The five-variable method and eyeballing the answer
◈ 9 cardsFind the unknown, pick the function, check the size — reduce any single-sum problem to rate, nper, pmt, pv, fv, and apply four sanity rules before you commit the number.
Every problem is the same table
A single-sum TVM problem has five variables — rate, nper, pmt, pv, fv — and exactly one is unknown. The method is mechanical: write the five in a column, fill in the four you are given (converting the rate and the periods together, Lesson 6.3), and the empty row names the function. pmt is 0 for every lump-sum problem; it stops being 0 in Module 7.
Worked example 1 — "how much today"
How much must Ossington Brewing deposit today, at 3.5 % compounded monthly, to have $40,000 in 5 years?
| rate | nper | pmt | pv | fv |
|---|---|---|---|---|
| 0.035/12 | 5 × 12 = 60 | 0 | ? | 40,000 |
The unknown is pv: =-PV(0.035/12,60,0,40000) = 33,586.83. Eyeball: less than 40,000 (a present value must be), and not by much — 3.5 % for five years is about 19 % of growth, so a deposit around 84 % of the target is right.
Worked example 2 — "what rate"
What annual rate, compounded semi-annually, turns $2,500 into $4,000 in 10 years?
| rate | nper | pmt | pv | fv |
|---|---|---|---|---|
| ? | 10 × 2 = 20 | 0 | −2,500 | 4,000 |
The unknown is rate: =RATE(20,0,-2500,4000) = 0.023778 per half-year. The question asked for an annual rate: × 2 = 0.047557, the nominal 4.7557 % compounded semi-annually (the effective rate would be 1.023778² − 1 = 4.81 %). Eyeball: 2,500 to 4,000 is a 60 % gain over 10 years; the Rule of 72 says a doubling at 4.76 % takes about 15 years, so 60 % in 10 is consistent.
The four eyeball rules
- FV > PV whenever r > 0. A future value smaller than the deposit, or a present value larger than the target, is a sign or period error — not deflation, not a discount.
- A doubling in under 5 years needs more than 14 % (72/5 ≈ 14.4). A
RATEanswer of 3 % for a five-year doubling is wrong before you check it. - A payment larger than the loan is a period error (Module 7): the annual rate was fed with monthly periods, or vice versa.
- The answer's units match the question's. Periods → years, periodic rate → annual rate, before you write it down.
Four stems, four functions
"How much will I have" → FV. "How much must I deposit today" → PV. "How many years" → NPER. "What return did I earn" → RATE. Each with pmt = 0, pv and fv of opposite signs, and the rate and period count adjusted together. That is the whole of single-sum TVM; Module 7 adds the payment.