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Advanced Loan Calculator

Calculate loan payments, interest rate, loan amount, or number of payments with different compounding frequencies. Free advanced loan calculator.

Loan details

$
%

Monthly payment

$575.73

48 monthly payments

Total payment

$27,635.15

48 payments

Total interest

$2,635.15

Amount paid above principal

Payment breakdown

  • Principal$25,000.0090.5%
  • Interest$2,635.159.5%

How this loan is calculated

Convert the quoted annual rate to a periodic rate, then solve the amortizing payment identity.

  1. Convert the annual rate to a periodic rate

    i=(1+rm)m/q−1i = \left(1 + \frac{r}{m}\right)^{m/q} - 1

    The nominal annual rate is 5.0%. Interest compounds 12 times per year and you pay 12 times per year. When those frequencies match, this is just the annual rate divided by 12.

  2. Apply the payment formula

    PMT=P×i(1+i)n(1+i)n−1\mathrm{PMT} = P \times \frac{i(1+i)^{n}}{(1+i)^{n}-1}

    P is $25,000.00 and n is 48. If the rate is zero, the payment is principal divided by n.

  3. Split principal and interest

    Total interest=PMT×n−P\text{Total interest} = \mathrm{PMT} \times n - P

    48 payments of $575.73 total $27,635.15. Subtract the $25,000.00 principal to get $2,635.15 of interest.

Quoted annual rates are nominal. The periodic rate used in the payment formula is equivalent-rate conversion i=(1+rm)m/q−1i = \left(1 + \frac{r}{m}\right)^{m/q} - 1, where m is compounding periods per year and q is payments per year. When compounding and payment frequency are both monthly, that is the same r/12r/12 used by a standard EMI.
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How the advanced loan calculator works

An amortizing loan is repaid with equal payments that cover interest first and the rest principal, so the balance reaches zero on the last installment. This calculator solves that identity for whichever unknown you leave out: the payment, the loan amount, the interest rate, or the number of payments. You can also set compounding and payment frequency independently. All math runs in your browser.

For a standard monthly installment with monthly compounding, the EMI calculator is the shorter path and includes an amortization schedule. If the loan is a 10-year fixed period that later resets annually, the 10/1 ARM mortgage calculator models the payment after the first adjustment. Once you have a payment, the 28/36 rule calculator checks whether it fits common front-end and back-end debt-to-income limits.

Periodic payment formula

The payment on a fully amortizing loan is the ordinary annuity formula. P is the principal, i is the interest rate per payment period, and n is the number of payments. If the rate is zero, the payment is principal divided by n.

PMT=P×i(1+i)n(1+i)n−1\mathrm{PMT} = P \times \frac{i(1+i)^{n}}{(1+i)^{n}-1}

Microsoft Excel documents the same relationship as PMT(rate, nper, pv). Its worked example of a $10,000 loan at 8% a year for 10 monthly payments is $1,037.03 per month. Wikipedia's 30-year mortgage example of $200,000 at 6.5% with monthly payments is $1,264.14.

Compounding versus payment frequency

Lenders quote a nominal annual rate r. If interest compounds m times per year and you pay q times per year, the rate that belongs in the payment formula is the equivalent periodic rate:

i=(1+rm)m/q−1i = \left(1 + \frac{r}{m}\right)^{m/q} - 1

When m and q are equal (both monthly, both weekly, and so on), this reduces to r/q. A 6% nominal rate compounded monthly has an effective annual rate of about 6.17%, which is Wikipedia's standard nominal-to-effective conversion. US consumer mortgages usually quote a monthly rate of APR/12, which this tool reproduces when compounding and payment frequency are both monthly.

Solving for principal, rate, or term

Rearrange the same identity to find how much you can borrow from a known payment:

P=PMT×(1+i)n−1i(1+i)nP = \mathrm{PMT} \times \frac{(1+i)^{n}-1}{i(1+i)^{n}}

The number of payments follows from taking logarithms. The payment must exceed the first period's interest, or the balance never falls.

n=ln⁡ ⁣(PMTPMT−iP)ln⁡(1+i)n = \frac{\ln\!\left(\frac{\mathrm{PMT}}{\mathrm{PMT} - iP}\right)}{\ln(1+i)}

The periodic rate has no closed-form solution, so this tool searches for the unique i that makes the present value of the payment stream equal the loan amount, then converts i back to a nominal annual rate with the same compounding convention. Total interest is always payment times n minus principal.

Frequently asked questions

What can I solve for?
Choose payment amount, loan amount, interest rate, or number of payments. Enter the other three plus compounding and payment frequency. Results update as you type.
How is this different from an EMI calculator?
An EMI is the monthly case of this formula: monthly compounding, monthly payments, and you solve only for the installment. Use the EMI calculator when that is all you need. Use this tool for weekly, biweekly, quarterly, or annual payments, or to back into the rate or term.
Why do compounding and payment frequency both appear?
The quoted annual rate is nominal. Compounding says how often interest is added. Payment frequency says how often you pay. When they differ, the periodic rate is i=(1+rm)m/q−1i = \left(1 + \frac{r}{m}\right)^{m/q} - 1, not a simple r/qr/q.
What if the interest rate is zero?
The payment is the loan amount divided by the number of payments, and there is no interest. Solving for the term is principal divided by the payment.
Why can the rate or term fail to solve?
The payment stream must be large enough to retire the loan. Total payments have to exceed the principal to imply a positive rate. Each payment has to exceed the first period of interest, or the balance grows and n is undefined.
Are the results stored?
No. Changing the fields only updates the page URL so you can copy and share your inputs.
Which currency is used?
Amounts are formatted in US dollars (USD).

Resources and references

The formulas and methods in this calculator were checked against these independent sources.