Financial mathematics: mortgages, payments and amortization schedules

When you borrow, you receive principal today and repay it over time. Financial mathematics describes this exchange: how much is due at each date, how much repays principal and how much pays for the money still borrowed. We begin with a small example and develop the payment formula and complete schedule.

First example: €115,000, 1.9% nominal annual rate, thirty years

A €115,000 fixed-rate loan is repaid over 30 years in 360 monthly payments. This is a French level-payment schedule, the first plan offered by the laboratory, which opens with these values and no intermediate rounding.

The monthly decimal rate is i = (1.9/100)/12 = 0.0015833333…; expressed as a percentage: 0.0015833333… × 100% = 0.15833333…%. With C = 115,000 and n = 360, R = C × i / [1 − (1+i)^(−n)] gives R = €419.3346912856….

The first interest charge is 115,000 × i = €182.083333…. Principal repaid is R − 182.083333… = €237.251357…, leaving €114,762.748642…. The table displays €182.08, €237.25 and €114,762.75. Interest in future payments totals 35,960.4888628… − 182.083333… = €35,778.41 rounded: this is not interest already accrued.

The 31.27% covers the whole thirty years; it is not an annual rate. This mode retains decimals during calculation and rounds only for display. Rounding every payment to cents instead changes the final adjustment and gives total interest of €35,960.92 for this example. The laboratory supports both conventions. The following sections explain the formulas and also use a short example rounded payment by payment.

Open the laboratory: compare four options, including both French variants. Change principal, rate, term and frequency, inspect each payment, compare charts and export a CSV.

1. Principal, rate and time: use consistent units

Let C be the principal, Y the years, m the payments per year and n = m × Y the total number of payments. A 20-year loan with monthly payments has n = 12 × 20 = 240 payments. The model uses a fixed rate and payments at the end of equal-length periods.

Convert the nominal annual rate, or TAN, into a decimal: 6% = 6/100 = 0.06. Calling this number j, the periodic rate is i = j/m. With a 6% nominal annual rate and monthly payments, i = 0.06/12 = 0.005. To display this as a percentage: 0.005 × 100% = 0.5% per month. Compute payment interest using 0.005, not 0.5 or 6.

An effective annual interest rate is a different input. If its decimal value is a, the equivalent periodic rate is i = (1+a)^(1/m) − 1. Conversely, a = (1+i)^m − 1. For instance, i = 0.005 produces a ≈ 0.061678: 0.061678 × 100% ≈ 6.1678%. A nominal rate cannot be converted into its annual effective equivalent simply by division.

The APR/TAEG concerns overall borrowing costs, including the fees covered by its calculation. The laboratory accepts either a nominal rate or an effective annual interest rate excluding fees; the latter is not the APR/TAEG. The Bank of Italy distinguishes payment calculations from additional financing costs.

2. Inside a payment: interest, principal and balance

Before payment k, the outstanding debt is D_k−1. Interest I_k accrues during the period. Payment R_k includes this interest and principal repayment Q_k. Every row follows three rules:

• I_k = D_(k−1) × i: interest on debt before payment.
• Q_k = R_k − I_k: principal actually repaid.
• D_k = D_(k−1) − Q_k: debt after payment.

With €10,000 outstanding, a monthly rate of 0.2% = 0.002 and a €200 payment, interest is 10,000 × 0.002 = €20, principal repaid is 200 − 20 = €180 and the new balance is €9,820. Do not subtract the entire payment from the debt: paying interest does not repay principal.

3. French amortization: level payments

This is the constant-payment schedule at a fixed rate. As debt falls, the interest portion decreases and the principal portion of the same payment increases. To determine R, require the present value of all future payments to equal C:

C = R/(1+i) + R/(1+i)^2 + … + R/(1+i)^n.

This geometric sum gives C = R × [1 − (1+i)^(−n)]/i. Solving for R gives R = C × i / [1 − (1+i)^(−n)]. When i = 0, use R = C/n directly instead of dividing 0/0. The power comes from making payments at different dates financially equivalent; each individual row charges interest on the outstanding principal.

Two French variants: what changes?

The laboratory presents them as two separate options, followed by Italian and principal-at-maturity schedules. Both variants use the same French payment formula; they differ in calculation precision and the treatment of the final payment.

First option: French without intermediate rounding

Payment R and every period’s components retain all available decimals. Amounts are rounded only for display in the table. The theoretical total repaid is n × R and theoretical interest is n × R − C. This is the first option, selected when the laboratory opens.

Second option: French payments rounded to cents

First, the theoretical payment is rounded to cents. At each date, interest on the outstanding debt is also rounded; subtracting that interest from the payment gives principal repaid. The final payment settles exactly the remaining principal plus the last interest charge. Total repayment is therefore the sum of actual scheduled payments, including the final adjustment, not necessarily n times the regular rounded payment.

The same inputs under both variants

Take €115,000, a 1.9% nominal annual rate, 30 years and 360 monthly payments. The periodic rate is i = (1.9/100)/12 and the theoretical payment is €419.3346912856…. The comparison gives:

The totals differ because of rounding applied during repayment; in this example, interest differs by €0.43. The first column reports theoretical payments and totals; the second follows our calculator’s cent-rounding convention. The two outstanding-balance curves may almost overlap.

The four laboratory options are therefore: French without intermediate rounding; French rounded to cents; Italian; principal at maturity. Italian and principal-at-maturity schedules retain cent rounding. A lender’s contract may use other rounding conventions.

Short French example with cent rounding: €12,000, three years, 6%, annual payments

Here m = 1, n = 3 and i = 0.06. Thus R = 12,000 × 0.06 / [1 − 1.06⁻³] ≈ €4,489.317753. The regular rounded payment is €4,489.32. Calculate and round each row to cents:

At the first date, I₁ = 12,000 × 0.06 = 720 and Q₁ = 4,489.32 − 720 = 3,769.32. At the second date, interest is calculated on 8,230.68 rather than 12,000. The last payment is adjusted by one cent to settle the balance exactly. Total interest is 720 + 493.84 + 254.11 = €1,467.95.

4. Italian amortization: equal principal

Divide C into n shares: Q = C/n. Before payment k, ignoring rounding, D_(k−1) = C − (k−1)Q; hence R_k = Q + D_(k−1) × i. Principal repayment remains equal while interest declines, so payments decrease.

In our example Q = 12,000/3 = €4,000. Payments are 4,000 + 720 = €4,720, then 4,000 + 480 = €4,480, and finally 4,000 + 240 = €4,240. Total interest is €1,440. Without rounding, a direct sum gives I_tot = i × C × (n+1)/2, since the opening balances form an arithmetic progression.

5. Principal at maturity: periodic interest and final repayment

In the bullet schedule used here, pay C × i each period and repay all of C at maturity. Principal is not gradually repaid. The example payments are €720, €720 and finally €12,720. Debt stays at €12,000 until the last payment. Without rounding, total interest is I_tot = n × C × i: here €2,160.

A small initial payment does not imply a small cost: the final principal must be available. American amortization with a sinking fund adds contributions to a separate fund and a fund return; that is a distinct model, not included in this bullet simulation.

6. A fair comparison: same terms, different cash flows

With equal principal, positive interest rate, term and frequency, the Italian schedule repays principal sooner than the French schedule and generates less total nominal interest in this comparison. It requires a higher first payment. Adding euros paid at different dates is not the same as comparing their present values. Real choices also depend on available cash, fees and contractual terms.

Try doubling the term in the laboratory while keeping principal and rate fixed: French payments fall, but total interest rises at a positive rate. Then try a zero rate: French and Italian schedules divide C equally, while the bullet schedule retains its final principal repayment.

7. Three exercises with explained answers

1. Debt €10,000, nominal annual rate 2.4%, monthly payments. If a payment is €200, find the first interest charge, principal repaid and remaining debt.

   Explained solution

   i = (2.4/100)/12 = 0.002, or 0.002 × 100% = 0.2%. Interest: 10,000 × 0.002 = €20. Principal: 200 − 20 = €180. Balance: 10,000 − 180 = €9,820.

2. A €6,000 loan has zero interest and 24 monthly payments. What is the French payment? And the Italian payment?

   Explained solution

   Without interest, each payment repays principal. Both schedules give R = 6,000/24 = €250. The bullet schedule instead pays zero at the first 23 dates and €6,000 at the last.

3. A €10,000 bullet loan lasts one year with a 2.4% nominal annual rate and monthly interest payments. Find the regular payment, final payment and total interest.

   Explained solution

   i = 0.024/12 = 0.002. Monthly interest is €20 because the principal does not fall. The first 11 payments are €20; the final one is €10,020. Total interest is 12 × 20 = €240 and total repayment is €10,240.

Reading the laboratory results

The laboratory offers four options: French without intermediate rounding, French rounded to cents, Italian and principal at maturity. The first retains decimals and rounds only displayed amounts; the other three use cent amounts. The rounded French schedule adjusts its final payment; Italian principal shares can differ by one cent. Contract schedules may use different calendars, day-count rules and rounding conventions. Fees, taxes, insurance, initial interest-only periods, early repayments and variable rates are not simulated. The schedule is an educational model, not a lender’s quotation.

Calculate your payment and complete schedule. For more on present-value formulas: OpenStax, Loan Amortization.