The big idea: If interest is added more often, the balance starts earning interest on that added interest sooner. That usually means more frequent compounding gives a larger final amount.
| Rate statement | How often interest is added | Result |
|---|---|---|
| 6% compounded yearly | Once per year | Slower growth |
| 6% compounded quarterly | 4 times per year | More growth than yearly |
| 6% compounded monthly | 12 times per year | More growth than quarterly |
Same nominal rate does not mean same final value: Two accounts can both say '6% per year', but if one compounds monthly and the other yearly, they will not end with the same amount.
Quick comparison
Which will be larger after one year: 1 000 at 8% compounded monthly?
Step by step
- Both have the same annual nominal rate: 8%.
- The monthly account adds interest 12 times, so the balance starts earning on earlier interest sooner.
Final answer
The monthly-compounded account will be larger.
Common mistake: Students often compare only the percentage and ignore the compounding frequency. In finance questions, the phrase 'compounded monthly' is never there by accident.
- present value (starting amount)
- future value (ending amount)
- annual nominal rate as a percentage
- number of compounding periods per year
- number of years
| Compounding | k | Interest added each period |
|---|---|---|
| Yearly | 1 | r/1 |
| Quarterly | 4 | r/4 |
| Monthly | 12 | r/12 |
| Daily | 365 | r/365 |
Worked example — monthly compounding
Find the value of $3 000 invested at 6% per year for 2 years, compounded monthly.
Step by step
- Write down the values.
- Substitute into the formula.
- Simplify the bracket and power.
- Calculate.
Final answer
The investment is worth about $3 381.11.
Two places students go wrong: Do not forget to divide the annual rate by k, and do not forget that the total number of compounding periods is kn, not just n.
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The big idea: The nominal rate is the rate the bank advertises per year. The effective growth is what really happens after the compounding frequency is taken into account.
For example, 12% compounded monthly does not mean the account simply grows by 12% once. It grows by 1% each month, and those monthly gains themselves start earning interest.
Worked example — one-year effective multiplier
Find the one-year multiplier for 12% nominal interest compounded monthly.
Step by step
- Monthly rate = 12% ÷ 12 = 1%.
- Monthly multiplier = 1.01.
- There are 12 months in one year.
- Calculate.
Final answer
The effective one-year multiplier is about 1.1268, so the account grows by about 12.68% over the year.
Why this matters: This is why two financial products with the same nominal rate can still produce different final values. The more frequent compounding creates a larger effective yearly growth.
The big idea: IB often asks which option is better. That means you must calculate both options and then write a conclusion that matches the context.
Worked comparison example
Which is better after 3 years for a $4 000 deposit: 5% compounded yearly, or 5% compounded quarterly?
Step by step
- Yearly compounding:
- Quarterly compounding:
- Compare the two final amounts.
Final answer
The quarterly-compounded account is better because it gives the larger final value after 3 years.
What 'compare' really means: Do not stop after writing the two answers. A comparison question needs a sentence such as 'Option B is better because...' using the actual values.
| Question wording | What IB wants |
|---|---|
| Find the value | One correct final amount |
| Determine which is better | Both values and a decision |
| Compare the options | A numerical comparison plus a conclusion in context |