EMI

A few years ago, if you would have asked me what EMI meant, I would have said: “Electromagnetic Induction”. No, this post isn’t about thaaaat EMI, it’s about Equated Monthly Installments.

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The other day, my sidey, Dr. Sambit Patra, (Doc became a Doctor is another big story 🙂 ) asked me how this particular website he was checking out calculated EMI. I tried to figure out the formula on my own, by starting from how the EMI payment scheme works. But I kept on hitting a dead-end.

Of course, my wounded pride wouldn’t let me google the answer out, but I was wasting precious time over nothing. So I googled. And among the first 100 links, all I found was about 1000 comments on 100 forums, all asking the same question, and no coherent answers.

The internet is more than ready for a proper mathematical explanation of EMI, and I have been looking for a good excuse to use WP’s $\LaTeX$ feature. So here goes.

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What are equated monthly installments (EMIs)?

In the normal course of things, whenever we take a loan, we borrow a principal ($P$) for a period of time ($n$ years) and we pay interest (annualised rate of interest, $i$) on the principal. The important thing to note is that: both these amounts, principal and interest, are payable only at the end of the $n$ years.

But paying back loans lump-sum has two problems:

1. It’s difficult for the loan-taker to pay-up so much money in one go.
2. The risk that that loan-giver will be taking, by waiting for the cash to come only at the end of the term, is a lot more than he is willing to take.

So we convert this one-time stream of cash into an equivalent stream of regular (which solves the problem of the loan-giver) and smaller (which solves the problem of the loan-taker) cash payments. The regularity could be as frequent (yearly, monthly, weekly) as the two parties want it to be; the more regular the payments, the smaller each payment will be.

If all the payments are equal, and the frequency of payment is monthly, then this regular flow of payments are called equated monthly installments.

How do I calculate the value of each EMI?

Now comes the tricky part. Suppose the value of each EMI is $x$.

Now when you pay up the first installment in the first month, what happens is that you are now paid interest on that installment at the rate of interest $i$, by the bank. It’s simple to understand why. You are paying the bank the money you owe it, before you have to pay them up. So essentially, you are giving them a loan of $x$ for the remainder period of the loan.

But this doesn’t mean that the bank will pay you interest now. What it will do is set it off against what you have to pay up at the end of the term.

So, if we apply the compound interest rule, $Amount = Principal(1 + i)^{n}$, where in this case, the principal is the first installment that you have paid, $x$ and the rate of interest remains the same, while the time is the remaining time on the loan, which is in months, $12n - 1$, the total amount at the end of the term will become:

$x(1+\frac{i}{12})^{(12n-1)}$

For each subsequent payment, the term will reduce by one month, till the last payment which will happen on the day, when the original lump-sum would have to be paid. So all the installments at the end, will add up to the following:

$x(1+\frac{i}{12})^{(12n-1)} + x(1+\frac{i}{12})^{(12n-2)} + \ldots +$
$x(1+\frac{i}{12})^2 + x(1+\frac{i}{12}) + x$

Now, if we use the summation formula of the finite geometric progression, where the starting term is $x$ and the multiplier is $(1+\frac{i}{12})$, then we get the following sum:

$\frac{x((1 + \frac{i}{12})^{12n} - 1)}{\frac{i}{12}}$

Now, if this amount sets off the loan exactly at the payable date, then the following equation must hold true:

$P(1+i)^n = \frac{12x((1+\frac{i}{12})^{12n}-1)}{i}$

After some nifty shifting around, we get the amount to be paid monthly:

$x = \frac{Pi(1+i)^n}{12((1+i)^n - 1)}$

(I have replaced $(1 + \frac{i}{12})^{12n}$ with $(1+i)^n$, and vice versa. This can be used, but it is strictly not correct, since I should have found out an $i^\prime$, such that $(1 + \frac{i^\prime}{12})^{12n} = (1 + i)^n$, and used $i^\prime$ instead.)

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So there you have it. Of course, you don’t need to do this on your own nowadays. Most statistical software have inbuilt functions (Excel has the function PMT()), and there are online calculators almost on every financial website. A good collection of financial calculators are available on calculator.com.

Disclaimer: Do check the final figures with your bank/loan institution. I am not to be blamed if you get it wrong 🙂

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2 responses to “EMI”

1. Avi

Sonny ,read ur post bfore u put this shit >>not parse..>>>jus wondering…if the rate of interest the bank sets aside on ur installment be same/different from at wat it is lendin u…>>> for banks borrowin/lendin rates are different(dats the whole idea behind a bank)..not sure though abt monthly installment stuff..>>

ps:nice post..nobody was anyways using ur formulas…

2. Not parse? What do you mean? It’s working for me…