The Math Behind French Amortization

French amortization is governed by one elegant formula. Given a loan amount L, an effective annual rate EAR, and a payment count n, the fixed payment P is:

P = L × r / (1 − (1 + r)−n)

where r is the periodic rate: the EAR expressed per payment interval. For monthly payments on a loan with 12 payments per year:

r = (1 + EAR)1/12 − 1

This conversion from annual to periodic rate is what makes the EAR the correct rate to use in the formula — not the nominal APR, and not a simple rate divided by the number of payments.

How each payment splits

Once P is known, the amortization schedule falls out naturally. For quota q (where q = 1 for the first payment):

interest_q = balance_q-1 × r
principal_q = P − interest_q
balance_q = balance_q-1 − principal_q

Because the balance decreases with each payment, the interest portion interest_q shrinks over time, while the principal portion principal_q grows. The payment P stays constant throughout.

This is the defining characteristic of French amortization: constant total payments, declining interest, and accelerating principal reduction.

Solving for the EAR from a payment

The formula above works forward from (L, EAR, n) to find P. The calculator also runs this backwards: if you know L, P, and n, it can solve for the implied EAR.

This requires a numeric solver (binary search in this case), since the EAR appears on both sides of the equation through the periodic rate. The solver starts with a wide search range and repeatedly bisects until it converges on the EAR that produces the given payment.

Why EAR and not APR?

The APR (Annual Percentage Rate) is a nominal rate — it does not account for intra-year compounding. If a loan quotes "12% APR" with monthly payments, the actual rate you pay is higher than 12%.

The EAR (Effective Annual Rate) does account for compounding. The relationship between APR and EAR for m payments per year is:

EAR = (1 + APR/m)m − 1

For 12 payments per year at 12% APR: EAR = (1 + 0.12/12)¹² − 1 ≈ 12.68%. This is what the calculator uses, and why it produces accurate schedules even for high-frequency payment structures.

The last payment adjustment

Because floating-point arithmetic and the annuity formula can produce tiny rounding errors, the final payment in a French schedule is usually adjusted slightly to ensure the balance reaches exactly zero. The calculator handles this by checking whether the last payment would produce a negative balance and clamping it to the remaining balance plus accrued interest.