As discussed in my previous post that the formula for calculating EMI is
E = P×r×(1 + r)^{n}/((1 + r)^{n} – 1)
Now let us try to derive the formula and use an example to create a sample calculation for a loan.
Suppose you take on a loan for P Rupees, the tenure of the loan is n months (for example, n=240 for a 20-year loan), the monthly rate of interest is r (usually calculated by dividing the annual rate of interest quoted by the bank by 12, the number of months in a year, and dividing that by 100 as the rate is usually quoted as a percentage) and ERupees is the EMI you have to pay every month. Let us use P_{i} to denote the amount you still owe to the bank at the end of the i-th month. At the very beginning of the tenure, i=0 and P_{0}=P, the principal amount you took on as a loan.
At the end of the first month, you owe the bank the original amount P, the interest accrued at the end of the month r×P and you pay back E. In other words:
or to rewrite it slightly differently:
Similarly, at the end of the second month the amount you still owe to the bank is:
or substituting the value of P_{1} we calculated earlier:
and once again expanding it and rewriting it slightly differently:
where “x^{y}” denotes “x raised to the power y” or “x multiplied by itself y times”. To make this look slightly simpler, we substitute “(1 + r)” by “t” and now it looks like this:
Continuing in this fashion and calculating P_{3}, P_{4}, etc. we quickly see that P_{i} is given by:
At the end of n months (that is, at the end of the tenure of the loan), the total amount you owe to the bank should have become zero. In other words, P_{n}=0. This implies that:
which means that:
We can simplify this further by noticing that we have a geometric series of n terms here with a common ratio of t and a scale factor of 1. The sum of such a series is given by “(t^{n} – 1)/(t – 1)”, which we substitute in the above equation to yield:
which can be rewritten as:
which can again be rewritten by substituting the value of t back as “(1 + r)” as:
and this is the formula for calculating your EMI. This formula can also be rendered more clearly as:
EMI Variation:
Month no. |
Outstanding amount |
Interest paid this month |
Principal paid this month |
EMI Payment for this month |
1 |
100,000 |
1,000 |
7,885 |
8,885 |
2 |
92,115 |
921 |
7,964 |
8,885 |
3 |
84,151 |
842 |
8,043 |
8,885 |
4 |
76,108 |
761 |
8,124 |
8,885 |
5 |
67,984 |
680 |
8,205 |
8,885 |
6 |
59,779 |
598 |
8,287 |
8,885 |
7 |
51,492 |
515 |
8,370 |
8,885 |
8 |
43,122 |
431 |
8,454 |
8,885 |
9 |
34,668 |
347 |
8,538 |
8,885 |
10 |
26,130 |
261 |
8,624 |
8,885 |
11 |
17,507 |
175 |
8,710 |
8,885 |
12 |
8,797 |
88 |
8,797 |
8,885 |
Raphael Lepisto
April 8, 2012 at 11:57 am
Really cool article, highly informative and professionally written..Good Job