Recently my pastor posed a problem to me: How to calculate his SECA offset? The answer can be found using algebra or using infinite series. (For the non-math inclined, skip to the bottom of this post for an answer.)

Background: Most of us who work for an organization in the United States are charged a 15.3% Social Security FICA tax. (This is 15.3% of our salaries.) Half (7.65%) of that tax is paid by our employers, and the other half is paid by us (in the form of a deduction from our paychecks). However, clergy are considered self-employed for Social Security tax purposes, which means that they have to pay all 15.3% themselves. To counteract this, many congregations pay their clergy an extra 7.65% of their compensation to cover the Social Security tax that is normally covered by employers. This is called a SECA offset. However, this extra bit is itself considered compensation, so it is also taxed by Social Security! This means that the SECA offset payment, if it is calculated this way, is actually less than needed to cover the intended part of the clergy’s Social Security tax. The congregation could then pay an extra part to make up that difference, but that extra part would also be taxed, which means they’re still underpaying… and so on forever, in a perpetual cycle.

Let’s take an example. Suppose a pastor makes a nice, round $50,000 in compensation. On top of this, the congregation pays the pastor a SECA offset of 7.65%, which is $3825. This means the pastor’s actual taxable income from the church is $53,825. The 7.65% Social Security tax on $53,825 is $4117.61, which means the congregation has actually underpaid by $4117.61 minus $3825, which is $292.61. The congregation could then give the extra $292.61 to the pastor, but that would itself count as income and thus raise the pastor’s Social Security tax even further. Again, this process could continue forever. Fortunately, looking at this problem the right way mathematically can solve it for us.

Let’s formulate the problem like this: **How much should a congregation pay its clergy in SECA offset if it wants that payment to be exactly 7.65% of the clergy’s total compensation?**

(*Infinite Series Solution.*) Turning to math, let’s let *A* denote a pastor’s original compensation. The first-order attempt at covering the SECA offset is . This, though, is taxable, so the congregation needs to include 7.65% of this amount, or . But then this extra is also taxable at 7.65%, so the congregation needs to include an additional , and so on. Continuing in this vein, we have what is known as an *infinite series *for the amount the congregation should pay the pastor in total compensation:

.

The shorthand way to write an infinite series like this one is with sigma notation:

.

This infinite series is a special class of infinite series known as a *geometric series*, . In our example, . Provided , geometric series satisfy the nice formula

.

This gives us our answer: The congregation should pay the pastor a total of

. Subtracting the original compensation of *A* from this, we’re left with for the SECA offset.

**This means the SECA offset should actually be 8.2837% of the original compensation if that payment is to be exactly 7.65% of the clergy’s total compensation.**

(*Algebra Solution.*) While the infinite series solution is the straightforward way of tackling the problem, the math is much easier if we set up an algebraic equation to solve the problem. Let’s let *x* denote the desired amount of SECA offset, with *A* the pastor’s original compensation. We want *x* to be exactly 7.65% of the pastor’s total compensation. The pastor’s total compensation, though, is . That gives us the following equation:

.

Solving this for *x*, we get , or , or , the same answer as before – but with less work.

(Of course, in practice there are other issues to consider, such as the fact that raising the pastor’s compensation by 8.2837% will raise his/her income tax and benefit payments as well. The answer to the problem as posed, though, which just focuses on the SECA offset, is 8.2837% of the original compensation amount.)