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Using the NFL Gambling Market as an Alternative Asset ClassKevin Meers, Sam Waters, and Zack WortmanStatistics 107Spring 2013A particular focus of modern portfolio theory involves diversifying risk through identifying and investing in assets that are uncorrelated with one another. In this pursuit, many investors have explored so-called “alternative asset classes” that are not traded on major stock markets. In this paper, we identify the market for gambling on games in the National Football League (NFL) as one such alternative asset. By modeling various betting outcomes, we develop multiple betting algorithms that provide returns from 7% to 20%, depending on what level of risk the investor would like to assume. These returns are, in theory, totally uncorrelated with market risk, and therefore valuable to any portfolio manager; further, they appear to dominate multiple market indices.Table of ContentsIntroductionMethodologyResultsDiscussionTablesFiguresReferencesAppendix2681216222526I. IntroductionLiterature ReviewA particular focus of modern portfolio theory involves diversifying risk through identifying and investing in assets that are uncorrelated with one another. Given the centrality of traditional securities markets in the contemporary investment world and the increasing interdependence of the global economy, an asset that is uncorrelated with worldwide economic performance could provide especial utility in risk diversification. While the performance of individual securities may depend on various asset-specific or industry-specific risks, virtually all can be affected by market volatility, and this shared risk factor problematizes efforts to minimize risk in a given portfolio. This shared association of stocks and bonds with market volatility incentivizes a search for alternative asset classes, and a variety enjoy heavy investment. Collectibles, from baseball cards to Impressionist paintings, are among the most public examples, and commodities and real estate can also offer investments that are (ideally) completely uncorrelated with market returns. In this project, we have chosen to explore the betting market of the National Football League as an alternative asset. The idea of sports gambling as an alternative asset class has existed for at least forty-five years, since Lyn Pankoff’s essay “Market Efficiency and Football Betting.” Pankoff hypothesized that, in sports betting, the collective judgments of individual bettors contained inefficient patterns that could be exploited for consistent individual profit and that betting odds were not necessarily a good barometer of intrinsic value (Pankoff 1968). In Pankoff’s view, football betting provided a unique lens for the analysis and understanding of market efficiency, as it allowed particular ease of access in searching for patterns and trends, given the clarity and availability of information (Pankoff 1968). A generation later, Joseph Golec and Maurry Tamarkin published “The Degree of Inefficiency in the Football Betting Market” in 1991. In their paper, Golec and Tamarkin identify several reasons why they hypothesize that analysis of football betting could produce a system that could yield consistent returns. First, much of the information regarding professional football games is inherently public, with various statistical measures of player and team performance (as well as additional factors like playing surface, kickoff time, etc.) easily accessible to all prospective bettors (Golec and Tamarkin 311). Second, the market is sufficiently large for robust statistical analysis, with many possible wagers and many bettors per wager (Golec and Tamarkin 312). Finally, the mix of professional and amateur bettors in the market suggests both that consistently profitable betting systems can exist and that the market is likely inefficient (Golec and Tamarkin 312). Ultimately, the authors conclude that significant and easily identifiable inefficiencies do exist in the NFL betting market, with bettors tending to underestimate the impact of home-field advantage and overestimate the impact of streaks and recent results (Golec and Tamarkin 312). Golec and Tamarkin note that both of these inefficiencies seemed to be disappearing, however, and future researchers or prospective systemized bettors would need to identify new, less obvious inefficiencies in the market. The notion of sports betting as an alternative asset class has gained traction since the publication of Joe Peta’s Trading Bases: A Story About Wall Street, Gambling, and Baseball, in which Peta outlines his creation of a baseball betting fund that produced a forty percent return in its first year by exploiting similar statistical inefficiencies to those that we attempted to identify in this project. Peta emphasizes a conceptual transition from viewing systemized sports betting as a form of entertainment gambling to a form of legitimized strategic trading (Peta 2012). Much as former Oakland Athletics General Manager Billy Beane utilized the capabilities of sabermetrics to more accurately assess value in baseball player performance, Peta uses those same capabilities to find value in the betting market. Peta uses rigorous statistical analysis to, in his own words, “quantify luck” (Peta 2012). Gambling TerminologyThere are two types of wagers one can traditionally make on an NFL game: the spread and the over-under. The spread is the amount of points “given” to the away team in order to draw equal volumes of bets on both teams. For example, if the spread is minus three, Vegas “gives” the away team three points, so the home team has to win the real game by more than three to “cover” and win the bet. An over-under simply states the predicted total points scored in a game by both teams combined. If the over-under is seventy, and the teams combine to score more than seventy, over would be the winning bet; if they score under seventy, under would be the winning bet. If, in the last two examples, the home team won by exactly three and the teams combined to score exactly seventy points, both bets would be a push and all bets would be nullified. Given this structure, we made four different types of bets when back-testing the strategy we devised using our model. These bets were “over”, “under”, “cover”, and “not cover”. DataIn our attempts to create a model that outperformed Las Vegas betting lines, we identified several factors that we felt might yield exploitative patterns, including the day of the week games were played, the playing surface (grass or turf), and the stadium type (outdoor, dome, or retractable roof). In addition to these background characteristics, our model depended heavily on a metric developed by Football Outsiders called Defense-adjusted Value Over Average (DVOA). DVOA aggregates the degree of success that a team had on all of its plays from scrimmage compared to the success of the average team on those plays. In evaluating each play’s success, it controls for the situation (down, distance, time, etc.) and opponent quality of that play. DVOA essentially describes the past performance of a given team at any point in the season, which we can use to predict how that team will perform in future games.We collected data on 5159 regular season games from the 1991 through the 2011 NFL seasons from Pro-Football- and to use in our model. These data include the Vegas spread, over-under, the result of the spread, the result of the over-under, the field type, stadium type, time of the game, week number, and offensive, defensive, and special teams DVOA for both the home and away teams. Even armed with these data, we anticipated that it would be extremely difficult to create a model that could consistently outperform betting lines set by professionals. We hoped, however, that the high proportion of football bets that are made by amateurs (or professionals who do not use statistical models) might create inefficiencies in the market that would be fairly easy to identify and significant enough to achieve healthy returns.II. MethodologyThe first component of this study uses R to model the probability of potential outcomes for lines and over-unders of all NFL regular season games from 1991 to 2011 using logistic regression. The second part conducts a simulation in R in which we place bets on all of these games based on the model’s predictions.We model four different binary outcomes: total points scored over the number set by Vegas (one if so, zero if not), total points scored under the number set by Vegas (one if so, zero if not), the home team does cover the spread set by Vegas (one if so, zero if not), and the home team does not cover the line set by Vegas (one if so, zero if not). We model each of these four outcomes separately rather than using two models with binary outcomes of over and under and cover and not cover because of the possibility of a third outcome in each case, “push”, in which the actual outcome falls right on the line or over-under set by Vegas so that the bet is nullified and money is returned. For each of the four outcomes, we investigate three different model types. All are logistic regressions. The first uses background characteristics which include week (1-17), field surface (grass or not grass), dome type (dome or not dome), roof type (retractable or not retractable), four dummy variables indicating scheduling outside of the traditional Sunday afternoon time slot (Saturday, Sunday Night, Monday, and Thursday), the Vegas spread, and the Vegas over-under. The second model includes these background characteristics along with Football Outsiders efficiency metrics. These metrics are offensive, defensive, and special teams DVOA for both the home and away teams. The third model includes all of the variables from the second model, along with additional interaction terms for all of DVOA variables.The betting simulation places $100 bets on all outcomes (over, under, cover, not cover) for which the model imputes a probability over 50%. We then calculate profits and expenditures for all bets made on each type of outcome to find our total returns across all games. We run the same simulation 16 times, with each simulation progressively cutting out an additional week in the beginning of the season. (So in the first simulation we make bets starting in Week 2, in the second we start in Week 3, in the third we start in Week 4, etc.) This enables us to see how returns and risk vary as we wait longer into each season to start making our bets. We reset the model for each of these simulations to only include the weeks being examined. We also examine profits in only recent seasons to see if the model’s returns hold up in more recent seasons to confirm that the market’s inefficiencies are still exploitable.III. ResultsSince we model four dependent variables (probability of over, under, cover, or not cover) separately, and we do so for 16 sets of weeks (Week 2-17, Week 3-17, Week 4-17,…, Week 17), we produced dozens of models over the course of our analysis. Rather than display and evaluate the results for every single regression, we will examine in detail only the four full models including interaction terms for each of the four dependent variables over the entire season (Weeks 2-17). These models are generally similar to those produced after subsetting the data by week and are more comprehensive than the models excluding DVOA terms, so explaining these models should convey the necessary understanding of all models used.P(Over)The results of this regression can be found in Table 1. This model estimates the probability of an over on the over-under line Vegas sets, given the background and DVOA variables for the teams involved for all weeks. The only regressors that were significant at the 5% level were away team offensive DVOA and home team special teams DVOA in primetime. The coefficients on both are positive, indicating that the expected probability of an over increases as away offensive DVOA increases and as home special teams DVOA increases. It appears that Vegas does not account enough for better away offenses and home special teams in its expectation for total scoring. It is important to remember that these results do not mean that these variables are positively associated with higher-scoring games. It means they are positively associated with higher-scoring games in comparison to the expectations set by Vegas’ over-under. This distinction is key and applies to all of the over-under and spread models used in this study.P(Under)The results of this model are displayed in Table 2. This model estimates the probability of an under on the over-under line Vegas sets, given the background and DVOA variables for the teams involved for all weeks. Over-under and away team offensive DVOA are significant at the 5% level. The coefficient on over-under is positive, indicating that as the number of total points that Vegas expects goes up, the probability of an under goes up. In other words, Vegas is setting its over-unders too high, on average. The coefficient on away offensive DVOA is negative, so the probability of an under increases as away offensive productivity decreases. As in the P(Over) model, Vegas underestimates the impact of the away offense on total points scored.P(Cover)The results of this analysis are found in Table 3. This model estimates the probability of the home team covering the spread that Vegas sets, given the background and DVOA variables for the teams involved for all weeks. The dummy variables for Monday and Thursday night games are significant at the 1% level with positive coefficients. This indicates that the home team is more likely to cover the spread on Monday and Thursday night games. The interaction variable between home defensive DVOA and away offensive DVOA is significant at the 5% level with a positive coefficient, but its interpretation is complex because a more negative defensive DVOA implies a stronger defense. Thus the probability of a cover increases when home defensive DVOA and away offensive DVOA share the same sign; this relationship implies a mismatch: either the home defense is strong and the away offense is weak or the home defense is weak and the away offense is strong. The probability of a cover decreases when there is no mismatch, i.e. when either both the offense and defense are strong or both are weak.P(Not Cover)The results of this model are available in Table 4. This model estimates the probability of the home team failing to cover the spread that Vegas sets, given the background and DVOA variables for the teams involved for all weeks. Over-under and away team offensive DVOA are significant at the 5% level. The dummy variables for Monday and Thursday night games are significant at the 1% level with negative coefficients. This indicates that the home team’s probability of not covering the spread decreases on Monday and Thursday night games. (This wording is confusing but necessary in order to interpret the model explicitly and account for pushes.) This is basically the same as saying that the home team covers more often on Monday and Thursday nights, which corroborates the results from regression III.In order to predict outcomes when betting only on a certain subset of weeks, we repeated the above modeling techniques while excluding data from the weeks for which we were not betting. These models are too numerous too display and analyze in full, but the general framework for analyzing these models is the same as seen above. After modeling all four outcomes over all of the necessary timeframes, we calculated returns and standard deviation for each of those timeframes. Figures 1 and 2 provide a graphical representation of how risk and return vary depending on when we start placing bets, and are explicitly stated in Table 5. Figure 1 shows how expected returns increase as the bettor delays his initial week of betting longer. The solid line represents mean expected return, while the dotted lines represent 67% and 95% confidence bands. Figure 2 plots risk against return with each dot representing a different week at which we began betting. We can see here that risk and return increase together in approximately linear fashion as we move deeper into the season. A bettor who starts betting in week two can expect a mean return of 7.0% and risk of 5.1%, while a bettor who starts in week 17 can expect a mean return of 20.3% and risk of 16.4%, so return and risk can vary greatly depending on how long the bettor waits to collect information and make his bets. Separate simulations for games played in 2000 and later, 2009 and later, and only 2011 produced similar results, indicating that these types of returns persist over time, and that the football betting market was still exploitable as of 2011.While these returns are in theory completely uncorrelated with the overall market, it is worth examining whether or not they actually are. To do so, we downloaded data on the S&P 500 since 1991 and correlated its yearly returns with the returns of each of our betting strategies. These correlations can be found in Table 6. The highest correlation is only 0.31, and most are much closer to zero. IV. DiscussionThe viability of sports betting as an alternative class has gained traction in recent years within the body of extant scholarly literature. Its lack of correlation with the overall market presents a unique opportunity for risk diversification in portfolio construction and possible inefficiencies in the betting market indicate the potential for healthy returns. We used logistic regression to model the outcomes of over-unders and spreads for NFL games, in an attempt to exploit these potential inefficiencies. We were able to predict outcomes with enough accuracy to earn healthy returns when back-testing our model, using Football outsiders per-play efficiency metrics and background variables like field surface and kickoff time.Mean returns depended on which week the bettor started implementing the model, with risk and return increasing as he waited longer to start. This is not surprising, because as we get deeper into the season, we collect a larger sample of information about each team. A team’s DVOA is a more accurate reflection of its true talent level in Week 17 than it is in Week 2. Returns were higher later in the season because of the model’s improved accuracy, but this came at the cost of higher standard deviation because funds were necessarily concentrated among a smaller number of bets. This trend allows for the creation of 16 different portfolios, each containing bets starting in a different week. Portfolios with bets starting in later weeks have higher expected returns and higher expected risk, while portfolios with bets starting in earlier weeks have lower expected returns and lower expected risk. The decision on which of these betting portfolios to invest in depends on the utility function of the individual investor. A more risk-averse investor would start betting at the beginning of each season, while a less risk-averse investor would wait to start betting until the latter half of the season. Either type of investor could incorporate his preferred betting strategy into his overall portfolio, using the football betting market’s independence from the overall market to diversify risk in his overall portfolio.One issue that cuts into the returns discussed above is what sports books refer to as the vigorish, or the vig. This is the fee that the book collects from the gamblers on each bet. Pinnacle, the top sports book, charges a vig of 5% on the amount expended for most bets. This dampens the impact of our returns discussed above, though one would still expect positive returns. Reasons for optimism remain, however, for the prospects of consistent healthy returns when using sports betting as an alternative asset class. Our model, while effective, is relatively simple compared to what a professional bettor could construct with more time and information. We used only background characteristics and one efficiency metric to model outcomes, but a wealth of data exists that could improve the model. Using scouting grades, individual player performance metrics, and individual player interaction terms would improve the model’s accuracy greatly. Incorporating news regarding player injuries, suspensions, playing time, and trades, would greatly improve the model’s efficacy as well. We might also produce a more accurate model if predicting betting outcomes in the playoffs. This paper demonstrates that returns increase as we wait longer into the season to start placing bets. Waiting until the playoffs to start betting allows even more time to pass and for us to collect even more information. The unique conditions of playoff football might also produce additional exploitable inefficiencies that would boost returns. It is encouraging that such a simple model yielded positive returns, when further research could yield a significantly more accurate model with much greater expected returns.Further, our method’s risk and return significantly outperforms and dominates the market indices “SPY” (the S&P 500) and “QQQ” (NASDAQ); it our algorithm almost dominates “TIP” (10-year Treasury Bonds), but TIP’s yearly returns have a standard deviation a half percentage point lower than our Week 2 and forward betting returns, so our strategy does not fully dominate TIP (though it does provide five percentage points higher return for that additional half percentage point of risk). These results are summarized in Figure 3.While there is significant room for improvement in our methods and models, our results clearly show that the NFL gambling market fulfills the requisite conditions for a viable alternative asset class. Returns based on NFL gambling are largely uncorrelated with major market returns. NFL betting returns are robustly positive, and we have provided limited evidence that our algorithms may outperform these market indices. We therefore highly recommend that portfolio managers invest significantly more time and energy into engineering models that beat Vegas and arbitraging the significant inefficiencies that we have found in this market.Table 1 P(Over) =Table 2P(Under) =Table 3P(Cover) =Table 4P(Not Cover) =Table 5Bets StartingReturnStandard DeviationWeek 27.00%5.11%Week 37.84%6.93%Week 47.11%6.73%Week 57.74%5.65%Week 68.01%6.44%Week 78.55%7.16%Week 88.41%6.53%Week 98.63%6.14%Week 109.21%7.07%Week 118.96%9.13%Week 129.42%8.77%Week 1312.10%9.62%Week 1412.16%10.11%Week 1513.15%12.61%Week 1615.02%11.10%Week 1720.25%16.44%Table 6Bets StartingCorrelation with S&P 500 Yearly ReturnsWeek 2-0.03Week 30.14Week 40.20Week 50.31Week 60.24Week 70.11Week 80.03Week 90.01Week 100.17Week 110.26Week 120.19Week 13-0.01Week 14-0.06Week 150.06Week 160.02Week 170.20Figure 1Figure 2Figure 3V. ReferencesWorks CitedGolec, Joseph, and Maurry Tamarkin. "The Degree of Inefficiency in the Football Betting Market." Journal of Financial Economics 30.2 (1991): 311-23. Web. 5 May 2013.Pankoff, Lyn D. "Market Efficiency and Football Betting." The Journal of Business 41.2 (1968): 203. Print.Peta, Joe. Trading Bases: A Story about Wall Street, Gambling, and Baseball (not Necessarily in That Order). New York: Dutton, 2013. Print.DataPro-Football-SoftwareRMicrosoft ExcelAppendix Imydata=read.csv("/Users/kevinmeers/Documents/Harvard/Junior Year/Spring/Stat 107/PROJECT/data.csv")primetime=mydata$Saturday+mydata$Sunday.Night+mydata$Monday+mydata$Thursdaymydata=cbind(mydata,primetime)adv.data=subset(mydata,Week>1)# Model Over/Under, P(Over)# SimpleOU.O=glm(mydata$Over~mydata$Week+mydata$Grass+mydata$Dome+mydata$Retractable.Roof+mydata$Saturday+mydata$Sunday.Night+mydata$Monday+mydata$Thursday+mydata$Spread+mydata$Over.Under,family=binomial("logit"))summary(OU.O)# FO StatsOU.O2=glm(mydata$Over~mydata$Week+mydata$Grass+mydata$Dome+mydata$Retractable.Roof+mydata$Saturday+mydata$Sunday.Night+mydata$Monday+mydata$Thursday+mydata$Spread+mydata$Over.Under+mydata$Home.O.DVOA+mydata$Home.D.DVOA+mydata$Home.ST.DVOA+mydata$Away.O.DVOA+mydata$Away.D.DVOA+mydata$Away.ST.DVOA,family=binomial("logit"))summary(OU.O2)# Full P(Over) ModelOU.O3=glm(mydata$Over~mydata$Week+mydata$Grass+mydata$Dome+mydata$Retractable.Roof+mydata$Spread+mydata$Over.Under+mydata$Home.O.DVOA+mydata$Home.D.DVOA+mydata$Home.ST.DVOA+mydata$Away.O.DVOA+mydata$Away.D.DVOA+mydata$Away.ST.DVOA+mydata$Home.O.DVOA*mydata$Away.D.DVOA+mydata$Home.D.DVOA*mydata$Away.O.DVOA+mydata$Home.ST.DVOA*mydata$Away.ST.DVOA+mydata$Home.O.DVOA*mydata$Away.D.DVOA*mydata$primetime+mydata$Home.D.DVOA*mydata$Away.O.DVOA*mydata$primetime+mydata$Home.ST.DVOA*mydata$Away.ST.DVOA*mydata$primetime,family=binomial("logit"))summary(OU.O3)P.Over=fitted(OU.O3)adv.data=cbind(adv.data,P.Over)# Standard bets: $100 on everything# Calculating "Over" returnsn=length(adv.data$Week)bet.o=rep(NA,n)profit.o=c()for(i in 1:n){bet.o[i]=sum(adv.data$P.Over[i]>0.5)if(bet.o[i]==1 && adv.data$Over[i]==1){profit.o[i]=100}else if(bet.o[i]==1 && adv.data$Push[i]==1){profit.o[i]=0}else if(bet.o[i]==0){profit.o[i]=0}else {profit.o[i]=-100}}profits.o=sum(profit.o)wins.o=length(profit.o[profit.o==100])win.pct.o=wins.o/sum(bet.o)# Calculating "Under" returnsbet.u=rep(NA,n)profit.u=c()for(i in 1:n){bet.u[i]=sum(adv.data$P.Under[i]>0.5)if(bet.u[i]==1 && adv.data$Under[i]==1){profit.u[i]=100}else if(bet.u[i]==1 && adv.data$Push[i]==1){profit.u[i]=0}else if(bet.u[i]==0){profit.u[i]=0}else {profit.u[i]=-100}}profits.u=sum(profit.u)wins.u=length(profit.u[profit.u==100])win.pct.u=wins.u/sum(bet.u)# Calculating "Cover" returnsbet.c=rep(NA,n)profit.c=c()for(i in 1:n){bet.c[i]=sum(adv.data$P.Cover[i]>0.5)if(bet.c[i]==1 && adv.data$Cover[i]==1){profit.c[i]=100}else if(bet.c[i]==1 && adv.data$Push[i]==1){profit.c[i]=0}else if(bet.c[i]==0){profit.c[i]=0}else {profit.c[i]=-100}}profits.c=sum(profit.c)wins.c=length(profit.c[profit.c==100])win.pct.c=wins.c/sum(bet.c)# Calculating "No Cover" returnsbet.nc=rep(NA,n)profit.nc=c()for(i in 1:n){bet.nc[i]=sum(adv.data$P.N.Cover[i]>0.5)if(bet.nc[i]==1 && adv.data$Not.Cover[i]==1){profit.nc[i]=100}else if(bet.nc[i]==1 && adv.data$Push[i]==1){profit.nc[i]=0}else if(bet.nc[i]==0){profit.nc[i]=0}else {profit.nc[i]=-100}}profits.nc=sum(profit.nc)wins.nc=length(profit.nc[profit.nc==100])win.pct.nc=wins.nc/sum(bet.nc)# Summaryprofit.t=profit.o+profit.u+profit.c+profit.ncexpended.t=(bet.o+bet.u+bet.c+bet.nc)*100adv.data=cbind(adv.data,profit.t,expended.t)yr.profit=c()yr.expended=c()yr.return=c()for(i in 1991:2011){yr.profit[i-1990]=sum(adv.data$profit.t[adv.data$Year==i])yr.expended[i-1990]=sum(adv.data$expended.t[adv.data$Year==i])yr.return[i-1990]=yr.profit[i-1990]/yr.expended[i-1990]}yrreturn1=yr.returnweek1=mean(yr.return)week1s=sd(yr.return)# Summary and Graphicsweeklyreturns=c(week1,week2,week3,week4,week5,week6,week7,week8,week9,week10,week11,week12,week13,week14,week15,week16)weeklysd=c(week1s,week2s,week3s,week4s,week5s,week6s,week7s,week8s,week9s,week10s,week11s,week12s,week13s,week14s,week15s,week16s)x=c(2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)low=weeklyreturns-weeklysdhigh=weeklyreturns+weeklysdlowest=weeklyreturns-1.96*weeklysdhighest=weeklyreturns+1.96*weeklysdpar(xpd=FALSE)plot(x,weeklyreturns,type="l",xlab="Week",ylab="Return",main="Returns By Week",ylim=c(-0.1,0.6))lines(x,low,col="blue",lty=2)lines(x,high,col="blue",lty=2)lines(x,lowest,col="red",lty=2)lines(x,highest,col="red",lty=2)abline(h=0)legend("topleft",legend=c("Mean Return","67% Confidence Interval","95% Confidence Interval"),col=c("black","blue","red"),lty=c(1,2,2))# Market ComparisongetSymbols("QQQ",from="1991-01-01")getSymbols("SPY",from="1991-01-01")getSymbols("TIP",from="1991-01-01")SP=as.numeric(yearlyReturn(SPY))NQ=as.numeric(yearlyReturn(QQQ))TY=as.numeric(yearlyReturn(TIP))mean(SP)mean(NQ)mean(TY)sd(SP)sd(NQ)sd(TY)means=c(mean(SP),mean(NQ),mean(TY))sds=c(sd(SP),sd(NQ),sd(TY))cor(yrreturn1,SP)cor(yrreturn2,SP)cor(yrreturn3,SP)cor(yrreturn4,SP)cor(yrreturn5,SP)cor(yrreturn6,SP)cor(yrreturn7,SP)cor(yrreturn8,SP)cor(yrreturn9,SP)cor(yrreturn10,SP)cor(yrreturn11,SP)cor(yrreturn12,SP)cor(yrreturn13,SP)cor(yrreturn14,SP)cor(yrreturn15,SP)cor(yrreturn16,SP)# Graphicslibrary(calibrate)fit=lm(weeklyreturns~weeklysd)plot(weeklysd,weeklyreturns,xlab="Standard Deviation",ylab="Return",main="NFL Betting vs. Market Returns",type="n",xlim=c(0,0.4),ylim=c(0,0.2))points(sds,means,type="p",col=c("red","blue","darkgreen"),pch=19)points(weeklysd,weeklyreturns,type="p")textxy(weeklysd,weeklyreturns,labs=x,cx=.75)lines(weeklysd,fitted(fit),col="black")legend("topright",legend=c("Betting Returns by Starting Week","S&P 500 Yearly Returns","NASDAQ Yearly Returns","10-Year Treasury Bonds"),col=c("black","red","blue","darkgreen"),pch=c(1,19,19,19)) ................
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