THE BOOK--Playing The Percentages In Baseball

I think there has to be a premium on higher WAR players.

Teams value WAR differently - we know this for a fact because the Yankees pay a 40% luxury tax on every dollar they spend. 

Teams that value each marginal WAR the most will, limited by roster spots, target the highest WAR players.  Once their rosters are set, the remaining mediocre, 1 year contract guys are left to be bid on by the teams that value each marginal WAR the least.

There were 53 1-yr deals according to the data.  FIVE went to the Redsox.  FIVE to the Dodgers.  2 to the MEts, and 1 to the Yanks.  I presume those are probably the 4 biggest markets.  That’s 13 players out of 53, for 4 of the 30 teams.

Look, it’s all nice to say something, but let’s back it up with SOME evidence.

If someone wants to breakup the data to show the small and large market teams pay differently, then do so.  Otherwise, I don’t buy anything other than a Yankee-bias (not a general large-market bias).

I’d say it’s far more likely that some teams have figured out that 1-year contracts are a good deal for a team, and that many teams use the 1-year deals to fill out their rosters because of the good return.

So they’re using expected WAR and actual dollars, right?  In the case of incentive contracts such as Smoltz last year, his base was not very high.  However, if he had reached his expected WAR, then he would have been paid more than he actually was paid.  Could a few incentive contracts like this be messing with the data by not matching up what teams were offering to pay for expected WAR?

Hello all, this is Donald, the author of the article on saberrattling. Thanks for noticing my article and initiating the discussion, Tom.

JB H, totally agree with you. I defined “big” market teams as the NY, LA, BOS and DET teams, and those averaged ~$3.5M per xWAR per year, while “small” market teams (ARI, COL, KC, MIL, MIN, OAK, TB, WAS) spent only $3M per win. Other teams fell in the middle of this range, at $3.26M per WAR.

Jordan, I have tried to take into account contract incentives where possible. For example with Smoltz, his base was $5.5M but I used a figure of $7M which is much closer to what he actually received. I suppose if he had provided 2.5 WAR as projected, he’d have gotten a bit more. But the point is I did try to correct for incentives, esp on the 1-year contracts.

I think the reason those short deals are cheaper are due to two main effects. First, you have to pay a premium for premium talent. Secondly, one-year deals are usually for older, unproven or injured players. Teams are often risking something, so they factor that risk into the contracts they offer.

Are you guys taking into account the time value of money for the long-term contracts? If baseball salaries rise 10% per year, then a one-year contract for $1m would be equivalent to a two-year contract worth $2.1m (receiving $1.1m next year is equivalent to receiving $1m this year). Running the numbers for Teix (counting his $5m signing bonus as part of 2009 salary) and using a 10% discount rate:

Year   Salary  Salary (present value)
2009    25.0    25.0
2010    20.0    18.2
2011    22.5    18.6
2012    22.5    16.9
2013    22.5    15.4
2014    22.5    14.0
2015    22.5    12.7
2016    22.5    11.5
TOTAL  180.0   132.3

The present value of salary is just salary/(1.1^(year-2009)).

I think that adjustment would bring the long-term contracts in line with the one-year deals.

"First, you have to pay a premium for premium talent.”

This is unsupported right now, as I’ve shown.  I can buy the 1-yr deals for older players getting a discount (Pettitte, RJ, etc).  But, once you get into multi-year deals, you don’t see that.

Take all the 2+ deals, where the WAR was under 2 for the first year.  Total WAR for the first and future years is 23 wins, and the total payout is 149MM$, or 6.5MM$ per win.  For all the other playrs, it’s 159 wins and 842MM$, or 5.3MM$.  So, teams got a DISCOUNT for premium players.

Or, if you don’t like that, let’s limit it to only the 2-yr deals, and set the threshhold at 2+ for the premium players and 1.5- for the rest.  The premium players have 32 wins over 2 years for 134.5MM$, or 4.2MM$ per win.  The lower players have 16.5 wins for 88MM$, or 5.3MM$ per win.

So, I reject the conclusion that there’s a premium paid for premium players.

The data is pretty clear here that this bias is limited to the 1-yr deals, and nothing else.

"I think the reason those short deals are cheaper are due to two main effects. First, you have to pay a premium for premium talent.”

85k of $3.3 mil/WAR is only 2.5%.  That seems both plausible and pretty small.

“Secondly, one-year deals are usually for older, unproven or injured players. Teams are often risking something, so they factor that risk into the contracts they offer.”

Even a crude estimate of expected WAR seems to factor in most of that.  If he’s in his decline phase, I’m knocking half a win a year off his expected production.  If I’m estimating future playing time for a previously injured player like Sheets, using a 5/3/2 weighted average, I get 90 IP.  If I’m estimating the production of an unproven player, I’m significantly regressing his previous production.

Dackle, I did do that, which is what I meant here:

“If we apply a 10% increase to the marginal dollar per win each year, then we get 4.7MM for the first year, and 10% higher after that, for the multi-year deals.  That’s pretty much what we expected, ignoring the economic collapse. “

That is, if I set the $ per win at 4.23 for the first year, and 4.23*1.1 for the second year and 4.23*1.21 for the third year, AND I chop off 0.5 wins each year for each player in their multi-year deals, I get:

$1.155 billion = total signed contracts
245 total undiscounted WAR
$1.156 billion = total value of $ per win for each year

As an example, here are Teix 8 years of WAR:
4.53 4.03 3.53 3.03 2.53 2.03 1.53 1.03

And here are his WAR salaries:
$19.2 $18.8 $18.1 $17.1 $15.7 $13.8 $11.5 $8.5

That’s based on using the 4.23MM$ in the first year, and bumping it up by 10%.

The sum total of those salaries is only 122.5MM$.

Now, if we EXCLUDE all 1-yr salaries, then the starting point is 4.7MM per win.

We get these salaries for him:
$21.3 $20.8 $20.1 $19.0 $17.4 $15.4 $12.7 $9.4

That totals 136MM$.

CC gets the opposite problem.  He comes in at 223MM$ with the 4.7 starting point and 201MM$ with the 4.23 starting point.

Overall, if we look at the 3-big signings for the Yanks (includnig AJ), then starting the first year at 4.57MM per year and increasing at 10%, then those three guys would earn 424MM$, compared to their salaries of 423.5MM$.

So, I don’t see any kind of premium here for the Yanks.

Dackie, good point about inflation. Certainly teams figure that into any long-term deal they make. I have not adjusted the value of the money for long-term deals. I simply took an average annual value and divided by xWAR; after all, the idea was to see what teams shell out for a FA available now in today’s market.

Tom, perhaps I should have been more clear with the word “premium”. I don’t mean players who are over 2 WAR, which is just above average. Perhaps we could look at this as players who generate at least 3.5 WAR (arbitrary but representative number). That gives us an average of $4.0M/xWAR. The average of the guys from 2.0 to 3.5 WAR is $3.3M/WAR, and the average for guys below 2.0 (who earned at least a $1M contract) was $3.2M.

First off, can we agree to exclude all 1-yr players?  Please?

Ok, so here’s the data for the 3.5+ guys, the 2.0-3.4, and the under 2.0 guys for multi-year deals:

For guys with at least 3.5WAR in their first year, that’s the 3 Yanks, Derek Lowe and Manny.  Their first year WAR is 23.7.  They signed for a total of 26 years.  Their expected WAR over those 26 years is 94.9.  Their salaries is 528.5MM$.

Using 4.7MM$ per win in the first year, and adding 10% each year, their WAR salaries is 552.7MM$.

So, they signed at a slight discount (4%).  There’s enough uncertainty in the WAR forecast that I’ll call that a wash.

For the 2.0-3.4 first-year WAR, they signed for 34 years, they have a total of 63.9 WAR (with 28.9 in their first year).  I have their WAR salaries as 322.4MM$, and they signed for 313.5MM$.

For the guys with under 2 WAR: WAR salaries of 112.2MM$, and actual salaries of $148.75MM$.  Those guys signed for a huge PREMIUM.

Donald, you are doing two major problems:

1. You are taking the salaries and dividing by years
2. You are only looking at the first year WAR

That’s wrong.  Because that implies that you are going to get the same WAR for the player throughout the life of the deal.

The correct (better anyway) way to do it is the way I’m doing it.

The Sabathia contract can’t really be looked at as 7/161.  It’s 3/69 with a player option for an additional 4/92.

It’s better to looked at it as 3/(69+negative value of the player option).  Negative value of the player option is the average negative surplus value when he doesn’t opt out times the chances he doesn’t opt out.  So maybe like 3/85.

Tom, I see what you’re saying about the lifetime value of a contract; but I never claimed that the xWAR value I list is going to stay constant over any contract (in fact, at the end of my original post I do an estimated contract for Matt Holliday and John Lackey, where they lose 10% of their WAR per year under contract!). So I am quite aware of player decline, thanks.

What I’ve done is a straight correlation study, so I can say that a player signing an n-year contract with this xWAR next year got this amount of money. I think you are right in that we should treat 1-year deals separately from multiyear contracts; if I partition it that way, I still only get $4M/xWAR for multiyear deals and $2.8M/xWAR for 1-year guys. That’s not too far off from your 45% discount you mentioned earlier.

Maybe the kind of comparison I have made is only valid when comparing only two-year deals, or only three-year deals, etc. I’ll have to think more about this.

Right, perfect.  We’re in synch on the 1-yr discount.

Now, show me where you get a premium for premium players.

Tango, I see what you’re doing now, that makes sense. The only problem I think is that when you forecast eight years out, even small changes to your model will significantly affect the outcome. For example, you set $/win at $4.23 in the first year, rising 10% per year, and Teix at 4.53 wins decaying at 0.5 wins per year. The result is a fair value of $122.5m. But when you make small adjustments to the salary inflation rate and Teix’s decay rate you get the following:

                     Salary inflation (%)
                8      9     10     11     12 
         0.3  151.2  156.1  161.1  166.4  171.9 
Decay    0.4  133.6  137.6  141.8  146.2  150.6
(WAR/yr) 0.5  116.0  119.2  122.5  125.9  129.4
         0.6   98.5  100.8  103.2  105.6  108.1
         0.7   80.9   82.4   83.8   85.3   86.8

So, you could plausibly argue that Teix is worth anywhere from $80m to $170m. Even a 0.1 win change in the decay rate and 1% change in the inflation rate results in a fair value range of $101m to $146m, so it’s hard to conclude whether the Yankees truly over or underpaid. Add in the uncertainty surrounding Teix’s true WAR rate and the $/win level, and it becomes very difficult to say whether the contract is expensive or cheap.

I have a feeling that one-year and multi-year contracts are being valued the same way, and that the difference in $/win stems from the way multi-year contracts are modelled. Some areas to look at might be adjustments for time value of money, WAR uncertainty, differences in aging decay rates, scarcity of available free agents at the position in the coming years etc.

Dackle, I agree with you totally.  This is why I always give a 0.5 win allowance at the starting point when judging contracts, because of the uncertainty.

And even the inflation rate of 10%, while true historically, can be anywhere from 0 to 15%.  So, the further out the years, the more sensitive.

I’m with you all the way here.

The best way to judge is based on the 1-yr only and 2-yr only and 3-yr only deals.  Once you’ve got those, then you kinda/hopefor that the rest of the deals follow the same kind of pattern.

Yeah, and the other thing is, there aren’t many eight-year contracts, so this problem is more of an exception than the norm.

Still, I think it is probably better to convert all long-term contracts to present value, rather than to project WAR and salary inflation into the future. If I convert Teix’s contract to present value using a variety of discount rates, I get:

Disc rate   Salary  Per year
    8%       139.8    17.5 
    9%       136.0    17.0
   10%       132.3    16.5
   11%       128.8    16.1
   12%       125.5    15.7

Then you can just divide by a simpler estimate of WAR—eg last three year total, or Marcel WAR for this year. In other words, if you eliminate estimated WAR over the life of the contract (requiring a somewhat subjective decay rate), then you’ve eliminated one dimension of uncertainty. In this case, if Teix can generate 4.53 wins, then at a conservative 8%, he’s worth $3.9m/win, and at an aggressive 12%, he’s worth $3.5m/win. The confidence range of $3.5-3.9m is a lot tighter than say C$80-170m if you project both WAR and salary inflation.

I don’t think I can agree with that.

Suppose that the drop in WAR is 3 wins per year.  Then, he’s out of baseball in 2 years.

What saves your process is that the WAR drop is more moderate.  However, I don’t know that it necessarily works.

What we do know is that a player is being paid for his future performance.  Therefore, you really have no choice but to estimate that future performance.

Dackle, I think the way I was treating it is along the lines of your last post. Thanks for articulating it well. You have corrected for inflation, which I didn’t do initially, but otherwise I think we’re thinking about this similarly.

Tom, I understand your point about estimating future performance, but I’m not sold that the best way to subtract 0.5 WAR per year (or 10% off per year or whatever method, for that matter). The player could sustain his production over the contract, or he could get injured and be out of baseball tomorrow.

As you say, let’s deal in what we do know. We know how much production we can expect out of a guy next year, and how much clubs are willing to pay for that and his future performance. Whatever the decline rate and the attrition risks are, teams have taken that into account already. So that’s why even though I don’t think dividing AAV by xWAR next year is a great measure (or even a really good one), I don’t think it’s an unreasonable estimate to figure it this way.

As for premium prices, let’s compare just contracts of the same length. Because of sample size, I think we have to stick to two- and three-year deals.

2-year contracts
xWAR premium: $5M (n=1)
xWAR above average: $3.4M
xWAR below average: $3.8M

3-year contracts
xWAR premium: (n=0)
xWAR above average: $4.1M
xWAR below average: $5.1M (n=3)

That last number is thrown off by the high-priced contract of Oliver Perez by the Mets. Without him, that number stands at $4.35M.

So Tom, you are right in that players who are low-WAR producers actually get paid more per win, just because their contracts are overall cheaper and it’s easier to tack on an extra $500k to secure the player for most teams than an extra $1M.

I still believe that “premium” players demand a bit more, but don’t have the data points to show that at this time, unfortunately. Good players get long-term deals, and I think we’ve established that we don’t quite know how to deal with those yet.

Let’s focus only on the 2-yr and 3-yr deals, and let’s presume both are efficient.

The 2-yr deals paid out 222.5MM$ for 42 years, and the 1st year WAR is being reported as 29.3 wins.

The 3-yr deals paid out 232.8MM$ for 27 years, and the 1st year WAR is being reported as 17.9 wins.

How can we create a model that makes both those two accurate?  That is, that the teams are paying properly.

If we start with no increase in dollars per win, then the 3-yr deal is valuing each win at 5.75MM$.

If we have a 5% increase in salary each year, the 3-yr deal is valuing each win at 5.55MM$.

If we have a 10% increase, the 3-yr deal is valuing each win at 5.35MM$.

What about the 2-yr deal? 

At a 10% increase, the 2-yr deal is valuing each win at 4.45MM$
At a 5% increase, it’s 4.6MM$
At no increase, it’s 4.65MM$.

As you can see, there’s no intersection.  Therefore, we can conclude that the 2-yr and 3-yr markets were also valued differently.  And if, you look at the signings, you can see that.  There were alot of 2-yr signings that echo the 1-yr signings.

Now, perhaps the 0.5 WAR drop is too much (not in reality, but in terms of the perceived drop in what they are actually paying for).

If we change the 0.5 to 0.25 instead, perhaps we will get our intersection (and this is basically what dackle is talking about, that the uncertainty in the future WAR estimates is too great).

I’ll look at that a bit later…

"I don’t think it’s an unreasonable estimate to figure it this way. “

I think it is unreasonable, but I am not here to convince you specifically.  Only to provide my best thinking on the matter.

***

“I still believe that “premium” players demand a bit more, but don’t have the data points to show that at this time, unfortunately.”

I can accept that you believe that.  And, I am glad that you acknowledge that you don’t have supporting evidence!  You’ll fit right in here…

***

I’ll tell you why you won’t find the evidence: because WAR is pretty much constructed to not let you.  If for example you DO find evidence, then all I have to do is move the replacement level baseline to the point where there is no premium!  I know, pretty sneaky of me.

As it stands, the market is paying as I’m describing it (as evidenced by all the offseason threads of the last few years), with my current WAR model.

But, please, feel free to keep looking, as I’m happy to tweak my model to fit the data so that there is no bias.

*I think it is unreasonable, but I am not here to convince you specifically.  Only to provide my best thinking on the matter.*

This is most ungracious of a sabermetric icon to say without addressing my points above why I think it’s valid. But it is not my goal to convince you, either. So we’re even.

Tango just walked us all step-by-step through all of his calculations how he felt it should be done, has explained his reasoning for doing it that way, and has fielded questions about his method.  He’s had several threads dealing with the same valuation methods where he gives detailed explanations on what he is doing and why.  He’s continuing to discuss it with you.  What more do you want?  I think it’s a bit ungracious to insult him and end the discussion for not getting exactly what you want even as he has defended his method and is still engaging in discussion with you on the matter.

Actually, I think the way to do it is the present value of the contract divided by the expected WAR over the life of the contract, because 4.53 expected wins next year isn’t the same as 36.2 (4.53 x 8) wins over eight years. I don’t think you can just divide the average annual value of the contract by next year’s expected WAR, because that’s not exactly what’s happening. The Yanks aren’t paying $180m in 2009 dollars for 4.53 WAR/year ($5m/win). In effect, they’re shelling out about $130m in 2009 dollars for say 22.2 WAR over the contract life ($5.9m/win).

The other complication is the longer the contract, the more the performance uncertainty. If you expect 2.0 WAR from Teix in year 8, that isn’t quite the same as 2.0 WAR next year for a player signing a one-year contract. For that player, it might be 2.0 wins +/- 0.5. For Teix in year 8, it might be 2.0 wins +/- 3.0 wins. Not sure though if that means the Yankees pay less for year 8 or Teix demands more.

Anyway, a one-year contract could indicate the terms were basically set by the team—ie the player took whatever he could get—thus a 45% discount.

Donald: because you have provided such fantastic data, I’m giving you wide latitude in your comments (something I reserve for the regulars, but you’ve got yourself alot of currency with me).  I’m reading it in the best possible light, and to that end, I’m just going to presume that your comment was a bit tongue-in-cheek.

As for addressing your method, I pretty much covered it with dackle in my post 19, didn’t I?  You made your post in post 20, and my only response is what I’ve already said in post 19.  We’re just going to go in circles.

If you have some new point to make that won’t be covered by my existing answers, then by all means, make it, and I’ll be more than glad to consider it.

I’m with dackle/25.

Alright, I’ll apologize for the snide remark. Tom, I do appreciate your input on this. I felt kind of dismissed, because I still can’t reconcile the gap in salary values right now..

I agree with Dackle that estimates of player decline and inflation make it very difficult to come up with a solid number on any contract lasting longer than 3 years or so. But can’t we assume that teams have already accounted for “depreciation” and attrition in their salary number already? If you consider the length of the deal and next year’s xWAR, why doesn’t that tell you the value of the deal the player will get?

My goal is not to “get what I want”, but to understand why the approach I’m taking is flawed. I understand the AAV is not the best way to calculate things, but it can’t be much off, can it? Maybe I’m just not seeing something very obvious.

"But can’t we assume that teams have already accounted for “depreciation” and attrition in their salary number already? “

Definitely correct.

***

“I understand the AAV is not the best way to calculate things, but it can’t be much off, can it? “

Yes, it is.

***

Think of it like this, WAR per year:
5, 4.5, 4, 3.5

Dollars per win, each year:
4, 4.4, 4.84, 5.32

Multiply the two and add it up.  That’s how much teams are going to pay.

If you can follow and accept that, then I think we can do business.

***

We can discuss as well that 3 wins in 4 years is not worth as much as 3 wins today, so it requires further discounting.  That’s another side-issue we don’t need right now.

It is because the teams have already accounted for those factors that we have to also account for them in determining what value they put on each marginal win.  The teams look at a player and estimate (via projections or gut feelings or however they are doing it) how much the player is worth to the team in wins, and then decide how much they are willing to pay for that (that may be a simplification, but it’s more or less the basis of this type of evaluation).  So if the team is expecting the player to decline in win value as well as for the market to rise, then that will affect the final value they give in a contract.

Basically, teams start with a value they are willing to pay for a win, and then work out the contracts they are willing to pay from there.  We have to do the reverse:  we start with the contracts they give and work back to the value per win.  To go back to their starting point, we have to go through the same process they went through.  If they start with a value of $4.5 million per win, and they are giving a 5 year deal, they will attempt to account for player decline and inflation in arriving at a final value for the contract, which means if they project the player at 3 wins in year 1, they will not be paying out 3 wins x 5 years x $4.5 mm per year.  They’ll adjust the future wins and future $/win to arrive at a different number than what that calculation would give you.  So instead of paying $68 million in total salary over 5 years, they might pay $52 million because of the expected decreases in player performance and the adjustments for inflation.

So if we know (or assume) that teams are accounting for that in their valuations, then we can’t just take the average value and divide it by the expected value of the first year, because running that calculation back to the value placed on a win assumes that the teams valued the player that way in the first place.  In the above example, we would take the $52 million and divide by 5 to get 10.4 million per year, and then divide by 3 to get an initial value per win of $3.5 million, which we know is way less than what the team started with in the above example.  That’s why we have to try to recreate the conditions the teams are using to value players in retracing their valuations back to value/win.  Because they have already accounted for it in arriving at their final valuations, we also have to account for those things in trying to retrace their steps back to the initial value per win.

There are several factors which can impact the premise of all these calculations: that contracts signed were efficient from both player’s and team’s side.
There were team-friendly contracts - like Pettitte and Wakefield. They wanted to negotiate with only one team, and signed contracts that were below their market value.
One-year incentive-based contracts should be counted as the projected value of the contract for projected WAR.
Some teams seem to allow wishful thinking to impact their reasoning - Mets with Perez, and (presumably) this year with Franceur.
Some teams improperly addressing defense - Dunn was payed for his bat, without taking into account that his D may turn him into replacement-level player.
WAR improperly addressing defense - consensus in baseball is that Teix is one of the best fielders at 1B, but UZR doesn’t think so; since !B is second-worst position for UZR, behind catcher, it is possible that Teix’s defensive value is higher than what WAR says.
I think that contracts for best players can actually give us the best information how teams value talent. Most of the time, if the team needs 2 WAR player, there are several options. But CC is only one. So, if a team wants to sign him, they need to offer close to the maximum they are prepared to pay him. So, they have to assess his value now, his decline, possibility of catastrophic injury, value of win now and 5+ years in future,… And, since we are talking about premium talent and very large contract, they have to do all those things well AND be ready to assume more risk than for some average player, because if they don’t, they won’t get him.
Also, with large contracts, it should be obvious which contracts were based on bad evaluation (Soriano, Lee - defense), which included premium based on other factors (Rodriguez - to show that Texas is serious and to serve as face of the franchise, next Jeter’s contract) and which represent team’s valuation (badly evaluated contracts also present team’s evaluation, but probably only from that one team).
With Yankees, luxury tax should be included. I’m not comfortable with 40% for every new contract, more like 40%*(affected salary)/(total salary). But, NYY contracts should represent only slight increase over what other teams were willing to pay.

Thanks for the explanations, I think I’m getting a grasp on my problem here. I think the way I’m defining what teams pay per win is different than actual money paid per win, and that’s causing the discrepancy (and frustration to Tom).

I was looking at this problem more from a “what can free agents expect to be paid” kind of angle. Given a contract length, player’s age, and current xWAR projection, you should be able to estimate what a club might be willing to offer. That’s different from how much teams are willing to pay per WAR. Sorry for any headaches I may have incurred.