Fundamentals of Business Mathematics and Statistics · Time Value of Money and Annuity - Simple and Compound Interest

Compound Interest and Compounding Periods: Formula and Sums

Updated 10 October 2026 · Fact-checked

Compound interest is interest earned on the principal plus the interest already added. Use A = P(1 + r/100m)^(mn), where m is the number of compounding periods in a year and n is the number of years. Divide the annual rate by m, multiply the years by m, then find CI = A − P.

Understand Compound Interest and Compounding Periods

Simple interest is always charged on the original principal. Compound interest is charged on the principal plus the interest added in earlier periods. So interest earns interest, and the amount grows faster than under simple interest.

The gap between two compounding dates is the compounding period. It can be a year, six months, three months or one month. At the end of each period, the interest is added to the principal. The next period's interest is worked out on this bigger amount.

The annual rate is quoted per year, but compounding happens more often. So you must convert both numbers. For half-yearly compounding, the rate per period is half the annual rate and the number of periods is twice the years. For quarterly, divide the rate by 4 and multiply the years by 4. For monthly, divide by 12 and multiply by 12.

The formula gives the amount (principal plus interest). The compound interest is the amount minus the principal. Many students stop at the amount and lose the mark, so check whether the question asks for A or for CI.

Sometimes the rate changes each year, for example 8% in year 1 and 10% in year 2. Then you cannot use one power. You multiply the growth factor of each year one after another.

Key formulas to remember

Compound amount (yearly compounding): A = P(1 + r/100)^n; P is the principal, r is the annual rate in % and n is the number of years.
Compound amount (m times a year): A = P(1 + r/(100 × m))^(m × n); m = 2 for half-yearly, 4 for quarterly, 12 for monthly. Rate per period = r ÷ m. Periods = m × n.
Compound interest: CI = A − P; Always subtract the principal if the question asks for interest.
Varying rates: A = P(1 + r1/100)(1 + r2/100)(1 + r3/100); Use one factor per year with that year's rate. Do not add the rates.
CI minus SI for 2 years (yearly compounding): CI − SI = P(r/100)²; Valid only for 2 years with yearly compounding. It is a quick way to find the principal when the difference is given.

How to solve Compound Interest and Compounding Periods questions

This method works for any compound interest question, whatever the compounding period.

1Note P, the annual rate r%, the time in years and how often interest is compounded.
2Find m, the number of periods in a year: 1 for yearly, 2 for half-yearly, 4 for quarterly, 12 for monthly.
3Find the rate per period = r ÷ m and the total number of periods = m × years.
4Write the growth factor per period as 1 + (rate per period ÷ 100).
5Raise the factor to the number of periods and multiply by P to get A. If rates vary, multiply the factors one year at a time.
6If the question asks for interest, subtract P from A.
7Check that A is larger than P and also larger than the simple interest amount for the same time.

Quickest way: Repeated multiplication with a clean factor

When to use it: Use it when there are 2 to 4 periods and the rate per period is a simple number such as 5%, 2% or 10%.

Convert to the rate and the periods per compounding period first.
Write the factor as a decimal: 5% becomes 1.05, 2% becomes 1.02, 10% becomes 1.1.
Multiply step by step: 1.1² = 1.21, 1.1³ = 1.331, 1.05² = 1.1025, 1.02² = 1.0404.
Example: ₹25,000 at 8% p.a. compounded quarterly for 6 months has 2 periods at 2%. A = 25,000 × 1.0404 = ₹26,010.
Look at the options. Remove any value that equals the simple interest answer or is smaller than P. Then match the last digits.

Common mistakes in Compound Interest and Compounding Periods

Using the annual rate and the number of years directly when interest is compounded half-yearly or quarterly.
Students remember A = P(1 + r/100)^n and forget to adjust it.
Fix: Always write m first. Divide the rate by m and multiply the years by m before putting numbers into the formula.
Reporting the amount as the compound interest.
The calculation ends at A and the question wording is skimmed.
Fix: Underline whether the question asks for amount or interest. If it asks for interest, subtract P.
Adding the rates when the rate changes each year, for example treating 8% and 10% as 18%.
Students try to reduce the question to one rate.
Fix: Multiply the yearly factors: P × 1.08 × 1.10. Each year's interest is on that year's opening amount.
Counting periods wrongly for part-years, such as 18 months with half-yearly compounding.
Time is not converted into the same unit as the compounding period.
Fix: Convert time to months and divide by the months in one period. 18 months ÷ 6 months = 3 periods.
Applying CI − SI = P(r/100)² to 3 years or to half-yearly compounding.
The shortcut is remembered without its conditions.
Fix: Use it only for 2 years with yearly compounding. Otherwise work out CI and SI separately.

Worked examples

Example 1

Find the compound interest on ₹40,000 for 1½ years at 10% per annum, compounded half-yearly.

Show the solution

P = ₹40,000, r = 10% p.a., time = 1.5 years, m = 2.
Rate per half-year = 10 ÷ 2 = 5%.
Number of half-years = 2 × 1.5 = 3.
A = 40,000 × (1.05)³.
1.05² = 1.1025 and 1.1025 × 1.05 = 1.157625.
A = 40,000 × 1.157625 = ₹46,305.
CI = 46,305 − 40,000 = ₹6,305.

Answer: Compound interest = ₹6,305 (amount = ₹46,305).

Example 2

₹50,000 is invested for 3 years. The rates of interest, compounded yearly, are 10% in the first year, 20% in the second year and 10% in the third year. Find the compound interest earned.

Show the solution

The rates differ each year, so multiply the yearly factors.
Year 1: 50,000 × 1.10 = ₹55,000.
Year 2: 55,000 × 1.20 = ₹66,000.
Year 3: 66,000 × 1.10 = ₹72,600.
Amount A = ₹72,600.
CI = 72,600 − 50,000 = ₹22,600.

Answer: Compound interest = ₹22,600.

Exam tips

Write the rate per period and the number of periods on your rough sheet before any multiplication. Most wrong options are built from errors in these two numbers.
Read the last line of the question. It may ask for the amount, the interest, or the difference between CI and SI.
Learn the powers of 1.1, 1.05 and 1.02 for 2 and 3 periods. They save time because the numbers in this paper are kept clean.
There is no negative marking, so if time runs out, eliminate options that are lower than P or equal to the SI answer and then guess.
For questions with an effective rate or a given amount, look at the related topics on nominal and effective rate and present value.

Practice questions from Time Value of Money and Annuity - Simple and Compound Interest

Compound Interest and Compounding Periods in other exams

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

Compound Interest and Compounding Periods: frequently asked questions

What is the compound interest formula for CMA Foundation?

The compound amount is A = P(1 + r/100)^n for yearly compounding. For m compounding periods in a year, use A = P(1 + r/(100m))^(mn). Compound interest is A − P.

How do I calculate compound interest half-yearly or quarterly?

Divide the annual rate by 2 for half-yearly or by 4 for quarterly. Multiply the number of years by the same number to get the periods. Then apply the compound amount formula and subtract P if interest is asked.

What is the difference between simple interest and compound interest?

Simple interest is always on the original principal, so it grows by the same amount each period. Compound interest is on the principal plus earlier interest, so it grows faster. For more than one period at a positive rate, compound interest is greater than simple interest.

What do I do if the rate of interest changes every year?

Do not use a single power. Multiply the principal by one growth factor for each year, using that year's rate. For example, 8% then 10% gives P × 1.08 × 1.10.