Between the persistence of Pete Rose, the ongoing turf war between Tribune Co. and the Wrigleyville neighborhood, and the deteriorating mental health of John Schuerholz, the most oft-reported story of this winter has been the apparent deflation in the market for free agents. Certainly, there are enough anecdotal examples to make a good case for the sky-is-falling crowd: Jeff Kent was signed rather cheaply, Frank Castillo took a pay cut of almost $4 million just for being his mediocre self, and the abundance of non-tenders suggest that teams expect that they can pay less for comparable talent by turning to the free-agent market than by accepting the terms of an arbitration settlement. Of course, there are counter-examples too; Jim Thome and Tom Glavine were signed to plenty generous contracts, St. Louis paid a premium to retain Woody Williams, and the Cubs seem to have pro-rated Antonio Alfonseca's contract over all 12 of his fingers.
What is in order is a systematic analysis of the free agent class of this winter as well as last, which takes into account not only the contracts the these players were signed to, but also how much value they are likely to provide to their new employers.
In order to evaluate the market for free agent baseball players as it stood one year ago, I gathered data from a variety of publicly available sources covering all free agent contracts signed during the 2001-02 off-season. To the best of my ability, I excluded extensions on existing contracts that had not yet expired, as well as arbitration settlements for players who had not accumulated enough service time to become unrestricted free agents. With these considerations in place, a total of 148 player contracts were evaluated.
My goal was to derive an annual net present value (NPV) on the guaranteed income associated with each free agent contract – that is, come up with a fair way to estimate their annual salary. Toward that end, the following rules were applied:
- Base salary, signing bonuses, player option years and buy-out payments on team option years constitute guaranteed income and were included in the NPV of the contract. Incentive payments and team option years were not included.
- For multiyear contracts, an annual discount rate of 5% was applied. Where it was available, I used year-by-year breakouts for base salary and signing bonus payments.
- In some cases, players were signed to minor league contracts and no further information was available on contract terms; in these cases, I estimated that the player had signed a one-year contract in the amount of $500,000.
In order to arrive at an annual salary, the total amount of income as determined under these rules was divided by the number of guaranteed contract years. This approach produced contract values ranging from $0.3 million per year (Pete Harnisch) to $16.3 million (Barry Bonds), with an average contract of about $2.3 million. The example below is for Jason Giambi, who by my reckoning makes about $14.4 million a year:
Table 1: Derivation of annual salary for Jason Giambi
|Year||Base Salary||Signing Bonus||
|Income||Discounted @ 5%|
|NPV of guaranteed income||$100,730,744|
|Annual salary (NPV/7 years)||$14,390,106|
The cost of a player, of course, is only one side of the equation; we must also estimate the return he can be expected to provide to the team signing him. This was accomplished by looking up his player card and evaluating his WARP (wins above replacement) for the three seasons 1999-2001. In order to derive a very basic estimate of his predicted value for 2002, which I have designated WARP', I applied the following formula:
WARP' = (WARP2001 * .5) + (WARP2000 * .3) + (WARP1999 * .2)
In other words, a reasonable proxy for the player's expected productivity is a weighted average of his productivity during the previous three seasons, with the heaviest emphasis given to the most recent year.
Certainly, another technique might have been applied to estimate a player's value; in particular, it would have been possible to roll our forecasting system backward one year, and run a projection for each free agent. I elected to stick with the WARP' "projection" because: i) although our forecasting system accounts for numerous subtleties that are not captured by the WARP' estimate, it is not clear that it would better represent how a major league team assesses player value; ii) the WARP' values account for defense, which our forecasting system does not.
Another, and potentially more serious, complication is that the WARP' prediction does not account for age-related decline for free agents signed to multi-year contracts. In this exercise, we have spread a player's compensation out evenly across all years, even when we know that his production is likely to decrease in the latter years of his contract. In the free market, a player might be willing to sacrifice income in the year immediately upcoming in exchange for a guarantee of income in future seasons when he is not likely to be as productive. It is partly in order to account for this that a low discount rate of 5% (close to a risk-free rate) has been applied to future contract years. However, this analysis may somewhat understate the market value for a player signed to a multi-year contract as compared to a player signed to a one-year contract.
Results for 2001-02 Free Agent Class
Figure 1 presents a plot of WARP' against annual salary for each of last winter's free agents. Pitchers and hitters were handled separately. My analysis suggests that teams are willing to pay substantially more for a pitcher of a given quality. It could be that the WARP' score is underrating the pitchers; it's also possible teams are predisposed to paying too much for them, based on a failure to account for the high variance in their performance, and a lack of appreciation for how much of run prevention is really the defense at the other eight positions.
There's a lot of information presented in the chart, but the most important thing to notice is the nature of the relationship between salary and performance, which is modeled by the blue (for hitters) and red (pitchers) lines. It is slightly non-linear; teams appear to be willing to pay a premium for top line talent. That is, a team will pay more than twice as much for a six-win player as it will for a three-win player. There are any number of ways to make sense of this, all of which have at least some basis in reality:
- Because of the talent pyramid, there is a scarcity of truly elite players, and therefore they are able to exert greater leverage when negotiating contracts.
- Elite players are perceived to provide other sorts of dividends, such as boosting attendance, and creating a signaling effect during the off-season.
- Major league teams are risk-averse, and expect that elite players will display less variance in their performance; they will pay a premium for this certainty.
- Teams aren't evaluating players properly, and these guys are overpaid.
In order to estimate the association between salary and productivity more precisely, I fixed the relationship between WARP' and annual salary such that a player with a WARP' of zero (e.g. a replacement-level player) would receive the minimum salary of $200,000. Regressing WARP' and the square of WARP' on annual salary with this constraint produced the following equations:
Hitters Salary (millions) = .0638 * (WARP')^2 + .4984 * WARP' + 0.2 Pitchers Salary (millions) = .1358 * (WARP')^2 + .7878 * WARP' + 0.2
Because of the non-linear component, these equations are somewhat difficult to appreciate intuitively, so I've provided the table below for that purpose.
Table 2: Model Salaries for 2001-2002 Free Agent Class, in Millions
The table illustrates both the escalation of salaries for elite players, and the gap between pitchers and hitters. (I haven't filled in the last row in the pitching column, because there weren't any data points in that area). Keep in mind that WARP' is measured in wins added. In the territory where free agents most commonly fall (between two and four wins added), teams were willing to pay a hitter about $0.88 million for the expectation of an additional win, and a pitcher $1.6 million.
This analysis also suggests that the free agent market is subject to a good deal of inefficiency. The diamonds and squares in Figure 1 represent individual free agent contracts, and a good idea of the market's efficiency can be gathered by looking at how closely they align to the regression line. In a perfectly efficient market, the data points would be lined up straight along the regression line; the further a dot strays from this line, the more out of line the player's salary in comparison to his expected productivity, and the more inefficient the market.
This characteristic is what the R2 of the regression measures. The R2 of the WARP' regressions was .7522 for hitters, and .5025 for pitchers, which implies that most of the variance in free-agent salaries is accounted by variance in their predicted performance, but certainly not all. This figure might have been somewhat higher if a more sophisticated metric for predicted performance were applied in place of WARP'. However, it is also quite probable that there is a good deal of systematic inefficiency in the free agent market, and that teams routinely overestimate or underestimate the value of certain types of players. Teams seem to struggle especially with assessing the value of pitchers.
The 2002-03 Free Agent Class
The same approach can be applied to evaluate contracts that have been signed this winter. As of this writing, I had information on 86 free agents for which contract terms had been disclosed, excluding Hideki Matsui and Jose Contreras, for whom we do not have an estimate of WARP' readily available.
The same valuation techniques were applied to this year's group of free agents. For each player, an annual salary was arrived at by the NPV approach described above, and an estimate of his value was calculated by deriving his WARP'. Figure 2 plots WARP' against annual salary for this year's free agent class:
Although it is not visually obvious, the slope of the line regressing WARP' on annual salary has changed; teams are in fact paying a bit less for comparable free agent talent this winter. Fixing the salary of a replacement level player at $300,000 (to account for the increase in the minimum salary per the new labor agreement), the results of our regressions for the 2002-03 free agents are as follows:
Hitters Salary (millions) = .0533 * (WARP')^2 + .4341 * WARP' + 0.3 Pitchers Salary (millions) = .1512 * (WARP')^2 + .5984 * WARP' + 0.3
A downward shift in the marketplace for free agents has taken place for all categories of free agents – hitters and pitchers, great players and marginal players, have all been affected. Table 3 presents the model salaries for this winter's free agents; the figure in parentheses represents the decline from how much we would have expected them to make last winter.
Table 3: Model Salaries for 2002-2003 Free Agent class, in Millions
|2||$1.38 (-5%)||$2.10 (-9%)|
|4||$2.89 (-10%)||$5.11 (-7%)|
|6||$4.82 (-12%)||$9.33 (-5%)|
Depending on the type of player, teams can expect to pay between 5% and 13% less than they would have last winter. To look at it another way, I estimate that the 86 free agents that I tracked would have made a collective annual salary of $240.4 million under the market conditions of last year. Instead, they will be paid $212.4 million, or about 11.7% less. After accounting for inflation, the difference in real dollar terms is between 13-14%.
This difference is only likely to increase as additional free agents are signed between now and the start of the regular season. The first batch of free agents to sign contracts–Jim Thome, Tom Glavine, Woody Williams–did so on relatively favorable terms. The next group, including Jeff Kent and Cliff Floyd, look like better bargains, while the most recently-signed free agents have often been available at substantial discounts. There is almost certainly a relationship between the point in time at which a free agent signs a contract and his leverage in the market, which gradually decreases as the snow thaws and the season approaches.
Moreover, contract lengths are also declining. Although the average contract length hasn't been that much affected, only six contracts have been signed this winter that extended for longer than three years; last year there were 11. The reported refusal of insurers to guarantee contracts for more than three years does seem to have had an effect on behavior.
Certainly, this analysis cannot provide a definitive answer as to the cause of the market downturn. The pattern of free-agent signings established this winter suggests that teams now employ a more modest estimate of the increase in revenue associated with winning an additional ballgame – that is, a player's marginal revenue product is lower. This would be consistent both with an anticipated decrease in baseball-related revenues (e.g. lower attendance and television contracts), as well as the disincentives created by the progressive taxation scheme imposed under the new labor agreement.
One additional piece of information is worth considering: the R2 of the regressions above. Variance in WARP' now accounts for about 72% of variance in salary for pitchers, and 85% for hitters; both figures are up considerably from last winter. The free agent market, although depressed, also appears to be behaving somewhat more efficiently. If WARP' is indeed a good benchmark for a player's true value, then teams are doing a materially better job of estimating the value of a player when negotiating his contract. It is possible that, among other things, teams are behaving more smartly than we are used to seeing; downward economic pressure has been observed to induce this trimming-the-fat effect in a variety of economic contexts.
The one hypothesis that probably can be ruled out on the basis of this evidence is that of an explicit attempt at collusion. A collusive market, although favorable from the perspective of ownership, does not operate efficiently. Because collusion imposes an artificial constraint on the number of bidders for any given player, the market is not as well equipped to estimate each player's value accurately. The fact that this year's market appears to be doing a better job of aligning salary with performance suggests that collusion is unlikely. There is a preponderance of evidence, however, that teams are managing their money more conservatively this winter.
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