Perpetuities and Annuities

(Editor's Note: This was originally posted here)

By now, you are all familiar with the Patriots trading Richard Seymour for Oakland's 17th overall pick in this year's draft. You also should know that the Patriots shipped their 28th overall pick for New Orleans' 2nd round pick and a first round pick in 2012. Let me put my Finance major to good work and try and explain what the Patriots are trying to accomplish.
Definitions:

Annuity - A fixed sum of money, paid to a party each year for a set length of time.
Perpetuity - An annuity with no end; a fixed sum of money paid each year for an indefinite length of time.
Coupon - The fixed sum of money.

My theory is that the Patriots have utilized Richard Seymour as the payment for a draft perpetuity. That's right- the Patriots will be reaping the benefits from Richard Seymour long after he's left the franchise and the league. Coach and GM Bill Belichick makes it clear that his main concern in the draft is the upkeep of the franchise. He never has the "win this year, or bust" mentality. He puts the team in the best position to compete this year, while still thinking of the health of the franchise's future. Belichick understands the value of building a team through the draft and he will continue to take draft picks to flesh out the roster.

 "But how is Seymour an endless supply of money?" you may be asking. Think of it this way:

1. The Patriots traded Seymour to Oakland in 2009 for the 17th pick in the 2011 draft.
2. The Patriots were free to trade the 28th pick in the 2011 draft because they already had a first round pick.
3. The Patriots traded the 28th pick for a 2012 1st round pick, as well as a 2nd round pick in 2011.

Do you see what happened? The trade of Seymour gave the Patriots additional ammo for the first round; the Patriots were able to trade out of the first round because they were able to pick a first round player. So the trade of Seymour netted the Patriots a bonus second round pick. This second round pick is the "coupon payment." However, this is an unnatural market and realize that the Patriots received just a 4th round pick in the 2007 Draft, to trade with San Francisco. Average the two (rough, I know), and my theory continues and states that the Patriots are in a position to receive a third round pick every year at no additional cost.

1. Seymour -> Oakland 2011 First + Patriots 2011 First -> 2x 2011 First (the Patriots' first and the Raiders' first)
2. 2x 2011 First -> 2012 First* + 2011 First + 2011 Second
3. 2012 First* + Patriots 2012 First -> 2013 First* + 2012 First + 2012 Third
4. 2013 First* + Patriots 2013 First -> 2014 First* + 2013 First + 2013 Third

In each year, the Patriots will receive the coupon payment of a third round pick, for however long the Patriots continue to trade. Of course, just like any investment, there is a time where the Patriots can cash in and turn a profit. To find this, we need to analyze the draft value chart as well as the perpetuity formula.

PV = C/r
PV = Present Value of Perpetuity
C = Coupon Payment
r = discount rate (rate of interest, etc)

So we know that the coupon payment is supposed to be a third round pick and in order to find the value, we'll look at the middle of the NFL Draft Value Chart. We'll use the mid-point of the third round because it's an average; the Patriots will receive a range of third round picks over the time frame of "forever" (remember: perpetuities have an indefinite time span), with the average landing in the middle of the round.

C = 190

Now, we need to find the discount rate. My belief is that the Patriots will utilize their earlier pick in the first round and will trade forward their lower pick. I have full confidence in Belichick's coaching ability, so I will say that the average pick that Belichick will trade forward will be in the second round of the playoffs (25-28), or a point value of 690. We'll say that the average forward first round pick will, like the third round of the draft, be found in the mid-point of the first round for a point value of 975. When the Patriots spend the mid-first pick, however, that is an incurred cost instead of spending their own late-first pick because the increase in first round position is removed from the price. For example, when the Patriots utilized the Raiders' pick, as opposed to their own pick, they were spending the value of the earlier pick. However, instead of reinvesting the added value of the mid-first pick (value that can be recovered by trading back down), the Patriots will be spending that added value. As a result, we must add the "cost" of spending the new pick over the Patriots' pick. Here's the math to find the discount value:

r = (975 - [975-160])/975 = 16.4%

With these two values, we find that the value of this perpetuity is:

PV = 160/0.164 = 975.6 points

So this perpetuity is equivalent to the 16th-17th overall pick in the NFL draft. Therefore, the perpetuity could be considered valued at 16.5th overall. As a result, if the pick that is traded is below the mid-point of the first round, then the Patriots should continue to reinvest and take advantage of the perpetuity. This is because the Patriots will be receiving the value of a 16.5th pick from a lower value pick, which can be considered "turning a profit." However, if the pick the Patriots are trading forward breaks into the top half of the draft, the Patriots could either trade down below the mid-way point, pick up additional and continue to turn the profit, or they could decide that the market has a fair value for the perpetuity and cash in.
If the pick is in the top half, then it is implied that both the Patriots and the team they traded with are in the bottom half of the league. If they continue to trade down, then they will only be receiving the benefits of the 16.5th overall for a higher cost and they will be losing value. Therefore, if both picks are in the top half of the draft, then the Patriots could cash in and make a sound financial decision. However, they do have the option of trading down and continuing the perpetuity.
I believe that the Patriots are putting themselves in a position where they will never be in the bottom half of the league and, as a result, will never be in a position where it makes financial sense to "cash in." Instead, the Patriots will continue to take advantage of this perpetuity and earn a coupon equal to a third rounder every year, all thanks to Richard Seymour.