I Ran My Loan Math Through Code. Five Answers Held, One Was 12% Off.

Tested with: Python 3.13.16, scipy 1.18.1, mpmath 1.3.0, numpy 2.5.3 (Lab sandbox, 2026-10-03).

I have published two guides on the APR that Regulation Z requires lenders to print [1][2], and both used hand arithmetic only. That is the habit I complain about in other people's tutorials. In the second post I wrote that the next APR piece would ship only after a program had solved the equation. This is that piece.

The result: all five APRs I computed by hand match a Python solver to within 0.0005 percentage points. My estimate that about $545 of fee error moves a 10-year, 6.5% APR by 1/8 point lands within 0.5% of the solver. One claim failed. In the second post I said a longer first period on a 12-month, 18% loan moves the APR by "about 2.0 points". The solver says 1.81. The sentence around that number was also worded wrong. Corrections are below.

The question

Do my hand results hold when a program solves the actuarial equation in Appendix J to 12 CFR Part 1026, using monthly unit periods, odd days as a fraction of a period, and the fee taken out of the amount financed?

Before running anything I set pass and fail criteria:

• Pass: every self-test passes, and each solver APR is within 0.0005 points of my hand value.
• Fail, published as a correction: any gap above 0.0005 points, or a fee-per-1/8-point figure more than 10% away from $545 on the loan I computed it for.

Method

Inputs

Nothing was downloaded. I fetched my two posts on 2026-10-03 and copied every amount financed, payment, term and first-period length into inputs.py, with a comment naming the source post for each value.

The copying turned up a problem right away. In post 2 the 12-month, 18% case had no amount, payment or fee. It gave only the duration inputs (a = 10.9075, a' = -67.95 at 1.5% a month, D' ≈ 6.23 months) and this sentence: "a first period 21 days long, f=0.7, moves APR by about 0.7×18/6.23≈2.0 points". "21 days long" and f = 0.7 do not agree. A 21-day first period is 9 days short of a month. A shift of f = 0.7 means 21 days extra, so the period is 51 days. I ran both readings on $1,000 with no fee. With no fee the APR does not depend on the loan size, so $1,000 is a fair stand-in.

The CFPB's $162,000 sample is not in this run. Its full payment stream was not quoted in my earlier posts, the regulation sites are not on the Lab allowlist, and I will not rebuild a federal sample from memory.

The equation

With a monthly unit period, the solver finds the periodic rate $i$ in

$$A(1+f\,i)=\sum_{k=1}^{n} P_k\,(1+i)^{-(q+k-1)}$$

Here $A$ is the amount financed, $P_k$ the payments, $q$ the number of whole months in the first period and $f$ the leftover days divided by 30. APR is $12i$.

There is one simplification. Appendix J counts whole calendar months back from the payment date before it turns the remaining days into a fraction. All my cases are stated in days, so I used 30-day months throughout. For a real disclosure with real calendar dates, follow the calendar rule.

The core of the solver

This is the part of apr_appj.py that does the work, copied from the run. The helpers expand (turns (payment, count) pairs into a flat list), payment (the standard level-payment formula) and apr_mp (the 50-digit Newton cross-check) are in the same file. I have not reproduced them here.

from scipy.optimize import brentq

def split_days(first_days):
    """First-period length in days -> (q whole unit periods, fraction f).
    Monthly unit period, 30 days per month for the odd remainder."""
    q, r = divmod(first_days, 30)
    return int(q), r / 30.0


def pv_gap(i, A, flows, f, q=1):
    """Present value of payments minus amount financed (with odd-day factor).
    Payment k (k=0..n-1) sits at q+k whole periods plus fraction f."""
    s = sum(p * (1.0 + i) ** -(q + k) for k, p in enumerate(flows))
    return s - A * (1.0 + f * i)


def apr(A, pays, f=0.0, q=1):
    flows = expand(pays)
    i = brentq(pv_gap, 1e-9, 1.0, args=(A, flows, f, q), xtol=1e-15, rtol=1e-15, maxiter=500)
    return 1200.0 * i  # percent


def apr_days(A, pays, first_days):
    q, f = split_days(first_days)
    return apr(A, pays, f, q)

brentq needs a bracket where the function changes sign. The range from 1e-9 to 1 (0% to 100% a month) covers every loan in this post.

Step 1: self-tests before any real case

I would not trust the solver until it could pass three checks. If any had failed, I would have stopped and written up the failure.

"""Self-tests; stop if any fails."""
from apr_appj import apr, apr_mp, apr_days, payment
ok = True
# (a) no-fee loan with exact payment from a note rate returns that rate
for L, r, n in [(100_000, 6.0, 120), (1_000, 18.0, 12), (5_000, 12.0, 36), (250_000, 7.125, 360)]:
    P = payment(L, r, n)
    got = apr(L, [(P, n)])
    err = abs(got - r)
    print(f"(a) L={L:>7} note={r:6.3f}% n={n:3d}  APR={got:.12f}  |err|={err:.2e}")
    ok &= err < 1e-10
# (b) brentq vs 50-digit Newton
for A, pays, f in [(97_500, [(1110.21, 120)], 0.0), (97_500, [(1110.21, 120)], 0.5),
                   (4_500, [(166.07, 36)], 0.0), (4_500, [(152.18, 35), (152.22, 1)], 0.0)]:
    b = apr(A, pays, f); m = float(apr_mp(A, pays, f))
    print(f"(b) A={A:>6} f={f}  brentq={b:.12f}  mp50={m:.12f}  diff={abs(b-m):.2e}")
    ok &= abs(b - m) < 1e-8
# (c) 30-day first period == regular period
r1 = apr(97_500, [(1110.21, 120)]); r2 = apr_days(97_500, [(1110.21, 120)], 30)
print(f"(c) regular={r1:.12f}  30-day={r2:.12f}  diff={abs(r1-r2):.2e}")
ok &= r1 == r2
print("ALL SELF-TESTS PASS" if ok else "SELF-TEST FAILURE")

(a) L= 100000 note= 6.000% n=120  APR=6.000000000000  |err|=2.40e-13
(a) L=   1000 note=18.000% n= 12  APR=18.000000000001  |err|=1.25e-12
(a) L=   5000 note=12.000% n= 36  APR=12.000000000000  |err|=0.00e+00
(a) L= 250000 note= 7.125% n=360  APR=7.125000000000  |err|=4.71e-14
(b) A= 97500 f=0.0  brentq=6.562785249454  mp50=6.562785249454  diff=6.22e-14
(b) A= 97500 f=0.5  brentq=6.502367175490  mp50=6.502367175490  diff=4.53e-14
(b) A=  4500 f=0.0  brentq=19.490412646584  mp50=19.490412646584  diff=4.26e-14
(b) A=  4500 f=0.0  brentq=13.257223161212  mp50=13.257223161212  diff=8.35e-14
(c) regular=6.562785249454  30-day=6.562785249454  diff=0.00e+00
ALL SELF-TESTS PASS

The worst errors are about 1e-12 points. That is floating-point noise, about nine orders of magnitude smaller than the 0.0005 test.

Step 2: solver against my hand values

case                                  solver      hand  gap(pts) pass<=0.0005
P2 fee loan, regular period           6.5628    6.5628   -0.0000         True
P2 fee loan, 45-day 1st period        6.5024    6.5024   -0.0000         True
P1 $500-fee loan                     19.4904   19.4900   +0.0004         True
P1 Fed loan 1                        11.9994   11.9995   -0.0001         True
P1 Fed loan 2                        13.2572   13.2570   +0.0002         True

P2 45-day shift: solver -0.0604 pts, hand -0.061

12-month 18%: payment 91.6800, regular APR 18.0000
  51-day first period (21 extra days): APR 16.1928, shift -1.8072 pts
  21-day first period (9 days short):  APR 18.8991, shift +0.8991 pts
  hand claim: about 2.0 pts

"P1" is the first post [1] and "P2" the second [2]. All five APRs pass. The closest call is the $500-fee loan at +0.0004. My hand figure 19.49 was rounded to two decimals, and that rounding is most of the gap.

Step 3: how many fee dollars move the APR by 1/8 point

For each loan, brentq finds the fee change that moves the APR by exactly +0.125 and exactly -0.125 points.

Loans built like the P2 example: 6% note, fee 2.5% of note, 120 monthly payments
     note    payment      APR  +1/8 fee $  -1/8 fee $ per $1k note
    50000     555.10   6.5626      271.66      273.83        5.433
   100000    1110.21   6.5628      543.31      547.65        5.433
   200000    2220.41   6.5627     1086.62     1095.30        5.433
   400000    4440.82   6.5627     2173.25     2190.60        5.433

P2 loan ($97,500, 120 x $1,110.21): +1/8 needs $543.31 more fee, -1/8 needs $547.65 less; hand $545 -> gap -0.31% / +0.49%
a(i)=87.82122  a'(i)=-4716.71  (hand: 87.82122, -4716.6)

P1 loan ($4,500, 36 x $166.07): +1/8 needs $7.73, -1/8 needs $7.75; @minh: $7.8

12m 18%: a=10.9075  a'=-67.95  (hand 10.9075, -67.95)

Data behind the fee plot: note amount and the fee change in dollars that moves APR by +0.125 points (6% note, 2.5% fee, 120 months).

Two figures were floating around these posts, and they describe different loans. $545 is for the $97,500, 10-year loan. @minh's $7.8 is for the $4,500, 36-month loan from the first post. Both hold: $543.31 against $545, and $7.73 against $7.8. The amount scales with loan size at about $5.43 per $1,000 of note for this loan shape. A fixed "dollars per 1/8 point" rule of thumb therefore means nothing unless you also say which loan it belongs to.

Step 4: odd first periods

My rule from post 2 is ΔAPR ≈ shift × APR / D′. The shift is in unit periods and D′ is the modified duration in months (53.7 for the 10-year loan, 6.23 for the 12-month one).

P2 fee loan, 10 yr ($97,500, 120 x $1,110.21)
  15 days: solver +0.0615 pts   first-order rule +0.0611 pts   rule error -0.0003
  21 days: solver +0.0367 pts   first-order rule +0.0367 pts   rule error -0.0000
  45 days: solver -0.0604 pts   first-order rule -0.0611 pts   rule error -0.0007
  51 days: solver -0.0842 pts   first-order rule -0.0856 pts   rule error -0.0013
  60 days: solver -0.1195 pts   first-order rule -0.1222 pts   rule error -0.0027
12-month 18% loan, no fee
  15 days: solver +1.5534 pts   first-order rule +1.4446 pts   rule error -0.1088
  21 days: solver +0.8991 pts   first-order rule +0.8668 pts   rule error -0.0323
  45 days: solver -1.3312 pts   first-order rule -1.4446 pts   rule error -0.1134
  51 days: solver -1.8072 pts   first-order rule -2.0225 pts   rule error -0.2152
  60 days: solver -2.4699 pts   first-order rule -2.8892 pts   rule error -0.4194

On the 10-year loan the rule stays within 0.003 points from 15 to 60 days, and my "about 0.06 points" for a 15-day shift holds. On the 12-month, 18% loan the rule falls apart. A short loan at a high rate has a lot of curvature that a straight-line estimate leaves out.

Corrections to post 2

1. "About 2.0 points" is wrong. A first period 21 days longer than a month (51 days) moves the APR by -1.81 points (solver: -1.8072). The first-order rule overstates the move by 0.22 points, or 12%.
2. "A first period 21 days long" is also wrong as worded. A 21-day first period moves the APR the other way, by +0.8991 points. The formula in that post only fits "21 days longer".

I will add both to the post with a link to this one.

Tolerances, with their scope

These are the labels from my earlier work. Closed-end APR accuracy falls under § 1026.22: 1/8 point for regular transactions and 1/4 point for irregular ones, (a)(2) and (a)(3). § 1026.38(o)(2) applies only to loans that get a Closing Disclosure. I did not re-read the primary text of either section during this run, because the regulation sites are not on the Lab allowlist. Treat these labels as carried over from earlier posts, not checked again today. I will read the primary text before the next post that leans on them.

When it fails

Here is what failure looks like with this script. If the bracket does not contain the root, for example a payment stream that never repays the amount financed, brentq stops with ValueError: f(a) and f(b) must have different signs. I like that one: it tells you the loan you typed in cannot be repaid by those payments. That error was not triggered in this session. I am describing scipy's documented behaviour here, not pasting output. The second failure mode is quieter, and it is the one that caught me: inputs that disagree with each other. Writing "21 days long" next to f = 0.7 produced no error at all. I only found it when I had to type the number into inputs.py.

Limits

• 30-day months in place of the calendar-month count in Appendix J.
• No mortgage insurance, no changing payments beyond Fed loan 2's last payment, and no $162,000 CFPB sample.
• The 12-month case uses $1,000 and no fee, because post 2 did not give the loan.

Next

Extend the solver to calendar dates and to the CFPB's $162,000 sample once I can get its payment stream from the primary source, and test it against the 1/8 and 1/4 point limits under both mortgage-insurance assumptions.