1. Figure out how much a win is worth,

2. Figure out how much an individual player contributed to his team's wins, and

3. Convert that number of wins into a dollar value.

We established that an average (i.e., .500) team in the Boston market would earn $186.3 million in revenue. It's important to note that this "average" team is actually very abnormal, in that every roster spot is held down by a league-average player. That is, all hitters generate the league-average amount of run production, and all pitchers prevent the league-average amount of runs from scoring. For this team, it's easy to figure out how much each league-average hitter contributes to revenue:

revenue produced = pct of plate appearances used x total revenue/2

...and each league-average pitcher:

revenue produced = pct of innings pitched x total revenue/2*

** - This formula assigns equal importance to pitchers and hitters, so each side of the ball is said to account for half of the total revenue. You could of course argue for slightly different percentages and adjust the weights accordingly. You could even add a defensive revenue number if you were sufficiently confident in your defensive metrics.*

But suppose we're not working with a league-average player. Take, for example, Mike Napoli's performance in 2011. Baseball Reference estimates that Napoli created 38 runs above average in 6.9% of his team's PAs. A lineup of nine Mike Napolis circa 2011 would produce 551 more runs than the average team, generating some stupid amount of additional revenue. A lineup of 93.1% average players and 6.9% Mike Napoli would be expected to finish with a run differential of +38. Since an average player with 6.9% of his team's PAs would generate $6.4 million in revenue, we can therefore say that Napoli generated an impressive $9.6 million in

*additional*revenue in 2011, meaning his performance was worth a total of $16 million to the Rangers. (For the record, Napoli earned $5.8 million that year, for an impressive return on the Rangers' investment.)

Of course, Napoli followed that up by producing 1 run above average in 2012, but never mind.

So now we know how to determine a player's worth using runs scored. Given past performances, we can then determine if a player earned his salary or, if we have projections for next year, we can determine whether a player is likely to earn his salary.

We can also work backwards to determine how many runs above average a player has to produce to earn his salary. Take Napoli's $13 million that the Sox owe him in 2013. In his career, Napoli has never accumulated more than 510 PAs (8.4%), so we'll use that as an optimistic assessment for playing time. Assuming the same revenue projections -- which is admittedly a little naive -- an average player using 8.4% of his team's PAs would generate $7.8 million in revenue. To break even in this scenario, then, the Red Sox seem to expect that Napoli will generate an extra $5.2 million, or

*23 runs above the average hitter*. If this seems like a lot, it is: last year, for instance, 23 RAA would have put you in the top 35 hitters in all of baseball, right behind Albert Pujols (+24) and a little ahead of Prince Fielder (+21).

Now let's investigate Shane Victorino. Like Napoli, Victorino also is due $13 million next year. Unlike Napoli, however, Victorino has been a full time player for the past five seasons, averaging 644 PAs (10.4%) over that span. An average player with this amount of playing time would be expected to generate $9.7 million, meaning Victorino only has to generate an extra $3.3 million in revenue, or 16 runs. Over his last five seasons, Victorino has averaged 18 RAA per year, meaning this is certainly feasible. And Victorino has an additional edge over Napoli: whereas Napoli is acknowledged to be an average defender at best, Victorino can save a few extra runs here or there with his glove.

In Victorino, the Red Sox have picked up an above-average outfielder in the field and at the plate with a proven record of performance; in Napoli, the Sox acquired a higher-risk player, with one season of sustained success at the plate and over 500 plate appearances. Assuming both players remain healthy, past performance suggests Victorino's will be the better of the two contracts.

Having said that, it's worth pointing out that there are several arguments for leniency when investigating these contracts. We have already established that this method ignores defensive value, and that a .500 Red Sox team will probably earn far more in revenue than our regression predicts for a typical team in a Boston-sized market. It's also important to note that these numbers are now two seasons old, and it's reasonable to expect that revenue will increase significantly when compared to the 2011 numbers I've used here. This will, of course, inflate the value of above-average players.

*As we get closer to the regular season, I'll revisit this post, using the predicted RAA to estimate what these and other key free agents should be worth this year.*

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