What Wall Street Can Teach Us About Baseball Players

A couple of years ago, I sat in the University of Virginia press box while Connor Jones, the school’s top pitching prospect, pitched a fairly good game. He had been throwing well the game before, and the game before that, but I distinctly remember this incident because the radio announcer who sat a few seats away from me commented on how fortunate UVa was to have such a pitcher. “You can’t find that kind of consistency nowadays,” he said. I wondered whether he was right, and if so, what that means for the game. Shouldn’t variance in a player’s performance affect his value?

Past Research

One way to measure fluctuation in player performance is through a comparison to the stock market. The performance of securities varies wildly, much like the performance of a baseball player, and often with little to no explanation. That doesn’t mean it’s not important or can’t be predicted. This year, I used an asset pricing theory model called the Capital Asset Pricing Model (CAPM) to measure the risk of baseball players. I treated them like securities to find this risk, likened to performance volatility.

This method of evaluation teases out the importance of consistent performance while also highlighting that a player with a highly variable performance may also have the potential to achieve the most. The model relies on a few major assumptions, one of which is that a team holds a large enough “portfolio,” or in this case, roster of players, that all unsystematic risk is diversified away. Unsystematic risk corresponds to an individual player.

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Based on this assumption, CAPM measures the systematic risk for each player, resulting in a variable called beta. The beta of a team can be plotted against the performance metric used to find it, like wRC+, to produce what I previously referred to as the team-specific replacement level. Individual player betas and performances can be compared to the replacement level to determine whether a player is considered a buy or a sell.

This graph is an example of CAPM using the Orioles and wRC+. The replacement level line is formed by connecting the point that corresponds to the risk of a risk-free/replacement level player (0) and his theoretical performance (86.7 wRC+) to the team’s risk (1) and team’s performance (96.5 wRC+). Then, by plotting the players’ risks and performances, we can see who falls above and below the line. This indicates which players are too risky for their return.

While CAPM makes conceptual sense, the model involves certain components that are not ideal. The beta of a player, for example, is measured in the context of his team and thus cannot be easily used to consider players in trade decisions. In addition, the model was simple and not adjusted for team specificity, and it should include more than one performance metric to determine beta.

The main goal remains to identify players who are considered risky. This is important for two reasons: that players, though this remains unproven, seem to follow a high-risk, high-reward standard, and that a team should take not only overall performance into consideration, but also the variance of that performance when making player valuation and acquisition decisions.

Improvements to the Model

Without sacrificing the decision-making threshold the model produces, improvements can be made to the way beta is calculated to compensate for the failures of CAPM as it stands. I propose a new way of calculating beta using a logistic regression.

Before the degree to which a player is risky becomes known, we want to know whether a player is risky in the first place. This question is suitable for a logistic regression, a method for analyzing data with one or more independent variables, but there are only two possible outcomes. In this case, the possible outcomes will be 1 (risky) and 0 (not risky).

The model uses a known set of independent variables and outcomes to predict outcomes for other observations. This is useful, because we want to predict whether a player is risky. Using this method to find a new beta resolves many of CAPM’s drawbacks when applied to baseball.

Multiple Independent Variables

Previously with CAPM, only one performance metric could be used at a time to determine a player’s risk. This means a player’s beta would vary based on which statistic was used and could be influenced by the innate variability of that statistic on its own. When using a logistic regression, there is no limit to how many variables can be tested in the model. Any statistic can be used, including other characteristic variables such as age. This is not to say all variables will be significant, but there are benefits to finding which variables are significant in predicting player risk in addition to being able to include more than one predictor variable.

Team Customization

In fact, being able to add more information to the model may benefit teams. Proprietary information can be evaluated and added to the model, including–but not limited to–medical information, scouting reports, and personality evaluations. Adding more specific information could add to the validity of the model and may also lessen the steepness of the slope of the team replacement level, which is believed to be softer in reality. There is also room for adjustment within the model based on a team’s interest and organizational philosophy.

Unsystematic Risk

While CAPM relies on the assumption that the investor holds a large enough portfolio that virtually all unsystematic risk is diversified away, this is not the case in a baseball team. A 40-man roster is large and has the capacity to hold many different types of players, but there is no way to diversify all of the unsystematic risk away from humans.

An example of this unsystematic risk when discussing players is injury. Even though it is statistically unlikely, it is still possible for every player to be hurt, and in that way, each player must still carry some unsystematic risk. The concept of measuring unsystematic risk is found in the Treynor-Black model. This model’s assumptions are similar to CAPM, yet it in addition to beta it employs alpha, a measure of unsystematic risk. This risk corresponds to the risk each individual player represents. The benefits of including this type of risk are vast. Incorporation of injury risk and personality traits could result.

However, the Treynor-Black model’s purpose is to measure the added value an investment manager brings to a portfolio. When applied to baseball on its own, this model would measure what the decision maker’s added value would be past a random choosing of which players to acquire. While this may be interesting, it is not the main goal of this research.

Instead of using the Treynor-Black model’s formula for alpha, we can consider that using multiple unadjusted performance metrics as predictor variables will measure a player’s total risk, both systematic and unsystematic, which is the most important to consider. However, if interest still lies in systematic risk, it is possible to adjust the performance predictor variables based on “risk-free” performance, or the expected performance of a traditional replacement-level player.

Method

Due to the sample size and the nature of these data, I used a robust logistic regression. To create the logistic regression model, I assigned ones and zeros to a random sample of existing players using an analysis of their performance variation based on standard deviations and their previous betas. Half of the sample was assigned values and was designated the training set. The other half was not and was designated the testing set.

When tested, the logistic regression model assigns each player a probability that he will take on a value of one. This probability value can be considered the new beta for a player, as it incorporates both types of risk and is contained on a 0-to-1 scale. The standard method for a logistic regression is to assign ones and zeros based on a cutoff of the probability value if p>0.5=1 and p<0.5=0.

To take team specificity a step further, teams can select their desired level of risk since this cutoff is adjustable and allows for a customized risk threshold based on team preference. If a team is looking for more high-risk, high-reward players, the cutoff can be set to probability>0.8=1, probability<0.8=0 and work from there. If a team is looking for a more conservative team profile, the cutoff can be set to probability>0.3=1, probability<0.3=0, and so forth.

This new beta can still be plotted against any performance metric but ensures the beta will not change based on which metric is being used. It also still allows for team replacement level, as the risk-free player’s beta will always be zero, and the team’s beta will always be 0.5. In addition, players from other teams can be plotted on the same graph and compared to one team’s replacement level, which could aid in decision making.

Conclusion

This method seems practical and has many benefits compared to the previous calculation of beta, but this is not without drawbacks. When testing logistic regression models predicted on season performance metrics, the results are limiting.

The previous process of calculating beta relied on a comparison of individual player variance to team variance, which points to the necessity of variance being included in predictor variables. Better results are achieved when including standard deviations as a variable in the model. Risk is a difficult thing to measure because it has so many meanings. Predicting team replacement level is also difficult, as it would ideally indicate a decision threshold.

A new model could be created solely for the purpose of determining a team’s decision threshold using past decisions. However, this likely would predict whether a team would make a decision instead of providing the optimal decision-making line. For example, using this methodology sacrifices the unique player-team beta that could be a necessity.

All in all, there is still work to be done to perfect the measure of risk and team decision-making, and I suspect going forth a better approach would be to keep them separate. Adding a logistic regression to help determine a new beta has been interesting, and further revision will be necessary to achieve the best results.