Methodology and Calculations of Dollars per WAR

Players with draft picks attached, like Curtis Granderson, get paid roughly $6.7 million less (via slgckgc).

Players with draft picks attached, like Curtis Granderson, get paid roughly $6.7 million less (via slgckgc).

In my last article, I provided the justification of dollars per WAR model’s use and documented many of its limitations. Today I will show how I calculate the approximate Dollars per WAR of the market, as well as some alternative methods I use, but I will begin by contrasting my methods with other methods.

My estimates tend to be much higher than the ones published on FanGraphs for several reasons. There is certainly value in using the FanGraphs’ method of prospective dollars per WAR estimates that develop alongside the market itself. The biggest difference in between my method and the one that Dave Cameron uses is that he uses projected WAR of available free agents, while I use retrospective WAR actually produced and estimate dollars per WAR after the fact. Given that we do not know how many WAR free agents will generate in the 2014 season, I need to make assumptions to produce an estimate of 2014 WAR that differs from how I approached 2013 WAR.

Why My Estimates Are Higher Than FanGraphs’

Some issues with Dave’s methods combine to produce a much lower estimate of free agent costs than I do. The cost of free agents is actually 16 percent higher retrospectively than using the “dollars” metric on the FanGraphs website. My estimate lines up much better with actual spending because I have the benefit of hindsight. There are four primary reasons why using prospective WAR makes FanGraphs’ estimates so much lower than mine, which I discussed at length here:

  1. Playing time projections for free agents are too high
  2. Players decline throughout their contracts, making $/WAR lowest in the first year
  3. Free agents who sign elsewhere give their new teams less bang for their buck
  4. The amount surrendered for free agents includes both the dollars and the draft picks foregone

Although Dave makes efforts to correct for the second item on this list, the sheer dropoff of many free agents (particularly ones who signed with other teams) makes this difficult to account for. It’s important to recognize that when using $/(Projected WAR) as your multiplier and projected WAR as your input does lead to reasonably accurate estimates of contracts. But the issue may be that if playing time projections are systematically biased for one class of players and not for others, you leave yourself open to mistakes by using a projected WAR based estimate when evaluating free agent contracts retrospectively.

How My Methodology Differs from Lewie Pollis’

Lewie Pollis wrote an excellent article recently, also using a retrospective WAR approach and got estimates similar to the ones that I will get today. Lewie and I differ in terms of two main aspects:

  1. Lewie uses only players who reached free agency, while I also use players who signed in advance
  2. Lewie does not incorporate the value of draft picks into his analysis

Incorporating draft picks is tedious, but the costs of draft picks are significant, and they tend to make my estimates 12 percent larger than they would be without accounting for them. However, my estimate is not going to be far off from Lewie’s anyway, because by using players who signed in advance I lower my estimate in the other direction.

Some may say that signing a player in advance allows for a discount from what the market price would be. I understand this argument, but I have two primary concerns.

One is that much of this “discount” comes from the “OPP Premium” (Other People’s Players) that I documented in the 2012 Hardball Times Annual, in which I showed that even players who re-sign during the offseason outperform players who sign with other teams, by 23 percent fewer dollars per WAR.

Additionally, so many superstars are re-signing with their old teams in advance of free agency that a large share of wins is purchased in this “shadow market” nowadays, and ignoring the prices in this market limits your sample. If Andrew Friedman signs the players on his own team who are likely to age well years in advance, and they are on team-friendly deals when they reach six years of service time, this is something we should evaluate to view him favorably, and when other teams wait to lock up their own players and spend more as a result, they should be viewed unfavorably.

Incorporating Draft Pick Compensation

The way I calculate draft picks is something that I have discussed in more detail here. In summary, I use Sky Andrecheck’s Draft Pick WAR Calculator, adjust for replacement level baseline, and approximate signing bonuses for players as well. Then I sum the WAR surrendered and bonuses saved for all players. The total bonus amount saved is subtracted from the total dollars spent on free agents, and the WAR foregone by losing draft picks is subtracted from the WAR accumulated by free agents.

The WAR accumulated by and bonuses spent on draft picks are both discounted using a 10 percent annual discount factor, and the draft picks foregone are treated as though they produce all of their WAR six years later than the free agent contract was signed (a rough approximation). The reason that I use a discount factor of 10 percent is that this is how the market tends to discount WAR in future seasons.

You can see this pretty clearly by comparing groups of signings. If you look at all players who signed for at least $10 million per season over the last eight years, those with no draft picks associated with them received approximately $6.8 million per WAR. However, the ones with draft picks associated with them received $5.8 million per WAR. However, using a 10 percent discount factor on future WAR, the cost to teams of their WAR was $6.8 million per WAR ($5.8 million worth of direct compensation and $1.0 million of equivalent value foregone on draft picks), equal to the amount for players with no draft pick compensation attached.

Since the draft pick compensation process in the latest CBA is simpler than the one in the previous agreement, it is actually much easier to estimate the value of a draft pick surrendered right now. Since the average pick surrendered (or foregone by re-signing a player who has rejected a qualifying offer) is about the 30th pick, we can estimate that the average WAR produced by the 30th pick before reaching free agency eligibility is about 2.05 WAR. This is worth only about 1.15 WAR in the present (discounted) and costs about a $2 million bonus about a year after the free agent deal is signed. Using the $/WAR estimates I get below, this makes the 30th pick worth about $6.7 million in today’s free agent market.

That means that analysts can simply tack on $6.7 million to the total cost of a free agent’s contract if signing him required surrendering or foregoing a draft pick, and can use this to compare to contracts without draft pick compensation included.

Results Using Actual WAR Retrospectively

Using these methods, I estimate the following costs per WAR for free agents during the 2006-13 years in which I have complete data sets. Note that even though the replacement level baseline for fWAR and rWAR is the same now, they still allocate WAR slightly differently between players eligible for free agency (with six years or more of service time) and those ineligible.

Cost per WAR, free agents, 2006-13
Year $MM / fWAR % growth ($MM / fWAR) $MM / rWAR % growth ($MM / rWAR)
2006 4.8 4.6
2007 5.6 17% 5.7 24%
2008 6.2 11% 6.4 12%
2009 6.4 4% 6.2 -3%
2010 6 -6% 6 -3%
2011 7.6 26% 8.2 35%
2012 6.5 -15% 7.1 -13%
2013 7.4 14% 7.8 9%
Total 6.2 6.4

The reason salary growth shrunk in 2010 was because of the deep recession of 2008-09 and its subsequent effect on spending, but you may notice that salary growth shrunk in 2012 after a steep climb in 2011. That is because the number of WAR produced by free agents (usually around 31 percent of all WAR produced) was unexpectedly low in 2011. There were only 250 fWAR and 237 rWAR produced by players with at least six years of service time in 2011, despite the fact that there are almost always about 310 of each.

Results Using Expected Cumulative WAR for Free Agent Eligible Players

Depending on the analysis for which you use a dollars per WAR estimate, it may be better to stick with the annual of average number of total WAR by players with six years of service time rather than that year’s realized total. This would lead to the following alternative estimates of dollars per WAR.

Expected Cost per WAR, free agents, 2006-13
Year $MM / fWAR % growth ($MM / fWAR) $MM / rWAR % growth ($MM / rWAR)
2006 5.0 5.1
2007 6.1 21% 6.2 21%
2008 6.3 5% 6.5 4%
2009 5.9 -6% 6.1 -6%
2010 6.1 2% 6.2 2%
2011 6.4 6% 6.6 6%
2012 7.0 9% 7.2 9%
2013 7.3 5% 7.5 5%
Total 6.2 6.4

Each method is appropriate for different circumstances. If you want to compare a player’s contract to other contracts signed in 2011, you want to use the softer grading curve that comes from comparing his deal to others signed in 2011. If you prefer to look at what determines the cost of WAR prospectively and how owners and GMs allocate money, the latter approach is more appropriate. For instance, in my next article I will look at what causes baseball salaries to grow over time, and for these it would be more appropriate to treat 2012 as a year in which owners opened the purse strings a little, rather than as a year where they tightened them.

Similar Methods of Estimation

Several other adjustments to this framework have some merit, and could be the appropriate model to use under some circumstances. For instance, players who were not in the first year of their contract in 2009 earned high salaries based on the expectations of economic growth that owners had at the time of their signing. As a result, they earned significantly more per WAR than players signed during the downturn. Of course, players earn more on average in the first year of their contracts every year, but it was especially true during 2009. If we use only the players on the first year of their contracts, the table would change to reflect this.

Since players in the first year of their contracts typically make $900,000 less than the entire set of players with six years of service time, I have added that amount back into the estimate to put it on an appropriate scale. In this case (again using expected WAR like the second version above), the table is as follows:

Expected Cost per WAR with First Year Adjustment, free agents, 2006-13
Year $MM / fWAR % growth ($MM / fWAR) $MM / rWAR % growth ($MM / rWAR)
2006 4.6 4.7
2007 6.7 46% 6.9 46%
2008 6.4 -4% 6.6 -4%
2009 5.8 -10% 5.9 -10%
2010 5.2 -10% 5.4 -10%
2011 6.6 26% 6.7 26%
2012 7.1 7% 7.2 8%
2013 7.5 7% 7.7 7%
Total 6.2 6.4

Lewie Pollis and Dave Cameron both prefer to include only free agent deals signed on the open market. Using only these deals, then adjusting for the $800,000/WAR difference between average deal signed in the previous offseason and the average deal overall, leads to the following estimates:

Expected Cost per WAR, offseason free agents only, 2006-13
Year $MM / fWAR % growth ($MM / fWAR) $MM / rWAR % growth ($MM / rWAR)
2006 5.2 5.3
2007 7.4 41% 7.6 42%
2008 6 -19% 6.1 -20%
2009 5.5 -9% 5.6 -9%
2010 5.3 -3% 5.4 -3%
2011 6.6 25% 6.8 25%
2012 6.3 -5% 6.5 -5%
2013 7.7 21% 7.9 22%
Total 6.2 6.4

Since these numbers really do tend to jump around a lot, I think that they are probably less useful than the methods that produce either of the first two tables above. However, I show them because I think that they are valuable for comparison, and certainly expose the issues surrounding the 2008-09 recession and its impact on revenue expectations.

Estimating 2014 Cost per WAR

This retrospective look is useful, but you are bound to be wondering what the cost per WAR is right now for 2014. This is actually pretty straightforward to estimate, especially by extending the methodology used in the second table above. Consider that:

  1. The total number of dollars expected to be spent on players with at least six years of service time in 2014 is about $2.145 billion (as of this writing on March 23), and the total will almost certainly be around $28 million more than that.
  2. Adding in the approximate WAR cost and bonus saved of signing the remaining free agent players who rejected qualifying offers, that means that the total draft pick WAR foregone for players this offseason (amortized over full deals, and adjusting for expected picks received at the end of a player’s contract) is about 30.2 WAR after discounting, and bonuses saved are worth about $46.4 million.
  3. Free agent eligible fWAR and rWAR totals will approximately match their 2006-13 averages

Therefore, by plugging these estimates in, I calculate that the costs of free agents in the 2014 season will be about $7.6 million per fWAR and $7.7 million per rWAR.

Estimating 1985-2014 cost per WAR

I do not have data on service time, contracts, or draft picks prior to the 2006 season, so estimating the cost per WAR in previous seasons is a little more challenging. However, since free agents earned about 75 percent of all salaries, using the salary database on Maury Brown’s bizofbaseball.com, I can estimate the cost per WAR for each for every season going back to 1985.

For draft picks, I just added 12 percent to the cost per WAR, since that was the average during recent seasons, and for total annual WAR averages I used an estimate based on production of players in their 30s (which matches the 2006-13 data well). This leads to the following estimates (note that 2006-14 values are based on the first table above):

Estimated Cost per WAR, 1985-2014
Year $MM / fWAR Year $MM / fWAR Year $MM / fWAR
1985 .68 1995 2.6 2005 4.7
1986 .80 1996 2.0 2006 4.8
1987 .68 1997 2.3 2007 5.6
1988 .79 1998 2.2 2008 6.2
1989 .94 1999 2.6 2009 6.4
1990 1.2 2000 3.1 2010 6.0
1991 1.6 2001 3.9 2011 7.6
1992 2.3 2002 3.9 2012 6.5
1993 2.4 2003 4.3 2013 7.4
1994 3.1 2004 4.2 2014 7.6

As I explained in my last article, these are only average costs per WAR and they do not pick up on all the subtleties involved in making intelligent free agent decisions. However, I have documented several areas of bias before that I would like to examine further with these updated numbers.

Revisiting the “Other People’s Players” Premium

The first is the “OPP Premium” explained in the 2012 Hardball Times Annual. In this article, I explained that players who signed with other teams cost more per WAR on a systematic basis than players who re-signed with their own teams. As I explained in that article, the OPP Premium for pitchers was very severe, while it was actually very mild for players. In that article, I used only multi-year deals and I did not adjust for the value of draft picks. Until recently, teams who re-signed their own players had to forego two draft picks (the other team’s pick plus a sandwich pick), which means that I systematically underestimated the cost of re-signed players ever so slightly in that piece. Consider the results now, separated by pitchers and hitters.

Revised OPP Premium
Group $ / fWAR, resigned $ / fWAR, OPP fWAR OPP Premium $ / rWAR, resigned $ / rWAR, OPP rWAR OPP Premium
Hitters 5.8 5.8 0% 5.7 6.0 5%
Pitchers 6.2 7.7 25% 6.2 8.5 36%

Amazingly, signing position players from other teams no longer appears to require a premium. However, this may be a misleading result. When you analyze only multi-year deals, the OPP Premium reemerges.

Revised OPP Premium, Multi-Year Deals Only
Group $ / fWAR, resigned $ / fWAR, OPP fWAR OPP Premium $ / rWAR, resigned $ / rWAR, OPP rWAR OPP Premium
Hitters 5.9 6.7 14% 5.8 7.0 21%
Pitchers 6.4 8.7 35% 6.5 9.2 41%

Since other people’s position players on one-year deals averaged only $3.2 million per WAR during the 2006-13 period according to my calculations, and given how frequently one-year deals for other people’s free agents include incentives that I was unable to incorporate into my analysis, I am extremely skeptical that I have the full data for these players. Assuming that hitters who sign one-year deals with new teams have more incentives built into their contracts, the OPP Premium is muted in the first table above. Using multi-year deals, which tend to have smaller incentives as a share of total compensation, the OPP Premium exists for both pitchers and for hitters. Regardless, it’s much larger for pitchers.

Revisiting Positional Differences in Cost per WAR

The last finding I would like to check is the comparison of $/WAR across different positions. The conclusions undoubtedly match those in my earlier work on the topic.

Cost per WAR, by position
Position $ / fWAR $ / rWAR
C 4.2 5.0
1B 8.0 7.1
2B 3.7 3.7
3B 4.8 4.8
SS 3.9 4.1
LF 8.8 10.6
CF 6.3 6.3
RF 6.3 6.3
DH 11.0 8.1
SP 5.7 7.0
RP 17.2 7.9
Total 6.2 6.4

By far, the most interesting finding is that although relievers are overpaid according to both versions of WAR, they are far more overpaid using FanGraphs’ version of WAR. Which method is appropriate probably depends on what you’re studying, but it certainly has an effect on whether you view free agent relievers as mildly overpaid or extremely overpaid.

The table also shows the same results that I found before, which are that outfielders, designated hitters and first baseman are overpaid on a Dollars per WAR basis, while other infielders and catchers are underpaid. Whether this is a systematic bias on the part of general managers or whether there is a market-based reason is less clear. It is certainly true that having fewer competitors on the steep part of their win curve with openings at a position will lead to lower bidding for such players, and given how often high-spending teams already have catchers and non-first-base infielders on their rosters may suggest that they can be had on the cheap in free agency.

Conclusions and Next Steps

There is nothing in baseball, economics or sabermetrics that can be boiled down to a simple number, so although I have provided reasonable estimates for retrospective analysis of cost per WAR in this article, simply linking to this piece will not end an argument. Recall that there are unique circumstances for any free agent decision, and that sometimes “overspending” is economically wise. Similarly, even paying a low cost per WAR is foolish for some teams as well. This simply shows the average cost per WAR in dollars and draft picks over the last several seasons and an estimate for what the average cost will be in 2014. In my next article, I will look at how we might see the cost per WAR to evolve in the future based on the relationship of baseball payrolls to different sources of revenue and the evolving economy.

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« Previous: Foundations of the Dollars-per-WAR Evaluation Framework
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Comments

  1. said...

    Great article Matt!

    Just one question, any reason for not including the arbitration salaries that come with a draft pick in your draft pick value calculations? I recently did an article at Breaking Blue on these calculations. I used a 9% discount factor similar to your ten percent factor, and I also included my very rough projection of $1.25M per WAR through arbitration.

    • said...

      I was going to ask a similar question. The 2.05 WAR for the 30th pick appears to be a net WAR (since Sky’s equation had the 30th pick closer to 3.6 WAR) , so I’m guessing that it already takes the pre-arb/arbitration salaries into consideration. However, this wasn’t 100% clear.
      Also, you mention that teams tend to discount future WAR by 10% annually. I get that this is how it’s valued on the open market, but is it necessarily the smart thing for teams to be doing? It seems that smart teams, especially those in smaller markets, can (and have) taken advantage of this apparent discount on future talent.

      • Matt Swartz said...

        Great questions, and you’re right I was too vague. Yes, I subtracted out the expected arbitration salaries as well as I could and that explains why the WAR totals are lower.

        Matt, you bring up an interesting point. I cannot say for sure whether 10% is “right,” but I’m not convinced it’s necessarily high or low. Winning is building a brand. The earlier you build the brand, the more fans that you bring in. The longer you delay winning, the more potential fans you could lose. Not only that, the growth rate of MLB franchises’ valuation and sale prices has grown so fast that I suspect the internal rate of return on investments in the MLB is very high, and winning has to be a major part of the investment expenditure. So 10% could very well be high or low, it’s tough to know. At this point, this is mostly descriptive rather than prescriptive.

  2. AJ said...

    I have a small question about $/WAR in general. In general, shouldn’t we consider ($-MinimumSalary)/WAR (since a replacement level player would make the minimum salary of $500,000). Since the team would have to spend half-a-million anyway, and they’d get replacement value anyway, they care about marginal value vs marginal cost.

    Am I correct? Obviously for Miguel Cabrera’s contract, the difference would be trivial, but for average players, I imagine it would be relevant…

    • Matt Swartz said...

      I left that part out of the article, but you are correct that I did that in the calculations and that excluding that would be a mistake.

      • AJ said...

        Thank you.

        I assume therefore that $/WAR (FanGraphs, Lewie Pollis) probably would as well.

  3. said...

    Great work, Matt! I just read Jeff Sullivan’s work on Fangraphs regarding Cabrera’s contract and posed this question to him; I’ll ask you as well: Is there a tool on any site to filter dollars-per-WAR for a player’s entire career? Your work tackles it more from a spending outlook – how much a team may want to spend, what the landscape may be for the free agent market, etc. From a player’s perspective, they seem not to care about which team pays the bills, and they probably don’t break down their earnings into free agent and non-free agent categories. They’d want to know how much they made playing baseball; to follow that, I’d like to know how much they made AND if they were worth it, relatively speaking. Any suggestions?

    • said...

      Baseball-Reference tries to count up how much a player made in their career at the bottom of each player page. It’s better for more recent years where data is more likely to be complete. I’m not sure about coming up with a value stat to compare it too.

      • Jeff said...

        Thanks, Matt. Yes, I’ve seen those numbers at BR. Looking at the Cabrera contract, and using Dave Cameron’s projection for the rest of Cabrera’s career, his total career WAR is expected to be 86. He will earn $408 million in his career, for earnings of $4.75 million per WAR. I looked at a few others, but was wondering if someone had already done that work for all players. Of course there would have to be an inflation adjustment to compare historical players.

  4. maqman said...

    Thanks for all the hard work Matt, I personally appreciate it. Being a little mathematically challenged some of it was a bit beyond my grasp but your conclusions paralleled my instinctive appreciation of WAR values. I have had reservations about Dave Cameron’s projections for a few years and liked Lewie Pollis’ more but yours are about as close to Gold Standard as I think it’s possible to get. I also admire your work on arbitration values for MLBTR. Well done you.

  5. Andy said...

    So Trout has produced roughly $130 million worth of WAR already. Within a few months, he will have justified the price of his entire six year contract, without even having begun playing those six years.

    Also, assuming his production holds up, by the end of those six years, his WAR total will justify not just his contract, but those of Pujols and Hamilton. The three contracts together total about $510 million.

    • Matt Swartz said...

      It might be helpful if you reread Part I. In that, I discuss the importance of not treating WAR created by pre-FA players as being worth as much as players who have reached FA. The FA process is designed so that many teams have a lot of cheap WAR that puts them on the cusp of the playoffs, and drives up the value of WAR. I cannot tell if you are suggesting that the numbers are too high, but that (the shape of the marginal win curve) may help explain it. Also, the $/WAR numbers are specifically constructed such that the sum of all FA-eligible WAR times $/WAR equals what those players cost. So by design it can’t be too high retrospectively.

      Another thing to note is that there is no retroactive performance that can “justify” new contracts. Trout’s past contributions have nothing to do with his future contributions or contributions of other players. But they sure can make a team like the Angels profitable despite overpaying on Hamilton and Pujols.

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