February 4, 2010 at 3:51 pm by Capitol Avenue Club under Atlanta Braves
I’ve finished the preliminary work and I’ve developed the beta version of the Time Adjusted Trade Value Calculator–one of the ultimate goals of this project. Before I get into formally introducing it, I want to spend a bit of time discussing the model that we’re working with. If we all understand and agree upon the model, something I think we can do, then discussions concerning this topic will be a lot more productive.
As I discussed in Part I of this series, when sabermetricians calculate trade value, they generally determine how many wins a player is worth, multiply that by the marginal value of a win, and subtract the player’s salary. The basic framework has NOT changed, that’s still what we’re doing. My model is basically the same thing that we’ve been using for years. All this model does is fine tune a few things so that it mirrors reality a bit better.
The first thing my model does to improve the current one is adjust for the time value of money. A dollar today is more valuable than a dollar in the future and that’s something that we sometimes ignore in trade value analysis.
The second thing my model does to improve the current one is adjust for the time value of wins. A win today is more valuable than a win in the future–something we’ve completely ignored. I discussed the theory of uncertainty of future winscapes at length in part I.
The third thing my model does to improve the current one is adjust for a team’s position on the win curve. Adding a win to an 85 win team has a lot more utility than doing the same to a 65 win team or a 105 win team.
So, the model is, quite simply, a player’s win value (adjusted for time) times the marginal value of a win (adjusted for position on the win curve) minus a player’s salary (adjusted for time) equals his trade value.
That’s it. That’s the model and it’s the exact same thing that we’re currently using. It’s only in the parenthetical statements, the adjustments, that my model deviates from that of what we’ve been using.
So, now that we have the model, we now spend many hours debating as to how to define each parameter in the model. I’ll present what I’ve done to define each parameter so far and we’ll let the discussion go from there. By the way, in no way am I asserting that these are the best ways to define each parameter, only the best ways I know of/can manage to. If you’re familiar with a better way to quantify something, SPEAK UP! I want input on this, I’m not smart enough to do this all by myself.
For a player’s win value, you can use whatever you want. It’s usually some forecast of the Wins Above Replacement metric, that seems to be what people are most comfortable with.
Adjusting that for time is a more open ended discussion. I’ve used the present value formula to calculate the time discount and made the rate of return adjustable. When you open the file, you’ll see a 6% discount rate. Let me explain how I came up with this figure. Basically, the question I attempted to answer was, “What are the chances a win is completely useless in the future”. I used Steve Sommer’s win curve to determine when adding an additional win is useless–about 80 wins. I accounted for variance by subtracting a standard deviation, then accounted for the noise of the data by subtracting the standard error of true talent level (measured by fWAR) regressed on actual wins. At that point, the probability of a team contending is so small, even if their true talent level jumps a standard deviation due to variance or error in measurement or whatever and the standard error of the win value metric vs. actual wins is accounted for entirely by over achievement, they still stand very little chance of contending. I used a χ²-distribution to calculate the probability of a team being that bad (a ~64.5 win team, 10.3 accounted for in SD of win values and 5.2 accounted for in standard error of win values vs. actual wins) in terms of true talent level. It’s 6%. Like I said, this is an open ended discussion and if you have any thoughts as to better ways to quantify this, please speak.
The marginal value of a win is another one that’s sort of open ended. I used Steve’s win curve with a $3.4 million per win base to calculate the marginal value of a win based on position on the win curve. Here is a table in which I’ve listed the marginal value of a win for each position on the win curve. The first column represents the range of true talent level, the second the marginal value of a win in million dollars:
You’ll note that when a team’s true talent level is between 83 and 88 wins the cost of a marginal win is calculated via function. This represents the “curve” part of the graph, the portion where the marginal value of a win is constantly changing depending on talent level. In essence, the MVOwinscape is a piece wise function in which the first derivative = 0 for everywhere except the 83-88 win range. I ran a polynomial best fit regression on the data between 83 and 88 wins, whose equation I used as my function to calculate the marginal value of a win for a team whose true talent level lies somewhere between 83 and 88 wins.
This, however, raises a big problem. Where does a team fall on the win curve in the future? I really don’t have a good answer (like I said, I’m not smart enough to do this all by myself). The only way I know how to handle this problem is to regress talent level to the mean, but how and over what time period? The how part I’m far from confident I’ve done correctly, I simply regress it in equal increments for lack of an idea of how to do it better. The over what time period part I’ve got a pretty good idea how to handle. Thanks to much assistance from Colin Wyers’ database articles, this research went a lot faster than anticipated, but what I’ve found is that teams’ current records have zero predictive value of their record five years from now and beyond. Therefore, I regress their record to the mean in five equal increments (once each year) and assume league average true talent level thereafter. This doesn’t mirror reality as well as I’d like, but it’s pretty much the best I can do given my mathematical skill set and lack of a crystal ball.
Player’s salary is the one thing there isn’t much debate on. It’s a noise-free variable, just enter his salary. Adjusting for the time value of money is simple, I just used the present value formula. The rate of return is adjustable, use whatever you think is correct. I’ve entered 8%. You’ll see a column called “PV (M)” in the TATVC (Beta), which represents the present value of a player’s future salary. I elected to keep the “Sal (M)” column intact for illustrative purposes.
So, let’s give this thing a try.
First of all, do NOT edit anything other than the green spaces. There is a second sheet in this document. It is extremely important not to edit anything on the second sheet. As always, I recommend saving two copies of the file, an archived copy and a working copy. I always edit the working copy, then paste the results into a third spreadsheet and only use the archive copy to recover a damaged working copy. The basic instructions for using this tool can be found at Beyond the Box Score. There are three additional parameters, true talent level, rate of return on money, and rate of return on wins. I’ve entered the details for Tim Hudson’s three year deal in the green boxes, I’ve estimated he’ll be a 3.5 win player in 2010, 3.0 in 2011, and 2.5 in 2012. I’ve entered his $9 million salaries for 2010-2012 and his $1 million buyout for 2013. I’ve estimated the Braves are an 85 or so win team, an 8% rate of return on money, a 6% rate of return on wins, and $5 million for draft pick compensation. So, we come up with a surplus of about $12.5 million, or, a top 76-100 hitter.
|Tim Hudson||True Talent Level:||85|
|Year||Sal (M)||PV (M)||Wins||Val (M)||Net (M)|
Our previous model had Tim Hudson’s trade value at $8.8 million. Adjusting for the time value of money, wins, and position on the win curve leads to a more valuable contract for the Braves. Consider, he’s an older player, and their contracts tend to be front-heavy in the win department, and the wins next year are more valuable than wins in 2012. Also, the Braves figure to be competitive during 2010 and if all goes well in 2011 and 2012, too, which makes the wins more valuable. Third, he’s paid $9 million each year, and $9 million in 2010 is worth more than $9 million in 2012.
The old model:
|Year||Sal (M)||Wins||Val (M)||Net (M)|
Let’s do another one, why don’t we? Mike Cameron, why don’t we?
The Red Sox true talent level is somewhere in the mid 90′s range. I picked 93. Using the same rate of return on money (8%) and the same rate of return on wins (6%), filling in his projected win values (3.6 (Fans projection) in 2010, -0.5 in 2011), and his $7.75 million average annual salary we see that his contract is worth about $13.6 million in excess value to the Red Sox.
|Mike Cameron||True Talent Level:||93|
|Year||Sal (M)||PV (M)||Wins||Val (M)||Net (M)|
The old model wasn’t too far off in this case, coming in at $13.1 million.
|Year||Sal (M)||Wins||Val (M)||Net (M)|
At this point I turn it over to you in the spirit of sabermetric peer review. Use this tool however you please, and if you have a suggestion to make it better, I’d like to hear it. Special thanks to Sky Kalman, Steve Sommer, and Colin Wyers whose work was instrumental in putting this together.