Banking and Insurance - Laws and Practice · Calculation of Interest and Annuities

Simple and Compound Interest: Formulas and Effective Rate

Updated 11 October 2026 · Fact-checked

Simple interest is charged only on the original principal: SI = P × r × t. Compound interest is charged on principal plus earlier interest: A = P(1 + r/m)^(mt). The effective rate converts a nominal rate with m compounding periods a year into its true annual yield: (1 + r/m)^m − 1.

Understand Simple and Compound Interest

Interest is the price of using money for time. A bank pays it on your deposit and charges it on your loan. The two basic methods differ in what the interest is calculated on.

Simple interest is calculated every period on the same original amount, the principal. Interest does not earn interest. So the interest is the same each year, and the total grows in a straight line.

Compound interest is calculated on the principal plus all interest added so far. Interest earns interest, so the total grows faster each period. The more often interest is added, the higher the final amount. Banks add interest quarterly on most deposits and monthly on many loans, so this matters in practice.

A rate quoted per year is the nominal rate. If it is compounded more than once a year, the money actually grows by more than the nominal rate. The effective rate is the single annual rate that gives the same result as the nominal rate with its compounding. Use it to compare products with different compounding frequencies.

In this paper, expect to compute interest, an amount or an effective rate from given facts, and to state which method applies. Always take time in years, matching the rate's period.

Key rules to remember

Simple interest: SI = P × r × t; r is the annual rate as a decimal (10% = 0.10); t is in years. For months, use t = months ÷ 12; for days, days ÷ 365.
Amount under simple interest: A = P + SI = P(1 + r t); Amount is the principal plus interest.
Amount under compound interest: A = P(1 + r/m)^(m t); m is the number of compounding periods per year. For annual compounding, m = 1 and A = P(1 + r)^t.
Compound interest: CI = A − P; Find the amount first, then subtract the principal.
Effective rate of interest: Effective rate = (1 + r/m)^m − 1; r is the nominal annual rate as a decimal. Multiply by 100 for a percentage. For annual compounding, effective equals nominal.
Rate and number of periods per compounding: Periodic rate = r ÷ m; number of periods = m × t; Use these two values in A = P(1 + i)^n, where i is the periodic rate and n is the number of periods.
Compound interest versus simple interest for two years: CI − SI = P × (r)² (annual compounding, 2 years); Works only for exactly two years with annual compounding. The difference is the interest on the first year's interest.

How to solve Simple and Compound Interest questions

Use the same routine for every question. It stops unit errors, which cost the most marks.

1Read the question and note whether interest is simple or compound. If it does not say, check whether the question mentions compounding or the term 'effective'.
2List P, the nominal annual rate r, the time in years t, and the compounding frequency m.
3Convert the rate to a decimal and the time to years. Convert months and days carefully.
4For simple interest, use SI = P × r × t and add P if the amount is asked.
5For compound interest, compute the periodic rate r ÷ m and the number of periods m × t. Then calculate A = P(1 + i)^n.
6Subtract P from A if compound interest is asked, not the amount.
7For an effective rate, compute (1 + r/m)^m − 1 and express it as a percentage.
8State the answer with the rupee sign and units, and add a one-line conclusion if the question asks for a comparison or advice.

Quickest way: Periodic-rate shortcut

When to use it: Use it when the question asks for an amount or effective rate with quarterly, half-yearly or monthly compounding and you have a calculator-free or time-limited setting.

Write i = r ÷ m and n = m × t first, before anything else.
Expand step by step for small n: multiply by (1 + i) once per period, rounding only at the end.
For the effective rate, compute (1 + i)^m directly. Quarterly at 8%: 1.02^4 = 1.08243216, so about 8.24%.
For a two-year, annual-compounding difference between CI and SI, use P × r² instead of full calculation.
Check reasonableness: compound amount must exceed the simple amount for the same rate and time when t > 1 year.

Common mistakes in Simple and Compound Interest

Using the annual rate and years directly when interest is compounded quarterly or monthly.
Students memorise A = P(1 + r)^t and apply it to every question.
Fix: Always compute i = r ÷ m and n = m × t first. For 12% quarterly for 2 years, i = 3% and n = 8.
Giving the amount when the question asks for compound interest.
The last step of subtracting the principal is skipped under time pressure.
Fix: Underline what is asked. Write CI = A − P as the final line.
Entering months as years, for example t = 6 instead of 0.5.
The period in the question differs from the rate's period.
Fix: Divide months by 12 before substituting. Write t = 6 ÷ 12 = 0.5 as a separate line.
Treating the nominal rate as the effective rate.
Both are called 'rate of interest' and look the same in the question.
Fix: If compounding is more than annual, the effective rate is higher than the nominal rate. Compute (1 + r/m)^m − 1 when asked to compare.
Applying the shortcut CI − SI = P r² to three years or to non-annual compounding.
The shortcut is remembered without its conditions.
Fix: Use it only for two years with annual compounding. Otherwise compute both amounts.
Rounding the periodic rate or intermediate powers too early.
Students want neat numbers.
Fix: Keep at least four to six decimal places in intermediate steps and round only the final answer to two decimals.

Worked examples

Example 1

A bank accepts a deposit of ₹2,00,000 for 2 years at 8% per annum compounded quarterly. Find the amount at maturity and the compound interest. Also find the simple interest on the same deposit at 8% for 2 years and the difference between the two.

Show the solution

P = ₹2,00,000; r = 8% = 0.08; m = 4; t = 2 years.
Periodic rate i = 0.08 ÷ 4 = 0.02. Number of periods n = 4 × 2 = 8.
A = 2,00,000 × (1.02)^8.
(1.02)^2 = 1.0404; (1.02)^4 = 1.0404 × 1.0404 = 1.08243216; (1.02)^8 = 1.08243216 × 1.08243216 = 1.171659381 (approx.).
A = 2,00,000 × 1.171659381 = ₹2,34,331.88 (approx.).
CI = 2,34,331.88 − 2,00,000 = ₹34,331.88.
SI = 2,00,000 × 0.08 × 2 = ₹32,000.
Difference = 34,331.88 − 32,000 = ₹2,331.88.

Answer: Amount at maturity ≈ ₹2,34,331.88; compound interest ≈ ₹34,331.88; simple interest = ₹32,000; compound interest is higher by ≈ ₹2,331.88.

Example 2

Bank A offers 12% per annum compounded half-yearly. Bank B offers 11.8% per annum compounded annually. Which is better for a depositor? Find the effective rate of Bank A.

Show the solution

Bank A: r = 0.12; m = 2.
Periodic rate = 0.12 ÷ 2 = 0.06.
Effective rate = (1.06)^2 − 1 = 1.1236 − 1 = 0.1236, that is 12.36%.
Bank B compounds annually, so its effective rate equals its nominal rate, 11.8%.
Compare: 12.36% > 11.8%.

Answer: Effective rate of Bank A is 12.36% per annum. Bank A is better for the depositor because its effective rate is higher than the 11.8% of Bank B.

Exam tips

Write the given data (P, r, m, t) in a small list before calculating. Examiners award marks for correct identification of values even if arithmetic slips.
Show the formula, the substitution and the result in separate lines. Descriptive papers reward visible method.
When asked to compare two products, compute effective rates and end with a one-line conclusion for the depositor or borrower.
Be ready for short theory questions: define nominal and effective rate, and state why compounding frequency changes the result.
State assumptions, such as a 365-day year for day counts, when the question does not specify.

Practice questions from Calculation of Interest and Annuities

Simple and Compound Interest in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Simple and Compound Interest: frequently asked questions

What is the difference between simple interest and compound interest?

Simple interest is calculated only on the original principal, so it is the same every period. Compound interest is calculated on the principal plus accumulated interest, so it grows faster. For more than one period at the same rate, compound interest is higher.

How do you calculate the effective rate of interest?

Divide the nominal annual rate by the number of compounding periods in a year, add 1, raise the result to the power of the number of periods, and subtract 1. For 12% compounded half-yearly, (1.06)^2 − 1 = 12.36%.

What is the difference between nominal and effective interest rate in banking?

The nominal rate is the stated annual rate without allowing for compounding within the year. The effective rate is the actual annual return or cost after compounding. They are equal only when interest is compounded once a year.

Does more frequent compounding always give a higher amount?

Yes, for the same nominal rate, positive principal and the same time, more frequent compounding gives a higher amount. The increase becomes smaller as the frequency rises.