Determining Which Free Agent Contract Is a Success or a Bust

I was born, raised and am currently residing in South Korea, where Chan Ho Park, Shin-Soo Choo, and some Korean-born Major League players spent their school days before moving to the US with a big dream. Many Korean MLB fans recognize that their free agent contracts with the Texas Rangers have been something that very far from what we call a success.  Choo is currently having a great year and has been rewarded with his first trip to the All-Star Game.

Spending more than a decade of time to observe the ongoing conversations about those free-agent contracts, I have been wondering if it is possible to view this matter rather mathematically and/or statistically. And I think I have found a way for it, if it is not an absolute solution. I have named it proW/sal, which stands for proportional fWAR divided by proportional salary.

The equation is rather simple: proW/sal = (fWAR of the selected player in a particular year/fWAR of the player whose fWAR was the highest in that year)/(salary of the selected player in a particular year/salary of the most expensive player of that year).

The following are the exemplary applications:

1. This can be used in comparing individual players’ performances in one year in their FA contracts. Alex Rodriguez won the AL MVP award in 2007 after having a monster season with 54 homers and 156 RBIs, batting stellar .314, .422, .645. This enabled him to sign another 10-year FA contract with the Yankees. Let’s compare his year to that of Albert Pujols, who was in the middle of the extension contract with the St. Louis Cardinals.

A-Rod recorded the highest fWAR of the year: 9.6. The most expensive player of the year was Jason Giambi, whose salary of the year was $23,428,571. Since A-Rod’s salary for that year was $22,708,525, his proW/sal would be (A-Rod’s fWAR in 2007/A-Rod’s fWAR in 2007)/(A-Rod’s salary in 2007/Giambi’s salary in 2007) = (9.6/9.6)/($22,708,525/$23,428,571), which would be rounded to about 1.07. This is a great number because this means A-Rod’s fWAR and salary made an equilibrium with the highest fWAR and the highest salary of the year.

Let’s look at Pujols in 2007. His fWAR was 7.7, which is amazing, but not quite like that of A-Rod. His salary, however, was only $12,937,813. So his proW/sal of the year would be (Pujols’ fWAR in 2007/A-Rod’s fWAR in 2007)/(Pujols’ salary in 2007/Giambi’s salary in 2007) = (7.7/9.6)/($12,937,813/$23,428,571), which is about 1.45.

Yes. Pujols’ fWAR was lower than that of A-Rod. But due to his low salary, Pujols can be considered to have had a way more number-efficient year than that of A-Rod.

2. This also can be used to determine the efficiency of an individual FA contract in a year or as a whole. Let’s look at Randy Johnson during his first six years as a Diamondback.

Johnson’s FA contract with the Arizona Diamondbacks in December 1998 is considered one of the most successive FA contracts ever. But how good was it? Let’s determine it by using proW/sal.

In 1999, Johnson’s first year as a Diamondback, the player who recorded the highest fWAR was Pedro Martinez(11.6). Randy was at 9.5. And the most expensive player of that year was Albert Belle, whose salary was $12,868,670, while that of Randy was only $9.7M. So his proW/sal in 1999 was (9.5/11.6)/($9.7M/$12,868,670), which is about 1.09.

In the same manner, Johnson’s proW/sal in 2000, 2001, 2002, 2003, and 2004 can be calculated. According to my calculation, they were 1.18, 1.65, 1.44, 0.43, and 1.41, respectively. The big jump was due to A-Rod’s record-breaking FA contract with the Rangers, as well as Johnson’s great performances.

So the average of those six years’ proW/sal would demonstrate the efficiency of Johnson’s first six years in Phoenix. It’s at 1.19–a huge number because this means he was either the best in fWAR in those years, while his salary was quite below that of the most expensive player of each year.

For comparison, the average proW/sal of Carlos Lee’s FA contract with the Houston Astros was 0.29, and that of Choo’s FA contract with the Rangers was 0.26 until last year. I would not even bother with Park’s contract with the Rangers.

This equation, however, is far from perfection. First, I have not considered about distinction between fWARs of pitchers and batters, as well as salaries of pitchers and batters. I wonder if it would be necessary to differentiate them so that the number becomes fairer.

In addition, this equation is yet to function properly in regards to the fWARs in negative numbers. Using this equation may improperly evaluate a player with a negative fWAR with a cheap salary. For example, let’s suppose a player, A, whose fWAR was -1.0, and whose salary was $1M, another player, B, with the same fWAR but $10M of salary. And let’s assume that the highest fWAR that year was 1.0, and the most expensive salary was $10M. Despite the same fWAR and the lower salary, A’s proW/sal would be -10, while that of B is only -1. I have not found a solution to this yet due to my lack of proficiency in mathematics and statistics.

Moreover, this equation only functions well within the comparison between/among FA contracts. If this gets to rookie players, the value skyrockets  to an absurd degree. For example, the proW/sal of Mike Trout in his Rookie-of-the-Year year of 2012 is about 569.63. Every FA contract will be viewed as a bust if it is compared to that.

So here it is. I hope people with much proficiency take a look at this and improve it.

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4 years ago

Really like the idea behind this. I wonder however if it would be best to compare to a pool of say the top 5 contracts to avoid the outlier problem like you had with ARods monster extension.

Ray Liotta as Shoeless Joe
4 years ago
Reply to  dbeattie

I was here to suggest the exact same thing. Another thought would be to make the comparisons position-specific. Or, at least, starting pitchers compared to starting pitchers, relievers to relievers, corners compared to corners, middle infield/center field, etc.

4 years ago

This is a really neat idea, kudos to you. As a mathematician, I have two suggestions for improvement on your method here.

1) I strongly recommend the usage of league averages instead of league maximums to help cut out the noise that can occur in outliers. For league average salary, I would limit the salaries to post-arb players only.

2) to fix your issue with negative WAR I have a couple possible solutions off the top of my head. You can either add a number to the numerator that has an absolute value unlikely to be surpassed by a negative WAR and would remain consistent season to season (Something like the highest single season WAR total of all-time). Or, you could throw the player war figure up into an exponent. That way your formula can always get smaller but can’t go negative. Both of these methods have their drawbacks though and I’m not sure either one is the “right” answer.

4 years ago

Great work. I enjoyed the read. I’m also going to use your formula. Hopefully you will be able to figure out the rest of it. Nice job !