Annuities, Perpetuities & Loan Amortisation: Interview-Ready Worked Examples
A ₹30 lakh home loan at 9% for 20 years feels like one number: an EMI of about ₹26,992. But in the very first month, roughly ₹22,500 is interest and only about ₹4,492 reduces the loan - the same payment quietly changes character every month.
- An annuity is a fixed payment received or paid at regular intervals for a finite number of periods.
- Present value of ordinary annuity = PMT × [1 - (1 + r)-n] / r.
- A perpetuity is a fixed periodic cash flow assumed to continue forever; PV = C / r.
- A growing perpetuity values cash flows growing at a constant rate; PV = C1 / (r - g), where r > g.
- Loan amortisation splits every EMI into interest and principal until the outstanding loan becomes zero.
- EMI formula = P × r × (1 + r)n / [(1 + r)n - 1]. Match r and n to the payment frequency.
- The biggest trap is using an annual rate with monthly EMIs without converting it to a monthly rate.
Big Picture: One Idea, Three Cash-Flow Shapes
All three topics are applications of the time value of money: a rupee today is worth more than a rupee later because money can earn a return. The only thing that changes is the shape of the cash flow - finite, infinite, or loan repayment.
Core Explanation: The Three Formulas You Actually Need
Annuities, perpetuities and amortising loans are not three separate chapters. They are three ways of arranging repeated cash flows on a timeline.
Annuity Worked Example: Value a Fixed Annual Cash Flow
Suppose you are promised ₹1,00,000 at the end of every year for 5 years. The required return is 10% per year. What is the present value?
The answer is not ₹5,00,000 because each future ₹1,00,000 is discounted back to today. The fifth-year payment is worth less today than the first-year payment.
Perpetuity Worked Example: Value a Cash Flow That Never Ends
Suppose a security pays ₹50,000 every year forever, and investors require an 8% return.
PV = C / r = 50,000 / 0.08 = ₹6,25,000.
If the cash flow is expected to grow at 4% forever and next year’s cash flow is ₹50,000, then:
PV = C1 / (r - g) = 50,000 / (0.08 - 0.04) = ₹12,50,000.
For a growing perpetuity, the discount rate must be greater than the growth rate. If r ≤ g, the formula breaks economically and mathematically.
Loan Amortisation Worked Example: Build the EMI Schedule
Now take the opening example: ₹30,00,000 home loan, 9% annual rate, 20-year tenure, monthly EMI.
Notice the insight: the EMI stays almost boringly constant, but the internal split changes every month. Early EMIs mainly compensate the lender for interest; later EMIs mainly repay the loan.
Loan Amortisation: What to Track
When a loan is analysed in finance, the EMI is only the start. You also check whether the borrower can afford it, how risky the collateral is, and how quickly the debt reduces.
Definitions You Can Say in One Breath
- Annuity: A finite series of equal cash flows paid or received at regular intervals.
- Perpetuity: A series of equal periodic cash flows assumed to continue forever.
- Growing perpetuity: A perpetual cash-flow stream growing at a constant rate, valued only when r is greater than g.
- Loan amortisation: A repayment process where each scheduled payment covers interest and reduces principal until the loan is fully repaid.
- EMI: A fixed periodic loan payment that combines interest due and principal repayment.
Case Study: Aavas Financiers and the EMI Logic of Affordable Housing
Aavas Financiers, an Indian affordable housing finance company, shows how amortisation converts a large home purchase into a monthly cash-flow decision.
Situation: Affordable housing borrowers often do not think in terms of net present value or loan duration. They think in terms of a monthly payment they can live with. For self-employed or informal-income households, that EMI must fit real cash inflows, seasonal income patterns and household obligations.
The move: Aavas focuses on underwriting the borrower’s repayment capacity and structuring loans as predictable EMIs. The primary driver is cash-flow fit: the loan works only if the monthly amortising payment is affordable. Supporting drivers include property collateral, local market knowledge, income assessment, collection discipline and conservative loan structuring.
Outcome or lesson: The business lesson is simple: an amortising loan is not just a formula in Excel. It is a risk-management design. For the borrower, it makes a large asset purchasable. For the lender, it creates annuity-like inflows, but with credit risk, prepayment risk and interest-rate risk attached.
How AI Changes Annuities, Perpetuities & Loan Amortisation
1. AI makes loan affordability more real-time. Lenders can analyse bank statements, salary credits, GST trails, account aggregators and transaction patterns to estimate repayment capacity faster. The finance logic is still EMI and DSCR, but the input quality improves.
2. AI turns amortisation into scenario simulation. Instead of one static schedule, teams can model floating-rate resets, part-prepayments, tenure changes and rate shocks quickly. This is useful for banks, NBFCs, treasury teams and retail borrowers comparing loan options.
3. AI increases the need for governance. Credit models that use alternative data must be checked for bias, explainability, data consent and compliance with Indian privacy and financial regulations. A faster model is not automatically a fairer model.
Use ChatGPT or Claude to generate an amortisation schedule for a sample loan, then verify the first three rows manually in Excel. For company prep, load an NBFC or bank annual report into NotebookLM and ask: “Where do interest-rate risk, credit risk and prepayment risk affect amortising loan cash flows?”
Interview Relevance
“A customer takes a ₹30 lakh loan at 9% for 20 years. Explain how EMI is calculated, why the interest portion is high initially, and how this connects to annuities.”
If numbers are given, do one small calculation aloud. Even calculating the first month’s interest correctly proves you understand amortisation better than most candidates.
Common Mistake
Candidates mix annual rates with monthly payments. That destroys the EMI. The fix: first convert the rate to the payment period and convert tenure into the number of payment periods.
What to Revise Next
Next, revise the return required for discounting and the interest-rate mechanics behind every formula here.