Loan Amortization Calculator: Daily Compounding Formula

A loan amortization calculator with daily compounding estimates a level monthly payment and how each payment is split between interest and principal, using the loan balance, annual rate, and term. The calculator below uses a standard monthly-payment formula with interest compounded monthly—not daily—so it is not a daily-compounding calculator; the distinction matters when matching a lender’s calculation.

Formula

For a fixed-rate loan with monthly payments, the level payment is calculated as:

Payment = P × r ÷ [1 − (1 + r)⁻ⁿ]

• P is the starting loan principal, in dollars.
• r is the monthly interest rate as a decimal: annual percentage rate (APR) ÷ 1,200. For an 8% APR, r = 8 ÷ 1,200.
• n is the number of monthly payments: term in years × 12.
• Payment is the monthly amount, in dollars, rounded to cents for this calculator.

Total interest is estimated as monthly payment × number of payments − principal. Because the payment is rounded to cents before this total is calculated, the total reflects the displayed payment. A lender’s final payment can differ slightly because of its own rounding and payment schedule.

Worked Example

Using the calculator’s default inputs—$20,000 principal, 8% APR, and a five-year term:

1. Convert the APR to a monthly decimal rate: 8 ÷ 1,200 = 0.0066667.
2. Find the payment count: 5 × 12 = 60 monthly payments.
3. Substitute into the formula: $20,000 × 0.0066667 ÷ [1 − (1.0066667)⁻⁶⁰].
4. The monthly payment is approximately $405.53, rounded to cents.
5. Total interest using that rounded payment is $405.53 × 60 − $20,000 = $4,331.80.

Assumptions

• The interest rate stays fixed throughout the modeled term.
• Payments are equal monthly payments made at the end of each monthly period.
• Interest is modeled monthly at APR ÷ 12, rather than accruing daily.
• Fees, taxes, insurance, PMI, escrow, and other charges are not included.

What Daily Compounding Changes

“Daily compounding” means interest is calculated using a daily rate and added to the balance according to the lender’s stated schedule. The exact method may use a 365-day or 366-day basis, a daily periodic rate, and the actual number of days between payments. Some loans accrue interest daily on the outstanding principal without adding unpaid interest to principal every day. That is daily accrual, and it is not necessarily the same as daily compounding.

A simple daily-compounding model with a nominal annual rate would use a daily rate such as APR ÷ 365 and a growth factor of (1 + APR/365)^days. But a monthly payment made against a changing balance means the number of days in each period and the payment dates matter. There is no single daily-compounding payment formula that will exactly reproduce every lender’s method without those details.

For an accurate comparison with a loan disclosure, check how the lender defines its rate, day-count basis, compounding or accrual method, payment dates, and rounding. If the lender provides an effective annual rate rather than a nominal APR, do not treat those figures as interchangeable: an effective rate already reflects compounding.

How to Read an Amortization Schedule

An amortization schedule shows the balance and payment allocation after each period. For a standard fixed-rate monthly loan, the interest portion in a period is generally the opening balance multiplied by the monthly rate. The rest of the payment reduces principal. As the balance falls, the interest portion usually falls too, while more of each level payment goes toward principal.

• Opening balance: principal remaining before that payment.
• Interest: the period’s interest charge under the loan’s stated method.
• Principal paid: payment minus interest, before any separately applied charges.
• Closing balance: opening balance minus principal paid.

This calculator reports the scheduled monthly payment and estimated total interest, not a full daily ledger. It assumes each scheduled payment is made on time and does not model extra payments or missed payments.

When to Use This Estimate—and Its Limits

Use the monthly formula to understand the basic payment arithmetic or to make an initial estimate when a loan has a fixed rate, a fixed term, and monthly level payments. Enter the loan amount, APR, and term as stated in the lender’s information. The rate is an input, not a claim about what a lender will offer; obtain the applicable rate and loan terms directly from the lender.

Common mistakes include entering 8 instead of 0.08 in a formula that expects a decimal rate, confusing APR with a daily rate, and entering months where the calculator expects years. This calculator expects APR as a percentage (for example, 8), and the term in years. Its formula converts those inputs internally. Rounding each period can also create a small difference from a lender’s final schedule.

The model is not suitable for variable-rate loans, irregular payment dates, interest-only periods, balloon payments, or loans where fees are financed into the balance unless those features are separately accounted for. It also cannot determine a payoff quote: accrued interest and charges through a specific date may affect that amount.

Frequently Asked Questions

What is the formula for a loan with daily compounding?
For a fixed balance and nominal annual rate, a basic daily growth factor is (1 + APR/365)^days. For a loan with regular payments, the balance changes after each payment, so an exact schedule also needs the lender’s day-count convention and payment dates.

Does this calculator compound interest daily?
No. It uses APR divided by 12 and a monthly amortization formula. It estimates a standard fixed-rate monthly payment and should not be presented as a lender-specific daily-accrual calculation.

Is APR the same as the interest rate used for daily compounding?
Not always. APR is a stated annual measure, while a daily periodic rate and the way interest accrues are defined by the loan terms. Check the lender’s disclosure for the applicable method rather than assuming APR ÷ 365 reproduces it exactly.

Why might my lender’s payment or total interest differ?
Possible reasons include daily rather than monthly accrual, actual payment dates, fees included in the balance, a different rounding method, or a final payment adjusted to clear the remaining balance.

Does making an extra payment change the amortization?
Usually, an extra amount applied to principal changes the outstanding balance and can alter later interest. The result depends on when the payment is made and how the lender applies it; neither is modeled here.

This result is an estimate, and the lender’s disclosure gives the exact figures.

Related

• Interest-Only Loan Calculator: Payment During and After the Interest-Only Period
• APR Calculator: The True Rate of a Loan With Fees
• Simple Interest Loan Calculator: I = P × r × t

in p | Loan amount | $ | 20000 | 1 | 100000000
in apr | Interest rate (APR) | % | 8 | 0.01 | 100
in yrs | Loan term | years | 5 | 0.0833 | 50
out pmt | Monthly payment | $ | p * (apr / 1200) / (1 - (1 + apr / 1200) ^ -(yrs * 12)) | 2
out int | Total interest | $ | pmt * yrs * 12 - p | 2