Three point shooting and efficient mixed strategies: A portfolio management approach

2.1An economic analysis of basketball: An overview

In this paper we identify the efficient mixed strategy for the NBA in aggregate by maximizing the Sharpe Ratio (SR), which is defined here as the expected points divided by the volatility of points. The optimal proportion of 3-point shots for the NBA is then identified from the efficient mixed strategy.

We then extend this aggregate analysis to the individual team level. In this extension we identify the efficient mixed offensive and defensive strategies by maximizing the SR for each team in each year. The efficient mixed strategies are computed from the offensive (shots made) and defensive (shots given up) distributions. For offense a larger SR is preferred to smaller because this implies that the expected points per unit of risk is higher. The opposite applies to the defensive SR, because a smaller SR implies that the expected points per unit of risk is lower for the team playing against the defense. Finally from the efficient mixed strategy a shot type pricing model is derived for both offense and defense. This provides measures of the risk adjusted price of each shot type by team. An immediate advantage of applying this theory to basketball is that these results are scalable to any number of shot types. A team could analyze different shot types such as corner 3-pointers as compared to other 3-point locations as well as the relative return on midrange jump shots (from 15 feet where they are possibly riskier while only yielding 2 points as a return).

In summary, adopting the above economic analysis of basketball results in the following parameters:

From (i) above, the unique mixed strategy is compared to the NBA actual aggregate statistics for 2- and 3-point shots. For (ii) and (iii) the 4-tuples are estimated using within season data and then applied to predict post season performance. Finally, to better understand our measures and their relation to basketball performance, we correlate our parameters to the factors identified by Oliver (2004) as capturing offensive and defensive performance. This will identify the empirical relationships between these theoretical parameters with modern basketball performance measures.

We first consider some simple statistics that describe the phenomena being analyzed.

2.2Offensive investments: Allocation of shot attempts to 2 and 3-pointers

Expected points is points per shot times the probability of successful shot attempts for each shot type. In Fig. 1 this is plotted over time for 2- and 3-point offensive shots. If coaches employ a risk neutral strategy, this would imply that only 2-point or 3-point shots are taken. The graph in Fig. 1 suggests that, for risk neutral teams, only 2-pointers would be used up to 1987 (expected points are higher from 2-point shots up to 1987), followed by only shooting 3-pointers from 1988 on. Actual NBA shot data do not reflect this extreme risk neutral prediction with all 2-point shots crossing over to all 3-point shots. Figure 2 depicts the interesting behavior for the variance of successful 2- and 3-point shots each year. This reveals that 2-point shot variance has been relatively stable over time until recently whereas 3-point shot variance has been declining. If risk is important and we equate variance to risk, as the riskiness of a 3-point shot declines the 3-point shot should be taken more frequently. In fact, this has happened. This is further reinforced from the observation that both the mean and variance of points are also responsive to the rule adjustments made during the 1994–95, 1995–96 and 1996–97 seasons. As noted above, the NBA shortened the distance of the line from 23 feet 9 inches and 22 feet at corners to a uniform 22 feet from the basket. In 1997–98, the NBA then reverted back to the original distance of 23 feet 9 inches and 22 feet at the corners. Figures 1 and 2 indicate that both the mean and the variance are sensitive to these rule changes between 1994 and 1997.

One advantage of invoking the predictions from the optimal mixed strategy is that the NBA can predict the impact from changing rules upon shot taking behavior, which is especially important if maintaining a particular 2- and 3-point shot ratio for fans is an important objective. However, before being able to make such predictions there is a third input that influences predictions from the optimal mixed strategy. This is the covariance (or correlation) between 2- and 3-point offensive shot investments. By using each team as an observation within each year the 2- and 3-point shot expected points correlations can also be computed each year. Figure 3 plots the results from this analysis of the 2- and 3-point data across teams, computing a Pearson product-moment correlation coefficient.

Combined, Figs. 1, 2 and 3 provide the basic building blocks for identifying the efficient mixed strategy for basketball as formally described next.

2.3Identifying the Efficient Mixed Strategy

The set of efficient 2- and 3-point shot strategies is identified by maximizing the Sharpe Ratio, where j = 1 (2-point shots) and 2 (3-point shots) and the maximization problem is defined as follows:

Subject to:

Where

ω = a vector of shot type weights (α₂, α₃) the proportion of 2- and 3-point shots in the mixed strategy and s^T the shot type points. Expected points from the mixed strategy is defined as: μ=sTωTp¯ . This expands as μ = 2 * α₂ * p₂ + 3 * α₃ * p₃ where p₂, p₃ are the probability of 2 and 3-point successful shots.

σ=ωT∑ω , ∑=variance covariance matrix of shot type points

For 2- and 3-point shots, the portfolio volatility (i.e., standard deviation) is: σ = (4α22σ22+9α32σ32+12α2α3rσ2σ3)0.5 where 4 equals2 points squared, 9 equals 3-points squared and 12 is 2*2*3 to cover the off diagonal terms.

For the problem defined above in equation (1), p, the probability of a successful shot computed from n shots is a random variable estimated from the data (dropping shot type subscript for expositional ease). The distribution for p is estimated by applying the central limit theorem to the large number of both 2 and 3-points shots each season, and the variance of p is normally distributed and equal to p*(1-p)/n. As a technical aside, we do not define variance in terms of being a random variable from the Bernoulli distribution, which assumes it is the outcome from a sequence of shot realizations generated from the Binomial (n, p) distribution. This is because invoking the Bernoulli distribution assumes that p is a parameter, i.e., a fixed real number, as opposed to being itself a random variable which must be estimated from the data. Finally, we compute the Pearson correlation coefficient, r, between 2- and 3-point shots from each team’s total shot statistics per regular season year. By assuming each team makes a tradeoff between using 2 and 3-point shots in their offensive strategy, we calculate the Pearson correlation coefficient using each team as an observation. The number of teams used to estimate r, range from 22 NBA teams in 1979 to 30 NBA teams currently, and finally covariance is then defined as rσ₂σ₃.

The solution to the above maximization problem, when applied to NBA aggregate data, identifies the efficient mixed strategy of 2- and 3-point shots for the NBA as a whole. Here we define efficiency for offense as the mixed strategy that maximizes expected points per unit of risk. For the NBA as a whole this is also equivalent to finding the minimum expected points given up per unit of risk. That is, the offensive and defensive problems are symmetrical. However, this symmetry does not extend to individual teams and both offensive and defensive performances are measured and analyzed separately for individual teams as discussed next.

1. Individual Team Level of Analysis

The game of basketball consists of both offense and defense and therefore to analyze basketball at the individual team level we first apply equation (1) to identify the two efficient mixed strategies, offensive and defensive for each team. In this paper, we solve for the mixed strategy that maximizes the Sharpe ratio (e.g., Elton and Gruber (1995)), and refer to this strategy as the “Efficient Mixed Strategy” (EMS). Central to this theory of investment returns is the idea that higher risk is associated with higher expected returns in a world where risk aversion is important. By applying investment theory to basketball, we relate the expected points from the 2- and 3-point shot types to the risk associated with each shot type for each team. The steepness of this relationship provides insight into the cost associated with a team’s tradeoff between risk and expected points. Formally the CAPM theory predicts the following general form for this relationship, referred to as the Security Market Line (SML) which is positively sloped and defined as follows (e.g., Bodie, Kane and Marcus (2013)):

Shot type betas are estimated in two equivalent ways, relative to each team’s efficient mixed strategy. First, beta is defined as the covariance between the average shot type payoff and the optimal mixed strategy payoff, scaled by the variance of the optimal mixed strategy payoff. The efficient mixed strategy is computed from the variance/covariance matrix, where covariance is estimated by computing the correlation between shot types estimated from the 30 teams, multiplied by the product of the shot type standard deviations. Equivalently, and simpler, shot type betas can be computed directly from the ratio of the average payoffs relative to the efficient mixed strategy as described by Equation (2).

To make the above theoretical concepts concrete it is instructive to first illustrate some of the important statistics that this analysis produces by applying it to the first and last place teams in the regular 2013 season. This will help to focus on some of the intuition behind the empirical results that follow.

2.4Example: Miami compared to Orlando 2013

During the regular 2013 season Miami attained the highest win percentage of 80.48 percent and Orlando the lowest, 24.39 percent. Based upon regular season team results, the SR, shot type betas and SML slope, for Miami and Orlando is as follows:

Miami 2013 Regular Season Offensive Performance

Sharpe Ratio = 16.33

Beta (2-point) = 0.96

Beta (3-point) = 1.15

Slope of the Shot Market Line (SML) = 1.03

Miami 2013 Regular Season Defensive Performance

Sharpe Ratio = 13.79

Beta (2-point) = 0.96

Beta (3-point) = 1.14

Slope of the Shot Market Line (SML) = 0.91

Orlando 2013 Regular Season Offensive Performance

Sharpe Ratio = 13.34

Beta (2-point) = 0.98

Beta (3-point) = 1.08

Slope of the Shot Market Line (SML) = 0.91

Orlando 2013 Regular Season Defensive Performance

Sharpe Ratio = 13.54

Beta (2-point) = 0.97

Beta (3-point) = 1.12

Slope of the Shot Market Line (SML) = 0.96

Major differences immediately appear between these two teams both offensively and defensively which can be interpreted as follows.

Consider first the Sharpe Ratio (SR) which imposes a price of risk constraint upon the efficient mixed strategy. A high offensive (low defensive) SR is preferred to a low offensive (high defensive) SR because the offensive SR measures the expected points per unit of risk and the defensive SR measures the expected points given up per unit of risk. For the case of Miami versus Orlando, Miami has a higher price of risk for its offensive mixed strategies. As a result, Miami will have a higher expected points per unit of risk than Orlando in its efficient mixed strategy.

Additional insight into the drivers of the above differences is provided by considering both the offensive and defensive SML’s. The analogy to the Security Market Line for basketball can be interpreted as follows. In regular CAPM all risky assets are predicted to lie on the Security Market Line in equilibrium. For basketball each risky shot type is predicted to lie on the Shot Market Line (SML). In the above example the slope of the SML is constructed for both offense and defense. For the case of offense a steeper SML implies that the expected points per unit of risk is higher for the 3-point shot compared to the 2-point shot. In addition, the steeper the SML the stronger the offense. Recall from Fig. 2 that the variance of 2-point shots has remained relatively stable for about three decades whereas the variance of the 3-point shot has been declining. The CAPM theory predicts the shot type beta is a function of shot type volatility (e.g., Bodie, Kane and Marcus (2013)) and as a result, the steeper the SML the better is the offensive 3-point shooting from an expected points perspective relative to the 2-point shot type. This difference between CAPM applied to financial securities as opposed to shot types, is that for financial securities the expected return pricing model includes a risk free rate, typically non-zero, whereas for the basketball shot types, the offensive shot pricing model originates at the origin (i.e., risk free rate equals zero). In both cases, however, steeper rays dominate flatter rays. The opposite applies to the defensive SML as it is defined in terms of points given up. In the above example, Miami compared to Orlando reinforces these insights.

4.1Results for NBA as a whole

In this section we present the results from solving each year for the aggregate offensive efficient mixed strategy for the NBA as a whole. The analysis proceeds by constructing the distribution of points from each shot type for the NBA as a whole by year. Figures 1–3 above depict the evolution of the 3-point shot and its use in the mixed strategy, over time. The negative correlation between 2- and 3-point shot successes is pronounced and reinforces this evolving use of the 3-point shot in NBA strategies. For example, as noted by players such as power forward David Lee of the Golden State Warriors: “The game is changing,” Lee said, “and I think one of the things is not telling ‘4s’ (power forwards) they’re going to be in the post all the time. Instead, teams are giving them the option to shoot mid-range shots and threes. Then the defense has to make the adjustment.”⁴

As a result, as the mixed 2- and 3-point strategy becomes more prevalent, the negative correlation in Fig. 3 is becoming more pronounced. The efficient mixed 2- and 3-point strategies allow for a consistency relationship to be tested between the predicted efficient strategies each year and the actual observed outcomes. These results are depicted graphically in Fig. 4.

Fig.4

Actual versus Optimal 3-Pointers (Actual Correlations).

Figure 4 reveals that, in the aggregate, NBA coaches have exhibited highly efficient 2- and 3-point shot selection strategies. That is, there is a high level of consistency between aggregate mixed strategy behavior and player abilities when reduced to means, variances and covariances. It can also be observed that both actual and predicted behavior is sensitive to the rule changes that took place over the 1990’s. As a result, analytical modeling combined with empirical analysis of basketball can provide predictions for likely outcomes of future potential rule changes.

The above analysis also provides insight into where the NBA is headed in terms of predicted proportions of 3-point shots. Clearly, from a fan perspective, the NBA probably wants to maintain an attractive balance between the proportions of 2-point and 3-point shots in the game. In investment theory the variance and covariance of shot types are the major drivers of the predicted proportion of shots for a risk averse world. Although Fig. 2 reveals that the NBA has gone down a steep learning curve associated with the accuracy of the 3-point shot from 1979 to 1995, this learning curve settles down post 1995. Between 1995 and 1997 rule changes largely accounted for observed changes in 3-point accuracy and post 1997 the evidence suggests that team strategy became a major driver of this behavior. This is implied from the observation that the correlation between top shooters (in terms of accuracy) and top 3-point shooters (in terms of attempts) in 2014–2015 is substantially higher (r = 0.417) than it was in 1998–1999 (r = 0.329). One of the leading 3-point shooters today is Stephen Curry (accuracy = 0.443, third in the NBA in 2014–15). Coincidently, his father Dell Curry was the most accurate 3-point shooter in 1998-1999 (accuracy = 0.476). While Stephen Curry led the NBA in 3-point shot attempts, Dell Curry was not even among the top 20 players in 3-point attempts. Although overall league shooting accuracy has not changed very much, the significant overlap between high frequency 3-point shooters and the most successful 3-point shooters today implies that mixed strategies have taken time to evolve differently from just accuracy alone. This is also reflected in the greater stability of correlation trends between 2-point and 3-point shots over time. In order to tease out the relative impact of each of these two important inputs we can systematically introduce the correlation results into the analysis. In the next section we compare the actual versus predicted behavior in Fig. 4 when using estimated correlations versus a baseline case of zero correlation (i.e., independence).

First we consider the baseline case under the assumption of zero correlation between shot types. For this analysis we estimate the optimal mixed strategies under the imposed condition that the two shot type distributions are independent. This will allow the effects of shot accuracy (shot expected points variance) versus mixed strategy (shot expected points correlations) to be teased out by comparing Figures 4 and 5.

Fig.5

Optimal Number of 3-point shots versus actual under independence condition.

The results from Fig. 5 reveal that in aggregate for the NBA trends in the predicted efficient strategies are consistent with trends in Fig. 4 but the gap between actual and predicted is widening which demonstrates the increasing importance of mixed strategies in the game. The increasing widening of the two series is likely to be a result of the strategy learning curve that coaches and players in the NBA have been ‘going down’ over time that was alluded to by David Lee’s comments above. Figures 2 and 3 reveal that both variance and correlations have substantially stabilized in the last few years. This implies that it is unlikely that the proportion of 3-point shooting is about to dominate the proportion of 2-point shots as some have speculated will happen⁵.

Next, we extend the aggregate analysis of the NBA to the analysis of individual team performances during the regular season. The objective of this section is to provide a descriptive analysis of the basketball performance in terms of the team level performance measures identified from the theory. We analyze the dependent variable, win percentage, as a fractional response variable. For this fractional response, the values lie in the closed unit interval [0,1]. The linear probability model applied to such a dependent variable cannot guarantee predicted values will not fall outside the unit interval (Wooldridge, 2002). For this reason, other methods such as a log-odds transformation of the dependent variable and NLS estimation are often proposed. It has been shown that such estimators, while estimable with OLS, are problematic. In particular, they do not deal with values at the boundary, 0 and 1, and the regression estimators β are not easily interpreted (Wooldridge, 2002). Papke and Wooldridge (1996) propose a method to address this problem. They show that using a generalized linear model with a Bernoulli log-likelihood function and robust estimates of the standard errors provides estimates that are efficient, allow for values at the unit interval boundaries of 0 and 1, and are directly interpretable. For these reasons, we apply the method proposed by Papke and Wooldridge (1996). In our estimates, we use the logit cumulative density function.

The results from Table 1 reveal that each year’s descriptive model fits the data well. Furthermore, from inspection of the individual coefficients the pair of offensive and defensive SML slopes each had significant t statistics at or around the 0.001 level with signs in the predicted directions. When we aggregate the data from 2007–2014, and estimate the model with fixed effects for year and team, we find all 4 offensive measures are likely to have an effect, while only the defensive SML has an effect over the 8 seasons, comprising 240 team seasons. In all these cases the models fit the data reasonably well, but the models are only descriptive, given they are estimated on the same within season data used to estimate the predictors themselves. Later even more insight is provided when we relate the set of economic variables identified by the theory to traditional basketballfactors.

4.2Out of Sample Analysis using the Post Season Championships

For the out of sample tests, the post season results from 2007 to 2014 were pooled to increase the sample size at the team level. In each year, the NBA champion is determined by a 16-team single-elimination tournament. The winner in each round in the tournament is the winner of a best-of-seven series. The tournament starts a few days after the end of the regular season. No adjustments to personnel occur between the regular season and post season. The loser of each best of 7 series exits the tournament. Given 16 teams, there are 15 series per year in each of the 7 years. The average series length is 5.6 games. The average annual tournament, composed of 15 series, has 83.8 games. Our analysis is at the game level. We use two dependent variables, (1) whether the home team won and (2) the point difference between the home team and road team. The predictor variables are offensive and defensive SML slopes, 2 and 3-point betas, and Sharpe ratios for the home team and the road team. In addition, year of the playoff and series (since games are nested under each 7 game series) are treated as random effects. We choose to use random effects for several reasons. We do not expect year to year variation to be systematically correlated with the effects of the predictors, so year to year variation is treated as random. Each series involves a set of games between the same teams. The observations within a series may be correlated. For this reason, observations within a series are modeled as random effects (Wooldridge, 2002; Angrist & Pischke, 2009).

These prediction models are multilevel models because you have games nested in series and series nested in year. We used the Stata command ME (multilevel estimation) to estimate the models. The first model, predicting the winner of a game in a series was estimated as a logit model using maximum likelihood. The logistic regression results for home game won are presented in Table 2. The fit is significant (Wald χ2 = 50.35, p < 0.001). The results again reinforce the descriptive within season analysis, and it is the offensive and defensive SML slopes that drive the results.

Next a more sensitive dependent variable, point differential, was analyzed to take into account the closeness of the game result. The prediction model for point differential is a multilevel mixed (having some fixed and some random effects) linear regression model that requires restricted maximum likelihood estimation. Restricted maximum likelihood first estimates the fixed effects and then estimates the random effects. The random effects are estimated with maximum likelihood without the fixed effects (Harville, 1977). The results are provided in Table 2. The linear regression of the point differential is significant (Wald χ2 = 77.92, p < 0.001). Again this analysis reinforced the same SML slope results where the four SML slope coefficients (offensive, defensive, home and away) are all highly significant (p < 0.01). The major conclusion from the individual team performance analysis is that SML slopes (offensive and defensive) are useful predictors of performance.

4.3Linking the Economic Variables to Basketball

In a popular and influential book, Oliver (2004) identifies four factors that determine success in the NBA. These are ranked as follows in terms of assessed order of importance for winning a game. The ranking is, Shooting, Turnovers, Rebounding and Free Throws per shot. In addition, the factors can be applied to both offense and defense. The variables for both offense and defense are:

It is useful to know how these four factors relate to the 4-tuple of economic variables identified in section 2 and we explore these relationships in a correlational analysis. It is noted however, that the four variables identified in our current paper’s economic analysis are derived from a coarse data set containing only 2- and 3-point shot distributions for both offensive and defensive performance. Refining the set of shot types as improved datasets become available in the future is likely to improve this type ofanalysis.

The results are provided in Table 3 and three estimates immediately draw our attention. First, offensive SML slope is extremely highly correlated (0.891) with the EFG% which is considered to be the major driver of success. However, recall from our analysis it was both offensive and defensive SML slopes that are the critical success drivers. As a result it is interesting to observe the defensive correlations. The EFG% of the opponent team is highly correlated (0.817) with the defensive SML slope. This is consistent with the fact that the higher the defensive SML slope the higher the expected points from the opposing team playing against the defense. Similarly, the EFG% correlation with the defensive SR is 0.374. Higher defensive SR is consistent with a higher expected proportion of 2-point shots being played against the defense. The other pair of interesting correlations are in relation to defensive team’s offensive rebound percentages. The correlation between defensive rebound percentages and the defensive SR is –0.25 and the defensive SML slope is –0.404. Again from a strategic defensive perspective this is reinforcing the fact that the opposing team is taking a higher proportion of 2-point shots against weaker defenses that in particular are weaker at defending against the opposing team’s rebounding successes. The above analysis also illustrates the advantage of the economic analysis that is decomposing expectations into both strategic and expected points components. For example, if a team tightens up its defensive rebounding this is likely to have both strategic and expected points implications. Strategic in the sense that the opposing offenses are likely to reduce the proportion of 2-point shots played against the defense and goals made against the defense are expected to decline.

The remaining two factors, TOV% and FT/FGA are not really captured (and probably should not be captured) in the data set analyzed in this paper.