Time Value of Money
Time Value of Money
The exam tests single-sum compounding and discounting, annuities, perpetuities, rate conversion, and loan amortization. Nearly every later topic-valuation, capital budgeting, cost of capital-depends on these mechanics.
Core relationships
- FV = PV(1+r)^n and PV = FV/(1+r)^n, where r is the periodic rate and n the number of periods.
- Ordinary annuity: PV = PMT × [1 − (1+r)^−n]/r; FV = PMT × [(1+r)^n − 1]/r.
- Annuity due = ordinary annuity × (1+r), because each payment occurs one period earlier.
- Perpetuity PV = PMT/r; growing perpetuity PV = PMT1/(r − g), valid only when r > g.
- APR to periodic rate: r = APR/m. EAR = (1 + APR/m)^m − 1; continuous compounding EAR = e^APR − 1.
- Loan payment: PMT = PV / annuity factor; interest each period = beginning balance × r, and the remainder reduces principal.
- Rule of 72: doubling time ≈ 72/(rate in percent).
- Uneven cash flows: discount each flow individually and sum; a timeline prevents date errors.
Worked example
$5,000 is deposited at 8% APR compounded quarterly for 3 years. The periodic rate is 2% and there are 12 periods, so FV = 5,000(1.02)^12 = 5,000 × 1.26824 = $6,341.20. The EAR is (1.02)^4 − 1 = 8.243%, above the 8% APR. Now value a 10-payment $1,000 ordinary annuity at 6%: PV = 1,000 × [1 − 1.06^−10]/0.06 = 1,000 × 7.3601 = $7,360.10. If payments came at the beginning of each year, multiply by 1.06 to get $7,801.71.
Traps
- Dividing APR by 12 when compounding is quarterly; always divide by the actual m.
- Assuming APR equals EAR-more frequent compounding pushes EAR higher.
- Treating an annuity due as ordinary, losing one period of interest.
- Using D0 where D1 belongs, or applying a growth perpetuity when g ≥ r.
- Calculator sign conventions: PV and PMT must have opposite signs to solve correctly.