Where your loan payment goes: interest, fees, and principal

SettleADebt R004 PASS-01 — A004

US consumer closed-end simple interest, amortization, and APR comparison

Disposition: The bounded investigation and six worked examples are complete. The legal APR calculation remains on hold because several operative Appendix J equations and example panels were unavailable as readable content. E05 therefore supplies a separately labeled numerical cash-flow calculation, not a compliant or legally determined APR.

Actual check date: October 27, 2026, America/Los_Angeles. Observed research-session clock readings ran from 12:39 PDT to 12:57 PDT on that date, corresponding to 19:39–19:57 UTC. Publication and compilation dates are separately recorded below.

This report treats the R004/A004 assignment as supplied. No prior R004 report, other chat, or dependent foundation was used. No external registry status was checked. No borrower, account, provider-status, property, license, or case-record investigation was performed. No files, subagents, contacts, applications, or publication actions were used. I do not claim independently verified UI model provenance or a separate independent review.

Principal findings

• Under the expressly defined monthly model, a hypothetical $10,000 loan at a 12% nominal annual rate over 12 months produces $661.86 of interest with cent posting and an adjusted final payment.
• Under the same original principal, rate, payment convention, and allocation assumptions, an additional $2,000 principal payment immediately after installment 3 reduces modeled interest to $491.29, saving $170.57 and producing a final partial payment in model month 10.
• Thirty days of hypothetical simple accrual on $10,000 at 12% produces $98.63 on a 365 denominator and $100.00 on a 360 denominator, when interest is rounded once at the end. That comparison establishes arithmetic, not which denominator a creditor may use.
• A hypothetical $300 upfront charge can leave the note balance at $10,000 while reducing amount financed to $9,700, if it is legally a prepaid finance charge. The corresponding numerical annualized cash-flow rate is approximately 17.8250%. This report does not label that result a legal APR.

All rates, fees, balances, and payment paths below are hypothetical parameters, not offers or actual borrower facts.

Reference convention: S001 means R004-P1-S001; C001 means R004-P1-C001. Every claim in the claim register belongs to A004.

1. The quantities that must remain distinct

The disclosure distinctions come from TILA §§1605, 1606, and 1638, Regulation Z §§1026.2, .4, .18, and .22, and their relevant official interpretations. uscode.house.gov+4

Fee classification comes before a legal APR calculation

A charge’s label, recipient, or timing is insufficient by itself. For example, the rule treats a qualifying application fee charged to all applicants differently from a loan fee charged only when credit is extended. Third-party charges require their own analysis, and specified insurance and security-related charges have conditional exclusions. “Mandatory” does not mean universally included, and “optional” does not mean universally excluded. The precise conditions are recorded in C009–C018. ecfr.gov+1

Disclosure rules also do not, by themselves, authorize a rate, fee, capitalization event, payment-allocation practice, or prepayment penalty. The legal obligation and applicable law remain controlling inputs. Consumer Financial Protection Bureau+1

2. Two separate interest models

2.1 Daily simple-interest model

For a period in which the interest-bearing balance and annual rate remain unchanged:

Plain text

interest = balance * annual_rate_decimal * accrual_days / day_count_denominator

Units are:

Plain text

dollars * annual decimal rate * days / days-per-rate-year = dollars

If balances, rates, or applicable denominators change, the stipulated simple-accrual model is segmented:

Plain text

total_interest = sum(balance_j * annual_rate_j * days_j / denominator_j)

This formula does not decide:

• Which event establishes the credited payment date: receipt, posting, value date, or another legally applicable event.
• Which endpoint is included.
• Whether a partial payment is immediately applied.
• Whether accrued interest is carried separately or validly capitalized.
• How a year boundary is handled.
• Whether interest is rounded daily, at payment, or only when posted.
• Whether an extra payment advances a due date or changes the required installment.

Those are unresolved contractual and legal inputs, not missing algebra.

The official commentary expressly distinguishes using a 360-day calculation tool from actually applying an annual rate divided by 360 across 365 accrual days. Its disclosure simplifications do not make those practices economically identical. A public generic creditor instruction also states a 365/366 convention, but only for the convention described in that document. Consumer Financial Protection Bureau+1

2.2 Fixed-payment monthly model

For E02 and E04, the model expressly uses equal monthly periods and a monthly rate.

Let:

Plain text

P = original principal in dollars
r = interest rate per model month, as a decimal
n = number of end-of-month installments
M = equal theoretical installment
B_k = principal immediately after installment k

With no fees, arrears, or capitalization complications:

Plain text

B_0 = P
I_k = r * B_(k-1)
principal_paid_k = M - I_k
B_k = B_(k-1) * (1 + r) - M

Repeated substitution gives, for r > 0:

Plain text

B_n = P * (1 + r)^n - M * [((1 + r)^n - 1) / r]

Setting the final balance to zero and solving for M:

Plain text

M = P * r / [1 - (1 + r)^(-n)]

This is a derivation from the stated recurrence, not a statutory formula for permissible interest.

For r = 0, use a separate equation:

Plain text

M = P / n
B_k = P - cumulative_principal_payments_through_k

Do not evaluate the positive-rate expression by dividing by zero.

When annual rate divided by 12 is appropriate

For this investigation:

Plain text

r = nominal_annual_rate / 12

is appropriate because the monthly model expressly defines the nominal annual rate that way.

It is not a general conversion for:

• An effective annual rate: the corresponding mathematical monthly rate would be (1 + y)^(1/12) - 1.
• A fee-inclusive legal APR used as though it were the note’s accrual rate.
• Daily simple interest merely because payments occur monthly.
• Irregular periods, variable rates, or a contract using another accrual convention.

For a 1% monthly rate:

Plain text

nominal annual rate = 12 * 1% = 12%
effective annual rate = 100 * [(1.01)^12 - 1]
                      = 12.6825030132%

That effective annual equivalent does not mean the E02 loan capitalizes unpaid interest monthly. E02 pays each month’s interest in full.

3. Worked-example register

E01 — Thirty days of simple accrual

Hypothetical inputs

Plain text

principal = $10,000
annual rate = 12% = 0.12
elapsed accrual units = exactly 30 days
principal unchanged throughout

The example stipulates 30 nonoverlapping accrual-day units. It does not assign real calendar dates, holidays, or a receipt/posting rule.

The required 365-versus-360 comparison is:

Plain text

unrounded difference = 100 - 98.6301369863
                     = $1.3698630137

difference between rounded end-of-period amounts = $1.37

Rounding changes the result

These are two different posting assumptions. Displaying a rounded per diem does not establish that a ledger rounds each day.

Authority boundary: Current official commentary discusses 360/365 differences. The generic creditor instruction in S013 explicitly states 365, or 366 in a leap year, but is dated December 17, 2010. It is not proof of a current universal practice or an agreed contract. A denominator of 360 also does not, by itself, establish a complete “30/360” date-counting convention. No complete Actual/Actual convention across year boundaries is established here. C033–C037, C045. Consumer Financial Protection Bureau+2

E02 — Fixed monthly amortization at a positive rate

Hypothetical inputs

Plain text

P = $10,000
nominal annual note rate = 12%
r = 0.12 / 12 = 0.01 per model month
n = 12 end-of-month installments
advance at t = 0
first installment at t = 1 month
original final installment at t = 12 months

Additional assumptions:

• No fees, changing rates, existing unpaid interest, or prepayment charges.
• Every scheduled payment is timely.
• Current interest is paid first; the remainder reduces principal.
• Interest is posted monthly to cents using nearest-cent, half-up rounding for these positive amounts.
• The ordinary payment is rounded to $888.49.
• The final payment is adjusted to the exact remaining cent balance plus that month’s posted interest.

These are model assumptions, not a finding about any real payment-allocation clause.

Step 1: Calculate the theoretical payment

Plain text

M = 10000 * 0.01 / [1 - (1.01)^(-12)]
  = $888.487886783416...

In an ideal calculation with fractional-cent payments and no intermediate rounding:

Plain text

12 * M = $10,661.854641400994...
total interest = $661.854641400994...

The analytic final balance is zero. Floating-point evaluation produced a residual of approximately 0.0000000000108 dollars, which is numerical evaluation noise, not a collectible residual.

The following schedule uses the separately identified cent-posting model, not fractional-cent cash payments.

Step 2: Calculate the first two installments

Plain text

Month 1 interest = round_cent(10000.00 * 0.01) = $100.00
Month 1 principal = 888.49 - 100.00 = $788.49
Month 1 ending balance = 10000.00 - 788.49 = $9,211.51

Month 2 unrounded interest = 9211.51 * 0.01 = $92.1151
Month 2 posted interest = $92.12
Month 2 principal = 888.49 - 92.12 = $796.37
Month 2 ending balance = 9211.51 - 796.37 = $8,415.14

Complete cent-posting schedule

Conservation and final-payment check

Plain text

11 * 888.49 + 888.47 = $10,661.86

principal repaid + interest paid
= 10000.00 + 661.86
= $10,661.86

final payment = 879.67 + 8.80 = $888.47

Blindly collecting twelve payments of $888.49 would collect $0.02 too much under this ledger. That difference is handled explicitly rather than silently erasing a balance.

The regulation’s commentary recognizes that collecting whole cents may require adjustment of the final installment. That disclosure observation does not prescribe this example’s particular ledger-rounding policy. C037–C040, C042, C046. Consumer Financial Protection Bureau

E03 — Zero-interest version of E02

Keep the same $10,000 principal, 12 monthly periods, and payment dates. Change only the interest rate to zero. There are no fees.

Plain text

M = P / n
  = 10000 / 12
  = $833.333333...

Choose eleven ordinary payments of $833.33, then adjust the final payment.

Plain text

11 * 833.33 = $9,166.63
10000 - 9166.63 = $833.37

Twelve payments of $833.33 would leave $0.04 unpaid. No division by the interest rate is used.

Zero interest does not by itself establish that a transaction is outside Regulation Z; the coverage rule also addresses written agreements payable in more than four installments, subject to the other coverage conditions and exemptions. Nor would a zero note rate establish zero credit cost if separately classified charges existed. They do not exist in E03. C002, C041, C047. Consumer Financial Protection Bureau

E04 — Same E02 loan, with extra principal after installment 3

Everything from E02 remains unchanged except this specified action:

Immediately after the regular third installment, apply an additional $2,000 directly to principal.

The model assumes that this application is permitted, immediate, and free of a penalty or other charge. It does not establish those rights for an actual loan.

There is no recast. The ordinary installment remains $888.49 on the original monthly sequence until a smaller final payoff payment.

The recurrence becomes:

Plain text

I_k = round_cent(0.01 * B_(k-1))
B_k = B_(k-1) + I_k - regular_payment_k - extra_principal_k

At month 3:

Plain text

balance after regular payment = 8415.14 + 84.15 - 888.49
                              = $7,610.80

balance after extra principal = 7610.80 - 2000.00
                              = $5,610.80

Complete accelerated schedule

Controlled comparison

Plain text

9 * 888.49 + 2000.00 + 494.88 = $10,491.29
10000.00 + 491.29 = $10,491.29

interest saved = 661.86 - 491.29 = $170.57
cash difference = 10661.86 - 10491.29 = $170.57

The $2,000 is included in total cash paid. It is accelerated principal, not itself an interest saving.

Neither this calculation nor the disclosure rules establish automatic advancement of due dates, an automatic recast, a prepayment right, or the absence of a penalty. Regulation Z’s prepayment disclosure distinguishes unpaid-balance finance charges from other finance-charge structures; it does not supply the entire substantive prepayment law. C043–C044, C048. Consumer Financial Protection Bureau+2

E05 — Upfront fee, amount financed, and numerical rate comparison

A. Legal-method status

The readable sources establish that the closed-end APR analysis concerns both amounts and timing; Regulation Z permits the actuarial method or the United States Rule. Appendix J’s readable prose describes nominal annualization and unit periods.

However, the retrieved current Appendix J page did not expose several necessary equation and example images as readable content. Direct image retrieval and the official annual-CFR PDF backup did not resolve that limitation. No result below is represented as a legal APR, a disclosed APR verified against Appendix J, or a compliance certification. C025–C030, C056. Consumer Financial Protection Bureau+3

B. Cash-flow assumptions

Use the E02 repayment stream:

Plain text

t = 0: gross loan advance of $10,000
t = 1 through 11 model months: payments of $888.49
t = 12 model months: payment of $888.47

Add a hypothetical $300 fee paid separately at t = 0. The note balance remains $10,000.

The two classification branches are conditional:

1. Included branch: the $300 is legally a finance charge and, because paid separately at consummation, a prepaid finance charge.
2. Excluded branch: the $300 actually satisfies an applicable exclusion—for example, a qualifying application fee charged to all applicants whether credit is extended or not.

These are alternative factual/legal conditions. The label is not an election available merely to obtain a preferred APR.

C. Accounting comparison

The included branch reconciles as follows:

Plain text

amount financed = 10000 - 300 = $9,700.00
finance charge = 661.86 + 300 = $961.86

amount financed + finance charge
= 9700.00 + 961.86
= $10,661.86 scheduled total of payments

all cash outflows
= 300.00 + 10661.86
= $10,961.86

The amount-financed treatment follows the structure of the official §18(b)(3) examples, including the need to avoid subtracting a charge twice. The treatment of the repayment schedule follows comment 18(g)-1. Consumer Financial Protection Bureau+1

Withheld-fee variation: If the included $300 were withheld from a $10,000 note instead, the consumer would receive $9,700 and later pay $10,661.86. Do not also add a fictitious second $300 cash payment.

Financed-on-top variation: If a $300 charge were added to a $10,000 advance, producing a $10,300 interest-bearing note, a new repayment schedule would be needed. The E02 $888.49 payment cannot be reused unchanged. If that $300 is a finance charge, the amount-financed calculation must exclude it without double subtraction; if it is a financed non-finance-charge item, its treatment differs. Actual classification remains unresolved.

D. Separately labeled numerical IRR calculation

Define:

Plain text

C_1 through C_11 = 888.49 dollars
C_12 = 888.47 dollars
q = numerical rate per equal model month
V = specified initial comparison value

Solve:

Plain text

F(q) = sum[C_k / (1 + q)^k, for k = 1 through 12] - V = 0

This is the present-value equation for the stated cash flows. It can also be derived by setting the terminal balance of a hypothetical accumulation recurrence to zero. It is presented here as finance mathematics, not as a verified implementation of Appendix J.

Convert the numerical monthly root as follows:

Plain text

nominal annualized percentage = 1200 * q
effective annualized percentage = 100 * [(1 + q)^12 - 1]

The very small difference between 12% and the first numerical annualized rate reflects the chosen cent-posted payment stream. It does not change the stipulated note rate.

In the excluded-fee branch, the consumer’s economic cash-flow calculation still starts with net $9,700 and therefore produces the same approximately 17.8250% annualized numerical rate. A conditional calculation using the $10,000 amount-financed input produces the other root. That difference illustrates why economic cash cost and legally defined disclosure inputs must not be conflated.

E. Numerical convergence and legal tolerances are different checks

The numerical computation used bisection on:

Plain text

0 <= q <= 0.05 per month

For V = $9,700:

Plain text

F(0) = +$961.86
F(0.05) = approximately -$1,825.10069028

For V = $10,000:

Plain text

F(0) = +$661.86
F(0.05) = approximately -$2,125.10069028

After 60 bisection iterations in ordinary floating-point arithmetic, the evaluated present-value residuals at the reported roots were approximately:

Plain text

V = 9700:  -$0.00000000000909
V = 10000: +$0.00000000001091

Both are below the stated numerical residual target of $0.00000001. This is an arithmetic check for these inputs, not general tool validation.

The relevant regulatory accuracy provisions are separate:

These rules do not convert the numerical outputs into compliant APRs. C027–C029, C049–C051. Consumer Financial Protection Bureau+3

E06 — Payment below accruing interest

Hypothetical inputs

Plain text

initial principal = $1,000
monthly rate = 1%
payment at each month-end = $5
observation period = 3 months
no fees

For both branches, payments are assumed to pay accrued interest before principal. The legal and contractual allocation rule is not established.

Branch A: Unpaid interest remains separate; no capitalization

Let U be accrued unpaid interest. Interest is calculated only on the unchanged $1,000 principal.

Plain text

new interest over three months = $30.00
cash paid = $15.00
unpaid interest = $15.00

ending principal + unpaid interest
= 1000.00 + 30.00 - 15.00
= $1,015.00

A general no-capitalization accounting recurrence, under the stipulated interest-first allocation, is:

Plain text

interest_paid = min(payment, old_unpaid_interest + new_interest)
principal_paid = min(principal, max(payment - old_unpaid_interest - new_interest, 0))
new_unpaid_interest = old_unpaid_interest + new_interest - interest_paid
new_principal = principal - principal_paid

Branch B: Unpaid interest is capitalized after each payment, only if legally and contractually permitted

This branch assumes permission solely to illustrate the mathematics. No source here establishes that permission for an actual loan.

Month 3’s unrounded interest is $10.1005, posted as $10.10 under this branch’s cent convention.

Plain text

total new interest = 10.00 + 10.05 + 10.10 = $30.15
cash paid = $15.00

ending balance
= 1000.00 + 30.15 - 15.00
= $1,015.15

The capitalized component is not new cash advanced to the consumer. There is no principal reduction in either branch.

Regulation Z’s discussion of actuarial and United States Rule APR computations distinguishes treatment of unpaid finance charges. Those calculation descriptions do not independently authorize servicing capitalization. C036, C052–C053. Consumer Financial Protection Bureau+1

4. Failure branches and gateways

5. Source register — R004-P1-S001 onward

Read-status convention: “Fresh” means content actually retrieved and read in this investigation. HTML sources are unpaginated unless otherwise stated. Multiple URLs for the same statute, regulation, commentary, or appendix are grouped as the same document family; they are not presented as independent corroboration.

Currentness limit: The official current pages were checked on October 27, 2026, but their displayed compilation dates differ. The eCFR captures displayed Title 12 as updated through October 23, 2026. The CFPB index stated that Part 1026 was most recently amended April 8, 2026; that is not an amendment date for every section. The statutory pages displayed the currency specified below. No claim is made that a separate comprehensive Federal Register and session-law audit closed every possible publication-to-compilation lag. ecfr.gov+3

6. Atomic claim register — A004

The mathematical rows identify derivations and model assumptions explicitly. They are not statutory assertions. Schedule-result rows refer to the complete displayed schedules, whose individual entries are reproducible from the stated recurrence.

6.1 Coverage, definitions, charges, and disclosures

6.2 APR, timing, and mathematical models

6.3 Calculated results and evidence status

7. Granular coverage register

“Answered” means answered within the expressly bounded reference/model scope. It does not mean an actual contract or charge has been approved.

8. Source-read and failure log

The successful fresh-read extents are recorded in the source register. The following table preserves the material failed, incomplete, or discovery-only paths.

9. Explicit self-check and remaining gaps

Self-check

Remaining research packets

The parent A004 dimensions remain open. The following packets preserve their scope without claiming they were completed here.

The immediate unresolved item for this pass is narrowly defined: recover and read the complete relevant official Appendix J equation content before converting E05 from conditional numerical finance mathematics into a legally evaluated APR calculation. Actual contract application would additionally require the permitted balance, accrual, allocation, fee, timing, and prepayment terms identified above.

END R004 PASS-01