When one takes a loan, a natural question that comes to mind is how much would the EMI (Equated Monthly Installment) be that one has to pay back to the bank every month. Finding out the EMI for different tenures of the home loan allows one to select the best tenure based on one's current and projected income and expenses and possibly other factors.
What I find is that most software engineers would rather construct or borrow elaborate spreadsheets for this purpose or use a webbased EMI calculator rather than calculating it themselves. Some brave souls ask around for the formula to calculate EMIs and then try to write a programme that calculates the EMIs for them. I find it particularly appalling that very few bother to figure it out for themselves, especially since it involves elementary algebra that most of us have surely learnt in high school. It either reflects the creeping sloth and sloppiness in our generation or the rote learning and regurgitating of formulae that our system of education seems to encourage. In any case, I attempt to show in this post how simple it really is to figure out how to calculate EMIs on your own and to hopefully encourage my fellow engineers into applying the basic mathematics they learn in school to reallife problems themselves instead of looking around to see if someone else has solved it for them.
For the sake of simplicity, assume that the loan is offered on a "monthly rest" basis. That is, the bank calculates the interest at the end of every month on the amount you still owe to the bank at the beginning of the month, adds it to the amount you already owe and then deducts your EMI from this to calculate the total amount you still owe to the bank at the beginning of the next month. Some banks offer loans on a "daily rest" basis, where the outstanding amount and the interest is recalculated every day, but you still pay back on a monthly basis. The older "annual rest" basis is no longer in use as far as I can tell. Note that it is easy to adapt the formula given here to the "daily rest" basis and that is, of course, left as an exercise for the reader.
Suppose you take on a loan for P Rupees, the tenure of the loan is n months (for example, n=240 for a 20year 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 E Rupees 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 ith 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:
P_{1} = P + r×P  E
or to rewrite it slightly differently:
P_{1} = P×(1 + r)  E
Similarly, at the end of the second month the amount you still owe to the bank is:
P_{2} = P_{1}×(1 + r)  E
or substituting the value of P_{1} we calculated earlier:
P_{2} = (P×(1 + r)  E)×(1 + r)  E
and once again expanding it and rewriting it slightly differently:
P_{2} = P×(1 + r)^{2}  E×((1 + r) + 1)
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:
P_{2} = P×t^{2}  E×(1 + t)
Continuing in this fashion and calculating P_{3}, P_{4}, etc. we quickly see that P_{i} is given by:
P_{i} = P×t^{i}  E×(1 + t + t^{2} + ... + t^{i1})
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:
P_{n} = P×t^{n}  E×(1 + t + t^{2} + ... + t^{n1}) = 0
which means that:
P×t^{n} = E×(1 + t + t^{2} + ... + t^{n1})
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:
P×t^{n} = E×(t^{n}  1)/(t  1)
which can be rewritten as:
E = P×t^{n}×(t  1)/(t^{n}  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:
Suppose you take a loan from a bank of 10,00,000 Rupees for 15 years at 8.5% annual rate of interest calculated on a monthly rest basis. In that case, P = 10,00,000, n = 15×12 = 180 and r = (8.5/12)/100 = 0.0070833333. Putting these values into the formula given above gives us E = 9847.40 (approximately).
When you write a programme to calculate your EMI using the formula given above, be careful to structure your computations to accommodate loss of precision and rounding errors. For more information, read "What Every Computer Scientist Should Know About FloatingPoint Arithmetic" (PDF) by David Goldberg or section 4.2, "Floating Point Arithmetic", in "The Art of Computer Programming, Volume 2: Seminumerical Algorithms" (3rd Edition) by Donald Knuth.
Update (20061219): Clarified how "r" is calculated from the quoted interest rate and added an example to illustrate the application of the formula. Added references to Goldberg's article and Knuth's book for the caveats to bear in mind while using floatingpoint arithmetic.
Update (20070201): The only intent of this post was to show how basic mathematics can be used to calculate EMIs. I am just an ordinary computer programmer and not a financial consultant. Please stop asking me for advice on your loan plans!

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Hey, I was looking exactly for this explanation to verify my auto loan. Thanks!
ReplyDeleteHi Mathew,
ReplyDeleteI used your EMI Formula, but coudnt come into good results,
Principal=650000
Interest=8.5%
Tenure=120 months(10 years)
Iam getting EMI as 5525000
but it is wrong , can you help me in calculating correct EMI
Thanks in advance,
my email id is smvikram[AT]rediffmail.com
I'm assuming that the interest rate you have given is 8.5% per annum.
ReplyDeleteIn that case, the monthly interest rate can usually be obtained by dividing this by 12. So the value of "r" for you is "(8.5 / 12) / 100" or about 0.0070833333. "P" is 650000 and "n" is 120. Putting these values into the given equation, your EMI should be about Rs 8,059/.
hi mathew,
ReplyDeletei just dont understand how a bank calculates the daily rest thing when actually u pay ur emi monthly i.e. ur principal is going to change on monthly basis only so how come the bank perfoems a calculation on daily basis??
See "Five minutes to understanding reducing balance". Usually, daily rest is slightly cheaper than monthly rest and is really useful while prepaying a part or the whole of your loan.
ReplyDeleteSuppose you take on a loan for 10,00,000 rupees for 10 years at 8.5% per annum.
If your balance is calculated on a monthly rest basis, "P" is 1000000, "n" is (10 years × 12 months/year) or 120 months and "r" is "(8.5 / 100) / 12"  the EMI then comes to about 12,398.57 rupees.
If your balance is calculated on a daily rest basis, "P" is still 1000000, "n" is (10 years × 365 days/year) or 3650 days and "r" is "(8.5 / 100) / 365"  the EMI (amount payable per day × 30 days/month) comes to about 12,202.24 rupees.
So you save about 196 rupees per month with a daily rest scheme and it is also fairer when you prepay in the middle of a month.
hi Mathew,
ReplyDeletei am getting exact amount as per my bank said by using your formula..thanks..
i want to know how to use the prepayment option wisely to lower my interest amount...
from arun
Arun: I'm just an ordinary software engineer  you'd be much better off talking to an appropriately qualified financial advisor for such stuff.
ReplyDeleteVery good explanation...I am back to my college days.
ReplyDeleteThanks matthew
ReplyDeleteaugustine
Hi Mathew,
ReplyDeleteGreat note on the EMIs. Is there a difference in calculating a home loan and an Auto loan? I tried an example on some EMI calculators online at sites like http://www.indiaautomobile.com/index.php?option=site&page=emi_calc
and http://www.driveinside.com/Research/EMICalculator.aspx.
The example I worked on was P = 1,80,000. R = 16% n = 36. The amoutn on the sites came to 6332. and the formula came to 6384. And for higher principal amounts, the difference was huge. So was wondering. Thanks.
hi
ReplyDeletemy loan is 1,00,000
rate of int is 12%
period = 5 years
compounded qurterly
but emi monthly
your formula is not fitting
thanks in advance
Anonymous (20061026): The first site that you link to as well as my formula both give the value 6328 for the values you have specified. Read the formula once again carefully and make sure you put in the value for the rate (0.16/12) and the period (36) correctly.
ReplyDeleteAnonymous (20061205): You haven't said what you expect the value to be. In any case, the different period for compounding and repayment would necessitate a bit tweaking as you can expect.
Great Job Mathew.
ReplyDeleteKeep posting more interesting and useful data like this.
Back to basics, very useful article. One clarification needs to be made wrt the rate of interest.
ReplyDeleteIf r is the rate of interest in percentage, then r in the formula is actually r/100.
Eg: if r=10.5%, then in the formula, r=0.105
Anonymous: Thanks, I've updated the post to clarify how "r" is calculated from the interest rate quoted by the bank.
ReplyDeleteHow to convert Diminishing Rate of Interest to Fixed Rate of Interest? I want to know the Fixed Rate of Interest equivalent to Diminishing Rate of Interest.
ReplyDeleteSuppose i want to take a home loan of 22,50,000 for 20 years.I have an option for downpayment of the entire payment at the begining of construction @9.5% p.a. OR follow a construction linked plan wherin i make the payments as the construction proceeds.The estimated time for completion of the building is 24 months.
ReplyDeletePlease advise which mode of payment (i.e whether a downpayment or a construction linked plan) works out to be more economical?
Also advise on the amount that would be payable on (monthly and at the end of the tenure )should either of the plans be used?
One problem with your post is that it still requires you to know the formula for a geometric series by rote. You could, of course, have derived it as so:
ReplyDelete(I'm going to use E(x, y)(f(i)) as the summation from i=x to y of f(i), that is for (i=x; i<=y; i++) sum += f(i)))
E(0, n)(t^i) = x
E(1, n+1)(t^i) = tx
E(0, n+1)(t^i) = tx + 1
E(0, n)(t^i) + t^(n+1) = tx + 1
x + t^(n + 1) = tx + 1
t^(n  1)  1 = tx  x
(t^(n  1)  1) / (t  1) = x
I'm sure you knew that, but I'm adding it here for the benefit of others.
Crap, I can't edit and I can't type.
ReplyDeletet^(n  1)  1 = tx  x
(t^(n  1)  1) / (t  1) = x
Obviously, those should be:
t^(n + 1)  1 = tx  x
(t^(n + 1)  1) / (t  1) = x
Could you please calculate for me the EMI for Rs 80,000/ loan taken @ 12% PA taken for 36 months?
ReplyDeleteI would be highly pleased if monthly EMI is shown for all the 36 months and how much more do i end up paying after 36 Months?
Thanks in Advance....
Jatin Shah
Jatin: The point of this post was to show you the mathematics behind deriving the formula for calculating EMIs. Once you know these, you should be able to calculate your EMI yourself.
ReplyDeleteHello Ranjit,
ReplyDeleteI found your blog, while I was searching for an EMI breakup calculator, and I was kind of impressed. Being an engineer, I felt a bit guilty for being like everyone else and hoping google would help me! Being more skilled in assembly and C (rather than C++/Java,etc), I started of in C; then wasnt impressed in the userfriendliness and did the same in Excel, and voila, I am pleased. Thanks for your post! It was truly inspirational:).
whoever, you are , your formula is great !!. Thanks !!
ReplyDeleteHi Mathew,
ReplyDeleteI would like to get the formulae to calculate the principal/interest breakup for each month for a monthly reducing interest. When i consulted a bank call centre, they were unable to give me the formulae, but instead sent me a mail showing a split up chart for the amount i asked for.
It is something like this:

Loan Amt: 1,00,000
Rate of interest: 14.5% reducing
Tenure: 60 months
EMI: 2,353

Month EMI Prpl Interest
0   
1 2353 1145 1208
2 2353 1158 1195
3 2353 1172 1181
..
......
58 2353 2270 83
59 2353 2297 56
60 2353 2325 28
Can you help?
asim.ma(at)gmail.com
Hi. I was able to get the EMI for a Principal and a tenure based on a calculator. However, I am keen to find out the Interest rate. I know that manipulating the formula for EMI that you gave should give me the rate of interest, but the (1 + R) to the power of N has me stumped. Thanks for any help.
ReplyDeleteHi,
ReplyDeleteI was googling for this formula and landed in your page. Was really useful.
Thanks!
Thank you very much the formula and most important of all  ITS DERIVATION........
ReplyDeletevery nice one...I think this is the only site where I saw someone taking pains to explain all the algebra behind that formulae...Appreciate all your efforts in doing that
ReplyDeleteOne of my freind have taken loan from a bank as commercial vehicle loan. The loan amount is 1050000/, tenure is 4 years, emi 26250/, total interet payable 157500/, total loan payable 1207500/, total emi payable 46. As per my calculation, 4 years (48 months) emi come to a total of rs.1260000/. The date of sanction is 30/09 and the first emi starts from 01/12. Please tell me the method as how they have calculated the above figures. my email id is vyapaar@gmail.com
ReplyDeleteHi Matthew,
ReplyDeleteAppreciate your efforts very much.
I was in the process of looking for formula and calculating by myself but your mail was a boon and reduced my work a lot.
Madan Mohan: You don't mention the rate of interest and whether it is a daily/monthly/annual rest loan. In any case, please read the article again and apply the formula given there. It should give you the answer.
ReplyDeleteVinay M: You're welcome.
Anonymous: Thanks.
Mathew,
ReplyDeletespledid description of EMI. You rock man. Keep going.
Love the explaination. Keep going dude you rock
ReplyDeletehey matthew,
ReplyDeleteI need to know the formula to calculate break up of principal and interest if i take a loan of say Rs 850000 but disbursed in parts @ 10% daily rests for a period of 10 yrs.
i have taken the loan in different amounts totalling to 850000 in five months time. Kindly help me
Mail me at frankremedios@gmail.com
Hi Mathew,
ReplyDeleteI'm Suzi from Malaysia. Had come accross ur blog while I'm searchin for daily rest mortgage calculator, but to no avail.I have applied for loan amounting $170000 with 5.99% per annum daily rest. Tenure is for 30 years. I know that I can save more on interest if I pay twice a month or more. Can u help me to come out with the formula on below item:
1) Monthly repayment that I should commit = R
2) amortisation schedule indicating both interest and principal value = I & P
3) How many years I can save if I pay more than I supposed to pay, i.e I pay $1000/mth instead of $900.
4) How much I can save if I pay $1000/month but 2 times (500 at one time)
Thanks in advance
Regrds,
Suzi
Hi Mathew,
ReplyDeleteI am from Brunei and stumbled on your blog. Can u work out the EMI if say the 1st year interest is 5.5%, the second 6.5% and the third to the 12th year is 7% on B$150000. 144 monthly payments. Thanks.
Nash
hi mathew.
ReplyDeleteyour formula is great but can you tell me how to calculate as how much principal amount and how much is the interest amount are we paying in each emi..
thanks
mohit
Mohit: Look at the formula for calculating "P[i]", which is the outstanding principal at the end of "i" months. For a given month "m", substitute "i = (m  1)" and "i = m" into this formula to obtain "P[m1]" and "P[m]"  the difference in these amounts is the bit of the principal that you paid off in month "m".
ReplyDeleteIf you substitute these values for "i" in the expanded formula and perform the subtraction on the expansions, you obtain "dP[m] = (P*r  E)*((1 + r) ^ (m  1))", where "dP[m]" is the amount of the principal repaid in month "m" and "x ^ y" denotes "x raised to the power y".
hi there!
ReplyDeletethis formula helps a lot.. tnx
but is there anyway i could
find P given E r and n?
or find r given P E and n
or find n given P E and r
thanks in advance!!!
Anonymous: Yes, of course you can find each of these quantities given the other three  after all, it is an equation involving four unknowns where you are providing the values for three.
ReplyDeleteFinding P given E, r and n is straightforward.
Given P, E and r, you can reduce the equation to the form "(1+r)^n = <something>" and you can then take the logarithms to some base on both sides to get the value of n.
Given P, E and n, it looks a bit difficult to calculate r (at least I don't remember a straightforward algebraic method to achieve this). You can get the answer numerically using Newton's method though.
Hi Mathew,
ReplyDeleteI am not able to calculate Interest rate , if all other factors are known. This is due to (1+r)^n. Can you give the suggestion for this.
Thanks,
Mohan
Hi Mathew,
ReplyDeletefirst,
Let me thank you for such a nice derivations. it is working perfectly.
Second,
Mathew, can you help me to find r when i have emi, p and n. it is needed because i want to calculate effective rate not a theoritical rate on the amt i actually received after processing and other charges.
:)
chukked@yahoo.com
Anonymous/Mohan/Chukked: I have posted the detailed answer for computing the interest rate given the other variables as a separate article titled "Calculating Interest Rates".
ReplyDeleteI need to know the formula to calculate break up of principal and interest if i take a loan of say Rs 850000 but disbursed in parts @ 10% daily rests for a period of 10 yrs. THE EMI IS MONTHLY @11233 BUT I PAY 11250
ReplyDeletei have taken the loan in different amounts totalling to 850000 in five months time. Kindly help me WITH the break up formula
Hi Mathew,
ReplyDeleteI came across your blog just two hours before my finance exam....was of immense help to me....thanks a lot for this wonderful job, and keep posting such nice stuff.
Calculation for other parameters:
ReplyDeleteE = Pk(1+k)^y / ((1+k)^y  1)
Calculating P is staight forward.
P = E((1+k)^y  1) / k(1+k)^y
Calculating y
E((1+k)^y  1) = Pk(1+k)^y
E(1+k)^y  E = Pk(1+k)^y
E(1+k)^y  Pk(1+k)^y = E
(1+k)^y ( E  Pk) = E
(1+k)^y = E/(EPk)
y log(1+k) = log(E/(EPk))
y = log(E/(EPk)) / log(1+k)
Calculating k
Still workin on. If someone solves do let us know.
I also published the same in my site
http://sudokusplashzone.com/mortgage/
Currently I am using newtons method to find rate in my program.
Thanks
Suriya.
Hi Mathew,
ReplyDeleteCould you let me know how to use this EMI formula on an excel sheet. I have varying interest rates for my home loan for the different years and it becomes constant after the 6th or 7th year
An immediate response will be highly appreciated
Thanks
Anonymous: I have almost no experience with Excel, so I can't help you here. Sorry. However, if you search, you can easily find this information elsewhere. For example, this link seems to have what you're looking for.
ReplyDeleteMathew, I took a housing of 22.6lakhs for 15 years at 11.25% interest. As I could calculate, EMI was Rs.26043/
ReplyDeleteBut when I got the breakup of EMI(principal+interest) for 1st 6 months, it was like this: 18381+7662, 4625+21418, 5358+20685, 4720+21323, 4765+21278, 6781+19262.
Can you please me figure out how this breakup was calculated? I took loan on 20th of a month.
Mathew,
ReplyDeleteI have made a multifeatured EMI calculator and have put it on my homepage:
http://manu.nitt.googlepages.com/EMI_calculator.html
In your freetime, use it and let me know your comments.
Thank you.
 Manu
Manu: That is a nice and useful page. Thank you.
ReplyDeleteMathew, I used the EMI calculator at manu.nitt.googlepages.com/EMI_calculator.html (above link).
ReplyDeleteI was able to calculate my EMI, total interest correctly.

For the breakups of my loan, I see that interest for 1st installment was calculated for 10 days ( as I took loan on 20th) and subsequently for every month.
Do you know any other calculator which can handle cases as mine?
Mathew, if I can take the liberty of answering the above question:
ReplyDelete@Gautam
Man it took me quite sometime to figure out you took it on a september 20th.
(you should have mentioned that first)
See, many banks start their installments on 1st day of the month next to date of loan issue.
For your case, outstanding P may be calculated as follow:
P[0]=2260000 ...given
R=0.1125*11/365 > interest calculated for 11 days (20th sep  30th sep)
EMI=26043
P[1]=P[0]*(1+R)EMI
P[1]=2260000*(1+.00339)26043
P[1]=2241619.34
For 2nd installment,
R=0.1125*31/365 (31 days in October)
P[2]=P[1]*(1+R)EMI
P[2]=2241619.34*(1+.00955479)EMI
P[2]= 2236994.56
and so on,
So, outstanding principal amounts for your loan will be:
2241619.34, 2236994.56, 2231636.12, 2226915.95, 2222150.69, 2215970.07, 2211100.23, 2205502.34 and so on.
That is, principal part of EMI will be 18380.66, 4624.78, 5358.45, 4720.16, 4765.26, 6180.61, 4869.85 etc.
The 6th installment will be 6180.61+19862.37. Probably you made a typo error.
And...
Now that you've given the idea, I've added that feature into my EMI calculator:
http://manu.nitt.googlepages.com/EMI_calculator.html
Let me know your comments.
hi.. that was a great work...!!
ReplyDeleteDo you have an idea of How to calculate the EMI for Diminishing Interest rate??
Hi Mathew,
ReplyDeleteI received loan from Bank Rs 50000 and Rate of Interest 9% for 5 years(60Months).
EmI= Rs 1037
If i am not pay EMI for 24 Months then
Prinical Amount=?
Pending EMI Amount?
Nice explanation.It was very helpful.Thanks
ReplyDeleteits very useful.thanks.
ReplyDeleteHi Mathew,
ReplyDeleteThe formula worked good and it was really useful. Thanks a lot man.
Regards,
Manoj
Mr. Mathew
ReplyDeleteGrt.... U helped me out. Thnx..
Mahendra.
Good work. I was looking exactly for the same. Thanks.
ReplyDeleteHi Mathew. It is very useful. Thank you on behalf of everyone. Santhosh
ReplyDeleteall i was looking for was the formula and i ended up also learning the derivation, thanks a lot mathew...
ReplyDeletePandurang
hi mathew  I would like to know how to calculate the outstanding Principal. For example i have taken a loan of 71287 at rate of 14% per annum( Monthly reducing)for tenure of 44 months. I have completed 18 instalments. Now how to calculate the outstanding principal.
ReplyDeleteAnonymous: You can treat your outstanding principal as a loan that has to be paid back in 44  18 (or 26) months at 14% per annum and with the same EMI as before. Using the EMIcalculation formula here, you should be able to work out the outstanding principal by substitution.
ReplyDeleteGuess too many comments already about how acurate this formula is... Will just say thank you..
ReplyDeleteMegha
Hi,
ReplyDeleteI went to your site and checked the calculator. Need a calrification:
1. When it's Monthly rest the interest and principal looks good.
2. When it's daily rest the interest become negative after guess 120 months. Can you shed some light on this.
3. For some reason when I use the formula in excel to calculte the Interest, principal and EMI I don't quite get the results correct.
1. Let's take a simple example where P, R, and N is given:
P: 10,00,000
R: 11%
Tenure: 240 Months
Rest: Monthly
2. If only EMI and rate of Interest is known eg.
EMI 46392.09 for 240 months and 11%how can I arrive at P, I on monthly basis for 240 months.
Thanks for all the info and sharing the same.
Nagendra
Nagendra: When it's daily rest, but you are paying back only every month, the precise value for the EMI depends on your bank.
ReplyDeleteA simple way is to make "r" equal to AnnualPercentRate/(100*365) and "n" equal to TenureInYears*365. "E" would then be your daily payback installment, which you can multiply by 30 to get your EMI.
Since you don't usually pay back every day, you should really tweak the above to accumulate compound interest till the monthly payback date.
Hi Mathew,
ReplyDeleteThank you so much. Its wonderful. I came up with an amortization chart in excel with the help of your formula.
Mathew,
ReplyDeleteI am ashamed of myself that I missed out on basic math to calculate this. Thanks for the posting. It made me remember my school days.
Nice Article.
ReplyDeleteIf you have MS Excel you can get this computed with PMT formula.
Hi Mr. Matthew,
ReplyDeleteYour Formula and its derivation for calculating EMI on Loan is very useful.
But can you tell me how to Calculate Converse of this, i.e, How to Calculate Final Return Value on Equated Monthly Payment(of RsP per month) at Annual Interest Rate of r% for xmonths(x/12years), where compounding frequency is ntimes/year(i.e. for quarterly,n=4 and halfyearly, n=2) for calculation of final return on EPF or Recurring Deposit or any such Fixed Recurring Deposit for a particular Timespan???
Will be glad if you can kindly derive a formula for above query as I seem to knot myself in long expressions and getting differing values for all.
Thanks In Advance
In continuation of Yesterday's Query, I have tried to calculate the Final Return Value on Fixed Investment made on a periodic basis such as return on EPF or Recurring Deposit.
ReplyDeleteLet us assume, a person invests RsP/month in Recurring Deposit for a tenure of n months at Annual Interest Rate of R% wherein Compounding Frequency is k(so for quarterly, k=4; halfyearly, its 2).
Now Formula for Compound Interest for Annual Compound Interest is
A=P(1+r/100)^n
While Compound Interest where Compounding Frequency is k,
Its
A=P(1+r/100k)^kt
So if a person invests say RsPv/quarter at Quarterly Rate of r(where r=R/100k) over n=kt intervals, then at end of 1st Interval,
P1=Pv+Pv*r=Pv(1+r)=Pv*t ………………….where t=(1+r)………………….(1)
i.e., Principal at end of 1st Interval is Amount Invested+Interest on that Principal.
Similarly,
P2=(P1+Pv)*r +P1 + Pv
i.e., Principal after 2nd Interval is interest for that interval on (Pv+P1) & Principal(Pv+P1)
or, P2=(P1+Pv)*(1+r)
or, P2=(Pv*t+Pv) * t…………………………………………..From (1) above
or, P2=Pv*t*(1+t)………………………………………………..(2)
Similarly,
P3=(P2+Pv)*r + (P2+Pv)=(P2+Pv)(1+r)=(P2+Pv)t………..From (1) above
or, P3=[Pv*t*(1+t)+Pv]*t=Pv*t*[1+t+t^2] ………..From (2) above
Similarly,
P4=(P3+Pv)*r + (P3+Pv)=(P3+Pv)(1+r)=(P3+Pv)t………..From (1) above
or, P3=[ Pv*t*(1+t+t^2)+Pv]*t=Pv*t*[1+t+t^2+t^3]
So at the end of nth Periodic Interval, we are left with
Pn=Pv*t*[1+t+t^2+t^3+…..+t^(n1)]
Now 1+t+t^2+t^3+…..+t^(n1) is a Geometric Series which adds up to
[t^n  1]/[t1]
So Pn=Pv*t*[t^n  1]/ (t  1)
Replacing t with (1+r), we get
Pn= Pv * (1+r) * [ (1+r)^n  1 ] / (r)
or,
Pn= (12/k)*Pv * (1+r) * [ (1+r)^n  1 ] / (r)
where r=R/100k and n=k*(number of years)
IMPORTANT NOTE(DEBUGGING NEEDED):
Notice that here for first interval, amount is (12/k)*Pv
So for 1st quarter, interest calculated is on 3000 even though 1000 is deposited in last month and average balance is only 1500 for that period.Or suppose, a person invests Rs1000/month in EPF with annual compounding.So average balance for that period is only Rs6000. So to get more accurate result, MAYBE we need to equate Pv to more reasonable value.So this derivation still needs some finetuning to readjust the quarterly deposit on which interest is calculated.
Example1:
Let us assume a person invests Rs1000p.m at Annual Rate of 7.5% per annum for 5years where compounding frequency is 4(quarterly)
So Pv=(12/4)*1000=3000
(n)=4*5=20
r=7.5/(4*100)=0.01875Total Investment=Rs1000*60months=Rs60000/
Return on Investment=Rs73341/
Same with monthly compounding,
Pv=1000
(k)=12
r=7.5/1200=0.0625
(n)=12*5=60
Return=??
Example2:
A Person is investing Rs4000p.m. in Recurring Deposit for 15years at Annual Rate of 8.25% (Compounded Monthly).
Pv=4000, k=12, r=8.25/1200=0.006875, n=15*12=180.
Amount Invested=Rs4000*12*15=Rs7.2lacs
Return=Rs1424955.73
Test to check above values:
The return can be calculated in two ways:Either we can assume amount to be deposited at beginning of compounding period or at the end of compounding Period.
What I have devised above calculates amount at beginning of Period.
Step1)Open any blank MSExcel Sheet,
Step2)Click on fx(below Insert and Print Preview button on righttop Panel Bar),
Step3)Category=Financial, Function=Fv
Step4)Rate=AnnualRate%/Compounding Frequency(k)
So if its 12%(Compound Quarterly), Rate=12%/4
Nper=Years*k=5*4
Pv=Fixed Periodic Amount=Monthly Amount*(12/k)
Type=Use Both 0 and 1. Return would be somewhere between these two.
Step5)Click OK
For daily reducing, you are multiplying 30 to the daily EMI. But shouldn't you multiply 30.46 (365/12) or exact number of days the month has.
ReplyDelete~Omprakash
Could you please help me to arrive at the rate of interest if I know the EMI, Principal and tenure.
ReplyDeleteEMI = 16000
Principal = 1830000
Tenure = 20 years
vale: See Calculating Interest Rates.
ReplyDeleteHi Mathew,
ReplyDeleteI appreciate your sincere efforts in giving this explaination which has cleared my long overdue doubts about EMI calculation.
However I think that there can be a simple and more intutive calculation based on present value:
Lets say:
pv_i = present value of your ith EMI
= EMI/(1+r)^i
Then the sum of present value of all the EMIs must be equal to the principal amount, i.e.,
P = pv_1 + pv_2 + ... + pv_n
= EMI*(1/(1+r)^1+1/(1+r)^2+...+1/(1+r)^n)
Now using basic algebra, (i.e. multiplying both sides by (1+r)^n), this will also lead to your formula.
Utkarsh: Yes, summed up PV of the EMIs would also lead to the same formula provided you assume that the discount rate is the same as the rate of interest charged by the lending bank. In that case, any fair repayment scheme (and not just EMIs) should have the same net present value of all the repayments as the lent amount.
ReplyDeleteI REQUEST YOU TO LET ME KNOW THE EMI ON RS.725000.NO OF YEARS 8,16 HALF YEARLY INSTALLMENTS(WITH 12%RATE OF INTEREST
ReplyDeleteHi Mathew,
ReplyDeleteHow can we calculate the outstanding balance at the end of any month? For eg: if we have taken a housing loan for 100,000 at Interest Rate of 10% for 120 months. How can we calculate the expected outstanding balance as at the end of 60 months or 70 months etc.
K NAGESWARA RAO: See the formula for P(i) in the post, which is the amount you still owe the bank at the end of "i" months.
ReplyDeleteA simpler way to look at it is to realise that the outstanding principal is the amount you have taken as a loan for the reduced tenure at the current rate of interest with the given EMI. You can then use the formula for calculating the EMI to calculate the outstanding principal.
simply Superb ! Your Explanation was simple and crystal clear and you made me feel guilty initially for not attempting to figure out the formula but later realized that i am after all a software engineer :)
ReplyDeleteThanks and Regards,
Bala.
Dear Mr. Mathew
ReplyDeleteBrilliant :), just opted out of taking a bank loan, will rather rely on my FD from state bank... thanks a lot...
It is Realy very helpful . Superb :)
ReplyDeleteRegards
Suchitra
Good Work, Congrats, Keep it up!
ReplyDeleteArun
Mathew, would you please tell me the formula  if interest charged monthly but repayment paid quarterly for declining balance method and what will be the formula if interest charged monthly but repayment paid quarterly. Your formula does not work in these case. my email adress: alan_majumder2003@yahoo.com
ReplyDeleteRegards
Gofran
Gofran: Yes, the formula won't work asis. You'll have to derive a new formula from first principles (as is done here) for such cases.
ReplyDeleteIf you are familiar with using a spreadsheet or writing a computer programme, you can also compute the sums iteratively.
Dear sir,
ReplyDeletePl. furnish the formula for calculating EMI in yearly diminishing method
raj.kl@rediffmail.com
Dr. Mr. Mathew
ReplyDeletePlease furnish me formula for calculating quarterly EMI, Half yearly EMI and Annual EMI, where interest is debited monthly.
My email ID is amarpalyam @gmail.com
what is the formula for calculating EMI quarterly?
ReplyDeletehi mathew, u have given a very nice and correct calculation. but for those who are weak in calculations, i have a better solution. try here : http://in.finance.yahoo.com/personalfinance/loans/emicalculator
ReplyDeleteits a very good emi calculator and u dont even need anything to calculate. just scroll the arrows for your loan amount, years and the rate of interest, and see your emi. simple, easy and correct
however i have a question to ask here. can u tell me if i opt for home loan, and take a construction linked plan from builder, then how do i do the calculations here. for ex: my total flat costs for 14 lakhs, and i pay 10% + 5% on my own (booking amount+ on possession) and rest is financed from bank for 20years. here,in the construction linked plan, we have slabs for construction 10% on excavation, etc etc and exact emi from bank starts after possession and during the construction period, we pay the interest to the bank on the amount the bank is paying to the builder. i understand this procedure, but i dont know the exact amount i will be paying each month during and after construction.
ReplyDeleteif u understand my question, then plz provide me a detailed chart of my payments.
kiran: Thanks for the link. The point of this post was to work out how to arrive at the formula for calculating EMIs using basic algebra.
ReplyDeleteAs for your question, typically you only pay the simple interest accrued on the amount paid by the bank to the builder till date  the EMIs start only after you take possession of the house. For example, suppose you've taken a loan at 12% and the bank has paid 200,000 so far to the builder. You will then pay 2,000 (1% of 200,000) every month till the next slab is paid out to the builder.
Of course, the details for your loan might vary (e.g. you might start paying EMI right from the start, your bank might compound interest daily, etc.).
Thanks for this very useful and simple formula.
ReplyDeleteall i need to do is to put it on excel and have the computation done by itself instead of using macro.
moreover, i can even tweak on the formula for me to make critical decision in making a loan.
Thanks a lot and more power!
Guys good Job with the derivation of the formula.
ReplyDeleteThere is even a simple excel formula which can do this. but not as precise.
the formula is
=PMT(A3,A7,A1)
where A1 is principle, A3 is r(interest rate per month) and A7 is no of installments.
@some1 how said need to know the G.P
ReplyDeleteIf go on with 2to 3 iterations, you'll land up in using PASCALS TRIANGLE and solve it from there.
I think it's not about mugging some formulas, it all about inquiry and logical thinking.
THERE'S LOT OF ROTE LEARNING GOING ON IN ENGG THESE DAYS.
thanks.
ReplyDeleteHi Mathew,
ReplyDeleteCould you help me to know about  What formula is used for calculating the reducing rate of interest??
For Example : I take a Car loan amount of Rs. 500000/ for 4 Yrs at the flate rate of 8.36% and my EMI amount is Rs. 14196/ and my total agreement value is Rs. 667200/. And finance executive told me that my reducing rate of interest is 16.00%.
So please tell me  which formula is used in excel for calculating the reducing rate of interest.
My email id is anujsirohi@hotmail.com
So please tell me
Hi Mathew.. Many thanks, I am a banking domain consultant and working for a core banking solution company, but i didnt know this derivation so far. Knowing it makes it much clearer.. Srinivas
ReplyDeleteHi, This is a gud post... But still I've a question...
ReplyDeleteSay for eg., if I take loan of 3 lacs for 3 yrs, but the rate of interest is 9% for 1st year and 10% for 2nd and 3rd year, then what formula is used to calculate EMI for 1st and then 2nd & 3rd years...
I would love to know this calculation... Bit curious...
Thanks in advance...
Regards
Sudarshan Gupta
sudarshanmca@gmail.com
Sudarshan: It's the same formula, except you use different principals, interest rates and periods. For the first year, you use the entire loan amount as the principal; for the second year, you use the stilloutstanding principal as the principal.
ReplyDeleteHi,
ReplyDeleteMathew, I understand the formula however i just want to ask for example the middle equation which is {( 1 + r )n } this is the only thing i dont understand pls help me, for example r=1, n=2 this means its done like this.. ( 1 + 1 )2 so 1 + 1 = 2 x 2 = 4??? sorry this is the only thing that bothers me how it is done pls help....
Thanks
Anonymous: Yes, if r=1 and n=2, the result would be 4. "r" is the rate of interest for the interest period expressed as a fraction. For example, if the *annual* rate is 8%, then "r" for a month would be (8 / 12) / 100 which is equal to 0.0066. "n" is the number of interest periods. If the loan is for 20 years, then "n" would be 20 * 12 or 240.
ReplyDeletehi mathew ,
ReplyDeletethanks for the derivation
u r simply superb abt this
HiMathew,
ReplyDeleteAfter searching for one week i came across your site and very satisfying explaining you have given.
I want to take a loan of Rs.2,50,000, Interest=13.75% , Tenure = 5 years and its a daily reducing interest.
As you explained, i did calculation to arrive at EMI for both monthly and daily reducing interest and below are the result
1. EMI(DailyReducing)
n= 5 (years) * 365 = 1825(days)
[250000*(13.75/365*100)*(1+(13.75/365*100))^1825] / [(1+(13.75/365*100)^1825)1]
If i solve this, i get Rs.189.45 as EMI per day.So per month (multiplying by 31) i get Rs.5873.06588 as monthly EMI
2.EMI(Mothly Reducing) =
n= 5 (years) * 12 = 60 (months)
[250000*(13.75/12*100)*(1+(13.75/12*100))^60] / [(1+(13.75/12*100)^60)1]
However when i solve the above the monthly EMI is coming to Rs.5784.711213 which is less that reducing EMI
As per your explaination in daily reducing rest, the EMI should be less(as per your example above). Or have i done any mistake in my calculation?
Please explain me.
Reply to my email ID.. vivek.urolagin@gmail.com
Thank you.
Vivek: You're taking a month as 31 days above, which is true only for some months in a calendar year. 60 months would equate to 1,860 days at 31 daysper month, whereas with the dailyrest calculation you're considering only 1,825 days; for a 30day month, the EMI with dailyrest comes to 5,683.5 according to your calculations, which is indeed lower than the EMI for the monthlyrest option.
ReplyDeletethanx a lot matthew...it was of gr8 help
ReplyDeleteThanks matthew..!!
ReplyDeleteMatthew thanks for making it so simple!!!
ReplyDeleteFor my car loan the finance guy in the showroom calculated EMI as, Rs.6572 for 200,000 principal amt with 36 months of tenure at 11.25%.
ReplyDeleteI called to bank guys and for the same above data they've calculated EMI as, Rs.6510
I got confused and searched and came here and Rs.6572 is matching as per the formula.
I followed other link "http://www.carwale.com/finance/calculateemi.aspx" and to my surprise for the above input data EMI is, Rs.6510 which is matching with bank guys calculation
which is correct????????????????
Anonymous: The formula given here calculates the
ReplyDeleteEMI payable in arrears, that is, what is due at
the end of each month. You'll get the same answer
from the site you referred to if you select the
"EMI in Arrears" option (not the default "EMI in
Advance" option).
Emi formula explained by mathew is superb.
ReplyDeleteI was wondering How the Emi was calculated.
Hi Mathew,
ReplyDeleteFirst of all, thanks a ton for your very usefull blog. For quite a long time, I have been searching the net to find out the calculation method of Housing Loan EMI. And your explanation was the best of all! Thank you very much.
.
Buddy, I have a complex situation here.. (should not be complex to you :)
I took a housing loan on Nov 1, 2009 of Rs.12,800,000 @ 9% PA for 15 years. And my EMI was 12983. And still I am paying the same EMI  12983. However, recently I came to know that my loan tenure has been increased from 180 months to 255 months. When I contacted the bank, I was told about the following change of interest made to my loan.
1. On 01112010, rate increased to 9.50%
2. On 01022011, rate increased to 10.50%
3. On 01052011, rate increased to 10.75%
4. On 01082011, rate increased to 11.75%
.
Could you please tell me if extension of loan tenure to 255 months, is correctly done by the Bank ? And if possible, could you pls explain me how the bank has done the calculation ?
.
Thank you very much..
Harikrishna
kghareesh@gmail.com
Hari, see my reply to Sudarshan above:
ReplyDeletehttp://rmathew.blogspot.com/2006/06/calculatingemis.html?showComment=1296530171942#c8398810267596404099
Here is a free EMI calculator. Really easy to use.
ReplyDeletehttp://www.emicalculator.in
pls calculate this p=30000 r13%, 1.1/2 years means 18 months
ReplyDeleteExcellent post Mathews!!...In the world where there thousands of automated EMI calculators on the web, people (even if engineers) seem less bothered about going to the core of the issue. Uve done an excellent job! Cheers! :)
ReplyDeleteSir, Can you suggest calculation for insurance linked deposit scheme..
ReplyDelete;) :) couldn't resist trolling you !
The formula is fantastic, I've linked this to my expression editor and running 100s of calculations fine tuning all along.
thanks esp for the mathematical formatting. the effort is greatly appreciated. thanks !
thanks...
ReplyDeleteHi Mathew,
ReplyDeleteI got your blog ref from http://emicalculator.net/, in this they are showing Loan Amount, Interest and Loan Term.. What will be the formula fee we will include the 4th factor i.e. One time charges or Processing fee which is paid to DSA or bank on disbuersment or upfront
i.e. Loan Amount is 1000000/, Rate of Int. is 6% PA, Term is 10 Yrs.and one time charges i paid is 10% of loan Amount
Please help me to delute these excess 10% in loan term
Thanks
 Tarun
Tarun, if the amount you're talking of is a fee,
ReplyDeletewhy would it enter into the EMI calculations?
Dear Mathew,
ReplyDeleteI just wanna to enter bcoz indirectly I am getting 900000/ Rs. after deducting 10% (100000/) from loan amount on disbuersmen. so I am utilizing 9 Lac rs. and paying interest fo 10 Lac and again repayment would be the same for 10 Lac rs. ... so i wanted to calculated the acutal Rate of Int. ... so please if u suggest me the way how we can delute 10% cost into the loan...
Thanks
Tarun, you have answered your own question: you're
ReplyDeletepaying interest on the original amount and that
amount is your total due  so it remains the
principal in the EMI calculation. The fee that you
paid is an additional expenditure.
Ranjit,
ReplyDeletePhenomenal blog!!
BTW, I see a common error in the way banks calculate monthly interest rates for monthly reducible balance loans. This is shown in my blog post (http://nitinchowdhary.blogspot.in/2012/03/homeloanwoes.html). My view is that the banks are taking us for a ride. Would love to hear your views about it.
Regards
Nitin
Thx it is very useful for me as I am planning for a business loan  hope its the same for home loan and personal loan
ReplyDeleteIt seems to me that the banks do not confuse/cheat/mislead/ creatively account (whatever you call it) at the time of setting the first EMI, but during their EMI reset. Like, if the rate is changed in year 5, they calculate the EMI keeping the original Principal in the equation. In fairness, at that point, the outstanding Principal must be considered to calculate the increased RoI.
ReplyDeletePaying the increased RoI on Principal already paid seems like inethical, if not outright criminal.
What say?
very useful blog
ReplyDeletevery nice and useful article...Well explained...thanks...
ReplyDeletevery nice one ... exactly what i was looking around for
ReplyDeleteGreat Job..that explains the cloud now :)
ReplyDeleteDear Sir
ReplyDeletei have taken housing loan form canarabank 4lakhs in the year 2005. that time intrest rate was 8.5% now it is 12% adn my emi also increased.
now my query is how much extra i have to pay if i take the 4lakh rupees.
asper my calculating iam getting nearly 9 lakh rupees i have to make repayment.
please clarify this imd.
Regds
saba
Shouldn't the formula be P1= P + r.P  (Principal part of E) as only part of the EMI is paid towards the principal. So the outstanding principal should be reduced only by that much amount.
ReplyDeleteAm i missing something here?
Anonymous: The "principal part of E" _is_ (E  r.P), since r.P is the amount of pending interest that you must pay back. So remaining principal "P1 = P  (E  r.P)", which is just "P1 = P + r.P  E" as above.
ReplyDelete