## Friday, 23 November 2012

### Deriving the formula for EMI and amortization schedule

emi

I am going to derive a simple math formula this time. The EMI (Equated Monthly Installment) disease is spreading. No, I’m not going to talk for or against it. I’m just going to derive the formula used to compute EMI and on what basis one decides the part of EMI that goes for principal and interest.

Let us say that you borrow L INR1 amount as loan at the rate of interest i% per annum for a period of n months. Let’s say we agree to pay every month an equal amount E (called EMI) to clear the loan in n months. Then what should be the E?

Since we are going to work in terms of month, we shall convert the rate of interest per annum to per month. Thus, the rate of interest per month is (i/12)%. This means for every 100 INR of the loan amount the lender charges an extra of i/12 per month. Equivalently, for each 1 INR of your loan L, the lender charges you an extra i/(12 × 100) per month. This means that at the end of the first month you owe the lender an amount which is the sum of the original loan L and the interest imposed in a month, i.e.,

L + L

 i 12 × 100

L

1 +
 i 12 × 100

Lr

where we have set r = 1 + i/(12 × 100) to simplify our notation. You will pay E at the end of first month and hence will owe an amount L1 = LrE. At the end of second month you will owe

L1 + L1

 i 12 × 100

L1

1 +
 i 12 × 100

L1r.

After paying E, you owe

 L2 = L1 r − E = (Lr −E)r − E = Lr2 − E(1+r).

Continuing this argument, we notice that at the end of n-th month, after paying E, you will owe

 Ln = Lr n − E(1+r+r2+…+rn−1).

Let us set S = 1+r+r2+…+rn−1 and get a formula for S in terms of r and n. In fact, S is the sum of the first n terms of a geometric series. Every term in the sum is a r multiple of its predecessor. Note that

rS − S=r + r 2 + r3 + … + rn−1 + rn − (1+r+r2+…+rn−1) =  rn −1
S=
 rn −1 r−1

and hence

Ln = L  rn − ES = L rn − E

 rn −1 r−1

.

If you want to finish your loan at the end of n-th month, you expect Ln =0. This gives that

E
 L rn (r−1) rn −1
=
L

1+
 i 1200

 n

 i 1200

1 +
 i 1200

 n
−1
.

This is the formula for the EMI that you pay for any kind of loan.

Those who have taken loan might have noticed that some part of EMI is deducted from principal and the remaining as interest. A natural question is, on any given month, how much from your EMI is deducted towards principal payment. Let us deduce this formula. Recall that at the end of first month the interest imposed is

L

 i 12 × 100

L (r−1).

This interest imposed, at the end of first month, is deducted from your EMI E and the balance is used as payment towards principal. Therefore the amount that goes as payment towards your principal loan amount is EL(r−1) at the end of first month. Continuing this way, one notices that out of your k-th month EMI E, an amount of rk−1 {EL(r−1)} is deducted towards principal payment and the rest is towards interest. Notice that r>1 and hence your principal payment increases with each month. Also, as derived above, the principal loan you owe after k-th month EMI is rkLE(rk−1/r−1).

1
the unit of currency is not an issue