amortization calculator

TheInfoList

OR:

An amortization calculator is used to determine the periodic payment amount due on a
loan In finance, a loan is the lending of money by one or more individuals, organizations, or other entities to other individuals, organizations, etc. The recipient (i.e., the borrower) incurs a debt and is usually liable to pay interest on that de ...
(typically a
mortgage A mortgage loan or simply mortgage (), in civil law (legal system), civil law jurisdicions known also as a hypothec loan, is a loan used either by purchasers of real property to raise funds to buy real estate, or by existing property owners ...
), based on the amortization process. The amortization repayment model factors varying amounts of both
interest In finance and economics, interest is payment from a debtor, borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is ...
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 An amortization schedule is a table detailing each periodic payment on an amortizing loan (typically a mortgage A mortgage loan or simply mortgage (), in civil law (legal system), civil law jurisdicions known also as a hypothec loan, is a ...
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 In mathematics and computing, a root-finding algorithm is an algorithm for finding Zero of a function, zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to ...
. The annuity formula is: $A = P\frac = Pi \times \frac \times \frac = \frac$ Or, equivalently:
$A = P\frac = Pi \times \frac = Pi \times \frac = Pi \times \left(\frac + \frac\right) = P\left\left(i + \frac \right\right)$ Where: * ''A'' = periodic payment amount * ''P'' = amount of principal,
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
of initial payments, meaning "subtract any down-payments" * ''i'' = periodic interest 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, $n = 30 \text \times 12 \text = 360\text$ Note that the interest rate is commonly referred to as an annual percentage rate (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 $r = 1 + i$. :$\;p\left(0\right) = P$ :$\;p\left(1\right) = p\left(0\right) r - A = P r - A$ :$\;p\left(2\right) = p\left(1\right) r - A = P r^2 - A r - A$ :$\;p\left(3\right) = p\left(2\right) r - A = P r^3 - A r^2 - A r - A$ This may be generalized to :$\;p\left(t\right) = P r^t - A \sum_^ r^k$ Applying the substitution (see
geometric progression In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero 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 ex ...
s) :$\;\sum_^ r^k = 1 + r + r^2 + ... + r^ = \frac$ This results in :$\;p\left(t\right) = P r^t - A \frac$ For $n$ payment periods, we expect the principal amount will be completely paid off at the last payment period, or :$\;p\left(n\right) = P r^n - A \frac = 0$ Solving for A, we get :$\; A = P \frac = P \frac = P \frac$

# 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 A student loan is a type of loan designed to help students pay for Higher education, post-secondary education and the associated fees, such as Tuition payments, tuition, books and supplies, and living expenses. It may differ from other types of l ...
and credit cards.