Earlier this week, several sources reported that Jason Bay has agreed to terms on a contract with the Mets. The contract is currently pending a physical, but is reported to run for four years and $66 million, with a vesting option of slightly more than $14 million. Previously, the Red Sox reportedly offered Bay a contract that would have paid $60 million over four years. For the moment, let’s ignore the vesting option (which reportedly vests easily-that’s a discussion for another day). Interestingly, Peter Gammons told WEEI that the Mets offer is “so back-loaded that [he had] been told by Mets people that it’s far less than what the Red Sox were offering in present-day value.” This statement set me off on a quest to make $6 million dollars disappear in the sands of time.

Back-loaded contracts were once all the rage. In the early half of the go-go Noughties, teams back-loaded contracts to postpone the full brunt of the pain. On first glance, this type of contract structure looks like the kind of shoddy amortization that felled the housing market a few years ago. However, when interest rates are high and investments offer strong returns, back-loaded contracts can be useful. Conversely, when interest rates are low and the economy is weak, they are less useful. To understand why, we must understand the concept of present value.

The value of money is a function of, among other things, the time at which it will be acquired. If a team defers payments to a player, it can spend more on current players. On the other hand, a player who receives deferred payment loses the ability to keep that money in his investment account, where it would likely earn interest. Payments later are effectively smaller than payments now.

The amount by which they are smaller depends on our choice of a discount rate. A discount rate is a combination of two separate but not unrelated factors: the expected rate of inflation over the time period in question and the rate of return on the next best investment. The sum of these two numbers is our discount rate. Inflation can be a tricky thing to predict, particularly under current economic conditions. However, one indicator macroeconomists use is the TIPS spread, which is the difference between the interest rates on a Treasury bond and its inflation-protected analog. Jason Bay’s contract is for four years, so we can use the spread on five-year Treasury bonds, which is currently 2.10 percent. This spread suggests an estimate of average inflation over the next five years is slightly more than two percent. In reality, the actual rate of inflation could be higher or lower.

The rate of return on alternate investments is a harder number to estimate. Although the economy has been slow, teams may use the money in any number of ways, including signing players that might put them into the playoffs. Let’s assume a rate of return of three percent, a modest return for a somber economic climate. That would give us an overall discount rate of five percent.

How would the present value of the various contracts be affected by this rate? First, let’s discount the Red Sox offer of $15 million per year for four years. In the first year, we won’t discount at all. In the second year, we’ll discount by five percent. In the third year, we’ll discount by five percent two times, and in the fourth year, we’ll discount by five percent three times. It looks like this: 15+15/1.05+15/1.05^2+15/1.05^3 = 55.8, so $55.8 million will be our baseline of comparison.

Now let’s imagine four different back-loaded contract structures from the Mets (the exact terms have not been reported).

           2010   2011   2012   2013  
Option 1   16.5   16.5   16.5   16.5
Option 2   13.5   15.5   17.5   19.5
Option 3   10.0   13.5   19.5   23.0
Option 4   10.0   10.0   21.0   25.0

And here are the comparisons with the Red Sox’ offer:

            Option 1   Option 2   Option 3   Option 4 
Mets         $61.4      $61.0      $60.4      $60.2
Red Sox      $55.8      $55.8      $55.8      $55.8
Difference    $5.6       $5.2       $4.6       $4.4

For Option 1, the net present value using a five percent discount rate is $61.4 million, which exceeds the Red Sox offer by $5.6 million. For Option 2, the present value is $61 million, which exceeds the Red Sox offer by just less than $5.2 million. For Option 3, we get $60.4 million, which is $4.6 million more. Finally, for the aggressively back-loaded Option 4, we get $60.2 million, which is still $4.3 million more than what the Red Sox were offering. It just doesn’t seem possible to get a contract back-loaded enough to make it worth less than the Red Sox offer unless the payments stretch out for nearly a decade.

However, you may be wondering about my choice of a discount rate. After all, it is the sum of two highly uncertain variables, and therefore it is also highly uncertain. What if we used a different discount rate? In the chart below, I have graphed the four options from above and discounted them each by four percent (blue), five percent (red), six percent (green), and seven percent (purple). I have also overlaid a transparency on each chart that reflects the Red Sox offer, discounted in the same fashion.

Bay Chart

Take a look at Option 4, the most back-loaded of the four. Compare the purple lines for both the transparency and the main series. Now look at the vertical area between the two. On the left, the Boston offer exceeds the New York offer; on the right, the two are reversed. However, the distance between New York and Boston is larger on the right than the difference between Boston and New York on the left. Thus, even assuming a higher discount rate and aggressive back-loading, it does not appear possible to make the contracts equal in value.

It is important to note that the usefulness of back-loading for teams is greatest when salaries are rising at a high rate from year to year, inflation is high, and the economy (and revenues with it) is strong. None of these descriptions fits the current economic climate. Although the Mets ownership, which reportedly invested money with Ponzi artist Bernie Madoff, apparently once believed in the possibility of consistent returns in the 10-13 percent range, that simply isn’t realistic today. Without returns like that, back-loading can lead to contracts that are albatrosses later on. Even in the early 2000s, when the economy was stronger, there were plenty of examples of back-loaded contracts that teams were happy to be free from. Given that Jason Bay is likely entering a slow decline, the Mets will breathe a sigh of relief when this back-loaded deal has run its course.