individual asset level. These models predict that if investors price assets using common sets of information or if they categorize assets into groups, then the prices on assets in these groups may be influenced by common factors. 7 In such cases, the price of an asset within a category is likely to co-move more 7 Earliest evidence of the existence of comovement in asset returns is documented by Lee, Shleifer, and Thaler (1991), who find that individual investors irrational behavior causes comovement between closed-

8 6 with prices of other assets in the same category than with prices of assets outside of the style. In addition, correlations in prices among same-category assets may be higher than correlations among these assets fundamentals would suggest. Numerous studies examine whether asset prices co-move depending upon memberships in, additions to, or deletions from indices. Barberis, Shleifer, and Wurgler (2005) examine stocks of companies that get added to the S&P 500 index and find that the cross-sectional variation in returns on these stocks can be better explained by variation in returns of other stocks in the S&P 500 index than by variation in returns on stocks in the complementary set (namely, stocks of non-s&p 500 companies). Greenwood (2008) focuses on the Nikkei 225 stock index, wherein a number of stocks are overweighted by a factor of 10 or more relative to their weights in a valueweighted index. The study finds a strong positive relation between the degree of overweighting and the degree of comovement of a stock s returns with returns on other stocks in the index, and a negative relationship between index overweighting and comovement with returns on stocks outside of the index. Boyer (2008) examines comovement in asset prices using the S&P / Barra Value and Growth indices where all stocks in the S&P 500 are divided into either a value group or a growth group based on a single variable: the book-to-market ratio. The study finds that a stock return s covariation with an index return increases significantly after the noted stock enters the index and decreases significantly after the stock leaves the index. Overall, these studies suggest that when stocks are added to and deleted from indices, the degrees to which stock returns co-move with index and non-index returns change due to investors behavior. 8 In addition to membership in indices, geographical and trading locations also appear to affect investors behavior. Bodurtha, Kim, and Lee (1995) show that prices of closed-end country end mutual fund discounts and small-capitalization stock returns. 8 In addition, Green and Hwang (2008) examine stocks that undergo splits and find that prices of similarly priced stocks move together, suggesting that investors appear to categorize based upon stock-price magnitude.

9 7 funds are affected by US stock-market movements even though home-country net asset values are not. Froot and Dabora (1999) study Siamese twin stocks that have proportional claims to the same underlying cash flows but trade in different locations. The returns for each twin stock co-move more with their local market index returns than with each other. Chan, Hameed, and Lau (2003) examine Jardine Group companies located in Hong Kong that delisted from the Hong Kong stock exchange and took the Singapore stock exchange as their primary location, and the authors find a similar impact of trading location on the stock returns. Pirinsky and Wang (2006) find that stock returns of companies exhibit a strong degree of comovement with returns of other companies headquartered in the same geographical area. Overall, these various findings of comovements among stock returns are suggestive of investors categorizing assets into groups. II. THE COLLEGE FOOTBALL WAGERING MARKET The college football betting market is well suited for investigating all types of investor behavior, for several reasons. First, each financial asset (i.e., each sports bet) in this market has a life of only one period. By design, the single-period payoff structure eliminates the possibility that comovement in assets prices is caused by comovement in cash flows in any of the future periods. Second, the single cash flow associated with a bet is easy to measure because a sports bet s payoff is unambiguously realized when the game is complete. Third, required rates of return on wagers are nearly constant across all wagers in our entire sample, because the risk surrounding the expected payoff remains the same across all bets. Therefore, fundamental value of a wager defined as expected future cash flow discounted at an appropriate required rate of return is likely to change only if the underlying cash flow changes. All of these features make the college football betting market an advantageous venue in which to test whether comovement in prices is related to comovement in fundamental values. While maintaining these advantages over securities markets, sports wagering markets are

10 8 also similar to financial markets in numerous ways. For one, the representative point-spread bettor is trying to maximize expected wealth, just as investors are. These bettors, while facing an expected net loss per wager, still maintain a value function in which (expected) wealth maximization factors predominantly. While entertainment (or consumption) value is still a part of the point-spread bettor s value function, it plays a much smaller role than in the odds betting market, wherein the winning net payoffs can range from quite small (on a bet on a heavy favorite) to quite large (on a bet on a huge underdog). A subset of bettors is certainly sentimental, serving as sports wagering markets analogue to sentimental investors in the financial markets. While some bettors may exhibit sentiment for certain teams, certain types of teams, or certain types of games, research generally shows that sentiment s effect on point spreads is negligible. 9 The small effect is likely due both to sentimental bettors being a relatively small clientele in the betting marketplace and to professional bettors standing ready to exploit any arbitrage opportunities that might arise, as informed traders and arbitrageurs do in capital markets. The markets are similar in other ways, as well. Point-spread information and stock-price information are equally widely distributed: any major newspaper that reports daily stock quotes will also report a long list of point spreads for a wide variety of games on any given day. Sports betting wise guys saturate the news services and gambling trade publications with their knowledge, while expert stock pickers offer their predictions for capital markets. Finally, bookmakers for sports bets are analogous to market makers for stocks and bonds. Thus, any findings using sports betting markets might be useful for wider financial audiences, because of these highlighted similarities. 9 In studies of football wagering markets, Avery and Chevalier (1999) and Durham and Perry (2008) find that betting strategies designed to exploit of the presence of any sentimental bettors are not profitable or, at most, marginally profitable.

11 9 III. MARKET MECHANICS, DATA, AND VARIABLES Point-spread wagering for a game works as follows. A bettor can place a wager on either the favored team or the underdog team at the point spread, where the point spread can take any value to the nearest half-point (including zero). 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. 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. An important participant in all sports wagering markets is the bookmaker, the agency that facilitates bets on all games. The point spread is the mechanism by which the bookmaker manages the amounts of dollars bet on each team in a contest. The bookmaker begins by setting an opening spread, usually (for the case of football) one week before the game is to be played; wagering on the game then commences. Bettors enter the marketplace and place their respective wagers. Given that the bookmaker s objective is to minimize, if not eliminate, any exposure to the ultimate outcomes of the games, the bookmaker will move the spread any time an imbalance emerges in the wagering across the two teams. 10 Betting continues right up until the game s opening kick-off; the spread that exists when betting ceases is called the closing spread. For any game for which its closing spread ended up being different from its opening spread, we can rightly infer that the bookmaker had to revise that game s point spread at least once during the 10 Avery and Chevalier (1999) suggest that the sports book might maintain a "balanced book" to remain credible with regulators and bettors, who are wary of possible fixed games. Or, a casino may mandate a "balanced book" in order to reduce agency costs associated with potential risk-taking behavior by the book managers. In their book on casino operations management, Kilby, Fox, and Lucas (2004, pp ) describe the bookmaker's objective.

12 10 week, in the face of uneven demand for wagers on the game s two contestants. An imbalance in wagers may emerge due to either (i) the arrival of new information that is relevant to the game s expected outcome, (ii) the arrival of sentimental bettors, (iii) the arrival of bettors who compartmentalize wagers, or (iv) randomness. For reasons developed above, we suspect that bettors might not focus on all of the different intricacies associated with each team upon which they can potentially bet. Instead, a simpler strategy for bettors might be to focus on broader categories of wagers and then view each category as a general type of investment. We hypothesize that bettors categorize betting opportunities according to an important attribute possessed by each team in the sample: namely, conference affiliation. Bettors, knowing the membership of each Division-I college football conference, might sort wagers into either those on major games or those on non-major games. 11 Alternately, bettors might sort wagers into either those on conference games or those on nonconference games. We contend that bettors categorize and that they must repeat their categorization decisions independently each week since all wagers have a one-week lifespan. If our contention is true and if, for a particular category of wagers, the amount of money that arrives is different than anticipated, then spreads on wagers within that category will co-move. Our study uses eight years of college football betting data. Our database consists of 4584 NCAA Division-I college football games from 1991 through 1998 every game during that period for which both opening spread and closing spread were posted. 12 For each observation, we have the home team, visiting team, opening spread, closing spread, and final score of the game. The data were compiled by Computer Sports World (CSW), which uses point spreads posted by 11 For ease of exposition throughout the remainder of this paper, we will refer to each game involving two major-conference teams as a major game and to any game involving at least one non-majorconference team as a non-major game. 12 This sample period is desirable because of the relative stability in conference memberships over this period. Only seven teams (out of 113 in our sample) changed conferences during this time. See footnote 14 for details.

13 11 the Stardust Casino Race & Sports Book. 13 We augment the CSW dataset with each team s conference affiliation in each year and we define the following conferences as the major conferences: Atlantic Coast, Big East, Big Eight, Big Ten, Big Twelve, Pacific 10, Southeastern, and Southwest. From , these eight conferences had contractual agreements for their teams to play in the five major New Year s Day football bowl games. The non-major conferences, thus, are the Western Athletic, Big West, Mid-American, USA, and Independents (i.e., all teams that are not affiliated with any conference). 14 For each observation in our study, we are interested in the following key variables: change in spread, forecast error, outcome against the spread (win, push, or lose), two indices that capture weekly average change in spread and weekly median changes in spread for major games, two indices that represent the same two measures of central tendency for non-major games, conference games, and non-conference games. We are also interested in variables that measure outcomes against spreads for the same four categories games. The point spread and actual outcomes are ordered from the home team s perspective; i.e., they represent a difference between the home and visiting teams scores. For example, a spread equal to 9 indicates that the home team is favored to win the contest by 9 points, a spread of 5 suggests that expectations are that the home team will lose by 5 points, and an outcome of 11 points indicates that the home team won by 11 points. Change in Spread: The point spread is sports betting s equivalent to stock price, making 13 Computer Sports World is a common data source for academic studies of sports gambling markets. For example, see studies by Dare and MacDonald (1996) and by Durham, Hertzel, and Martin (2005). 14 We classify Notre Dame as a major-conference team, even though Notre Dame is an Independent. Also, Florida State (1991), Penn State (1991, 1992), and South Carolina (1991) were Independents for short parts of our sample period before joining major conferences; we classify these teams as majorconference teams for our entire study. Four teams (Houston, Texas Christian, Southern Methodist, and Rice) were part of the Southwest Conference from and then joined non-major conferences in Arkansas switched from the Southwest Conference to the Southeastern Conference in We compiled conference memberships by using The Official NCAA Football Records Book, published by the National Collegiate Athletic Association.

14 12 change in spread our analog to change in price of a stock. We note that most of the studies previously mentioned within this paper focus on returns as the variable of interest. Durham and Santhanakrishnan (2011) demonstrate a positive monotonic relationship between realized return and the change in point spread for a wager, thereby allowing us to employ change in spread as a reasonable proxy for realized return in our market. The change in spread for a wager is calculated as the closing spread minus the opening spread. The intra-week change in spread is a measure of excessive desired wagering on one of the teams in the wager since the spread should not change otherwise. A positive intra-week change in spread indicates that, as reflected by the spread, the home team became more favored (or less of an underdog) during the course of betting; a negative intra-week change suggests the opposite. Forecast Error: The forecast error for a wager is calculated by taking the difference between the game s actual outcome and the closing spread for the game, where actual outcome is simply calculated as the difference between the home team s points scored and the visiting team s points scored. A positive forecast error suggests either that a favored home team won the game by an amount greater than the closing spread or that a home-team underdog lost the game by an amount less than the closing spread or won the game outright. 15 A forecast error of zero indicates that the favored team won the game by an amount exactly equal to the spread. A negative forecast error indicates either that a favored home team won by less than the spread or lost the game outright or that a home-team underdog lost the game by more than the spread. Outcome Against the Spread (OATS): Because bettors are interested in the ultimate payoff on a bet, and because a bet s payoff is a function of the sign of the forecast error for the bet, we 15 As an example, an outcome of 11 points with an accompanying point spread of 9 results in a forecast error of +2: the favored home team won by two points greater than the spread. Or, as another example, an outcome of 3 points with an accompanying spread of 5 points also results in a forecast error of +2: the underdog home team lost by two points less than the spread.

15 13 also construct a dummy variable to represent whether a bet on the home team wins, ties, or loses against the spread. This variable is a dummy transformation (1, 0, 1) of the forecast error (greater than, equal to, or less than zero). In other words, it equals 1 if the home team covered the spread, 0 if the home team pushed, and 1 if the home team failed to cover the spread. Major Index: For a given wager on a major game in a given week, MAJOR is calculated as the average of point-spread changes on all other major-game wagers in that week; i.e., this index calculation excludes the point-spread change on the specific wager for which the index value is being calculated. For a given wager on a non-major game in a given week, the MAJOR index variable is calculated as the average of point-spread changes on all major-game wagers in that week. 16 We also alternately specify MAJOR for each wager in a given week, using the median point-spread change on all major-game wagers during that week. 17 Non-Major Index: For a given wager on a major game in a given week, NON-MAJOR is calculated as the average of point-spread changes on all non-major-game wagers in that week. For a given wager on a non-major game in a given week, NON-MAJOR is calculated as the average of point-spread changes on all other non-major-game wagers in that week. As an alternate specification, we use the median point-spread change for all non-major-game wagers in that given week for the non-major index. Conference Index and Non-Conference Index: The indices CONF and NON-CONF are calculated, for each wager in each week, in similar fashion to the major and non-major indices. These indices are alternately specified as means and as medians. 16 For certain weeks within our sample period, the number of observations that fall into a particular category of wagers might be quite small, thereby increasing the impact that a single outlying value for Change in Spread may have on our construction of this index and the three following indices that are also based on change in spread. Therefore, we choose to windsorize our entire dataset, using the 5th- and 95thpercentile values for Change in Spread ( 3 and +3, respectively) as our cutoffs. 17 We use medians as a second alternative by which to specify these same four indices, in order to avoid any undue impact that an outlier value may have when we construct these indices using averages.

16 14 Major Outcome Against the Spread: For any given wager in a given week, the M-OATS index variable is specified alternately as the average outcome against the spread and as the median outcome against the spread, for all major-game wagers in that week. Non-Major Outcome Against the Spread, Conference Outcome Against the Spread, and Non-Conference Outcome Against the Spread: NM-OATS, C-OATS, and NC-OATS are calculated, for each wager in each week, in similar fashion to the major and non-major outcomeagainst-the-spread indices. Table 1 reports descriptive statistics for the two key variables used in our study: change in spread and forecast error. The mean and median changes in spread are and ; changes in spreads range from a 95th-percentile value of to a 5th-percentile value of The mean and median forecast errors are and ; 90% of our observations forecast errors lie between and IV. METHODOLOGY We follow a methodology employed by Pirinsky and Wang (2006), a method that is similar to that of Barberis, Shleifer, and Wurgler (2005), to examine whether bettors categorize bets. We examine two types of categorization: first, wagers on major games and wagers on non-major games, and, second, wagers on conference games and wagers on non-conference games. We choose the first categorization because games involving major-conference teams are likely to garner more pre-game attention among viewers, from media outlets, and in expert analysts prognostications. The motivation for the second categorization is that whether a game is a conference game or a non-conference game is probably the single most common distinction used in college football. If the valuation of wagers is influenced by such categorizations, we would expect to find that spreads for one category of wagers are more likely to be related to spreads of similar wagers.

17 15 Therefore, we examine whether change in spread for a wager is influenced more by change in spread for similar wagers than by change for dissimilar wagers. As explained earlier, change in spread is our study s analog to percent returns used by both Pirinsky and Wang (2006) and Barberis, Shleifer, and Wurgler (2005). If spreads for wagers are determined solely by fundamental factors, then we would anticipate that these spreads will be influenced by change in spread for any other wagers only if the underlying cash flows for a particular category of wagers are similarly moving together. Since a bet s payoff is a function only of whether the wager won, pushed, or lost and is not a function of the magnitude of the forecast error we model the underlying cash flow using an ordered probit analysis, wherein the dependent variable reflects whether a bet won, pushed, or lost. Our two explanatory variables capture whether wagers on similar games won, pushed, or lost and whether wagers on dissimilar games won, pushed, or lost during the same week. The payoff on a bet is our analog to earnings, which are used by Pirinsky and Wang (2006) as a proxy for underlying cash flow for a stock. V. RESULTS AND ANALYSIS If bettors do, indeed, separate bets into the two different style categories suggested above, then we might expect to find different intra-week price, or point-spread, patterns across the different styles of wagers. As our first test for evidence consistent with style investing, we segment our dataset into subsamples of wagers, depending on the conference affiliation(s) of each game s participants, and then calculate mean values for changes in spreads and forecasts errors for these different subsamples. Table 2 reports mean and median values for changes in spreads and forecast errors (along with the number of observations used to calculate, as well as statistical significances). Panel A of Table 2 reports these statistics for the two different categories of wagers: wagers on major games and wagers on non-major games. The mean change in spread of and the mean

18 16 forecast error of for wagers on major games are statistically significant at the 1% and 10% levels, respectively. The mean change in spread of and the mean forecast error of are both significant at the 10% level for wagers on the complement set of games. Panel B reports the same set of statistics for the next categorization: wagers on conference games and wagers on non-conference games. The mean change in spread equals for conference games and is statistically significant at the 1% level. The mean forecast errors for conference and non-conference games are and , both significant at the 10% level. Taken as a whole, the statistically significant differences in these key variables across matchup types suggest that bettors may be categorizing wagers along these lines. We thus proceed to investigate for further evidence that bettors do, indeed, categorize. A. Categorization of Games into Major- and Non-Major-Conference Games In our next tests, we follow the leads of both Pirinsky and Wang (2006) and Barberis, Shleifer, and Wurgler (2005), using an ordinary least squares regression analysis to estimate the following equation separately for each category of wager: CHANGE = + majormajor + non-majornon-major +. (1) For each category, we regress the vector of point-spread changes on wagers in that category on vectors of the two weekly indices described in section II above: MAJOR and NON-MAJOR. More specifically, we run three regressions per category: one for the overall time period, one for the earlier half of the sample ( ), and one for the latter half ( ). Furthermore, for each of these six regressions, we alternately specify the two indices using medians, as explained earlier in our descriptions of the two indices. Table 3 reports results for the twelve regressions. The first trio of regressions uses the subsample of wagers on major games, with the indices specified as averages. One coefficient is positive and statistically significant: is the coefficient on MAJOR for the latter-year re-

19 17 gression. This positively-signed coefficient suggests that the spreads on major-game wagers are positively related to the major index of spreads. The next set of regressions is for wagers on nonmajor games: none of the coefficients from any of the three regressions emerges as being statistically significant. The third set of regressions (or the first set in Panel B) is for the wagers on major games, but now using medians to construct the indices. Three coefficients are statistically significant within these regressions and all three of them are coefficients on MAJOR (0.6206, , and ). The final regressions are for the non-major-game subsample of wagers, using medians for the indices. The coefficient on NON-MAJOR is statistically significant for all three regressions: , , and for the overall, , and regressions, respectively. The six positively-signed coefficients suggest that the spreads in each category of wagers are positively related to spreads of other wagers within the same category. From the twelve regressions, seven similar-wager coefficients (six of which emerge when our indices are specified as medians) end up being statistically significant and positive, suggesting comovement among spreads within categories. In addition, all twelve dissimilar-wager coefficients end up being insignificantly different from zero, suggesting no comovement in spreads across categories. At this point, we consider it important to acknowledge that we performed the same twelve regressions, utilizing an alternate specification that was advanced by both Dare and McDonald (1996) and Dare and Holland (2004). We understand the authors argument that we potentially must account for two well-documented fixed effects: the home-team effect and the favorite effect. Since we order the point spreads for all of our games from the home teams perspectives, we still must be careful to separate the favored-team effect from the effects that our indices of interest might have on the dependent variable. Thus, we re-specify regression equation (1) to in-

20 18 clude a dummy variable and two interaction terms to capture any potential favored-team effect. CHANGE = + favorite + majormajor + non-majornon-major + favorite majormajor + favorite non-majornon-major +. (1a) The relationships in which we are primarily interested remain those between our dependent variable and the two non-interaction explanatory variables. Our findings on all of these variables' coefficients are exactly the same under this new specification as under the original specification for which we report results in this paper. All coefficients under this new model remain statistically significant and of the same sign, as compared to their counterparts under the original models. As mentioned earlier, theoretical models of categorization behavior predict that if investors categorize, then the comovement among prices of assets within a category may different from the comovement among the underlying cash flows of the same assets. We thus next turn our attention to the question of whether cash flows actually co-move positively, negatively, or not at all within styles, across styles, or both. To address this issue, we employ an ordered probit analysis on the data, using the following specification: Prob ( OATS = 1 ) = F( 1 + moatsm-oats + nmoatsnm-oats) (2) Prob ( OATS = 0 ) = F( 2 + moatsm-oats + nmoatsnm-oats) F( 1 + moatsm-oats + nmoatsnm-oats) Prob ( OATS = 1 ) = 1 F( 2 + moatsm-oats + nmoatsnm-oats) where the dependent variable reflects whether the home team covers the spread, pushes, or fails to cover the spread. As discussed earlier, the outcome against the spread for a given bet is the variable that determines whether a bettor wins, ties, or loses on that bet. Thus, outcome against the spread is an appropriate proxy for the underlying cash flow associated with a bet. The explanatory variables are the Major Outcome Against the Spread index and the Non-Major Outcome Against the Spread index.

21 19 Table 4 reports results for our twelve probit analyses. Results for the subsample of wagers on major games appear in the first halves of Panels A and B, the difference being that the explanatory indices are specified as averages in Panel A and as medians in Panel B. The coefficients on the major-game index (M-OATS) are statistically significant and negative in all six regressions, suggesting that cash flows on major-game wagers co-move negatively. Results for the subsample of non-major games also shows six statistically significant, negative coefficients on the similar-game index (i.e., on NM-OATS), suggesting that cash flows on non-major-game wagers also co-move negatively. Taken together, our results from Tables 3 and 4 reveal that positive comovement across assets prices is not accompanied by positive comovement across those assets underlying cash flows; in fact, the accompanying comovement in cash flows is negative. (Pirinsky and Wang (2006) also find negative comovement in underlying cash flows.) As suggested by Barberis and Shleifer (2003), the dissimilar comovements are consistent with the notion that bettors group wagers into the two different categories: wagers on matchups involving two major-conference teams and all other wagers. B. Categorization of Games into Conference and Non-Conference Games We next perform an identical series of tests for a second categorization of wagers, namely wagers on games involving two teams from the same conference and wagers on games involving two teams from different conferences. We begin with the following regression: CHANGE = + confconf + non-confnonconf +. (3) We run regressions separately for all wagers on conference games and for all wagers on nonconference games, for the overall time period, for the earlier half of the sample, and for the latter half. Table 5 reports the regression results. Our general finding largely driven by our results in Panel B is that the changes in spreads are positively associated with changes in spreads for wa-

22 20 gers within the same category, suggesting comovement among spreads for wagers within the same category. We next examine whether cash flows actually co-move within categories, across categories, or both, employing ordered probit analyses on the data: Prob ( OATS = 1 ) = F( 1 + coatsc-oats + ncoatsnc-oats) (4) Prob ( OATS = 0 ) = F( 2 + coatsc-oats + ncoatsnc-oats) F( 1 + coatsc-oats + ncoatsnc-oats) Prob ( OATS = 1 ) = 1 F( 2 + coatsc-oats + ncoatsnc-oats) where the dependent variable is the same as described for equation (2). The explanatory variables are the Conference Outcome Against the Spread index and the Non-Conference Outcome Against the Spread index. We perform probit analyses separately for each category of wagers, with the explanatory indices alternately specified as averages and medians, for the overall time period and for the two time subsamples. Table 6 reports the probit results, results that are consistent across specifications, wager subsamples, and time periods. In all twelve analyses, we find that underlying cash flows do not co-move positively with cash flows of wagers within the same category. (In fact, for all specifications, the underlying cash flows actually co-move negatively.) Together, the results from Tables 5 and 6 show that while point spreads co-move positively within categories of wagers (and do not co-move at all across categories of wagers), such comovement is not accompanied by any positive comovement in underlying cash flows within categories. These findings support the idea that bettors categorize wagers into bets on conference and on non-conference games. VI. CONCLUSION The objective of this paper is to examine whether bettors categorize bets on games in col-

23 21 lege football wagering markets. Researchers have a growing belief that investors in capital markets categorize assets, leading to comovement in asset prices within categories. This study adds to the body of knowledge by testing this proposition in the college football betting market, an experimental setting that is advantageous for various reasons. First, a sports wager has a life of only one period, with the single-period payoff structure eliminating the possibility that comovement in assets prices is caused by comovement in cash flows in any of the future periods. Second, the cash flow from a sports bet is easy to measure because the bet s payoff is unambiguously realized when the game is complete. Third, required rates of return on wagers are nearly constant across all wagers since the risk surrounding the expected payoff is the same for every bet. Furthermore, bettors are likely to be constrained by attention scarcity because of the number of games played each week. In order to simplify, bettors may categorize wagers based on teams conference affiliations. We employ an amalgam of the methodologies utilized by Pirinsky and Wang (2006) and by Barberis, Shleifer, and Wurgler (2005) to examine whether variation in spreads of wagers is explained more by variation in spreads of similar wagers than by variation in spreads of dissimilar wagers. Overall, we find that changes in spreads are related to changes in spreads for similar wagers while they are unrelated to changes in spreads for dissimilar wagers (though these findings almost exclusive apply to when we specify the similar- and dissimilar-wager indices as medians, as opposed to as averages). In addition, for all categorizations, for both specifications of indices, and for all time periods, we find that variation in underlying cash flows for wagers is negatively related to underlying cash flows of similar wagers, similar to Pirinsky and Wang s findings for earnings. In aggregate, positive comovement across assets prices is not accompanied by similar comovement across those same assets underlying cash flows. In a frictionless market with rational investors, any comovement across asset prices is ex-

24 22 pected to be associated with comovement across underlying cash flows of those assets. Yet, if investors group wagers into different categories, asset prices within a group may move together even when the underlying cash flows exhibit no comovement. Our evidence is consistent with this latter alternative, suggesting that investors in our marketplace (i.e., bettors) categorize bets on games based on teams conference affiliations. Our investigation of categorization adds to the growing body of literature on whether investors and bettors behave rationally when transacting in capital markets and wagering markets. Future research might investigate the reasons for why bettors appear to categorize. Possible reasons include bettors sentimental attachment for certain conferences teams, disparity in information availability across conferences, or bettors desire to simplify their investment (wagering) choices.

25 23 BIBLIOGRAPHY Avery, C., Chevalier, J., Identifying Investor Sentiment from Price Paths: the Case of Football Betting [1999] Journal of Business 72 at pp Barberis, N., Shleifer, A., Style Investing [2003] Journal of Financial Economics 68 at pp Barberis, N., Shleifer, A., Wurgler, J., Comovement [2005] Journal of Financial Economics 75 at pp Bodurtha, J., Kim, D., Lee, C., Closed-End Country Funds and US Market Sentiment [1995] Review of Financial Studies 8 at pp Boyer, B., Comovement Among Stocks with Similar Book-to-Market Ratios [2008] unpublished working paper, Brigham Young University. Chan, K., Hameed, A., Lau, S., What If Trading Location is Different from Business Location? Evidence from the Jardine Group [2003] Journal of Finance 58 at pp Corwin, S., Coughenour, J., Limited Attention and the Allocation of Effort in Securities Trading [2008] Journal of Finance 63 at pp Dare, W., Holland, S., Efficiency in the NFL Betting Market: Modifying and Consolidating Research Methods [2004] Applied Economics 36 at pp Dare, W., MacDonald, S., A Generalized Model for Testing the Home and Favorite Team Advantage in Point Spread Markets [1996] Journal of Financial Economics 40 at pp Denis, D., McConnell, J., Ovtchinnikov, A., Yu, Y., S&P 500 Index Additions and Earnings Expectations [2003] Journal of Finance 58 at pp Durham, G., Hertzel, M., Martin, J., The Market Impact of Trends and Sequences in Performance: New Evidence [2005] Journal of Finance 60 at pp Durham, G., Perry, T., The Impact of Sentiment on Point Spreads in the College Football Wagering Market [2008] Journal of Prediction Markets 2 at pp Durham, G., Santhanakrishnan, M., Point-spread Wagering Markets Analogue to Realized Return in Financial Markets [2011] unpublished working paper, Portland State University. Fox, C., Ratner, R., Lieb, D., How Subjective Grouping of Options Influences Choice and Allocation: Diversification Bias and the Phenomenon of Partition Dependence [2005] Journal of Experimental Psychology 134 at pp Froot, K., Dabora, E., How Are Stock Prices Affected by the Location of Trade? [1996] Journal of Financial Economics 53 at pp Green, C., Hwang, B., Price-Based Return Comovement [2008] scheduled for forthcoming publication in Journal of Financial Economics. Greenwood, R., Excess Comovement of Stock Returns: Evidence from Cross-Sectional Variation in Nikkei 225 Weights [2008] Review of Financial Studies 21 at pp Kahneman, D., Attention and Effort. Englewood Cliffs, NJ: Prentice-Hall. Kalay, A., Kronlund, M., Stock Splits Information or Liquidity [2009] unpublished working paper, University of Chicago.