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How Good Does Yu Darvish have to be to be Worth It?

We won't find out if the Jays have even won the bidding rights to negotiate with Darvish for another couple days, but with all the speculation that the Jays have won, I have inevitably started thinking past just hypotheticals, to whether this is something we should want the Jays to be doing. For me, this means thinking in terms of cost/benefit analysis rather than merely looking at how he could improve the team, so in this article I'm going to flesh this line of thinking out. First, I want to clear that this is purely from a baseball perspective. There's been talk of additional revenues this deal could bring in, and it's certainly both interesting from a business point of view (not to mention relevant to the baseball side), but there's just too many unknown known unknowns to include in what I want to discuss.

The standard way to assess a free agent contract is to project the player's production (usually in terms of current true talent WAR), assign a value to that production using the market value of a win, and apply aging factors, salary inflation to get the expected value of the future production, and compare that to the guaranteed money the plaer received.

In the case of Darvish, it's quite a bit more complicated - we really don't know his true talent level in MLB, there's the whole issue of accounting for the posting fee, and we have no real idea about what it will actually take to sign him, either in terms of guaranteed years or average annual salary value (AAV). Moreover, while I've watched some of the video of Darvish available, I'm certainly not a professional scout and it's not my forte (though I am trying to learn much more about this area). Various prospect experts have weighed in, but we still don't have a very concrete idea of what can be expected in any transition from NPB. So what I want to do is turn the question in its head, and determine how good Darvish's true talent level has to be in order to justify a range of possible contracts.

I have seen figures as low as $10M for the AAV (though I doubt it could be this low), and claims that Darvish wants $20M per season, so these are the salary ranges I'm using. In terms of contract length, I will be considering from 3 to 7 years - I absolutely can't see Darvish accepting a guarantee of any fewer, and I absolutely can't see AA going any longer. The other thing I am not considering is options, because frankly they're too hard to model in corectly, so you would have to value them and I am unaware of a Black-Scholes model for baseball options. Also, they would distract from the actual analysis.

The first issue is how the posting fee should be figured into this type of analysis. I approach this by asking what the posting fee gets you - six years of control of the player, even if the contract isn't that long. So, it seems approprate to amortize the posting fee over the 6 years, meaning the rumoured $48M fee would work out to $8M per year. The only exception would be if the contract is more than 6 years in guaranteed length, in which case the fee should be amortized across the length of the agreement. The other way to treat the posting fee is just to dump the entire cost in the first year cost. I don't think this is as appropriate from a baseball point of view, but is more consistent from a Net Present Value (NPV) perspective, so I'll return to it at the end of the article.

Next, we have to make a variety of assumptions in order to complete the analysis. They are:

  • The value of a marginal win in 2012. I assume $5M per win in my base case, which is what Fangraphs is currently using.
  • Salary inflation in subsequent years. I assume a base case of 5% per year, though it has been higher in recent years (closer to 7-10%, but I prefer to err on the side of caution).
  • Aging factors - Darvish is 25, so I assume constant production from 2012 through his age 27 season in 2014, and a decline of 0.5 WAR/year thereafter

These assumptions, along with the AAV and contract length, allow us to set the surplus value of the contract equal to zero (since we wouldn't expect the surplus value on a free agent contract to be positive) and solve for the true talent WAR in 2012 - essentially telling us how good he has to be to justify the contract. For a 5-year deal with an AAV of $15M (the midpoints of my range), it look like this:

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So basically, if Darvish were to get a contract of $75M over 5 years, with a $48 posting fee, he'd have to produce at a true talent level of 4.5 WAR next year. By point of reference, for a pitcher to put up that kind of value in last season's run environment, he's have had to pitch 200IP at an ERA of roughly 3.30. This is, however, only one estimate of the contract, so below is a sensitivity table showing what level of production (WAR) would be needed to justify contracts across the ranges presented. For those who prefer to think in terms of ERA, the table on the right converts the WAR level to the equivalent ERA (assuming 5.30 AL replacement starter ERA, 200 IP and 10 runs/WAR):

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Note: The dollar figures in this, and all further tables, are Darvish's salary only, though the posting fee is taken into account.

Now, maybe you think that the new CBA is going to make free agent salaries increase, since teams can't spend as much in other areas, so the 5% inflation number is too low. So let's change it to 10% and see how the results differ:

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It's pretty clear that the increased inflation means that less production is required, however, it was suprising to me that the difference wasn't much bigger. A further argument that some make is that $5M is too much to pay for a win, so I'll show what happens it you change to $4M per win (with 5% inflation):

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The final analysis I want to do is to see what happens if you include the posting fee as a cost in 2012, and then nothing afterwards. This is obviously going to make shorter deals look really ugly, but again, I think it's an interesting approach, especially for the 5-6 year range where the deal is likely to end up, because it looks at things on more an a NPV basis, which is important from a business analysis perspective:

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As expected, the production level has to be much higher. Now, I've only looked at a few of the ways you could change the assumptions, but I don't want to make a million charts. So, what I've done is included a link to download my Excel file. It's got two pages, first to change assumptions and look at specific scenarios and the second to compile the senitivity tables. Both have simple but effective macros to automate everything. So if you think that the marketing revenues will effectively reduce the posting fee, you can go ahead and do that, or reduce the annual salary to account for it. Changes can also be made to any of the assumptions I've discussed above.

This brings me to the end, and normally a few concluding thoughts would be in order. However, since the purpose of this was more to explore various scenarios, there's only observation that I want to make. It's pretty clear that whichever team wins the bidding, assuming they also sign Darvish, they will be paying him as a #1/2 type, and he will have to produce accordingly to live up to the contract.

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