A 25-year-old who invests $5,000/year at 8% accumulates roughly $1,398,000 by age 65. Wait until 35 and the same $5,000/year for 30 years yields $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 ~$868,000 instead of $2,571,729 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).
- Real rate = (1 + nominal) / (1 + inflation) - 1, always slightly less than nominal minus inflation.
- NPV > 0 means accept; IRR > required return means accept. When they conflict, trust NPV.
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. Rent, insurance premiums, and lease payments trigger BEGIN mode. Loan payments and bond coupons use END mode.
The full lesson (about 4,295 words, 29 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.
Learning objectives
- B.12
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