The Loan Payment Formula

How lenders turn a principal into a fixed monthly payment

Finance FundamentalsLoans & AmortizationFree preview

⏱️ About 15 min

You borrow $20,000 for a car at 6% a year over five years. Where does the fixed $386.66 monthly payment the lender charges come from?

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The big idea: A loan payment is the level annuity payment whose present value, discounted at the periodic rate, exactly equals the amount borrowed.

🎯 By the end, you'll be able to

State the loan payment formula and identify each of its inputs.
Compute a monthly payment from principal, annual rate, and term.
Explain why a higher rate or a shorter term raises the payment.

📎 Helpful to know first

The Module 1 lessons on present value and annuities.

A loan is an annuity in reverse

When you borrow a principal $P$, you agree to repay it with a stream of equal payments. Those payments are an annuity, so their present value must equal the principal you receive today. Solving the annuity present-value formula for the payment gives the loan formula.

\[ PMT = P \times \dfrac{r}{1 - (1 + r)^{-n}} \]

Reading the formula

$P$ is the principal borrowed, $r$ is the periodic rate (the annual rate divided by 12 for a monthly loan), and $n$ is the number of payments. The denominator discounts the entire payment stream back to today.

⚠️ Solving the annuity for the payment

The payment is just the annuity formula solved for $PMT$; a loan is an annuity the borrower sells to the lender in exchange for cash today.

🎮 Loan Payment Explorer LIVE

Predict first: Predict first: if you double the term from 5 to 10 years, does the monthly payment roughly halve?

Monthly payment as principal, annual rate, and term change; total interest is shaded.

📝 Worked example: You borrow $20,000 at 6% APR for 5 years, with monthly payments. What is the monthly payment?

1. Periodic rate: $r = 0.06/12 = 0.005$; number of payments $n = 60$.
2. Discount term: $(1.005)^{-60} = 0.741372$, so $1 - 0.741372 = 0.258628$.
3. Numerator: $P \times r = 20000 \times 0.005 = 100$.
4. Divide: $PMT = 100 / 0.258628 = 386.66$.

✓ Monthly payment = $386.66

✏️ Practice: A $250,000 mortgage at 5.4% APR over 30 years, with monthly payments. What is the monthly payment?

💡 Hint

Use $r = 0.054/12 = 0.0045$ and $n = 360$; $PMT = 250000 \times 0.0045 / (1 - 1.0045^{-360})$.

Answer

Payment = $1,403.83

Check your understanding

1. In the loan payment formula, raising the periodic rate $r$ while holding $P$ and $n$ fixed makes the payment:

A higher rate means more interest per period, so the level payment must rise to still retire the balance.

2. The loan payment formula comes from setting the present value of the payment stream equal to:

The payments are an annuity whose present value must equal the amount borrowed today.

✅ Key takeaways

The payment formula is $PMT = P \times r / (1 - (1+r)^{-n})$.
$r$ is the per-period rate (annual divided by 12 for monthly loans) and $n$ is the number of payments.
A higher rate or a shorter term raises the required payment.

➡️ The payment is fixed, but its split between interest and principal changes every month - that split is the amortization schedule, next.

Want to test yourself on this? Try the Finance Fundamentals Aptitude test →