- Amortization calculator
An

**amortization calculator**is used to determine the periodic payment amount due on aloan (typically amortgage ), 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. Theamortization schedule is a table delineating these figures across the duration of the loan in chronological order.**The formula**The calculation used to arrive at the periodic payment amount assumes that the first payment is not due on the first day of the loan, but rather one full payment period into the

loan .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 , net of initial payments, meaning "subtract any down-payments"

* "i" = periodicinterest rate

* "n" = total number of payments:For a 30-year loan with monthly payments, $n\; =\; 30\; ext\{\; years\}\; imes\; 12\; ext\{\; months/year\}\; =\; 360\; ext\{\; months\}$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.

**Derivation of the formula**The formula for the periodic payment amount $A$ is derived as follows. For an amortization schedule, we can define a function $p(t)$ that represents the principal amount remaining at time $t$. We can then derive a formula for this function given an unknown payment amount $A$ and $r\; =\; 1\; +\; i$.

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

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

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

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

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

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

**Compounding**When the compounding period is the same as the payment period (e.g., when interest is compounded monthly and payments are also monthly), then $;i$ can simply be calculated by taking the annual interest rate ($;i\_\{\; ext\{annual$) and dividing it by the number of payments per year. In some situations, however, the compounding period and the payment period are not the same, as in the case where payments are made biweekly or weekly but interest is compounded monthly. In Canada, this situation is quite common for mortgages, where interest compounds semi-annually while payments are usually monthly or biweekly. In these cases, $;i$ can be calculated by the following formula:

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

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 $;i\_\{\; ext\{annual$ divided by the number of payments per year.

**Other uses**While often used for mortgage-related purposes, an amortization calculator can also be used to analyze other debt, including short-term loans,

student loans and credit cards.**ee also***

Amortizing loan

*Amortization schedule

*Amortization (business)

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Amortization**— or amortisation is the process of decreasing, or accounting for, an amount over a period of time. The word comes from Middle English amortisen to kill, alienate in mortmain, from Anglo French amorteser , alteration of amortir , from Vulgar Latin… … Wikipedia**Amortization (business)**— For other uses of Amortization, see the Amortization disambiguation page. Amortization is the distribution of a single lump sum cash flow into many smaller cash flow installments, as determined by an amortization schedule. Unlike other repayment… … Wikipedia**Amortization schedule**— An amortization schedule is a table detailing each periodic payment on a amortizing loan (typically a mortgage), as generated by an amortization calculator. While a portion of every payment is applied towards both the interest and the principal… … Wikipedia**Mortgage calculator**— Mortgage calculators are used to help a current or potential real estate owner determine how much they can afford to borrow on a piece of real estate. Mortgage calculators can also be used to compare the costs, interest rates, payment schedules,… … Wikipedia**Amortizing loan**— In banking and finance, an amortizing loan is a loan where the principal of the loan is paid down over the life of the loan, according to some amortization schedule, typically through equal payments.Similarly, an amortizing bond is a bond that… … Wikipedia**Weighted-Average Life**— The Weighted Average Life (WAL) of an amortizing loan or amortizing bond, also called average life, [ [http://www.pimco.com/LeftNav/BondResources/Glossary/ PIMCO glossary] ] is the weighted average of the times of the principal repayments : it s… … Wikipedia**List of real estate topics**— This aims to be a complete list of the articles on real estate. NOTOC compactTOC # *72 hour clause A *Abstract of title *Acknowledgment *Acre A measure of land *Ad valorem tax *Adjustable rate mortgage (ARM) *Administrator/Administratrix *Adverse … Wikipedia**Index of real estate articles**— Property law Part of t … Wikipedia**Casio CFX-9850GB PLUS**— is an upgraded version of the Casio CFX 9850G graphing calculator.OverviewThe calculator weighs about 190 grams including batteries, and measures about 19.7mm x 83 mm x 176 mm. It is capable of performing a vast array of scientific calculations,… … Wikipedia**Annual percentage rate**— Parts of total cost and effective APR for a 12 month, 5% monthly interest, $100 loan paid off in equally sized monthly payments. The term annual percentage rate (APR), also called nominal APR, and the term effective APR, also called EAR,[1]… … Wikipedia