# Formula for time taken to pay off loan with annual interest and monthly repayments

I am trying to find how to work out the period of time it will take to pay off a loan. There are plenty of online calculators out there but I cannot find the formula they use. I want to recreate the formulas in excel.

I have found this site which gives variations of the compound interest formula. It states

In order to work out calculations involving monthly additions, you will need to use two formulae - our original one, listed above, plus the 'future value of a series' formula for the monthly additions.

Giving the formulas (for contributions at the end of a month) as;

Compound interest for principal:

``````P(1+r/n)^(nt)
``````

Future value of a series:

``````PMT × (((1 + r/n)^(nt) - 1) / (r/n))
``````

Total:

``````(P(1+r/n)^(nt)) + (PMT × (((1 + r/n)^(nt) - 1) / (r/n)))
``````

Where

``````A   =  the future value of the investment/loan, including interest
P   =  the principal investment amount (the initial deposit or loan amount)
PMT =  the monthly payment
r   =  the annual interest rate (decimal)
n   =  the number of times that interest is compounded per unit t
t   =  the time (months, years, etc) the money is invested or borrowed for
``````

I have the principle, the annual interest rate and the monthly payment. I assume `n`, the number of times that the interest is compounded is 12, for months in the year (online calculators seem to support this).

I worked out the 'future debt' using this formula and if I manually change the time, I can tell that for example;

``````P   = -20000
r   = 1.1
PMT = 400
``````

that the time `t` to pay off the loan would be roughly 4 years, 3 months (4.25).

However, this is pretty fiddly having to change the time in two places in the formula and 'eyeballing' when the future debt (total) is close to zero.

I believe I would like a variation of the total formula above to be something like

``````t = ...
``````

Can someone rearrange the formula for this? (or provide the correct one)

I have checked a number of questions but most just want to find out the future value using a fixed time.

• (This is why you learn Algebra in high school.) Jul 25 at 9:22
• To compute the number of periods needed to pay off a loan, given an interest rate, payment amount, starting balance and ending balance, use the `=NPER()` function. Jul 25 at 9:27
• Perhaps this answer will help : What is the formula for loan payoff date? Jul 25 at 10:32

In your formula ("Compound interest for principal"), `P(1+r/n)^(nt)` - it is implied that `r` is a nominal interest rate compounded according to the frequency `n` per annum. So a specific nominal rate for the particular compounding frequency, rather than an effective annual rate.

The future value of a series with deposits made at the end of each period is obtained by summing multiple versions of the above formula (with `PMT` instead of `P`): The formula for a loan with payments at the end of each period can be obtained similarly. Instead of compounding forward to a future value it discounts the payments back to the present (initial) value of the loan: `s`. This formula can be expressed for `t`, the number of years to pay off the loan: For example, with 1.1% nominal interest compounded monthly

``````s = 20000
r = 0.011
n = 12
PMT = 400

∴ t = -(Log[1 - (r s)/(n PMT)]/(n Log[(n + r)/n])) = 4.26713 years
``````

Also in Excel, as suggested by RonJohn, documented here, the same result: • Thank you, in libreoffice calc the formula is `=NPER(C6/12,C7,-C5,,0)`
– myol
Jul 25 at 14:20