How Long a Loan Takes to Repay

Works out how long a loan takes to clear once the monthly payment is fixed. It is the fixed-instalment formula solved for the number of payments.

A fixed-instalment loan is made of four quantities: the amount borrowed, the annual rate, the term and the monthly payment. Fix any three and the fourth follows. This calculator fixes the monthly payment and finds the term.

Solving the fixed-instalment formula for the number of payments $n$ gives this.

n = \dfrac{-\log\left(1 - \dfrac{P i}{M}\right)}{\log(1 + i)}

$n$ is the number of payments
$P$ is the loan amount
$M$ is the monthly payment
$i$ is the monthly rate, the annual rate (%) divided by 100 and again by 12

A logarithm appears because in the original formula $n$ sits up in the exponent of $(1 + i)^{n}$ , and a logarithm is what brings an exponent back down.

Example

With a loan of 30000000, an annual rate of 1.5% and a monthly payment of 100000, the monthly rate is 0.00125. The product $P i$ is 37500, so the bracket becomes $1 - 37500 \div 100000 = 0.625$ .

Working that through gives about 376.24 payments, or about 31.35 years. The total repaid is about 37623786, of which about 7623786 is interest.

Some loans never finish

If the monthly payment is too small, the calculation has no answer. That happens when $M$ is at or below $P i$ .

$P i$ is the interest due in the first month, 37500 in this example. Paying only 30000 would not even cover it, and the missing 7500 would be added to the balance. The next month you would owe more than you borrowed, and the interest would be larger still. No number of payments ever clears it.

The boundary case $M = P i$ behaves the same way. Every yen paid goes on interest and the principal never moves. In the formula the bracket becomes 0, its logarithm runs to negative infinity, and $n$ diverges.

Small changes in payment move the term a lot

A modest change to the monthly payment shifts the term sharply. Raising 100000 to 120000 in this example cuts 31.35 years down to 25.00 — over six years saved for an extra 20000 a month.

Going the other way is steeper still: drop to 80000 and the term stretches to 42.19 years. The closer the payment gets to the 37500 of first-month interest, the slower the principal falls.

Points to watch

The answer is not a whole number of payments. 376.24 means 376 full payments followed by a part payment. Real contracts round the count to a whole number and adjust the final payment.

The calculation assumes the rate never changes, and it excludes guarantee fees, arrangement fees, insurance and tax.