Time value of money is ONE equation with FIVE variables. Solve for any one given the other four, and you have solved 60% of the quantitative methods section of CFA Level 1 + most DCF questions on finance interviews. Most candidates memorise formulas instead of internalising the lever. And lose the ability to set up novel problems. The trick: learn the financial calculator workflow and the five variables; formulas become a corollary.
By the end, you will know the five TVM variables, the two cash-flow patterns (single sum / annuity / perpetuity / uneven), the annuity-due vs ordinary distinction, and the mental model for any new TVM question.
Five variables
N. NUMBER OF PERIODS (years, months, quarters. Match to payment frequency).
I/Y. INTEREST RATE per period (convert annual rate if compounding is not annual).
PV. PRESENT VALUE (cash outflow today = negative; cash inflow = positive, per sign convention).
PMT. PERIODIC PAYMENT (equal cash flow each period, 0 for single-sum cases).
FV. FUTURE VALUE (cash flow at end of N periods).
The five variables satisfy ONE equation (or two, depending on cash flow pattern). Give four of the five; solve for the fifth. Financial calculators (HP 12C, BA II Plus) have buttons for each; enter four, press CPT + the fifth button. Learn this workflow before memorising formulas. The calculator does the arithmetic.
Four cash-flow patterns
SINGLE SUM. PV to FV or vice versa. FV = PV x (1 + r)^N. PMT = 0. Simplest pattern.
ANNUITY. Series of EQUAL payments. PV and FV factor formulas apply: PV of annuity = PMT x [1 - (1+r)^-N] / r.
PERPETUITY. Infinite annuity. PV = PMT / r. Used for bond pricing, dividend discount models.
Most exam questions fall into one of these four patterns. Identify the pattern before setting up the calculation. Uneven cash flows can look overwhelming but reduce to a sum of single-sum PVs; the calculator handles them via the NPV function.
Ordinary annuity vs annuity-due
ORDINARY ANNUITY: payments at END of period. Most mortgages, most bonds, most standard patterns. ANNUITY-DUE: payments at BEGINNING of period. Rent-payments are annuity-due (rent paid at start of month); retirement annuity is often annuity-due (payment received at start of retirement year). Annuity-due PV = Ordinary PV x (1 + r). Exam routinely gives same numbers in both flavours to check this distinction.
On financial calculators: "BGN" mode toggles to annuity-due; default is "END" (ordinary). Switching modes is a common exam question. Do you remember to switch for an annuity-due problem? The modest 1+r factor adjustment is one of the most-tested subtleties.
The general mental model
For any TVM problem: (1) draw a TIMELINE showing all cash flows, (2) identify the pattern (single / annuity / perpetuity / uneven), (3) identify what is given and what is unknown, (4) set up calculator with four of five variables, (5) solve for the fifth. This sequence works for every TVM question regardless of surface complexity. The timeline step is the most skipped; skipping it causes most computational errors.
Ordinary annuity vs annuity-due
Ordinary annuity: payments at END of period. Annuity-due: payments at BEGINNING. Annuity-due PV = Ordinary PV x (1 + r). Exam often gives same numbers in both flavours to check mode-switching on the calculator.
Ordinary vs annuity-due
Ordinary annuity payments at END of period. Annuity-due at BEGINNING. Annuity-due = Ordinary x (1+r). Exam tests mode-switching routinely.
Match period frequency
N and I/Y must be in same period units. Monthly compounding = N in months, I/Y as monthly rate. Annual rate / 12 for monthly conversion. Mismatched frequency = wrong answer.
Perpetuity. Bond pricing and dividend discount
Preferred stock paying Rs. 100 dividend forever, required return 8%: PV = 100 / 0.08 = Rs. 1,250. Perpetuity pattern used for infinite cash flow valuation in Gordon dividend model, PVGO calculations, terminal value in DCF.
Key Takeaways
TVM = 5 variables: N, I/Y, PV, PMT, FV.
Single sum, annuity, perpetuity, uneven are the four cash-flow patterns.
PMT = 0 to collapses to PV-FV relationship (single sum case).
Annuity-due = Ordinary x (1+r); calculator BGN/END mode matters.
Perpetuity PV = PMT / r; used for infinite cash-flow valuation.
Read Next
Exam foundations done. The final set of articles concerns everyday money decisions. Debt, EMIs, budgeting. Where compounding works AGAINST you if mismanaged.
