The time line is just a number line for money. Mark today as period 0 (the leftmost point). Mark each future period (1, 2, 3, ...) along the axis. Above each period, draw an arrow up for money coming to you, or down for money leaving you. The magnitude (size) of each arrow is the dollar amount. That is it. The visual carries 90% of the work; the algebra is just compressing the picture into a single number.
| Real-world scenario | Period 0 (today) | Periods 1 to N | Timeline shape |
|---|---|---|---|
| Tax refund of $3,000 arriving in 1 year | (nothing) | Up arrow at period 1: +$3,000 | Single future sum |
| Buying a $40,000 car with cash today | Down arrow: −$40,000 | (nothing) | Single present sum |
| $2,000/month rent for 12 months | (nothing) | 12 down arrows of $2,000 each | Annuity (12 periods) |
| An endowment paying $40,000/year forever | (nothing) | Up arrows of $40,000, every year, no end | Perpetuity |
| Mortgage: receive $300K today, pay $1,799/mo for 360 months | Up arrow: +$300,000 | 360 down arrows of $1,799 each | Single sum + annuity (mixed) |
Sign convention: arrows above the axis are inflows (positive cash flow to you); arrows below the axis are outflows (negative cash flow from you). The same transaction has opposite signs from the two parties' perspectives. When a bank lends you $300,000, the bank draws a down arrow at period 0 (money leaves them) — but you draw an up arrow at the same period (money arrives to you). This sounds obvious. It causes more student confusion than any other single thing in TVM. Always ask: whose perspective am I drawing?
Period 0 is now. Period 1 is the end of period 1, not the start. This convention is universal — every textbook, every spreadsheet function (PV, FV, PMT, NPV), every CFA exam problem assumes it. A 5-year investment runs from period 0 to period 5. A loan whose first payment is due immediately (rare — usually called an annuity-due) has its first arrow at period 0, not period 1. Most loans, leases, and bond coupons pay at the END of each period, so the first arrow lands at period 1.
Three structures, three formulas, every problem. (1) Single sum: one arrow somewhere on the line — use PV = FV/(1+r)ⁿ to slide it left, or FV = PV·(1+r)ⁿ to slide it right. (2) Annuity: equal arrows over a fixed stretch — use the PV-annuity or FV-annuity formula. (3) Perpetuity: equal arrows forever — use PV = C/r. Mixed problems (mortgages, bonds, leveraged buyouts) decompose into these three. Once you can identify which structure you have from the timeline, choosing the formula is automatic.
Sit with the ideas.
You are taking out a $300,000 mortgage at 6% for 30 years and want to draw the cash flows from your perspective as the borrower. Which arrows should the timeline show?