The two anchor points on the timeline
Two values for the same money — one if it sits today, one if it sits later. They're connected by the discount rate r we just met. Money in the future is the future value (FV). Money today is the present value (PV). Two names for two ends of the same arrow.
The arrow points forward when you're growing today's money into the future:
Read this slowly. PV is what you have now. (1+r) is the multiplier per period — earn r percent and you have 1+r times what you started with. Do that n periods → (1+r)^n. That's just exponents from Track 1, lesson 5. Multiply PV by that and you've forward-projected to year n. FV is the result.
Going the other way — the arrow runs backward too
Same equation, different direction. If you know FV and want PV, just rearrange. Track 1 lesson 1 — the seesaw move. Divide both sides by (1+r)^n:
That's discounting — pulling future money back to today. (1+r)^n in the denominator. Bigger r means a bigger denominator, which means a smaller PV. Bigger n (longer wait) also means bigger denominator, smaller PV. Both observations are exactly the numerator–denominator play from Track 1 lesson 3 in action.
Concrete numbers, no calculator needed for the picture
You have ₩1,000 today. Stick it in a 5% account for 10 years. What's the FV?
FV = 1000 × (1.05)^10 ≈ 1000 × 1.629 ≈ ₩1,629. About 60% bigger than the start.
Reverse: someone offers you ₩1,629 in 10 years, with current rates at 5%. What's that worth today?
PV = 1629 / (1.05)^10 ≈ 1629 / 1.629 ≈ ₩1,000. The math is symmetric. You don't need to remember two formulas — they're the same equation read in opposite directions.
What changes if r or n changes
Let's see the levers in action with the same ₩1,000 today:
- 5% for 10 years → ₩1,629 (factor: 1.629)
- 10% for 10 years → ₩2,594 (factor: 2.594) — doubled the rate, more than 1.5x the result
- 5% for 20 years → ₩2,653 (factor: 2.653) — same rate, doubled the time, similar result
- 10% for 20 years → ₩6,727 (factor: 6.727) — both doubled, result more than 4x
Notice: doubling the rate or the time doesn't double the result — it does much more. That's the exponent at work. Compounding is non-linear, and that non-linearity is where finance gets its power (and where retirement planning has to live with it).
The takeaway
One equation, two views. FV = PV × (1+r)^n takes today's money to the future. PV = FV / (1+r)^n brings future money to today. Same arrow, different direction. Every other TVM equation in finance — annuity, perpetuity, Gordon Growth — is built on top of this single relationship. So burn it in: FV and PV are connected by the discount factor (1+r)^n.