The big idea: Each deposit grows for a different number of periods.
The earliest deposits have the most time to earn interest.
| Deposit | Time available to grow |
|---|---|
| First deposit | longest |
| Middle deposit | medium |
| Last deposit | shortest |
Not all deposits grow equally long: This is why an annuity is not just payment × number of periods.
Future value: The future value of a savings annuity is the total value at the end after all the deposits and their growth are combined.
Small-scale example
Three deposits of $100 are made yearly into an account earning 10% per year.
Which deposit contributes the most at the end?
Step by step
- The first deposit has the most time to grow.
- The last deposit has the least time to grow.
Final answer
The first deposit contributes the most.
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Main approach: In IB, these questions are often handled using TVM or annuity functions on the calculator, but you should still understand what the variables mean.
| Variable | Meaning |
|---|---|
| PMT | regular deposit |
| N | number of periods |
| I% | interest rate |
| FV | future value |
Savings annuity sign sense: For calculator workflows, deposits may be entered with one sign and the final amount with the opposite sign, depending on convention.
IB-style question — savings annuity future value [3 marks]
Dev sets up a regular savings plan paying a nominal annual interest rate of 5.4%, compounded monthly. He deposits $250 at the end of each month for 6 years. Find the future value of the plan, correct to the nearest cent.
Step by step
- Identify the TVM inputs. Monthly for 6 years gives N = 6 × 12, with no opening balance.
- Enter the deposit as a negative PMT (money out), then solve for FV on the GDC.
- Read the future value from the solver.
Final answer
FV = $21 202.37
More than one way to save: IB may ask which savings plan gives the larger final value.
You compare the final accumulated amounts, not just the deposit sizes.
Reasoning example
Which might produce more after many years: smaller monthly deposits at a better rate, or larger deposits at a weaker rate?
Step by step
- You cannot decide from one feature only.
- You must compare the final values after the same time period.
Final answer
Calculate both plans and compare the future values.
See how examiners mark answers
Access past paper questions with model answers. Learn exactly what earns marks and what doesn't.
Two questions. One linked answer.: First, you're given a lump sum that grew to a larger amount over several years. You find the interest rate that caused the growth.
Then that money moves into a fund that pays out a fixed amount every month. You find how much is left after a number of years.
The balance you found in the first part becomes the starting amount in the second.
IB-style worked example: Try this question yourself first, then check the steps below.
Part (a) — find the interest rate
Lucas deposited $75 000 into a savings account with a nominal annual interest rate of I% compounded monthly. At the end of 6 years, the balance had grown to $102 000.
(a) Find the value of I.
Step by step
- No regular deposits — PMT = 0.
- N = 6 × 12 = 72, PV = ±75 000, FV = ±102 000, PMT = 0, P/Y = C/Y = 12
- Cursor on I%, press ALPHA + ENTER to solve.
Final answer
I = 5.13 (5.1272...)
Part (b) — find the remaining balance
Lucas withdraws the $102 000 and places it in an annuity earning 5.7% nominal annual interest, compounded monthly. At the end of each month he receives a payment of $650.
(b) Find the amount remaining after 10 years. Express your answer to the nearest dollar.
Step by step
- Use the $102 000 from part (a) as PV. The rate is now 5.7% — not the rate from part (a).
- N = 10 × 12 = 120, I% = 5.7, PV = ±102 000, PMT = ∓650, P/Y = C/Y = 12
- Cursor on FV, press ALPHA + ENTER to solve.
- Round to the nearest dollar.
Final answer
$75 315
3 things to check before submitting:
- PMT = 0 in part (a) — do not enter a monthly payment for the growth phase.
- The rate in part (b) comes from the question, not from part (a).
- Round your final answer to the nearest dollar — the last mark is lost if you leave it unrounded.