Category: tax

Category: math

If you are self employed (SE), or burn the midnight oil to make extra money, then you are familiar with form 1099-MISC. When you file tax, besides income tax, you need to pay SE tax on IRS schedule SE, a simplified version is shown here:

   1a Net farm profit or (loss) from Schedule F
    b ... enter the amount of Conservation Reserve Program payments ...
   2 (*) Net profit or (loss) from Schedule C
   3 Combine lines 1a, 1b, and 2 
   4 (*) Multiply line 3 by 92.35% (0.9235)
   5 (*) Multiply line 4 by 15.3% (0.153). 
   6 (*) Deduction for one-half of self-employment tax. Multiply line 5 by 50% (0.50).

where 1a and 1b are rare form of income most of us do not have, so basically to figure out the SE tax, you multiple the net profit by 92.35%, and then 15.3%.

The percentage 15.3% is the SE tax, also called FICA tax to fund the social security and medicare. For employees, these two taxes appear in Box 4 and 6 of your W-2. If you are not highly paid individual who are affected by additional medicare tax as added by the Affordable Care Act (ACA, aka Obama Care), then you can verify that the social security tax withheld, and medicare tax withheld are 6.20% and 1.45% respectively, or 7.65% together. Not shown on W-2, but the employer also contribute the same amount. As a SE person, you are employer and employee at the same time, so you pays the double amount: 7.65% * 2 = 15.3%. This explains the FICA tax rate, but where does 92.35% (0.9235) comes from?

Apparently, the percentate comes from:

1 - 7.65% = 92.35%

i.e. the percentage left after one-half of the SE tax (employer portion) deducted.

However this is incorrect because you deduct the employer’s portion based on the net income, but actually pay the same percentage based on the amount after deduction. Let the net income to be N, the employer’s portion you deduct is:

N * 7.65%

The employer’s portion that you pay is:


These two amounts should be equal but are not equal.

The correct percentage X that you deduct should be the tax you actually paid, or:

N*(1-X)*7.65% = N*X

OR X = 7.65%/(1+7.65%) = 7.11%.

Using the formula for Taylor series expansion, and ignore the second order and higher, we have:

X = 1 - 7.65% = 92.35%

So we can say the percentage that IRS used is first order approximation of the Taylor series, although there is no such a need as this is a constant that only need to be compute once. Fortunately this “approximation” favors the taxpayer.

By the way, the 92.35% of the net earnings from the Sch. C business is what listed as “Your Taxed Social Security Earnings” and “Your Taxed Medicare Earnings” in “Your Social Security Statement”, and this ought to be what is used to compute the “quarter of coverage”.

In computing the retirement contribution of SE person, however, the IRS choose to face the mathematical reality.

When setting up a SE retirement plan, you need to set a plan contribution percentage up to 25% based on the plan compensation. While this is easy to compute for your employees as the plan compensation is the net income without adjustment. For the SE person to make his own contribution it is “more complicated” because the net income should exclude the contribution amount to arrive plan compensation. For example, if your plan contribution rate is 10%, and your net income is 100. If you use the plan rate then your contribution is 100*10% = 10, and the plan compensation is 100 - 10 = 90, so the contribution percentage becomes 10/90 = 11.11% which violates the preset plan contribution percentage. In the words of IRS:

your plan compensation and the amount of your own plan contribution/deduction depend on each other - to compute one, you need the other (this is a circular calculation).

To solve this problem, IRS proposes to use a reduced rate which, of course is listed in a table found in Publication 560 together with worksheets, something that IRS does best. The table is copied below for all percentages from 1% to 25%:

Plan rate (R) Your rate (r)
1 .009901
2 .019608
3 .029126
4 .038462
5 .047619
6 .056604
7 .065421
8 .074074
9 .082569
10 .090909
11 .099099
12 .107143
13 .115044
14 .122807
15 .130435
16 .137931
17 .145299
18 .152542
19 .159664
20 .166667
21 .173554
22 .180328
23 .186992
24 .193548
25* .200000*

Let us see how to use the table. Use the same sample as above, if the plan rate is 10%, your rate should be reduced to 9.0909%. So your contribution is 100*9.0909% = 9.0909. The plan compensation is then 100 - 9.0909 = 90.9091, and then plan contribution rate “magically” equals to 9.0909/90.9091 = 10% within the margin of errors, which agrees with the plan policy.

How does this “magic” occur? The “circular calculation” is actually called an equation in mathematical terms. Let N be net income, C be your contribution, R be the plan contribution rate, and P the plan compensation. True that Plan compensation depends on contribution:

P = N - C

True also the contribution depends on the plan compensation:

C = P * R

We can resolve this circular relationship mathematically, and get

r = C/N = R/(1+R)

where r is your rate defined as the ratio of contribution over the net income. We can easily verify, as an example, that when plan rate is 10%, the personal rate is 10%/(1+10%) = 9.0909%, which agrees with IRS table.

We can take first derivative, or rewrite the C/N = 1/(1+1/R), or use the IRS table to show that the personal rate is monotonically increasing function of R, and the maximum personal rate is 20%.

Here we assume the net income has been adjusted for SE tax with one-half of SE tax deducted. As we recall from discussion above, the SE tax is unadjusted net income times (1-7.65%) and times 15.3%, so the net income adjusted for SE tax is unadjusted net income times (1-(1-7.65%)*7.65%), so the maximum personal contribution rate over unadjusted net income is:

20% * (1-(1-7.65%)*7.65%) = 18.587045%

which is documented in Wikipedia, which says:

the contribution limit for self-employed persons is more complicated; barring limits, it is 18.587045% (approximately 18.6%) of net profit

Now resolved the “complication”.