The monthly payment formula
An amortising loan is repaid through a fixed monthly payment that covers the month's interest and then reduces the principal. The formula reads payment = C × t / (1 − (1 + t) to the power −n), where C is the amount borrowed, t the annual rate divided by twelve, and n the number of payments. At a zero rate it collapses to the amount divided by the number of months.
On €200,000 at 3.6% over 25 years, it gives €1,012 a month. What that payment is made of changes from start to finish: the first contains €600 of interest and €412 of principal, the last €3 of interest and €1,009 of principal. The same sum changes nature.
That is also why the outstanding balance falls slowly at first. On this loan it is still €172,959 after five years and €122,096 halfway through the term, a little over 60% of the amount borrowed once half the payments have been made.
- Amount borrowed: the principal, excluding the deposit and the notaire fees.
- Nominal annual rate: the contract rate, divided by twelve and applied each month.
- Term: the number of years, converted into monthly payments.
- Arrangement fees: paid up front, they stay out of the payment and go into the APR.
How to use the calculator
The four fields are enough for a complete answer: the amount, the nominal rate, the term and the arrangement fees. The payment, the paydown curve and the four figures under the chart recompute on every keystroke.
Arrangement fees are the one field that leaves the payment alone. They change the APR and the total cost of credit, because the sum actually received is smaller than the principal borrowed while the repayments stay the same.
Borrower insurance has no field, because the model does not compute it. On a mortgage it often runs to a line of the same order as several tenths of a point of rate, and it adds to everything shown here.
All four parameters fit in the page address. The "Copy link" button gives a URL that reopens the same loan, figures included.
Reading the result
The headline figure is the monthly payment, insurance excluded. The four boxes under the chart say what it costs over time: the APR, the total interest, the total cost of credit, which adds the arrangement fees to the interest, and the total repaid, which adds the principal to the rest.
On the default values, €200,000 at 3.6% over 25 years with €2,000 of fees, the payment is €1,012, interest reaches €103,602, the total cost of credit €105,602 and the total repaid €303,602. Put another way, the borrower repays a little over one and a half times what was borrowed.
The term moves those two figures in opposite directions. The same amount at the same rate gives €1,440 a month over 15 years for €59,129 of interest, €1,170 over 20 years for €80,854, €1,012 over 25 years for €103,602, and €909 over 30 years for €127,345. Between 20 and 30 years, the payment drops by €261 and the interest rises by €46,491.
APR and nominal rate
The nominal rate is the contract rate. The APR is the single rate at which the present value of the payments equals the sum actually received, that is the principal minus the fees paid up front. The two never quite coincide.
A gap remains even with no fees at all, because the APR is an actuarial rate annualised over twelve monthly periods. On the default loan with no arrangement fees, a 3.60% nominal rate gives a 3.66% APR. Adding €2,000 of fees takes it to 3.76%.
That is what makes two offers with different headline rates comparable. An offer at 3.60% with no fees gives a 3.66% APR and €103,602 of total cost. An offer at 3.40% with €4,000 of fees gives a 3.65% APR and €101,166 of total cost. The headline rates are two tenths of a point apart, the APRs one hundredth.
Frequently asked questions
How is a loan's monthly payment calculated?
The payment on an amortising loan is C × t / (1 − (1 + t) to the power −n), with C the principal borrowed, t the annual rate divided by twelve and n the number of payments. On €200,000 at 3.6% over 25 years, so 300 payments, it comes to €1,012 a month.
What is the difference between the nominal rate and the APR?
The nominal rate is the one written in the contract and is what the payment is computed from. The APR is the actuarial rate that makes the present value of the payments equal the sum actually received, arrangement fees deducted. On a €200,000 loan at 3.60% over 25 years, the APR is 3.66% with no fees and 3.76% with €2,000 of fees.
How much interest does a loan cost in total?
On €200,000 at 3.6%, total interest reaches €59,129 over 15 years, €80,854 over 20 years, €103,602 over 25 years and €127,345 over 30 years. The amount borrowed and the rate stay the same: only the length of time the principal stays owed changes.
What does paying off early save?
Paying €200 extra each month on the default loan brings the term down from 300 to 229 payments and removes €27,137 of interest, roughly a quarter of the total. The effect comes from the principal falling faster, so next month's interest falls too. The model counts no early-repayment charge, which most contracts provide for.
Do arrangement fees change the monthly payment?
No. They are paid up front and the payment is still computed on the full principal at the nominal rate. They change the APR and the total cost of credit, since the sum actually made available is smaller than the principal borrowed.
Is borrower insurance included?
No. Neither borrower insurance, nor guarantee fees, nor notaire fees are modelled. The payment shown is therefore below the one that will be debited on a real mortgage, and the APR shown below the contractual APR.