# How do I calculate monthly payments given a certain loan amount, APR, and loan term in years?

I have a homework assignment to create a calculator that calculates a monthly payment, given a certain loan amount, APR, and loan term (in years). What is the formula I need to use?

• Have you searched for "loan payment formula? I get many results that give the fairly simple formula. Commented Oct 30, 2020 at 17:02

Using the following variables

s = principal
r = periodic rate
n = number of periods
d = periodic payment

Standard loan equation - formula derived by induction

$s=\sum_{k=1}^{n}\frac{d}{(1+r)^k}=\frac{d-d(r+1)^{-n}}{r}$

$\therefore d=rs(\frac{1}{(1+r)^n-1}+1)$

d = r s (1/((1 + r)^n - 1) + 1)

If your APR is quoted in the USA it will be a nominal rate. For a loan with monthly repayments and a "nominal APR compounded monthly" the periodic rate r = APR/12

E.g.

APR = 5% = 0.05
r = 0.05/12 = 0.00416667

If your APR is quoted in Europe it will be an effective annual rate. To convert to a periodic monthly rate r = (1 + APR/100)^(1/12) - 1

E.g.

APR = 5% = 0.05
r = (1 + 0.05)^(1/12) - 1 = 0.00407412

See Investopedia - Present Value of an Ordinary Annuity for more insight into how this is calculated.

For example, the Investopedia loan illustration, with 5 annual payments and APR of 5% (nominal compounded annually is the same as effective annual rate.)

Using the same example of five \$1,000 payments made over a period of five years, here is how a present value calculation would look. It shows that \$4,329.58, invested at 5% interest, would be sufficient to produce those five \$1,000 payments.

s = 4329.58
r = 0.05
n = 5

d = r s (1/((1 + r)^n - 1) + 1) = 1000
• Just to check my math. Given s=\$4329.58, r=0.05/12 n = 5*12 = 60 months, then d = \$81.70. Is that correct? Commented Oct 30, 2020 at 18:14
• @moonman239 Hi, yes correct for a nominal APR compounded monthly. The type of nominal rate should always be quoted, because 5% APR compounded quarterly is not the same as 5% compounded monthly, or daily. The correct answer \$81.70 compares OK to the ballpark estimate \$1000/12 = \$83.33, just as a rough check. Commented Oct 30, 2020 at 18:22