The setting
A loan of principal P is repaid in n equal payments of PMT, one at the end of each period, with interest rate r per period.
Present value
The principal must equal the present value of all the payments, each discounted by the interest that accrues until it is paid:
A geometric series
Factor PMT out of the sum. What remains is a geometric series with ratio 1/(1+r), whose sum has a closed form:
Substituting and solving for the payment:
The second form comes from multiplying numerator and denominator by (1+r)^n.
A worked example
Take P = 100,000, r = 1% per month and n = 12 months. Then (1+r)^n = 1.126825, so PMT = 100,000 × 0.01 × 1.126825 / 0.126825 ≈ 8,884.88. The total paid is 12 × 8,884.88 = 106,618.56, so the interest is about 6,618.56. The numbers are illustrative.
Limits
- It assumes a fixed rate, equal payments and payment at the end of each period.
- Fees, insurance and indexers are not included, so the effective cost differs.
- A formula is not a credit decision: the institution decides.
Background reading
Works by other authors that treat the underlying methods. They are not Menlo Tech publications.
- Standard result of financial mathematics; see any introductory text on the mathematics of finance.
Related
Case studies