Last week, we took a look at option years, which are commonly appended to the end of player contracts. The conclusion was that a team (or player) can negotiate an option such that the expected outcome—that is, the weighted mean of all possible outcomes—is a net monetary plus. While a clever team can’t guarantee a moneymaking scheme, it can tilt the scales in its favor by properly pricing the buyout. For the rest of this article, we’re going to look at things from the team’s point of view, so when we talk about profit, it’s the team that is making the money.

Properly pricing the buyout requires a projection of a player’s performance, which I’ve done using Net Win Shares Value as a valuation tool. This projection is not simply a number, it is a range of numbers with different probabilities. That is, the team might guess that there was a 50% chance the player would be worth X, but also a 10% chance that the player might be worth Y or Z. A team may use anything from one of the common systems (Marcel, ZiPS, PECOTA) to phrenology; complicating matters further is that the team with the rosiest projection of a free agent is often the team that bids the highest.

In the absence of more detailed information (or a harder-working writer), a decent assumption is that whatever the projection system, the team thinks the player will have either no value, positive value, or negative value. Mathematically, we’ll model this as a bell curve with an average of zero, greater than zero, or less than zero. The team might be very confident of this projection, not very confident, or someplace in between; we can model this mathematically as a bell curve with a very small standard deviation, very high standard deviation, or a moderate standard deviation.

Last time we described how we could use this projection, which we called pNWSV, to compute the expected profit or loss as a function of the buyout price. Using the projection distributions described above, I did this very calculation. Let’s take a look at some plots:

Not sure how to read the plots? The blue line on the first plot represents the case where the team is very certain that the player will be worth $1 million using Net Win Shares Value. The green line on the last plot represents the case where the team is somewhat confident that the player will be worth $0. Keep in mind that the following discussion only applies to a bell-curve shaped range of possibilities for a player’s projected Net Win Shares Value. With that in mind, what do the plots tell us?

First, the team never loses money when there is no buyout. No matter how ridiculous the option price might be, the team assumes no risk if there is no buyout. Manny Ramirez has a pair of $20 million options appended to the end of his megacontract with the Red Sox, but there is no buyout if the team declines them. Options without buyouts exist for reasons other than purely economic; they probably serve as a de facto no-trade leverage.

Second, the higher the price of the buyout, the more expected profit or loss goes to the average projected Net Win Shares Value. What that means is that if the player projects to have positive value, the team will always come out ahead regardless of the buyout price. It’s a better and better deal as the buyout price goes to zero.

Third, even if a player projects to having negative value, it is still possible for a team to price the option such that it can be expected to come out ahead. This is particularly true when the team is very uncertain about its projection. This uncertainty plays a big role in options at the end of long-term contracts, as projections become murkier with each additional year. Looking at the middle plot, even a buyout of a few million dollars for a player projected to have negative value (red line) can be expected to net the team some profit. That may seem counterintuitive, but we’re seeing the effect of a team protecting itself against the probability that the option is a no-brainer. It’s an insurance policy that can protect a team in the event that the player becomes a huge bargain based either on performance or inflation in the market.

###### Case Study: Eric Chavez

Before the 2004 season, the Oakland Athletics signed Eric Chavez to a six-year, $66 million contract (aside: there is apparently a clause in Chavez’s in contract stipulating that general manager Billy Beane must care for his star third basemen’s two dogs during certain games, which I am not valuing as part of his contract). In 2011, the A’s have a team option worth $12.5 million or a buyout for $3 million. Does the option year favor the player or the team?

First we need to reasonable projection for Chavez’s performance. Simple-mindedly, let’s assume that the A’s thought he would contribute nine Win Shares Above Bench through his age-28 season and then lose one WSAB every year throughout the life of his contract. This means that in 2011, his option year, he might be projected for four WSAB.

In order to convert four WSAB in 2011 into Net Win Shares Value, we can use Dave Studeman’s Net Win Shares Value Calculator. In order to do so, we need to convert $12.5 million in 2011 to its 2007 value. Assuming steady 10% inflation in player salaries, the option year is $8.5 million in today’s dollars, making his projected 2011 Net Win Shares Value -$1.3 million.

Whether this option is fairly priced depends on the uncertainty involved in the projection. The buyout is $2 million (in 2007 money) and is more or less set at the same amount that Chavez is projected to be overpaid by, assuming some uncertainty in the projection. This option is pretty fairly priced—Billy Beane didn’t get the best of Chavez’s agent at the time (coincidentally Dave Stewart, a former teammate of Beane).

###### Case Study: Lance Berkman

The Astros signed heavy hitter Lance Berkman to a six-year, $85 million extension prior to the 2005 season. There is an option for the 2011 season worth $10.2 million (in 2007 dollars) with a $1.4 million buyout.

In 2004 he had 19 WSAB in his age-28 season; let’s use our simple-minded projection system to knock off one WSAB every year until the option year. Using that methodology, we can say that he will be worth $4.3 million in Net Win Shares Value. Given the positive projection, the Astros are likely to come out ahead regardless of the buyout price—which is pretty low anyway. In this case, because Berkman is projected to be underpaid in his option year anyway, no buyout can tilt the deal in favor of Berkman.

Keep in mind that there are significant caveats in this analysis. All of my assumptions are open for criticism; the one that I think is least accurate is the assumption that the distribution of probabilities in a projection is shaped like a bell curve. A better distribution is probably one with a long tail on the negative end, accounting for the potential that the player will not play at all due to injury, ineffectiveness or maybe alien abduction. This is a more complex calculation which is possible but slightly more difficult. Unfortunately for you, I’m kind of lazy. Furthermore, my “projection” system was pretty simplistic so I would advise against putting too much stock in the final conclusions regarding Chavez and Berkman. Still, as I said last week, I’m a process-oriented type and I’m pretty pleased with this process.