TVM Rocket — Cash-Flow PV
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Present Value — Intuition First
Money later is worth less than money today. PV tells us the fair price today for future cash flows by shrinking each future payment using the rate r.
Cash‑Flow Stream (finite)
Different amounts for a few years
Discount each yearly payment back to today and add them up.
Formula: PV = Σ Ct/(1+r)t
0
1
$125.96
2
$106.54
3
$130.55
Example: r = 10.01%, payments: $125.96, $106.54, $130.55
PV = $125.96/(1+0.1001)^1 + $106.54/(1+0.1001)^2 + $130.55/(1+0.1001)^3 = $300.59
Perpetuity (level)
Same amount every year forever
Because payments never stop, the PV is like price = payment ÷ rate. First payment arrives in 1 year.
Formula: PV = C / r
0
1
$94.73
2
$94.73
3
$94.73
4
$94.73
5
$94.73
⋯
$94.73
⋯
Example: C = $94.73, r = 3.04%
PV = C/r = $94.73 / 0.0304 = $3116.12
Annuity (level)
Same amount every year for n years
End‑of‑year payments. Like a short perpetuity that stops after n years.
Formula: PV = C · (1 − (1+r)−n) / r
0
1
$136.64
2
$136.64
3
$136.64
4
$136.64
5
$136.64
6
$136.64
7
$136.64
8
$136.64
Example: C = $136.64, n = 8, r = 10.12%
PV = C·(1 − (1+r)−n)/r = $136.64·(1 − (1+0.1012)−8)/0.1012 = $725.79
Common pitfalls
- Use decimal r in formulas (e.g., 8% → 0.08).
- Assume end‑of‑year timing unless stated. If payments start today (annuity‑due), multiply annuity PV by (1+r).
- Perpetuity payments must start in 1 year for PV=C/r to apply.
Correct answers raise the rocket!
Lesson demo — quiz starts when you click Start.