Amortization Calculator

Home »Finance »Amortization Calculator

Amortization Calculator

This calculator shows the total interest paid, total principal paid, and balance remaining over a selected payment range for a fixed-rate loan or annuity.

Amortization Calculator

No result yet — fill in inputs and Compute.

Related Calculators:

TVM Calculator Mortgage Calculator

Calculator Resources

This section provides essential information and guidance to help you make the most of the calculator.

Overview of Amortization

Amortization refers to the process of gradually paying off a debt over a fixed period of time through regular, equal payments. Each payment consists of both principal and interest. In the early stages of a loan, a larger portion of the payment goes towards interest, while a smaller portion reduces the principal balance. As the loan matures, more of each payment goes towards the principal, and less towards interest.

This calculator is designed to analyze fixed-rate loans and annuities, showing how payments are allocated and how the remaining balance decreases over time.

Payment Entry and Sign Handling

For user convenience, you can enter the payment amount (PMT) and other monetary values as either positive or negative numbers. The calculator will automatically treat payments as amounts paid out (outflows), so you do not need to worry about the sign when entering your values.

Future Value (FV)

FV is the target balance at the end of the term. Enter 0 for a loan that fully amortizes. Enter a positive amount for a balloon payment (e.g. a 7-year loan amortized over 30) or a lease residual. The calculator uses FV to check whether the entered PMT is consistent with N, I/Y, PV; if the deviation exceeds rounding, a note appears under the result.

Understanding the variables is crucial for effective use of the calculator:

N (Number of Periods): This represents the total number of payment periods over the entire life of the loan or annuity. For example, a 30-year mortgage with monthly payments would have: $ N = 30 \times 12 = 360 $ periods.

I/Y (Interest Rate per Year): This is the stated annual interest rate for the loan. The calculator converts this annual rate into a periodic rate based on the compounding frequency.

PMT (Payment Amount): This is the regular, equal amount paid each period (e.g., monthly mortgage payment). Enter this value to analyze totals for a given payment schedule. If you need to solve for the required payment to fully amortize a loan, use the Time Value of Money (TVM) Calculator .

PV (Present Value): This represents the initial principal amount of the loan or the lump sum value at the beginning of the annuity. For a loan, it's the amount borrowed.

FV (Future Value): The loan balance remaining at the end of the total number of periods (N). For a fully amortizing loan enter 0. For a balloon payment or lease residual enter the expected ending balance. The calculator uses FV to verify the entered PMT is consistent with the other inputs.

P/Y (Payments per Year): This indicates the number of payments made annually. Common values include 12 for monthly, 4 for quarterly, 2 for semi-annually, and 1 for annually.

C/Y (Compounding Periods per Year): This specifies how many times the interest is compounded annually. This is crucial for determining the effective interest rate per payment period. Common values are 12 for monthly, 4 for quarterly, 2 for semi-annually, and 1 for annually.

P.T (Payment Timing): This setting determines whether payments are made at the beginning (BGN) or end (END) of each period. "END" is common for loans and is the default. "BGN" is typical for annuities due.

P1, P2 (Payment Range): These inputs define a specific range of payments (from P1 to P2) for which you want to see the accumulated interest, principal paid, and the remaining balance. For example, to see the breakdown for the first year of a monthly loan, you would set P1 = 1 and P2 = 12.
To view the interest paid (INT) and principal paid (PRN) for, or balance remaining (BAL) after a particular payment, set both P1 and P2 to that payment number (e.g., P1 = P2 = 24 to view the totals for the 24th payment).

This section shows the main formulas used in amortization calculations, so you can understand how your results are determined.

1. Periodic Interest Rate ($i$):
If compounding and payment frequencies are the same ($C/Y = P/Y$), then:
$$ i = \frac{I/Y}{100} \div P/Y $$ If $C/Y$ and $P/Y$ are different, first compute the interest per compounding period:
$$ i_1 = \frac{I/Y}{100} \div C/Y $$ Then, convert to the equivalent interest per payment period:
$$ i_2 = (1 + i_1)^{C/Y \div P/Y} - 1 $$ Use $i$ or $i_2$ as the periodic rate in all subsequent calculations.

2. Loan Payment Formula ($PMT$):
The formula for the fixed payment on a fully amortizing loan is:
$$ PMT = \frac{PV \cdot i}{1 - (1 + i)^{-N}} $$

($PV$ = loan principal, $i$ = periodic rate per payment period, $N$ = total number of payments)

3. Remaining Loan Balance after $n$ Payments:
The balance after $n$ payments is: $$\text{Balance}_n = FV_{\text{Loan Principal}} - FV_{\text{Payments Made}}$$ Where:

$FV_{\text{Loan Principal}} = PV \cdot (1 + i)^n$
$FV_{\text{Payments Made}} = PMT \cdot \frac{(1 + i)^n - 1}{i}$

Plugging these in: $$\text{Balance}_n = PV \cdot (1 + i)^n - PMT \cdot \frac{(1 + i)^n - 1}{i}$$

4. Interest and Principal Portion of Each Payment:

Interest portion for payment $n$: $$ \text{Interest}_n = \text{Balance}_{n-1} \times i $$
Principal portion for payment $n$: $$ \text{Principal}_n = PMT - \text{Interest}_n $$

BGN (annuity-due) note. The formula above is the END convention (ordinary annuity). In BGN mode the first payment lands at $t=0$, so $\text{Interest}_1 = 0$ and $\text{Principal}_1 = PMT$. For $n \geq 2$, $\text{Interest}_n = \text{Balance}_{n-1} \times i$ where $\text{Balance}_{n-1}$ is the balance just after payment $n-1$ (matches the TI BA II Plus and numpy_financial.ipmt convention).

5. Totals Over a Payment Range ($P1$ to $P2$):

$$ \text{Total Interest} = \sum_{k=P1}^{P2} \text{Interest}_k $$
$$ \text{Total Principal} = \sum_{k=P1}^{P2} \text{Principal}_k $$

6. Payment Timing (BGN/END):
If payments are made at the beginning of each period (BGN), the payment amount is adjusted by multiplying by $(1 + i)$.