Valuing Nolan Arenado’s New Contract

Hey everyone, and welcome to the convergence of two recurring segments. It’s the highly awaited crossover between “Can You Believe the Cardinals Got Nolan Arenado for That?” and “Let’s Value Gimmicky Contracts,” two columns I almost assuredly enjoy writing more than you enjoy reading.

Let’s get the deferred money part out of the way first, because while it’s obviously very important to the Rockies and Cardinals, it has nothing to do with Arenado’s decision-making. He’s getting his cash, and whether the check says Monfort or DeWitt, the cash still spends the same. It won’t affect his decision on whether to rip the whole contract up.

As Jeff Jones reported, Arenado agreed to modify his contract as part of the trade. In 2021, he was due $35 million. Now, he’ll receive $15 million this year, paid directly by the Rockies. He’ll also receive $20 million in deferred compensation, regardless of whether or not he opts out. If he’s still under this contract, that money will be sent to the Cardinals, who will then pay it to Arenado. If he opts out, the Rockies will pay him the $20 million directly.

Finally, if Arenado doesn’t opt out, the Rockies will be on the hook for the $16 million he’s due in 2027. That’s the new year that the Cardinals agreed to as part of the trade, and while it’s unclear exactly why the Rockies chose to pay that part rather than some pro-rated portion of earlier salaries, here we are.

For the Cardinals, this is a great fit. They’d been acting as though cash was a key constraint this year, and getting a year of Arenado at no cost (literally, no monetary cost!) does a good job of making the short-term books work. In the long run, they’re paying him nothing for one year (if he opts out after 2021), $35 million over two years (if he opts out after 2022), or $164 million over seven years.

With that covered, let’s talk about Arenado’s options. The structure is as straightforward as it gets for these kinds of things. After 2021, Arenado will have the option to walk away from the entire deal and become a free agent. If he opts to remain in St. Louis, he’ll get another chance to wash his hands of the deal after 2022. If he still wants to stay, then he’ll be under contract until after the 2027 season.

When Arenado signed his extension two years ago, I covered a probabilistic way of thinking about the value. In the interim, a few things have happened. First, Arenado had a down 2020. Second, Dan “Dr. ZiPS” Szymborski gave me a long-term forecast for Arenado that beats my generic aging expectations. Finally, I’ve added a few bells and whistles to the option model in the interim. For the most part, though, we’re just running it back.

To calculate the value of an opt out, we need a few things. First, a central projection for how good a player will be in the future. ZiPS is all over that. Here’s the next five years of Arenado’s projections:

I’ll assume a standard aging curve after that, which gets us a central tendency for how his career will go.

Next, we need to apply variance. I’ve found in previous studies that projections for players who fit Arenado’s mold — mid-career and projections above 2 WAR — vary with a standard deviation of roughly 1.4 WAR from year to year. We’ll start with projections as a baseline, but to figure out the value of an option — the right but not obligation to do something — we’ll have to simulate 10 million or so different futures, then work out what happens on average.

Finally, we need to figure out how to translate Arenado’s projections into a potential new contract. Converting WAR to dollars misses some team-building effects, and as Craig Edwards showed last year, it’s not as simple as applying a linear conversion. Still, it’s my model, and it’s just for Arenado, not for the dang league as a whole. Let’s start with $8 million per win.

While we’re varying projections, we also need to vary the cost of a win. I assumed that the cost of 1 WAR will increase by an average of $250,000 per year, but with a standard deviation of $800,000. In plenty of years, the cost of a win will go down, even if the overall cost increases over time. There might be, say, a global pandemic, or a shortened season with no fans… wild nonsense like that.

Finally, we get down to brass tacks. When the player reaches his opt out, there’s a simple calculation. Take his new median projection, age it down as appropriate, and come up with a WAR projection for the remaining years of the contract. Multiply that number by the cost per win that we simultaneously calculated, and you have the contract that Arenado would sign, in a perfectly efficient market, if he opted out. I added one quick sanity check: if it’s within $10 million dollars of breaking even, Arenado won’t leave. That represents the uncertainty of finding a new contract, as well as the benefits of familiarity.

If there were only one opt out, our calculation would be simple. After one year, we simply apply an aging penalty and a random change in projection. Then, we use that to price out a new contract. With only one year before an opt out, things are simple like that.

In Arenado’s case, the odds are stacked in favor of him declining the opt out. The average situation (no change in projection) sees him with a $101.5 million contract after this year. He’d need to raise his 2022 projection by roughly 1.5 WAR to merit a contract that would be worth opting out for. How often does that happen? Roughly 14% of the time.

Let’s look at it more thoroughly. Here’s how much he projects to make in each scenario, including the $35 million he’s making in 2021 regardless:

In that sense, the opt out already in Arenado’s contract is “worth” $6 million. It could be worth more, though. Replace our forecasts with the ZiPS forecasts that ignore 2020 (they used 2019’s rate stats again), and instead things get a little spicy:

Hey, look! If Arenado were projected to be a bit better — and again, injury has at least something to do with why he’s not — then his option would be worth more. Neat stuff!

That’s not why you’re here, though. Or, maybe it is, but that’s not what Arenado’s contract looks like anymore. In agreeing to head to St. Louis, Arenado gained an extra opt out after the 2022 season.

It’s slightly tricky to model nested opt outs like this, where the second one can only be exercised if the first isn’t. In practice, the decision won’t be overly complex. After 2021, Arenado and his agents will model something that looks a lot like the calculations I did above, valuing his existing contract based on his current projections and including a 2022 opt out. That number will come out to something higher than $180 million, because some amount of the time, he’ll opt out after 2022 and get a raise.

With that number in hand, they’ll approximate what he could get on the open market and compare the two. If he thinks he can get more now, on the open market, than the value of his existing contract inclusive of the opt out, he should leave. Otherwise, he should stay and wait the extra year to see what happens.

Unfortunately, that doesn’t work well in my framework. In each of my 10 million simulations, computer Arenado would need to run 10 million simulations to work out the value of his option using this Monte Carlo method. That comes out to one hundred trillion simulations, and it doesn’t even add much precision for our trouble. Instead, I’m just setting the cutoff higher; Arenado will need $15 million in prospective gains to pull the trigger on opting out after year one.

How does this change things? Less than you’d think. I previously calculated that Arenado would opt out 14% of the time; he now opts out 12.4% of the time in year one. With more time to vary, and thus a higher percent chance of improving enough to opt out, he also has an 8.2% chance of opting out after 2022. All together, that looks like this:

In the scenarios where Arenado opts out after 2022, he is, on average, quite good. This makes sense, because he’s only opting out in the best 8% of scenarios. If he muddles along, he’s staying. If he has a late-career renaissance, it only stands to reason that he’ll be more valuable. All told, the two opt outs add a projected $13 million to the value of the deal — not bad!

Once more, let’s plug in a more optimistic view of Arenado and see what shakes out. This is Dan’s pre-2020 version, where he’s projected for 5 WAR in 2021:

Again, the better Arenado is now, the higher the chance he opts out, and the more the second opt out is worth to him. In the base case ZiPS projections, a second opt out added roughly $6 million to Arenado’s expected earnings. In this pre-injury case, a second opt out would add more than $17 million.

That’s the gory math of the way this complex contract works. In practice, however, Arenado seems likely to take the Clayton Kershaw route. Kershaw, too, had the choice of opting out of his contract early and becoming a free agent. He used that leverage to get the Dodgers to offer him a new contract, avoiding free agency but monetizing his ability to leave. Even if Arenado improves this year and next, that seems like the most likely eventuality.

What’s an opt out worth? In Arenado’s case, it’s real money in expectation. Add that to the extra contract year that he snagged as part of the deal, and it’s easy to see why the player’s union was okay with him taking deferrals in 2021. Remember, though: even in the scenario where he’s at his best, he still leaves less than half the time. It’s valuable despite being unlikely, and that’s the magic of options.