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Risk vs. Reward

[Hey, danup here--Extra, extra: Khalil Greene is a Cardinal, in case you don't check the fanposts. Current reports have the trade being for "two pitchers, neither of whom was on the major league roster at the end of last season"; according to the mothership one is a right-handed reliever. Now, back to your regularly scheduled analysis.] 

WARNING: Lots of numbers forthcoming!!!!!!!

I want to thank both Joe Sheehan (subscription required) and Derrick Goold for providing primers for today’s discussion, which is not all that different, BTW, from the discussions of the last couple of days. (Meet the new boss – same as the old boss!) There’s been lots of discussion the last few days about the Cards’ decision to not offer arbitration to either Looper or Springer, but I wanted to put all this into a financial context.

First of all, I’m going to use as my basis Nate Silver’s study a few years back on the value that draft picks offer teams. This is critical to our analysis of the Cards’ decision since that decision could ultimately lead to the Cards receiving draft picks as compensation. I’ll add as disclaimers to our study here that BP uses numbers for WARP and for the value on the free agent market of a win that aren’t generally accepted from other sources. For example, they set the bar for replacement level lower than other sources do and, consequently, have determined that the marginal value of a win is less than the $4.5 – 5 million generally recognized at other sites. Still, if we use their values alone, they’ll translate and work for our analysis.

In 2005, Silver determined that the marginal value of a win was $2.14 M. If we assume an 5% inflation rate over three seasons, then the BP number for this season becomes $2.48 M. BP determined that Springer was worth 2.0 WARP and Looper was worth 4.4 WARP. This puts Springer’s estimated performance value for ’08 at $4.96 M and Looper’s at $10.91 M. Using Justin Inaz’s numbers Springer would be worth about $2.5 M and Looper would be worth approximately $9 M.

Silver’s study determined the average WARP of players drafted between picks 1-7, 8-15, 16-25, and 26+. Now, Springer was a type A free agent. If he had been offered arbitration and signed elsewhere, the Cards would have received either 2 26+ picks (a sandwich pick + a first rounder if drafting between 26 and 30 or a second rounder if drafting between 1 and 15) OR a 26+ pick and a 16-25 pick. Not knowing who would sign him, we could determine that there was a 2/3 likelihood of the Cards receiving 2 26+ picks and a 1/3 likelihood of the Cards receiving a 26+ pick and a 16-25 pick.

Silver determined that, 3 years ago, the net present value of a player chosen between #16 and #25 was worth $8.94 M and the net present value of a player chosen 26th or later was worth $3.24 M. We have to increase that value by 5% per year as we did the value of free agent contracts. After 3 years, the value of a 16-25 pick became $10.36 M while the value of a 26+ pick became $3.75 M.

So, at what point should the Cards offer Springer salary arbitration? If we do, and he signs elsewhere, there is a 2/3 chance that the Cards will receive a net return (from the draft choices) of $7.5 M ($3.75 M x 2) and a 1/3 chance that the Cards will receive a net return of $14.11 M ($10.36 M+ $3.75M). Therefore, the expected net return if the Cards offer arbitration and Springer signs elsewhere is $9.70 M. However, as many of you pointed out, there was a distinct possibility that Springer would have accepted the Cards’ arbitration offer. He would have had only a week to make that decision but clearly, it was something the Cards feared. Why?

Springer received $3.5 M last year and some believed that he might receive up to $4.5 M in arbitration this year. Was he likely to be worth $4.5 M? If you believe BP’s numbers and he pitches as well as he did last year, he would be worth $4.96 M again – that’s better than $4.5, obviously. That makes the decision to offer him arbitration a no-lose situation. You might point out that there’s an opportunity cost involved – if we give Springer $4.5 M, that’s $4.5 M we don’t have to give to someone else (a 2B, SS or SP, for example). However, we could always trade Springer for that player or to get the salary off the books, thus freeing up the money to be used on the free agent market. Therefore, he is still worth an additional $460,000 to the team and it’s a no-lose situation.

This year’s PECOTA projections aren’t up yet but, as of last year, PECOTA had him projected at 1.1 WARP for 2009. It’s safe to say that that number will be a little higher based on the fact that he pitched above his projections for 2008 so we’ll use 1.5 WARP as a projection. Using BP’s numbers, 1.5 WARP is worth $3.72 M. If we’re paying him $4.5 M, we’re losing $0.78 M by signing him to the 1 year contract. Let’s just call it an even $1 M loss to the team by resigning Springer – whether we keep him or trade him, b/c we’re paying $4.5 M for a $3.5 M asset.

If there is a 50-50 chance that Springer accepts arbitration, there is a 50% chance that the team takes a $1 M loss and a 50% chance that the team gains $9.70 M. If you could pay $1 to participate in a coin flip where, if you lose, you lose the dollar and if you win I’ll give you $9.70, would you participate? That’s a pretty good deal. It’s a no-brainer, isn’t it? Still, some of you have pointed that Springer was likely to accept arbitration. What would the likelihood have to be in order to make the arbitration offer a bad one for the Cards?

In order for this to be a bad offer for the Cards, there had to be a 90.7% likelihood that Springer would accept arbitration. If the likelihood that he would decline was even as great as 10%, it was worth it to offer arbitration. Apparently the Cards decided that his likelihood of accepting was at least 90.7% but, remember – this assumes that Springer would receive $4.5 M in arbitration. The Cards likely would have offered something near what he received last year -- $3.5 M -- leaving a 50-50 chance that he’d receive $4.5 M and a 50-50 chance he’d receive $3.5M. The expected cost of the contract would’ve been $4 M – putting the Cards in the hole just $500,000. If that’s the case, there would have had to have been a better than 95% likelihood that Springer would accept. This is a gamble the Cards had to take. The worst-case scenario is that the Cards are set back $1 M. The best case scenario is that they end up +$9.7 M. It was worth the risk, IMHO.

Now let’s look at Looper. Goold believes that it was likely that Looper, too, would have accepted – or at least that the Cards believed that. His rationale – the economy sucks right now (for baseball players, too) and Looper would have decided he’d be better off w/ a 1 year deal this year and a 3 year deal next year than a 3 year deal today. Maybe, but again, maybe not.

Now, Looper is known to be seeking a 3 year deal so it may seem more likely that he would decline the offer than it would be for Springer. However, b/c he’s a type B free agent, the potential payoff to the Cards of him declining arbitration is much lower. This time, we know that the pick would be a 26+ pick worth $3.75 M to the team – considerably less than Springer’s $9.70 M. Moreover, Goold believes that it was possible that Looper would receive an arbitration award of as much as $10 M using Carlos Silva’s contract as a baseline. I’m skeptical. Looper received $5.5 M last year and he wasn’t THAT good. Let’s say a 50% chance of $10 M and a 50% chance of $7.5 M – for an expected award of $8.75 M.

What will Looper be worth next year? He was worth anywhere from $9M to $11M this year. Unfortunately, PECOTA doesn’t help us much as it treated Looper’s marginally successful ’07 campaign as a fluke. ZIPS has Looper projected to be marginally better than he was in 2007 and BP has him at 3.2 WARP for 2007 so let’s go w/ 3.5 WARP. That makes his expected value in the neighborhood of $8.68 M – about what his expected award is. Yes, maybe the Cards don’t want to pay it but they’re going to have to pay someone to be the 5th starter. That cost will likely be $9M or so. If Looper accepts and Carp is healthy and able to pitch, they could trade Looper and his 1 year contract – freeing up that money to spend on a middle infielder.

Let’s go with the worst-case scenario – Looper receives a $10 M arbitration award leaving the Cards losing $1.32 M on the gamble ($10 M payout - $8.68 M value). If there’s a 50-50 chance that he accepts arbitration, there’s a 50% chance the Cards lose $1.32 M and a 50% chance the Cards gain $3.75 M. It’s still a great deal for the team. In fact, it’s a good deal for the team as long as there’s less than a 74% chance that he accepts. If the likelihood that he accepts arbitration is 74% or higher, we shouldn’t offer – in the worst-case scenario.

However, his expected award is $8.75 M and his expected value is $8.68 M – a net loss to the team of just $0.07 M. Using that as our baseline, as long as the likelihood that Looper accepts arbitration is less than 98.17%, the Cards should make the offer. Again, the downside is a loss to the team of $70,000. The upside is $3.75 M. This is more of a no-brainer than the Springer offer. For each, there is very little downside and quite a bit of upside. Sure, it’s possible that both Looper and Springer would have accepted and flamed out but the likelihood is that both offers would have worked out to the team’s advantage. They are, to put it bluntly, offers that should have been made.

In the Sheehan article referenced at the very beginning, he says that

There’s is simply no reason, even given the externalities present, for teams to be as risk-averse as they were in this process. The risk isn’t great enough, and the reward is considerable.

I couldn’t agree more.