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An amortization calculator is used to determine the periodic payment amount due on a loan (typically a mortgage), based on the amortization process.

The amortization repayment model factors varying amounts of both interest and principal into every installment, though the total amount of each payment is the same.

An amortization calculator can also reveal the exact dollar amount that goes towards interest and the exact dollar amount that goes towards principal out of each individual payment. The amortization schedule is a table delineating these figures across the duration of the loan in chronological order.

While normally used to solve for A, (the payment, given the terms) it can be used to solve for any single variable in the equation provided that all other variables are known. One can rearrange the formula to solve for any one term, except for i, for which one can use a root-finding algorithm.

The formula is:

$A\; =\; Pfrac\{i(1\; +\; i)^n\}\{(1\; +\; i)^n\; -\; 1\}\; =\; frac\{P\; *\; i\}\{1\; -\; (1\; +\; i)^\{-n\}\}$

Where:

- A = periodic payment amount
- P = amount of principal, net of initial payments, meaning "subtract any down-payments"
- i = periodic interest rate
- n = total number of payments

- For a 30-year loan with monthly payments, $n\; =\; 30\; text\{\; years\}\; times\; 12\; text\{\; months/year\}\; =\; 360text\{\; months\}$

Note that the interest rate is commonly referred to as an annual percent (e.g. 8% APR), but in the above formula, since the payments are monthly, the rate $i$ must be in terms of a monthly percent. Converting an annual interest rate (that is to say, annual percentage yield or APY) to the monthly rate is not as simple as dividing by 12, see the formula and discussion in APR. However if the rate is stated in terms of "APR" and not "annual interest rate", then dividing by 12 is an appropriate means of determining the monthly interest rate.

- $;p(0)\; =\; P$

- $;p(1)\; =\; p(0)\; r\; -\; A\; =\; P\; r\; -\; A$

- $;p(2)\; =\; p(1)\; r\; -\; A\; =\; P\; r^2\; -\; A\; r\; -\; A$

- $;p(3)\; =\; p(2)\; r\; -\; A\; =\; P\; r^3\; -\; A\; r^2\; -\; A\; r\; -\; A$

We can generalize this to

- $;p(t)\; =\; P\; r^t\; -\; A\; sum\_\{k=0\}^\{t-1\}\; r^k$

Applying the substitution (see geometric progressions)

- $;sum\_\{k=0\}^\{t-1\}\; r^k\; =\; 1\; +\; r\; +\; r^2\; +\; ...\; +\; r^\{t-1\}\; =\; frac\{r^t-1\}\{r-1\}$

We end up with

- $;p(t)\; =\; P\; r^t\; -\; A\; frac\{r^t-1\}\{r-1\}$

For $n$ payment periods, we expect the principal amount will be completely paid off at the last payment period, or

- $;p(n)\; =\; P\; r^n\; -\; A\; frac\{r^n-1\}\{r-1\}\; =\; 0$

Solving for A, we get

- $;A\; =\; P\; frac\{r^n\; (r-1)\}\{r^n-1\}\; =\; P\; frac\{i\; (1\; +\; i)^n\}\{(1\; +\; i)^n-1\}$

- $;i\; =\; left(1\; +\; frac\{i\_\{text\{annual\}\}\}\{c\}right)^\{c/p\}\; -\; 1$

where c is the number of compounding periods per year and p is the number of payments made per year. The purpose of this formula is to calculate what the interest rate would have to be at each payment point in order to get the same effective annual rate for compounding at the compounding frequency. You will notice that if c and p are the same, then the formula simplifies to $;i$ being equal to $;i\_\{text\{annual\}\}$ divided by the number of payments per year.

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Last updated on Wednesday September 17, 2008 at 19:32:56 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday September 17, 2008 at 19:32:56 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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