Amortization calculator
Encyclopedia
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 schedule calculator is often used to adjust the loan amount until the monthly payments will fit comfortably into budget, and can vary the interest rate to see the difference a better rate might make in the kind of home or car one can afford. 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 annuity formula is:
Where:
This formula is valid if i > 0. If i = 0 then simply A = P / n.
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
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.
is derived as follows. For an amortization schedule, we can define a function 
that represents the principal amount remaining at time 
. We can then derive a formula for this function given an unknown payment amount 
and 
.
We can generalize this to
Applying the substitution (see geometric progression
s)
We end up with
For
payment periods, we expect the principal amount will be completely paid off at the last payment period, or
Solving for A, we get
or
After substitution and simplification we get
Loan
A loan is a type of debt. Like all debt instruments, a loan entails the redistribution of financial assets over time, between the lender and the borrower....
(typically a mortgage
Mortgage loan
A mortgage loan is a loan secured by real property through the use of a mortgage note which evidences the existence of the loan and the encumbrance of that realty through the granting of a mortgage which secures the loan...
), based on the amortization
Amortization (business)
In business, amortization refers to spreading payments over multiple periods. The term is used for two separate processes: amortization of loans and amortization of intangible assets.-Amortization of loans:...
process.
The amortization
Amortization (business)
In business, amortization refers to spreading payments over multiple periods. The term is used for two separate processes: amortization of loans and amortization of intangible assets.-Amortization of loans:...
repayment model factors varying amounts of both interest
Interest
Interest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets. It is most commonly the price paid for the use of borrowed money, or money earned by deposited funds....
and principal into every installment, though the total amount of each payment is the same.
An amortization schedule calculator is often used to adjust the loan amount until the monthly payments will fit comfortably into budget, and can vary the interest rate to see the difference a better rate might make in the kind of home or car one can afford. An amortization calculator can also reveal the exact dollar amount that goes towards interest
Interest
Interest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets. It is most commonly the price paid for the use of borrowed money, or money earned by deposited funds....
and the exact dollar amount that goes towards principal out of each individual payment. The amortization schedule
Amortization schedule
An amortization schedule is a table detailing each periodic payment on an amortizing loan , as generated by an amortization calculator. Amortization refers to the process of paying off a debt over time through regular payments...
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 loanLoan
A loan is a type of debt. Like all debt instruments, a loan entails the redistribution of financial assets over time, between the lender and the borrower....
.
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
Root-finding algorithm
A root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f = 0, for a given function f. Such an x is called a root of the function f....
.
The annuity formula is:
Where:
- A = periodic payment amount
- P = amount of principal, netNet (economics)In economics, net means after deductions. A related concept is gross, meaning before deductions.Nett is an alternative spelling used in British English.-Usage:...
of initial payments, meaning "subtract any down-payments" - i = periodic interestInterestInterest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets. It is most commonly the price paid for the use of borrowed money, or money earned by deposited funds....
rate - n = total number of payments
This formula is valid if i > 0. If i = 0 then simply A = P / n.
- For a 30-year loan with monthly payments,
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
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 is derived as follows. For an amortization schedule, we can define a function that represents the principal amount remaining at time . We can then derive a formula for this function given an unknown payment amount and .We can generalize this to
Applying the substitution (see geometric progression
Geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression...
s)
We end up with
For
payment periods, we expect the principal amount will be completely paid off at the last payment period, orSolving for A, we get
or
After substitution and simplification we get