Amortizing Loan Payments and the Interest-Only Distinction
Summary
The document asks how to calculate interest on a loan, then presents the periodic payment formula for a fully amortizing loan. It derives the formula by tracking how the balance grows with interest and falls with each payment, and explains that the balance is zero after the final payment. It also separates each payment into interest and principal components and gives a monthly-payment example using a stated loan amount, rate, and term.
The method applies to a mortgage-style loan with regular payments that repay both principal and interest. It does not calculate a true interest-only loan, where scheduled payments cover interest while principal remains outstanding; the document instead redirects to amortization. Its example assumes a fixed rate and monthly periods, and the answer’s description of periodic rate as annual rate divided by total number of periods should be interpreted carefully: the usual conversion divides by periods per year. The linked amortization table is not included in the supplied text.
Key ideas
- An amortizing payment combines interest and principal repayment.
- The payment formula is derived by setting the balance after the final period to zero.
- The interest portion of a period’s payment is calculated from the outstanding balance and periodic rate.
- A true interest-only loan does not repay principal through its scheduled interest payments.
Tags
Full text
# Interest only loan formula
# Interest only loan formula
Forgive me I am not good at economics, I have following values.
- amount I want to lend.
- the current bank interest rate.
- the amount of years the loan will be for
Please tell me the formula to calculate how much interest only amount that I will have to pay back over the life of the loan.
## Answer by Mayou (score 0, accepted)
https://quant.stackexchange.com/a/8741
If this is a mortgage-type loan, the most "straightforward" way of knowing the total interest amount is to use an amortization table. The table lists the periodic payment, as well as the interest portion and principal repayment portion, for each payment period.
Here is the derivation of the formula for the period payment (interest + principal repayment). Let n be the total number of periods (frequency * maturity), R be the annualized interest rate, r be the periodic interest rate, X be the periodic payment and P be the amount of the loan. Let P(n) be the remaining face value at time t = 1,2,....,n. We have:
$$P(0) = P\\ P(1) = P(0)(1+r) - X = P(1+r) - X\\ P(2) = P(1)(1+r) - X = P(1+r)^2 - X(1+r) - X\\ P(3) = P(2)(1+r) - X = P(1+r)^3 - X(1+r)^2 - X(1+r) - X\\ P(n) = P(1+r)^n - X\displaystyle\sum\limits_{i=0}^{n-1} (1+r)^i = P(1+r)^n - X\displaystyle\frac{r^n-1}{r-1} $$
Since the remaining balance would be 0 by the end of time n (you would have paid off your loan), $P(n) = 0$
Therefore, we can compute the periodic payment X as:
$$ X = P \displaystyle\frac{r(1+r)^n}{(1+r)^n - 1} $$
Note that r is the periodic interest rate, i.e. $ r = \displaystyle\frac{R}{n} $. The reason why P is timed by $(1+r)^n$ is that the amount borrowed is subject to time value of money.
Let's now take an example. Suppose you are borrowing an amount of $1,000 at 6% for 1 year, with monthly payments. We have:
$$n = 1 * 12 = 12$$ $$P = 1,000$$ $$R = 0.06$$ $$r = \displaystyle\frac{R}{n} = \displaystyle\frac{0.06}{12} = 0.005$$
The periodic payment X is: $$X = 1,000 * \displaystyle\frac{0.005*(1+0.005)^{12}}{(1+0.005)^{12}-1} = 86.07 $$
The interest payment portion of X at n = 1 is: $Interest = P * r = 1,000 * 0.005 = 5$ The principal repayment portion of X is: $X - Interest = 86.07 - 5 = 81.07$
For complete calculations, here is the amortization table:Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.