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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.

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closed as off-topic by jeff m, chrisaycock Aug 13 '13 at 20:14

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This is a site for professionals in quantitative finance. –  chrisaycock Aug 13 '13 at 20:14
    
Although this question is pretty basic and doesn't belong in this site, I still provided you with an answer. Please refer to the answer below. –  Mariam Aug 13 '13 at 20:44

1 Answer 1

up vote 0 down vote accepted

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:

enter image description here

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Thanx @Mariam I am not that much interested in finance and economics, for me this pretty basic thing is too much. –  Anfal Aug 13 '13 at 23:53

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