Mortgage guides

Chapter 1: How your monthly principal-and-interest payment is calculated | For curious minds

Sources and calculations checked: 2026-10-11

Start with practical takeaways, then explore how present value and geometric series lead to the principal-and-interest payment formula.

Terminology guide: check the meanings used in these articles

You do not need to understand these formulas to apply for a mortgage. This article is for readers who are curious about numbers and want to know where the monthly payment comes from. If you enjoy math, come explore with me.

Just want the takeaway?

Your monthly principal-and-interest payment depends on the loan amount, interest rate, and repayment term. For a fully amortizing fixed-rate loan with on-time payments, the combined monthly payment of principal and interest stays the same. Compared with the previous payment, the next payment includes less interest and more principal.

Knowing this gives you a useful starting point for comparing loan options. The derivations below are optional; you can also talk to me about your homebuying plans.

Enjoy math? Let’s start with a question

There is a math problem behind principal-and-interest payments that I find fascinating: interest accrues each month, and each payment pays down some principal. How do we choose a fixed payment that brings the balance to exactly zero at the end?

There are two ways to derive the same monthly payment formula:

Calculate present value: discount each future payment back to today, then find the payment whose total present value equals the amount borrowed.
Track the remaining balance: add interest and subtract each payment, then use a geometric series to find the payment that brings the balance to zero.

The first approach discounts future payments back to today; the second tracks the balance forward through time. Both reach the same result. Below, R denotes the remaining loan balance.

Which payment are we calculating?

We consider a fixed-rate, fully amortizing loan with payments made at the end of each month. We calculate principal and interest, assuming monthly interest, on-time payments, no extra principal payments, and no payment-by-payment rounding.

This is not your entire housing expense. Property taxes, homeowners insurance, and any applicable mortgage insurance require separate consideration. Include HOA fees in your housing budget if applicable. The CFPB explains the difference between principal and interest and the total payment.

What determines the payment?

Symbol	Meaning
P	Initial loan principal, not the home’s price
r	Monthly interest rate
n	Number of monthly payments
M	Fixed monthly principal-and-interest payment

Divide the contractual annual rate, written as a decimal, by 12 to get r. The loan term in years multiplied by 12 gives n. Do not substitute APR, which includes other financing costs.

First derivation: what are future payments worth today?

A payment of M one month from now has a present value of $\frac{M}{(1 + r)}$ . The same payment two months from now has a present value of $\frac{M}{{(1 + r)}^{2}}$ .

Why divide? At this interest rate, one dollar today corresponds to $(1 + r)$ dollars a month later. Discounting reverses that calculation.

Using the same monthly rate, the present values of all payments add up to the principal:

\begin{array}{rcl} P & = & \frac{M}{(1 + r)} + \frac{M}{{(1 + r)}^{2}} + \dots + \frac{M}{{(1 + r)}^{n}} \end{array}

The discounted payments form a geometric series with common ratio $q = \frac{1}{(1 + r)}$ :

\begin{array}{rcl} P & = & M \cdot (q + q^{2} + \dots + q^{n}) \\ = & M \cdot \frac{q (1 - q^{n})}{1 - q} \\ = & M \cdot \frac{1 - {(1 + r)}^{- n}}{r} \end{array}

Solving this equation for M gives the monthly payment formula. This is the present value of an ordinary annuity: equal payments occur at the end of each period, and the principal equals their present value.

We have expressed the principal as the present value of future payments. Next, we can check the result from a different direction: track the balance month by month until the loan is paid off.

Second derivation: follow the remaining balance

Let Rₖ be the balance after payment k, starting with $R_{0} = P$ . Each month, add interest and subtract the payment:

\begin{array}{rcl} R_{1} & = & P (1 + r) - M \\ R_{2} & = & R_{1} (1 + r) - M \\ = & P {(1 + r)}^{2} - M ((1 + r) + 1) \\ R_{3} & = & P {(1 + r)}^{3} - M ({(1 + r)}^{2} + (1 + r) + 1) \end{array}

After n payments:

\begin{array}{rcl} R_{n} & = & P {(1 + r)}^{n} - M (1 + (1 + r) + \dots + {(1 + r)}^{n - 1}) \end{array}

The bracketed terms form a geometric series with common ratio $(1 + r)$ :

\begin{array}{rcl} 1 + (1 + r) + \dots + {(1 + r)}^{n - 1} & = & \frac{{(1 + r)}^{n} - 1}{r} \end{array}

To pay off the loan exactly, set $R_{n} = 0$ and solve:

\begin{array}{rcl} 0 & = & P {(1 + r)}^{n} - M \cdot \frac{{(1 + r)}^{n} - 1}{r} \\ M & = & \frac{P \cdot r \cdot {(1 + r)}^{n}}{{(1 + r)}^{n} - 1} \\ = & \frac{P \cdot r}{1 - {(1 + r)}^{- n}} \end{array}

The remaining-balance approach gives the same formula as the present-value approach. Requiring the final balance to be zero and equating the present value of payments to the principal are two ways to describe the same repayment plan.

Two derivations, one payment formula

Both derivations lead to the following monthly payment formula:

\begin{array}{rcl} M & = & \frac{P \cdot r}{1 - {(1 + r)}^{- n}} \end{array}

Superscripts denote powers. Interest rate alone is not enough: the amount borrowed and repayment term also determine the payment.

What else matters when choosing a mortgage?

With the same principal and interest rate, on-time payments, and no extra principal payments, a longer term lowers the monthly principal-and-interest payment but increases the total interest paid over the full term. The lowest payment does not automatically mean the best fit.

If you’re preparing to buy in the Chicago area, I’d be happy to explain these numbers and their tradeoffs in plain language.

Next: Chapter 2 — How to calculate your monthly principal-and-interest payment budget

Talk to Chris about your homebuying plans

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