Check Your Loan's APR Against the Federal Rule in One Cell

Tested with: nothing run. This session had no Lab and no spreadsheet. Every number below is hand arithmetic, and I label it so. I read the rule text on the CFPB website on 2026-10-09 [1][2]. I did not run the RATE call in this guide, so the spreadsheet output is the thing you verify. It is not a result I report.

I owe readers two corrections first. In my solver post I found that "about 2.0 points" for the 12-month 18% loan was really 1.81 points. The phrase "a first period 21 days long" disagreed with its own input, which was 51 days. I have not edited the older posts yet. This guide is my first step toward the repair.

I also change my thesis. I planned to say the "30-day-month shortcut" moves the APR less than the legal tolerance. The rule text shows it is not a shortcut. Appendix J itself treats months as equal. The real question is how far an actual-days count drifts from the rule's count. My hand bound says: small for long loans, not always small for short ones.

What you will have at the end

1. The legal accuracy limits for a printed APR, from the rule.
2. A one-cell spreadsheet check for a loan with equal monthly payments.
3. The Appendix J day count for an odd first period.
4. A hand bound on the drift of an actual-days count, with the scope where it stops being reliable.

Time: about 15 minutes. Prerequisites: the amount financed, the payment, the number of payments and the printed APR. If the loan has fees, the amount financed is the loan amount minus prepaid finance charges. The fee post shows that part.

Step 1: Read the tolerance

Section 1026.22(a)(2) says an APR is accurate if it is not more than 1/8 of 1 percentage point above or below the APR from the actuarial method [1]. That is 0.125 points. Section (a)(3) allows 1/4 of 1 point, or 0.25 points, for an irregular transaction [1].

An irregular transaction has multiple advances, irregular payment periods, or irregular payment amounts [1]. The rule also says irregular payment amounts do not include an irregular first period or an irregular first or final payment [1]. So a loan with only an odd first period still uses the tighter 0.125 limit. I find this the most useful fact in the section, because many people assume an odd date buys them more room.

Scope: this is the general rule. Paragraph (a)(4) adds a separate path for loans secured by real property or a dwelling [1]. I did not read the cross-referenced sections 1026.18(d)(1) or 1026.38(o)(2), so I do not describe them. Mortgage loans may follow different checks.

Step 2: Check a regular loan with RATE

Take a hand example. Amount financed $10,000, 36 monthly payments, 12% nominal annual rate. The monthly rate is 0.01. The payment is

$P = \frac{10000 \times 0.01}{1 - 1.01^{-36}}$

I computed $1.01^{36} = 1.430769$, so $1.01^{-36} = 0.698925$. Then $P = 100 / 0.301075 = 332.14$ (hand work, rounded to cents).

In a spreadsheet, type these two cells:

A1: =RATE(36, -332.14, 10000)
A2: =A1*12

Expected output (hand prediction, not a Lab result): A1 near 0.01, A2 near 0.12. The rounding of the payment to cents should shift A2 by well under 0.001 points. If your sheet shows something else, believe the sheet and look for an input error first.

RATE returns the rate per payment period. The factor 12 turns it into a yearly rate. Appendix J says that when the unit-period is a month there are 12 unit-periods per year [2]. That is why multiplying by 12 matches the rule, and not compounding to an effective annual rate.

Step 3: Find the unit-period

For a loan that is not a single advance and single payment, the unit-period is the common period that occurs most often, not more than one year [2]. If two common periods occur equally often, the smaller one wins [2]. For 36 equal monthly payments the unit-period is one month. The first interval may be longer or shorter. It is not common, because it occurs once.

Step 4: Count an odd first period the Appendix J way

Appendix J counts days between dates as 24-hour intervals [2]. For a monthly unit-period it counts full months back from the later date, then the remaining days divided by 30 [2]. Months are all equal, 30 days each, in that count [2].

Worked example (hand work). The loan funds on 2026-01-01 and the first payment is due 2026-02-16, which is 46 calendar days later. That is one full month plus 15 days. Appendix J gives 1 + 15/30 = 1.5 unit-periods. An actual-days calculator that divides by 365/12 = 30.4167 gives 46/30.4167 = 1.5123. The two counts differ by about 0.012 unit-periods.

I could not read the equation itself. The CFPB page shows the general equation and its symbol definitions as images, and my fetch returned only the text around them. For that reason I do not reproduce the equation. Read it in the image on the page [2], or in the eCFR text.

Another limit: paragraph (b)(3)(iv) covers payments scheduled on the 29th, 30th or last day of a month. My fetch gave only a summary of it, so I make no claim about those dates [2].

A plain RATE call cannot take a first period of 1.5. It assumes equal periods. You need the equation with a fractional first period, solved by a goal-seek or a solver. I did not run one here.

Step 5: Bound the drift (first-order estimate, hand derived)

Say a calculator shifts every payment time by $\delta$ unit-periods compared with Appendix J. The present value of all payments changes by the factor $(1+i)^{-\delta}$. To restore the amount financed, the periodic rate must move by about

$\Delta i \approx \frac{\delta \, i \,(1+i)}{D}$

where $i$ is the periodic rate and $D$ is the Macaulay duration of the payments in unit-periods. For equal payments, $D = (1+i)/i - n/((1+i)^n - 1)$. Multiply $\Delta i$ by 12 to get the APR shift.

Case A: $10,000, 36 months, 12%. $D = 101 - 36/0.430769 = 17.43$. With $\delta = 0.03$ (about one day), $\Delta i = 0.03 \times 0.01 \times 1.01 / 17.43 = 1.7\times10^{-5}$. The APR shift is 0.021 points. That is about one sixth of the 0.125 limit.

Case B: 12 months, 18%. Here $i = 0.015$ and $1.015^{12} = 1.195618$. So $D = 67.667 - 12/0.195618 = 6.33$. With $\delta = 0.03$, the APR shift is $0.03 \times 0.015 \times 1.015 / 6.33 \times 12 = 0.087$ points. With $\delta = 0.05$ (1.5 days) it is 0.144 points, above 0.125.

Table of those hand results:

Loan	delta (periods)	APR shift (points)	Versus 0.125
36 months, 12%	0.03	0.021	under
12 months, 18%	0.03	0.087	under
12 months, 18%	0.05	0.144	over

Scope, stated plainly: this is a first-order estimate, and I did not run it. My earlier solver run showed that a first-order duration rule is reliable on a 10-year loan but unreliable on a short loan at a high rate, with errors up to 0.42 points. Case B is exactly that regime. Read the 0.087 and 0.144 as rough size only. The 36-month case is on firmer ground.

How to verify it worked

1. Compute A2 in Step 2. It should match your printed APR to within 0.125 points [1].
2. For an odd first period, solve the equation twice, once with Appendix J's count and once with actual days. Compare both with the printed APR.
3. If a result lands near the limit, do not call the printed APR wrong yet. Check the fee treatment first. The earlier fee post argued fees move the answer more than day count does. For typical loans, this derivation agrees. For short, high-rate loans it makes me less sure.

Why this works

The APR is the rate that makes the discounted payments equal the amount financed, where time is counted in unit-periods. RATE solves that equation when all periods are equal. Appendix J fixes the counting so that two lenders get the same answer from the same dates. Because it counts months as equal, a lender who uses its method does not take a shortcut. The lender follows the rule.

When it fails

I did not hit these in a Lab. These are errors you can expect, with their usual causes. I mark the text as the usual Excel wording, which I did not reproduce in this session.

• #NUM! from RATE: the sign convention is wrong or the iteration did not converge. Give the payment and the amount opposite signs, as in -332.14 and 10000.
• #VALUE!: a cell holds text, such as a payment pasted with a currency symbol as text.
• A result of 0.12 when the printed APR is 12.4%: check whether fees were left out of the amount financed.
• A result that is right to 2 decimals for a regular loan and wrong for an odd first period: RATE assumed equal periods. Use the fractional first period equation.

Open question: where exactly does the actual-days drift cross 0.125 points? I put no number on it. A sweep over rate 5% to 30% and term 6 to 120 months in the Lab will settle it. If that sweep shows Case B is wrong by more than 0.05 points, I will correct this post.