Financial Management
Compounding and discounting
Simple interest applies to principal only; compound interest applies to principal plus accumulated interest.
Definition
Simple interest grows a balance linearly: the same money is earned each period on the original principal. Compound interest grows geometrically because each period's interest is added to the base and itself earns interest later.
Because lenders quote nominal rates with different compounding frequencies, the effective annual rate (EAR) converts any nominal rate into one comparable annual figure.
Intuition
Compounding is interest earning interest. The gap between simple and compound growth is tiny over one year and enormous over thirty.
When to use it
- Comparing products or loans that quote different compounding frequencies.
- Converting a nominal rate into the rate that actually applies to a year.
- Deciding between simple and compound assumptions in an exam question.
When not to use it
- As a substitute for a full timeline when contributions or withdrawals vary.
Formula
Simple interest
I = P × r × t
Interest on the original principal only.
Formula
Compound interest
FV = PV × (1 + r ÷ m)^(m·t)
Compounding m times a year for t years.
- m
- Compounding periods per year
- t
- Number of years
Formula
Effective annual rate
EAR = (1 + r ÷ m)^m − 1
Converts a nominal rate into a comparable annual rate.
Worked example
Comparing two quoted rates
A deposit pays 12% nominal compounded monthly. What is the effective annual rate?
- 1
Periodic rate
12% ÷ 12 = 1% per month (0.01).
- 2
Compound for a year
(1 + 0.01)¹².
- 3
Convert to a rate
1.01¹² = 1.126825, so EAR = 0.126825.
EAR = 12.68%, not 12%. The extra 0.68 points come from monthly compounding.
Common mistakes
- Comparing nominal rates without converting them to the same frequency.
- Using the EAR as if it were a monthly or quarterly periodic rate.
- Assuming continuous compounding unless the question says so.
How it connects
is part of
The source is a component of the target idea.
- Time value of money: Compounding is one half of the time-value relationship.
relates to
Useful to study together, in either order.
is an advanced form of
The source extends the target once the basics are secure.