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The Maths Behind French Amortisation

French amortisation 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.

Understanding the annuity formula

The formula may look intimidating, but it has an intuitive interpretation. The numerator L × r represents the interest you would pay in the first period if no principal were repaid. The denominator (1 − (1 + r)−n) is a scaling factor that spreads this cost across all payments whilst ensuring the loan is fully repaid by the end.

As the number of payments n increases, the denominator approaches 1, meaning the payment approaches the pure interest amount. This is why extremely long loans have low monthly payments but pay enormous total interest. Conversely, as n decreases, the denominator shrinks, increasing the payment but reducing total interest.

How each payment splits

Once P is known, the amortisation 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, whilst the principal portion principal_q grows. The payment P stays constant throughout.

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

Practical example: step by step

Let us work through a concrete example. Suppose you borrow £200,000 at 6% EAR for 30 years with monthly payments.

Step 1: Convert EAR to periodic rate

r = (1 + 0.06)1/12 − 1 ≈ 0.004868

Step 2: Calculate total payments

n = 30 × 12 = 360 payments

Step 3: Apply the formula

P = 200,000 × 0.004868 / (1 − (1 + 0.004868)−360)
P ≈ £1,198.40 per month

Step 4: First payment breakdown

interest_1 = 200,000 × 0.004868 ≈ £973.60
principal_1 = 1,198.40 − 973.60 = £224.80

Over the loan lifetime, you pay approximately £231,424 in interest on top of the £200,000 principal.

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.

Using APR directly in the annuity formula would systematically understate your payments and produce an impossible schedule where the loan never fully repays.

Beyond monthly: other payment frequencies

The formula works for any payment frequency. For quarterly payments (4 per year), the periodic rate conversion becomes:

r = (1 + EAR)1/4 − 1

For biweekly payments (26 per year):

r = (1 + EAR)1/26 − 1

More frequent payments reduce the periodic rate, which slightly lowers total interest because principal is repaid sooner. However, the effect is modest compared to choosing a shorter loan term.

Common calculation mistakes

  • Using APR as EAR: This produces payments that are too low and a schedule that never fully amortises.
  • Simple division for periodic rate: Dividing APR by 12 ignores compounding and produces slightly incorrect results.
  • Ignoring the last payment adjustment: Without adjusting the final payment for rounding errors, the schedule may show a small remaining balance or overpayment.
  • Wrong payment count: For a 30-year loan with monthly payments, n = 360, not 30.

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.

Why this matters

Understanding the maths behind your loan empowers you to make informed decisions. You can verify lender calculations, compare offers accurately, and model scenarios like extra payments or refinancing. The formulas are not just academic exercises — they represent real money that you will pay or save over the life of your loan.