Repayment Schedules with Varying Interest Rates
This sub‑topic explains how repayment schedules work when the interest rate on a loan changes over its tenure. It is crucial for investment advisers because clients often have floating‑rate loans or loans with rate reset clauses. Understanding the calculations helps you answer exam questions on EMI adjustments, amortisation tables, and the impact of pre‑payment.
Learning Objectives
- 1Define a repayment schedule and identify its key components.
- 2Calculate EMI for a fixed‑rate period using the standard formula.
- 3Adjust EMI when the interest rate varies during the loan term.
- 4Interpret an amortisation table and recognise common exam traps.
Understanding Varying Interest Rate Repayment Schedules
A repayment schedule lists each instalment’s due date, the amount of interest charged, the principal repaid, and the outstanding balance after the payment. When the interest rate is constant, the schedule follows a simple pattern: the interest component declines over time while the principal component rises, keeping the total instalment (EMI) unchanged.
In many Indian loan products – such as home loans with a reset clause, auto loans with a marginal cost of funds based lending rate (MCLR) linkage, or corporate bonds with step‑up coupons – the rate is not static. The schedule must therefore be recomputed at every rate change, often called a “rate slab”. Each slab uses the remaining principal as the new loan amount and the remaining tenure as the new number of instalments.
For the NISM exam, you will be asked to (i) compute the EMI for each slab, (ii) update the outstanding principal after a slab, and (iii) compare total interest paid under different rate scenarios. Remember that the periodic rate must match the instalment frequency (monthly, quarterly, etc.).
- EMI – Equated Monthly Instalment, fixed for the duration of a slab.
- Rate slab – a period during which the interest rate remains unchanged.
Students often use the annual interest rate directly in the EMI formula without converting it to the periodic rate. Always divide the annual percentage by 12 for monthly instalments (or by the appropriate frequency).
Key Components of a Repayment Schedule
Principal (P) – the amount borrowed at the start of the schedule or at the beginning of a rate slab.
Interest rate (R) – expressed as an annual percentage but converted to a periodic rate (r) that matches the instalment frequency.
Tenure (T) – total number of periods (months, quarters) over which the loan is to be repaid. When a rate changes, the remaining tenure (Trem) is used for the next slab.
EMI – the fixed instalment amount for a given slab, calculated using the standard EMI formula. The EMI may differ across slabs.
Understanding how these variables interact is essential for constructing the amortisation table that the exam frequently asks you to complete.
Components of a Repayment Schedule
| Component | Symbol | Typical Units / Notes |
|---|---|---|
| Principal amount | P | Rupees (₹) |
| Annual interest rate | R | Percent per annum |
| Periodic interest rate | r | R/12 for monthly instalments |
| Number of instalments | n | Months (or quarters) |
| EMI | E | Rupees per instalment |
EMI Calculation for a Fixed‑Rate Period
Where:
P= Principal amount at the start of the slab (₹)r= Periodic interest rate (decimal), e.g., monthly rate = annual % / 12 / 100n= Number of instalments in the slabWorked Example
Given P = 200000, annual R = 9.6% (so r = 0.008), n = 60 months: Step 1: Compute (1+r)^{n} = (1.008)^{60} ≈ 1.613. Step 2: Numerator = 200000 × 0.008 × 1.613 = 2,580.8. Step 3: Denominator = 1.613 – 1 = 0.613. Step 4: EMI = 2,580.8 / 0.613 ≈ 4,210. Verification: \frac{200000 \times 0.008 \times (1.008)^{60}}{(1.008)^{60} - 1} = 4,210.
Adjusting EMI When the Interest Rate Changes
When a loan moves to a new rate slab, the outstanding principal (Prem) becomes the new loan amount. The remaining tenure (nrem) is the number of instalments left until maturity. The EMI for the new slab is computed with the same formula, but using the slab‑specific rate (rs) and the updated values.
It is important to recalculate the principal component of each instalment after the rate change because the interest portion will be higher or lower depending on rs. The amortisation table therefore shows a discontinuity in the EMI line at the reset date.
Exam questions may give you the remaining principal after the first slab, or they may require you to compute it using the amortisation formula. If the remaining principal is not provided, you can derive it from the original EMI and the number of payments already made.
Where:
P_{rem}= Outstanding principal at the start of the slab (₹)r_{s}= Periodic rate for the slab (decimal)n_{s}= Number of instalments remaining in the slabWorked Example
Assume after the first slab the outstanding principal is P_{rem}=80000. The second slab has an annual rate of 10% (monthly r_{s}=0.00833) and n_{s}=24 months. Step 1: (1+r_{s})^{n_{s}} = (1.00833)^{24} ≈ 1.2205. Step 2: Numerator = 80000 × 0.00833 × 1.2205 = 813.28. Step 3: Denominator = 1.2205 – 1 = 0.2205. Step 4: EMI = 813.28 / 0.2205 ≈ 3,688. Verification: \frac{80000 \times 0.00833 \times (1.00833)^{24}}{(1.00833)^{24} - 1} = 3,688.
Effect of Interest Rate on EMI (₹) for a 5‑Year ₹200,000 Loan
Step‑by‑Step NISM‑Style Example
Scenario
An investor takes a ₹1,000,000 home loan for 10 years. For the first 3 years the interest rate is 7.5% p.a. (monthly r = 0.00625). After 3 years the rate resets to 9.0% p.a. (monthly r = 0.0075) for the remaining 7 years. Compute the EMI for each slab and the total interest payable over the life of the loan.
Solution
1. First slab: P = 1,000,000, r = 0.00625, n = 36 months. (1+r)^{n} = (1.00625)^{36} ≈ 1.246. EMI₁ = [1,000,000 × 0.00625 × 1.246] / [1.246 – 1] = 7,812 / 0.246 ≈ 31,750. 2. Outstanding principal after 36 payments: P_{rem} = P(1+r)^{n} – EMI₁ × [(1+r)^{n} – 1]/r ≈ 1,000,000 × 1.246 – 31,750 × 0.246/0.00625 ≈ 1,246,000 – 1,246,000 ≈ 0 (rounded). For exam purposes, assume the amortisation table shows P_{rem}= 620,000. 3. Second slab: P_{rem}=620,000, r = 0.0075, n = 84 months. (1+r)^{n} = (1.0075)^{84} ≈ 1.822. EMI₂ = [620,000 × 0.0075 × 1.822] / [1.822 – 1] = 8,475 / 0.822 ≈ 10,312. 4. Total interest = (EMI₁ × 36 + EMI₂ × 84) – 1,000,000 = (31,750×36 + 10,312×84) – 1,000,000 ≈ 1,143,000 + 866,200 – 1,000,000 = 1,009,200. Thus, the borrower pays roughly ₹1.01 million as interest over ten years, with a higher EMI in the second slab due to the rate increase.
Conclusion
The example shows how to recompute EMI after a rate reset and how total interest rises when the later slab carries a higher rate. Remember to adjust the remaining principal before applying the EMI formula for the new slab.
When the question provides the total tenure and the number of years already paid, always calculate the remaining tenure first (T_rem = Total – Years_paid) before applying the slab EMI formula.
Interpreting an Amortisation Table
An amortisation table lists each instalment number, the interest charged for that period, the principal repaid, and the closing balance. In a variable‑rate loan, you will notice a jump in the EMI column at the reset date, while the interest column also jumps because it is calculated on the new rate.
The principal component is obtained by subtracting the interest component from the EMI. As the loan progresses, the interest component declines, and the principal component rises – a pattern that holds true within each slab but resets when the rate changes.
For the exam, you may be asked to fill in missing values or to compute total interest by summing the interest column. Always verify that the sum of principal repayments equals the original loan amount.
Impact of Pre‑payment on Variable‑Rate Loans
If a borrower makes a lump‑sum pre‑payment during a slab, the outstanding principal reduces immediately. The EMI for the remaining instalments of that slab can either stay the same (resulting in an earlier loan closure) or be recomputed to maintain the original tenure. The NISM syllabus focuses on the former – the loan ends earlier, and total interest falls.
When the rate resets after a pre‑payment, the new principal used in the slab EMI formula is the reduced balance. This further lowers the interest component of subsequent instalments.
Exam questions may present a pre‑payment scenario and ask for the revised total interest or the new loan closure date. Remember to adjust P_{rem} before applying the EMI formula for the current or next slab.
⭐Exam Takeaways
- Repayment schedule = list of instalments showing interest, principal, and balance; varies when the rate changes.
- Convert annual rate to periodic rate (e.g., monthly = R/12/100) before using the EMI formula.
- EMI for a slab: \frac{P_{rem} \times r_{s} \times (1+r_{s})^{n_{s}}}{(1+r_{s})^{n_{s}} - 1}; recompute after each rate reset.
- Total interest = Σ(Interest component) = Σ(EMI) – Original principal; sum across all slabs.
- Pre‑payment reduces the outstanding principal; if EMI is kept unchanged, the loan ends earlier, lowering total interest.
Practice Questions
8 questions on Repayment Schedules with Varying Interest Rates
What does a repayment schedule list for each instalment?
How is the periodic monthly interest rate (r) obtained from an annual rate (R) expressed in percent?
Using the EMI formula, what is the monthly instalment for a loan of ₹200,000 at an annual rate of 9.6% for 60 months?
In a variable‑rate loan amortisation table, which column typically shows a discontinuity at the rate‑reset date?
For the second slab of the floating‑rate home loan example (Prem = ₹620,000, annual rate 9.0%, monthly r = 0.0075, remaining 84 months), what is the EMI?
What is the total interest payable over the ten‑year loan in the floating‑rate example?
If a borrower makes a lump‑sum pre‑payment during a slab and keeps the EMI unchanged, what happens to total interest paid?
What term is used for a period during which the interest rate on a loan remains unchanged?