| TVM symbol | Meaning |
|---|---|
| N | N = (number of times interest is added per year) × (number of years). Examples: Monthly = 12 × years; Quarterly = 4 × years; Yearly = 1 × years. |
| I% | interest rate per year |
| PV | present value / initial amount |
| PMT | payment per period (0 if no regular payments) |
| FV | future value / ending amount |
| P/Y and C/Y | payments per year and compounding periods per year |
Sign convention: If money leaves your pocket at the start, PV is usually entered as negative.
If money comes back to you later, FV is positive.
Opposite signs stop calculator errors.
Quick setup example
A saver invests $4 000 now.
How would PV and FV usually be signed?
Step by step
- The saver pays money out now, so PV is negative.
- The future amount comes back later, so FV is positive.
Final answer
PV is usually negative and FV positive.
N, C/Y and P/Y — what to enter: N is the total number of compounding periods, not the number of years.
C/Y is how many times interest is added per year. P/Y always matches C/Y in IB.
- Half-yearly → C/Y = 2, so N = years × 2
- Quarterly → C/Y = 4, so N = years × 4
- Monthly → C/Y = 12, so N = years × 12
IB-style worked example: Try this question yourself first, then check the steps below.
Part (a) — find the amount after 5 years
Pietro invests €1500 at a nominal annual interest rate of 2.75%, compounded half-yearly.
Find the amount after 5 years.
Step by step
- Total compounding periods (N):
- Nominal annual rate stays as given:
- €1500 leaves your pocket → negative sign:
- No regular payments in this question:
- Half-yearly means the interest is added twice a year — so set:
- Solve FV → answer (positive, comes back to you):
Final answer
€1719.49
Part (b) — find the interest rate
Tommaso invests €1500.
He wants it to increase to 1.5 times the initial amount in 5 years, compounded quarterly.
Find the nominal annual interest rate r%.
Step by step
- Target amount = 1.5 × 1500. Calculate it first:
- Quarterly means the interest is added four times a year — so C/Y = 4.
- Total compounding periods (N):
- €1500 leaves your pocket (invest) → PV is negative. €2250 comes back → FV is positive:
- No regular payments:
- Compounding settings:
- Solve I% → this gives the nominal annual rate r:
Final answer
r = 8.19
3 things to check before submitting:
- C/Y must match the compounding frequency — half-yearly = 2, quarterly = 4, monthly = 12.
- N = years × C/Y (total periods, not just years).
- PV and FV must have opposite signs when both are given — the mark scheme checks this.
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Regular deposits — when PMT ≠ 0:
- PV = lump sum put in at the start.
- PMT = fixed amount added every period.
- Sign rule: PV and PMT must have the same sign — both negative if money leaves your pocket.
IB-style question: A student deposits USD 1000 into a savings account on 1 January.
The account pays 4% annual interest, compounded monthly.
At the end of each month, the student also deposits USD 100.
- (a) Find the balance after 2 years, correct to 2 decimal places. [3 marks]
- (b) Find the number of complete months for the balance to first exceed USD 5000. [2 marks]
Part (a) — find the balance after 2 years
What is FV after 24 months?
Step by step
- N = 24 (2 years × 12 months).
- I% = 4. The GDC divides by 12 per month automatically.
- Both the opening deposit and the top-ups leave the pocket — negative.
- P/Y = 12, C/Y = 12. Cursor on FV → solve.
Final answer
FV = 3577.43
Part (b) — solve for N
After how many complete months does the balance first exceed 5000?
Step by step
- Same setup as part (a). Now N is unknown and FV is known.
- The target comes back to the student → positive.
- PV = −1000, PMT = −100 unchanged — money still going out.
- Cursor on N → solve.
- 36 months is not enough. Round UP — the question says 'first exceed'.
Final answer
37 complete months
Two things students get wrong:
- PMT sign: if PV is negative, PMT must also be negative. Both represent money leaving your pocket.
- Rounding N: the GDC gives 36.5, not 37. Always round up for 'first exceed' or 'how many complete months'.
| Error | Why it happens | Fix |
|---|---|---|
| Wrong sign on PV/FV | Calculator expects cash-flow direction | Use opposite signs |
| Wrong N | Using years instead of periods | Convert years to total periods |
| Wrong P/Y or C/Y | Forgetting frequency settings | Match them to the question |
| Answer looks unrealistic | Setup mistake or wrong units | Check with rough mental estimate |
Sanity-check the answer: If 5% interest for a few years gives a final value smaller than the starting value, something has gone wrong.
Quick common-sense checks catch many calculator-entry errors.