THE BOOK--Playing The Percentages In Baseball

Tuesday, November 15, 2011

How do you (quickly) compare contracts of differing lengths and amounts?

By Tangotiger, 05:48 PM

The non-quick way is to do what I did here. That chart shows how to infer the wins you are paying for, based on years and length of contract. For example, in that chart, it says that if you sign someone who is a true talent 4 WAR player entering the 2008 season, any of these salaries would be equivalent:

yrs    total    perYear
1    $17.6      $17.6 
2    $34.5      $17.3 
3    $50.5      $16.8 
4    $65.2      $16.3 
5    $78.0      $15.6 
6    $88.7      $14.8 
7    $96.5      $13.8 
8    $100.7      $12.6

Notice what happens if you simply add the number of years to the average annual salary:

yrs    total    perYear    function
1    $17.6      $17.6      $18.6 
2    $34.5      $17.3      $19.3 
3    $50.5      $16.8      $19.8 
4    $65.2      $16.3      $20.3 
5    $78.0      $15.6      $20.6 
6    $88.7      $14.8      $20.8 
7    $96.5      $13.8      $20.8 
8    $100.7      $12.6      $20.6

Here it is for a 2-WAR player entering 2008:

yrs    total    perYear    function
1    $8.8      $8.8      $9.8 
2    $16.1      $8.1      $10.1 
3    $21.4      $7.1      $10.1 
4    $24.3      $6.1      $10.1

It starts to break down when you get to true 5 WAR players or above, but it only breaks down if those superstars sign a 3-yr deal or shorter:

yrs    total    perYear    function
1    $22.0      $22.0      $23.0 
2    $43.8      $21.9      $23.9 
3    $65.1      $21.7      $24.7 
4    $85.6      $21.4      $25.4 
5    $104.9      $21.0      $26.0 
6    $122.6      $20.4      $26.4 
7    $138.2      $19.7      $26.7 
8    $151.1      $18.9      $26.9 
9    $160.5      $17.8      $26.8 
10    $165.7      $16.6      $26.6

In all cases, the “function” column is pretty stable.

So, there was a discussion about using the contract a player signed to get his value. And of course, how to compare players of differing years.

Therefore, I will offer that we simply do:
Annual Average Salary + Years Signed

A 3yr/36MM$ salary (value of 3+12 = 15) is equivalent to a 5yr/50MM$ salary (value of 5+10=15).

Nice and simple.

If a player is negotiating with a team, if he says “I need one more year”, then the team can just say “Then you have to take 1MM$ a year less”.

Nice and simple (as a starting point).

• Sabermetrics • Finances

#1 David MIck (see all posts) 2011/11/15 (Tue) @ 19:31

I love how simple that is. It’s just my speed.

#2 Xeifrank (see all posts) 2011/11/15 (Tue) @ 20:26

So let me see if I get this right. Let’s take Matt Kemp’s new contract.

8/$160M, $20M per

this is the equivalent to…

7/$147M, $21M per
6/$132M, $22M per
5/$115M, $23M per
4/$96M, $24M per

then the algorithm breaks down from here?

#3 (see all posts) 2011/11/15 (Tue) @ 20:29

I’ve been wanting to see more time-value analysis of these deals, but I’m not sure how to derive a discount rate (inflation of $, inflation of $/win, owner’s expected return) and I can’t decide how to account for what the owner would pay instead. Any ideas?

#4 Bill Waite (see all posts) 2011/11/15 (Tue) @ 21:18

How about a simple formula that takes into account the idea that the effect of inflation is proportional to the size of the salary and that the cost of the extra year depends on the aging curve?

Something along the lines of:

Salary*(1 + .05*years + .01*(finalAge-30)^2)

Those particular weights (which I just made up off the top of my head) would set a 5/50 contract with a 30-year player equivalent to a 3/36.3 contract. With a 25-year-old player, 3/32.6. 35 years old, 3/37.7.

It would also set Matt Kemp’s equivalencies at:

7/$153 ($21.85/yr)
6/$142 ($23.75/yr)
5/$128 ($25.6)
4/$109 ($27.3)

#5 MGL (see all posts) 2011/11/15 (Tue) @ 21:44

Nice job! Amazing how you come up with stuff like this!

#6 David Cameron (see all posts) 2011/11/15 (Tue) @ 22:00

This formula says that 4/116 is equivalent to 10/230. I’m pretty sure Albert Pujols would sign the latter contract and not even bother countering the former one.

At high AAVs, the extra year is clearly worth more than $1 million per season.

#7 Telnar (see all posts) 2011/11/15 (Tue) @ 22:43

Implicit in [6] seems to be an assumption that -0.5 WAR/yr isn’t the right aging adjustment for periods as long as 10 years. Do we have evidence one way or the other on this?

#8 Tangotiger (see all posts) 2011/11/16 (Wed) @ 00:02

Dave: I did say this is a simple quick way to do it. And by design, rules of thumb break down at the extremes.

Giving me Pujols, signed through age 42, at near ARod-level, is about as extreme as you can come.

It’s like my -0.5 win aging. It’s less if you are under 27, and more if you are 37 or older.

These are quick short-cuts, that allows us to move the conversation in some forward direction. Maybe not completely forward in a straight line, but forward nonetheless.

#9 Tangotiger (see all posts) 2011/11/16 (Wed) @ 00:37

Let’s try to come up with a more reasonable aging pattern for Pujols:

wins      $perWin      Value 
 6.50      $5.00      $32.5 
 6.00      $5.25      $31.5 
 5.45      $5.51      $30.0 
 4.85      $5.79      $28.1 
 4.20      $6.08      $25.5 
 3.50      $6.38      $22.3 
 2.75      $6.70      $18.4 
 1.95      $7.04      $13.7 
 1.10      $7.39      $8.1 
 0.20      $7.76      $1.6

We start him at 6.5 WAR (just for illustrative purposes), and make him drop progressively more each year.

We start at 5MM$ per win, and increase by 5% each year.

So, we get the values above.

That gives us this:

yr    salary    avg    function
1     $32.5      $32.50      $33.5 
2     $64.0      $32.00      $34.0 
3     $94.0      $31.35      $34.3 
4     $122.1      $30.53      $34.5 
5     $147.6      $29.53      $34.5 
6     $170.0      $28.33      $34.3 
7     $188.4      $26.91      $33.9 
8     $202.1      $25.27      $33.3 
9     $210.2      $23.36      $32.4 
10     $211.8      $21.18      $31.2

It works pretty nicely through 8 years.

You just have to be careful with ANY rule of thumb.

#10 Tangotiger (see all posts) 2011/11/16 (Wed) @ 01:03

By the way, this works if every subsequent year you add is worth 2MM$ less than the year before.

For example, you get 20MM$ the first year, the value is 21 “points” (1+20).

You get 18MM$ the second year, that’s a total of 38MM for 2 years, or 19 a year, or 21 “points (2+19).

You get 16MM$ the third year, that’s an average of 18MM$ a year for 3 years, or 21 “points”.

You get 14MM$ the 4th year, that’s an average of 17MM$ a year for 4 years, and again, 21 “points”.

***

So, we can see when things breakdown. It won’t work for a guy in his mid 20s, because he’s not aging that bad. Indeed, he may even get MORE money the more years.

And we can see that it’ll break down for a guy in his late 30s, as the more years he adds, he’s going to get way less than a 2MM$ drop for each subsequent year.

In the Pujols example above, we see that his year 8, 9, and 10, gets far bigger than a 2MM$ drop from the previous year.

***

So, as long as you can imagine the player you are looking at getting 2MM$ for each subsequent year, then this equation is fine.

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THE BOOK--Playing The Percentages In Baseball

Tuesday, November 15, 2011

How do you (quickly) compare contracts of differing lengths and amounts?

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