Build and reconcile a level-payment loan schedule

Solve the payment from present value

A fixed-rate, end-of-period, fully amortizing loan is an ordinary annuity viewed from the principal date. For Cedar Works:

PV = $48,000 at month 0
i  = 0.06/12 = 0.005 per month
n  = 48 end-of-month payments

Start with the ordinary-annuity present-value relation and divide both sides by its factor:

PV = PMT × [1 - (1+i)^(-n)] / i
PMT = PV × i / [1 - (1+i)^(-n)]
     = $48,000 × 0.005 / [1 - (1.005)^(-48)]
     = $1,127.281394…

Showing the rearrangement matters: the payment is the unknown cash flow whose present value equals opening principal, not a number supplied by the calculator.

At zero rate, payment is simply principal divided by count. At a positive rate, that shortcut underpays because it ignores interest on the outstanding balance.

Roll each row forward

Use the prior closing balance as the next opening balance:

interest = opening principal × periodic rate
principal reduction = total payment - interest
closing principal = opening principal - principal reduction

The total payment is level; its components are not. Interest generally falls as opening principal falls. Principal reduction rises by the same amount because the total remains constant.

Four independent checks

1. Every row satisfies opening plus interest minus payment equals closing.
2. Every next opening equals the prior closing.
3. Principal reductions across all rows sum to opening principal.
4. Final principal is zero within the analytical numerical tolerance.

Also confirm that total payments minus opening principal equals total interest. Agreement across these controls is stronger than checking only the final cell.

Analytical versus contractual rounding

This module retains full precision across all 48 rows and rounds only displayed amounts. A real loan system may round each month's interest and principal to cents and adjust the last payment. Neither schedule should be called wrong until its declared rounding policy and contract are inspected.

Accounting and finance interpretations

An accounting learner can trace the difference between cash paid, interest for the period, and liability reduction. That does not establish recognition, effective-interest requirements, current classification, or disclosure under a particular standard. A finance learner can trace total contractual cash under the bounded assumptions. That does not establish affordability, suitability, credit quality, refinancing risk, or whether borrowing is advisable.

Exit check

A $10,000 opening balance carries 1% monthly interest and a $900 payment. The first row has $100 interest, $800 principal reduction, and $9,200 closing principal. Explain why subtracting the full $900 from principal would be wrong and identify the equation that catches it.