Lesson 1.3: Loan amortisation schedules and the maths of the EMI
Learn how an EMI is calculated, how each instalment splits into interest and principal, how the balance (your EAD) falls, and why flat rates mislead.
By the end of this lesson you will be able to build an amortisation schedule for a loan, calculate the monthly instalment (EMI), split any instalment into interest and principal, find the outstanding balance at any month, and convert a flat rate into the reducing-balance rate it really is. You will also see why this arithmetic belongs in a credit risk course: the outstanding balance is the exposure at default from Lesson 1.1, and the schedule is the benchmark against which every missed payment in Lesson 1.2 is measured. You need only percentages and a calculator. The formulas are explained step by step and then checked on one running loan.
Amortisation is the process of paying a debt down in regular instalments, so that each payment covers the interest owed for the period and also repays a part of the original amount, the principal. It sounds mechanical, and it is, which is why it is so useful. If you know the amount, the rate and the tenure, every number in the schedule is fixed. A credit analyst can tell at a glance whether a loan is behaving as the schedule says it should.
The running example
Throughout this lesson we use one loan: ₹2,00,000 for 24 months at 12 per cent a year, repaid in equal monthly instalments. The monthly rate is 12 ÷ 12 = 1 per cent. We will compute its instalment, build its schedule, test it against a prepayment and compare it with a flat-rate version.
Before any formula, think about what an equal instalment has to do. The lender gives you ₹2,00,000 today and wants it back with interest over 24 months, in 24 identical payments. Money today is worth more than money later, so each future payment is worth a little less than its face value today. The instalment must be the amount which, once its future payments are brought back to today's value, adds up to exactly ₹2,00,000.
The EMI formula
An equated monthly instalment, or EMI, is the fixed amount paid each month. With a loan amount P, a monthly interest rate r (as a decimal) and n months, it is:
EMI = P × r ÷ (1 − (1 + r)^−n)
The bracket is the heart of it. The term (1 + r)^−n is the factor that brings a payment n months away back to its value today. The expression 1 − (1 + r)^−n, divided by r, is the combined present value of one rupee paid each month for n months. The EMI is then the amount that makes the whole stream equal to P.
For the running loan, r = 0.01 and n = 24.
1. (1.01)^24 = 1.269735, so (1.01)^−24 = 1 ÷ 1.269735 = 0.787566.
2. 1 − 0.787566 = 0.212434.
3. EMI = 2,00,000 × 0.01 ÷ 0.212434 = 2,000 ÷ 0.212434 = ₹9,414.69.
Over 24 months the borrower pays 24 × 9,414.69 = ₹2,25,952.6, so the total interest is about ₹25,953. Notice that this is much less than simple interest of 12 per cent on ₹2,00,000 for two years, which would be ₹48,000. The reason is that the borrower does not owe ₹2,00,000 for two years. The balance shrinks every month, and interest is charged only on what is still owed. That is what reducing balance means.
Predict first: in the first instalment of ₹9,414.69, how much is interest and how much is principal? Think about what is owed in month 1 and what 1 per cent of it is.
In month 1, the borrower owes the whole ₹2,00,000, so the interest is 1 per cent of that, ₹2,000. The principal repaid is the rest: 9,414.69 − 2,000 = ₹7,414.69. The balance drops to 2,00,000 − 7,414.69 = ₹1,92,585.31.
Building the schedule
Every month repeats the same three steps: interest equals the opening balance times r, principal equals the EMI minus the interest, and the closing balance equals the opening balance minus the principal. Here are the first three months of the schedule.
Month · Opening balance · Interest · Principal · Closing balance1 · 2,00,000.00 · 2,000.00 · 7,414.69 · 1,92,585.31
2 · 1,92,585.31 · 1,925.85 · 7,488.84 · 1,85,096.47
3 · 1,85,096.47 · 1,850.96 · 7,563.73 · 1,77,532.74
Look at the pattern. The EMI is constant, but the interest falls every month, because it is charged on a smaller balance, and the principal rises by exactly the amount the interest falls. In month 1 principal is 79 per cent of the instalment. By month 24, interest is only about ₹93 and principal is about ₹9,321. This is why paying a loan off early in its life saves much more interest than paying it off near the end: in the beginning, most of what you pay is interest.
If you want the balance at any month without building every row, there is a direct formula. The balance after k payments equals the present value of the payments still to come: balance after k months = EMI × (1 − (1 + r)^−(n − k)) ÷ r. After 12 months, 12 payments remain. (1.01)^−12 = 0.887449, so the bracket is 0.112551, divided by 0.01 gives 11.2551, and the balance is 9,414.69 × 11.2551 = about ₹1,05,963.
That number deserves a second look. Half the tenure has passed, but 53 per cent of the original amount is still owed (1,05,963 ÷ 2,00,000 = 0.53). The borrower has repaid only 47 per cent of the principal in half the time, because the early instalments were mostly interest.
Why this is credit risk, not just loan maths
The balance column is the exposure at default from Lesson 1.1. If the loan defaults after month 12, the lender's exposure is about ₹1,05,963, not ₹2,00,000 and not the sum of future payments. If it defaults after month 3, the exposure is ₹1,77,533. The same borrower, defaulting at different times, creates different losses, even with the same loss given default. That is why a lender's expected loss depends on when in the loan's life defaults tend to happen. Vintage analysis in Module 4 studies exactly that timing.
The schedule is also the benchmark for delinquency. The lender knows, for every loan on every date, what the balance should be and what has been paid. A loan whose actual balance is higher than the schedule's balance is behind. A loan that is repaying faster than the schedule has made extra payments. Days past due, from Lesson 1.2, is the measured gap between schedule and reality.
Try the calculator with your own loan.
With the default inputs you should see an instalment of about ₹9,415, total interest of about ₹25,953 and a balance of about ₹1,05,963 after month 12, which is 53 per cent of the loan. Try a longer tenure of 48 months. The instalment falls, but the total interest rises, and after 24 months the borrower still owes about 56 per cent of the loan. A longer tenure lowers the monthly burden and raises both the cost and the lender's exposure.
Flat rate versus reducing-balance rate
In Indian retail lending, and in many two-wheeler, consumer-durable and small-ticket loans, you will see a flat rate quoted. A flat rate applies the stated percentage to the original amount for the whole tenure, with no allowance for the falling balance. Take the same ₹2,00,000 for 24 months at a flat 8 per cent a year.
Interest = 2,00,000 × 0.08 × 2 years = ₹32,000.
Total repayable = 2,00,000 + 32,000 = ₹2,32,000.
EMI = 2,32,000 ÷ 24 = ₹9,666.67.
An 8 per cent flat rate sounds cheaper than 12 per cent reducing. It is not. The EMI of ₹9,666.67 is larger than the ₹9,414.69 charged at 12 per cent reducing. To find the real, reducing-balance rate, we ask: at what annual rate does a 24-month loan of ₹2,00,000 give an EMI of ₹9,666.67? Solving this by trial gives a monthly rate of about 1.22 per cent, which is a yearly rate of about 14.7 per cent. The 8 per cent flat rate is, in effect, an annual rate of nearly 15 per cent. A rough rule of thumb is that a flat rate is worth about 1.8 times its face value as a reducing rate, though the exact multiple depends on the tenure.
Why the difference? Because the flat-rate lender charges interest on the whole ₹2,00,000 even though the borrower is steadily repaying it. In the last month the borrower owes only about ₹9,500, but the flat rate still charges interest as though ₹2,00,000 were owed.
This is a clear example of why analysts must always translate quoted rates to a common basis before comparing them. The same applies to fees. A processing fee of 2 per cent charged upfront reduces the money the borrower receives while the EMI stays the same, which raises the effective rate again. The honest comparison is the annual percentage rate, or APR, that includes all charges. Regulators in many markets now require lenders to disclose it.
What changes the schedule
The schedule above assumes nothing goes wrong and nothing changes. Real loans have variations, and each one affects the balance and therefore the exposure.
Part-payment or prepayment. If the borrower pays more than the EMI, the balance falls faster and the loan ends earlier, saving interest.
Floating rates. If the interest rate changes, the lender either adjusts the EMI or extends the tenure. The balance path changes.
Moratorium. A pause in repayments lets interest build up, so the balance after the pause is higher than the schedule said.
Bullet and balloon loans. The borrower pays interest only, or a small instalment, with the principal due at the end. The balance stays high until the last day, so exposure is large throughout.
Restructuring. A lender rewrites the schedule for a borrower in trouble, which resets the dates and the delinquency count. Analysts must track restructured loans separately.
Test one of these on the running loan. Suppose the borrower pays an extra ₹2,000 a month on top of the EMI.
With the defaults, an extra ₹2,000 a month closes the loan in about 19.3 months instead of 24, saving about 4.7 months and about ₹5,300 of interest. For a risk analyst, the lesson is that prepayment shortens the life of a loan, which lowers its exposure, but it also removes the good borrowers early. The loans that stay are on average riskier, which is called adverse selection by attrition. A portfolio's delinquency rate can rise simply because the best customers left.
Building a schedule in Python
The formulas above translate into a few lines. The function below returns a full schedule as a table.
python
import pandas as pd
def schedule(principal, annualrate, months):
r = annualrate / 12 / 100
emi = principal r / (1 - (1 + r) -months)
rows, balance = [], principal
for m in range(1, months + 1):
interest = balance r
princ = emi - interest
balance -= princ
rows.append((m, round(interest, 2), round(princ, 2), round(max(balance, 0), 2)))
return emi, pd.DataFrame(rows, columns=["month", "interest", "principal", "balance"])
emi, sch = schedule(200000, 12, 24)
print(round(emi, 2)) # 9414.69
sch.iloc[[0, 11, 23]] # months 1, 12 and 24
Two checks are worth making every time. The balance at the last month should be zero, apart from a rounding difference of a few paise. And the sum of the principal column should equal the loan amount. If either fails, the formula or the rounding is wrong.
A worked example: tenure and total cost
Tenure is the lever borrowers and lenders pull most often, and the schedule shows what it does. Take the same ₹2,00,000 at 12 per cent a year over three tenures.
Tenure · EMI · Total paid · Total interest12 months · ₹17,770 · ₹2,13,237 · ₹13,237
24 months · ₹9,415 · ₹2,25,953 · ₹25,953
36 months · ₹6,643 · ₹2,39,144 · ₹39,144
Tripling the tenure from 12 to 36 months cuts the monthly burden by almost two thirds, from ₹17,770 to ₹6,643, but it triples the interest too. A borrower who picks the longest tenure to make the instalment affordable pays ₹25,907 more than one who can manage 12 months. From the lender's side there is a second effect. A longer tenure keeps the balance high for longer, so the exposure at default stays large through more of the loan's life, and there are more months in which something can go wrong.
A worked example: the same default at three dates
Return to the 24-month loan and suppose the borrower stops paying after a given month, with a loss given default of 60 per cent. The loss depends heavily on the date.
Default after month 3: balance ₹1,77,533, loss = 0.60 × 1,77,533 = ₹1,06,520.
Default after month 12: balance ₹1,05,963, loss = 0.60 × 1,05,963 = ₹63,578.
Default after month 18: balance ₹54,563, loss = 0.60 × 54,563 = ₹32,738.
The earliest default costs more than three times the latest, for the same borrower and the same recovery rate. This is why lenders watch early-life defaults so closely. A loan that defaults in its first few months has barely repaid any principal, and often signals fraud or a weak underwriting decision rather than bad luck. Vintage curves in Module 4 are built to show this pattern.
Common mistakes
Using the original amount as the exposure for the whole life of the loan. The balance falls, and exposure at default depends on when the default happens.
Comparing a flat rate with a reducing-balance rate. Convert to a common basis first. A flat 8 per cent is about 14.7 per cent reducing for this loan.
Forgetting that early instalments are mostly interest. The balance falls slowly at first, so early foreclosure saves less than people expect.
Mixing the yearly and monthly rate. Divide the annual rate by 12 for monthly instalments, and use the number of months, not years, as n.
Ignoring fees. Upfront charges raise the effective rate. Use the APR for comparisons.
Treating restructured loans as fresh loans. A new schedule resets the clock and hides earlier delinquency unless it is flagged.
Key takeaways
An EMI is the fixed payment that makes the present value of all instalments equal the loan: EMI = P × r ÷ (1 − (1 + r)^−n).
Each month, interest equals the opening balance times the monthly rate, and the principal is the rest of the instalment. Early instalments are mostly interest.
The outstanding balance is the exposure at default, and it falls slowly at first. For a 24-month loan at 12 per cent, 53 per cent is still owed after 12 months.
A flat rate charges interest on the original amount for the whole tenure, so it is much more expensive than the same number quoted as a reducing rate. An 8 per cent flat rate is about 14.7 per cent reducing.
Prepayment, rate changes, moratoriums and restructuring all change the schedule, so a credit analyst tracks the schedule against what actually happened.
Prepayment removes the good borrowers early, so portfolio delinquency can rise even when nobody gets worse.
Check yourself
1. Compute the EMI for ₹1,00,000 over 12 months at 12 per cent a year.
r = 0.01. (1.01)^−12 = 0.887449, so the bracket is 0.112551 and the EMI is 1,000 ÷ 0.112551 = about ₹8,885.
2. How much of the first instalment is interest on a ₹5,00,000 loan at 9 per cent a year?
The monthly rate is 0.75 per cent, so the interest is 5,00,000 × 0.0075 = ₹3,750.
3. A loan of ₹3,00,000 is offered at a flat 10 per cent for 3 years. What is the EMI, and is the real rate 10 per cent?
Interest is 90,000, the total is 3,90,000 and the EMI is ₹10,833. The real reducing rate is much higher, about 18 per cent.
4. Why does a borrower still owe 53 per cent of a 24-month loan at the halfway point?
Early instalments are mostly interest, so only a little principal is repaid at first. The principal share rises over time.
5. Why might a portfolio's delinquency rate rise even if no borrower gets worse?
Good borrowers prepay and leave, so the remaining pool is riskier on average.
Try this
Take a real loan offer, such as a two-wheeler loan quote, a personal loan advertisement or a home loan calculator on a bank's site. Set aside forty minutes. If you do not have one, use a made-up loan of ₹3,00,000 for 36 months.
1. Write the amount, the stated rate, whether it is flat or reducing, the tenure and any upfront fee.
2. Compute the EMI by the formula and check it against the lender's own figure. If you used a flat rate, compute the EMI from the flat formula and use the flat-rate simulator to find the reducing-balance rate it equals.
3. Build the full schedule in a spreadsheet or with the Python function. Check that the closing balance is zero.
4. Mark the balance after 6, 12 and 24 months. For each, write what the lender would lose if the borrower defaulted then and the loss given default were 60 per cent.
5. Use the extra-payment simulator to find how much an extra payment of 10 per cent of the EMI saves.
6. Write a short note answering: at which point in the loan's life is the lender's exposure highest relative to what the borrower has paid, and what does that mean for when it should worry most about defaults?
A good result is a one-page sheet with the schedule, three exposure values and a plain-language note on the true cost. The next lesson fixes the other half of every default model: what exactly counts as a default, and over what period.
Tags: Credit Risk Analytics Amortisation EMI Loan maths Beginner