A 25-year-old who invests $5,000 at the start of every year, earning 8%, accumulates roughly $1,398,000 by age 65. Wait until 35 and the same $5,000 a year for 30 years yields about $611,000. That ten-year delay cost $787,000, on only $50,000 less in contributions.
You use a dollar today because you can invest it. That single idea drives every retirement projection, loan amortization, education funding plan, and investment analysis.
HIGH-FREQUENCY: The five-variable framework appears on virtually every TVM question. Identify the four knowns and one unknown before touching your calculator.
- PV (present value), current worth of a future sum or payment stream
- FV (future value), value at a future date, grown at a specified rate
- N (number of periods), total compounding or payment periods
- I/Y (interest rate per period), rate of return or discount rate per period
- PMT (payment), periodic cash flow (zero for lump-sum problems)
A single dollar compounds by multiplying itself by the growth factor once per period; reversing that chain discounts future money back to today.
Common mistakes
- Wrong annuity mode on the calculator. Having BEGIN mode set when the question describes end-of-period payments, or vice versa. The wrong-mode answer is off by exactly (1 + r), and that answer is always among the choices. Trap: Ordinary annuity at 7% gives $52,970. Annuity due trap answer: $52,970 x 1.07 = $56,678.
- Failing to adjust for inflation in retirement problems. When the question says "in today's dollars," inflate the payment to retirement-date dollars first, then apply the real-rate annuity factor. The common trap is using the nominal 7% rate with today's-dollar payments and never inflating, which yields ~$929,000 instead of $2,671,511 in David's example.
- Mixing compounding and payment frequencies. A monthly-payment mortgage at 6% annual uses 0.5% monthly and 360 periods, not 6% and 30. Forgetting to divide the rate and multiply the periods produces a wildly incorrect payment.
Bottom line
- Every TVM problem has five variables (PV, FV, N, I/Y, PMT); know four, solve for one.
- Ordinary annuity = end of period; annuity due = beginning. Wrong mode gives an answer off by exactly (1 + r).
- Retirement lump sums: read the stem's timing. Beginning-of-period withdrawals, the usual capital needs assumption, are an annuity due (BEGIN); end-of-period withdrawals stay in END.
- Real rate = (1 + nominal) / (1 + inflation) - 1, always slightly less than nominal minus inflation.
Exam shortcut
Before starting any TVM problem, mentally confirm two things: which variable am I solving for, and are payments at the beginning or end? The wrong annuity mode gives an answer off by exactly (1 + r), and that trap answer is always a choice. BEGIN = Before Everything Goes In Naturally.
The full lesson (about 5,602 words, 37 min read) adds 8 worked examples, all 5 common mistakes, a self-check, free in the app.
Learning objectives
- B.12
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