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EMI Calculations

25 Mar

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 Pi 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 P0=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: 

P1 = P + r×P – E 

or to rewrite it slightly differently:

P1 = P×(1 + r) – E 

Similarly, at the end of the second month the amount you still owe to the bank is:

P2 = P1×(1 + r) – E 

or substituting the value of P1 we calculated earlier:

P2 = (P×(1 + r) – E)×(1 + r) – E 

and once again expanding it and rewriting it slightly differently:

P2 = P×(1 + r)2 – E×((1 + r) + 1) 

where “xy” 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:

P2 = P×t2 – E×(1 + t) 

Continuing in this fashion and calculating P3, P4, etc. we quickly see that Pi is given by:

Pi = P×ti – E×(1 + t + t2 + … + ti-1

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, Pn=0. This implies that:

Pn = P×tn – E×(1 + t + t2 + … + tn-1) = 0 

which means that:

P×tn = E×(1 + t + t2 + … + tn-1

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 “(tn – 1)/(t – 1)”, which we substitute in the above equation to yield:

P×tn = E×(tn – 1)/(t – 1) 

which can be rewritten as:

E = P×tn×(t – 1)/(tn – 1) 

which can again be rewritten by substituting the value of t back as “(1 + r)” as:

E = P×r×(1 + r)n/((1 + r)n – 1) 

and this is the formula for calculating your EMI. This formula can also be rendered more clearly as:

So in the entire formulation of EMI calculations we found out that the Bank calculate the interest earned on the loan every month and add to the total amount own for the next month. This chain keeps on happening until the total amount owe’d to the bank is zero.

EMI Variation:

Since in our derivation we assumed that the bank is calculating the interest on a monthly basis. But the bank can use it on annual basis or the more advanced version of daily basis. The annual version was used by the bank before and now more often the EMI is calculated by using either monthly or daily interests.
Now let us take an example
Here the loan amount is 100000, which is lent at a interest rate of 12% with a loan tenure of 12 months.
The monthly EMI is calculated at the annualized rate of 12% 
P=100000
R=12/12=1%
N=12
E=100000*1/100(1+1/100)^12 divided by ((1+1/100)^12-1)
by using a scientific calculator i found out that 101/100^12=1.126825
By using the above calculations 
E=Rs.8,885 per month
Now let us calculate the interest component and the principal component paid every month 
Interest paid every month = 1% of the principal(rate of interest is 1% per month)
Principal component= EMI- interest component
For eg: for first month:
Interest =1% of the amount due=1%of 100000=1000
EMI=8885
So Principal component repaid=EMI-interest repaid
=8885-1000=7885.
Using the above understanding we now can prepare a chart of the interest compoent and the principal component repaid.

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

Here we observe that for the initial tenure of the loan ,less of the principal is repaid then the interest component and in the later stages less of the interest is repaid and more of the principal component.
So from this observation we can deduce that for a given amount of principal and rate more is the tenure of the loan more is the total interest we are paying for the loan.

Simple Reason Time is money !….

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1 Comment

Posted by on March 25, 2012 in Entertainment

 

One response to “EMI Calculations

  1. Raphael Lepisto

    April 8, 2012 at 11:57 am

    Really cool article, highly informative and professionally written..Good Job

     

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