Definitions

# Amortization calculator

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.

## 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:

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

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, $n = 30 text\left\{ years\right\} times 12 text\left\{ months/year\right\} = 360text\left\{ months\right\}$

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 $p\left(t\right)$ 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 $r = 1 + i$.

$;p\left(0\right) = P$

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

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

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

We can generalize this to

$;p\left(t\right) = P r^t - A sum_\left\{k=0\right\}^\left\{t-1\right\} r^k$

Applying the substitution (see geometric progressions)

$;sum_\left\{k=0\right\}^\left\{t-1\right\} r^k = 1 + r + r^2 + ... + r^\left\{t-1\right\} = frac\left\{r^t-1\right\}\left\{r-1\right\}$

We end up with

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

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

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

Solving for A, we get

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

## 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 ($;i_\left\{text\left\{annual\right\}\right\}$) 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:

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

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 $;i_\left\{text\left\{annual\right\}\right\}$ 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.