| Variable | Meaning in annuity questions |
|---|---|
| N | total number of payment periods |
| PMT | regular payment or deposit |
| PV | starting amount / loan amount |
| FV | ending balance or future value |
| I% | interest rate |
Choose the unknown: Read the question carefully: are you solving for the final value, the payment, or the number of periods?
Setup example
A saver deposits $200 each month for 5 years. Which TVM entries are immediate?
Step by step
- Monthly means N = 5 × 12 = 60.
- PMT = 200, and the period frequency settings should match monthly.
Final answer
N = 60, PMT = 200, monthly settings.
Match the period frequency: Monthly payments mean monthly periods. This must match the calculator settings.
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Repayment setup example
A $12 000 loan is repaid monthly over 4 years. What is N?
Step by step
- Monthly over 4 years gives 4 × 12 periods.
Final answer
N = 48
Periods, not years: Do not type N = 4 if the payments are monthly. N must count payment periods.
The scenario: Lena buys furniture for €7 200. The shop offers two loan options.__LINEBREAK___Option A: 3-year loan at 12 % per year, compounded monthly. No deposit. Option B: 3-year loan at r % per year, compounded semi-annually. 15 % deposit paid upfront (before any repayments begin). Semi-annual repayments of €1 280.
Part (a) — find the monthly repayment [7]: (i) Find the monthly repayment under Option A. (ii) Find the total amount paid. (iii) Find the interest charged.
Part (a) solution
Option A: 3-year loan at 12 %, compounded monthly.
Step by step
- (i) TVM Solver — N=36, I%=12, PV=7200, FV=0, P/Y=C/Y=12 → solve PMT:
- (ii) Total paid — Lena makes 36 monthly payments of €239.13 each. Multiply to find the full amount she hands over:
- (iii) Interest charged — Lena only borrowed €7 200, but she paid back €8 608.68 in total. The extra amount is the cost of borrowing — this is the interest:
Final answer
PMT = €239.13 | Total = €8 608.68 | Interest = €1 408.68
Part (b) — find the interest rate r [5]: (i) Find the amount Lena borrows under Option B. (ii) Find the value of r.
Part (b) solution
Option B: 15 % deposit paid upfront, semi-annual repayments of €1 280.
Step by step
- (i) Lena pays 15 % of the price before the loan starts. That reduces how much she needs to borrow:
- (ii) TVM Solver — N=6, PV=6120, PMT=−1280, FV=0, P/Y=C/Y=2 → solve I%:
Final answer
Borrowed = €6 120 | r = 13.86 %
Part (c) — which option is cheaper? [2]: State which option Lena should choose. Justify your answer.
Part (c) solution
Compare the total cost of each option.
Step by step
- Total A (no deposit):
- Total B (all 6 repayments + the upfront deposit):
Final answer
Lena should choose Option A. €8 608.68 < €8 760, so Option A costs €151.32 less. Both totals must be shown for full marks.
Part (d) — real future value [4]: Lena chooses Option B. The shop invests every payment at 0.4 % per month. Inflation is 0.15 % per month.__LINEBREAK__Find the real future value of all Option B payments at the end of the 3-year period.
Part (d) solution
Use the real rate = investment rate − inflation rate.
Step by step
- Real monthly rate:
- Grow the deposit (paid at month 0) for 36 months:
- Effective real semi-annual rate:
- TVM: N=6, I%=1.5094, PV=0, PMT=1280 → grow the 6 repayments:
- Total real future value:
Final answer
Real future value ≈ €9 177
Easy marks to lose:
- Part (b): use P/Y = 2, not 12. Semi-annual ≠ monthly. Check the compounding frequency every time.
- Part (c): show both totals. Writing only 'Option A' earns 0. Both numbers must appear.
- Part (c): add the deposit to Total B. Total B = (repayments × n) + deposit, not just repayments.
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Interpret the answer: If the calculator gives a non-integer number of periods for a 'full months' question, you may need to round up.
Interpretation example
A calculator gives N = 26.4 months. How many full monthly payments are needed?
Step by step
- 26.4 months means 26 full payments are not enough.
- So you need the next whole payment count.
Final answer
27 full monthly payments