Here’s a question that sounds trivial but isn’t: would you rather have $100 today or $100 a year from now? Everyone picks today — but ask them why and most people shrug and say “impatience.” The real answer is sharper than that. Money you hold today can be invested and grow, so $100 now isn’t equal to $100 later — it’s strictly bigger, because it has a head start. Money has a clock, and time literally has a price. Nail that one idea and loans, bonds, mortgages, stock valuations and DeFi yields all start running on the same engine.
Why Money Has a Clock
Before you read — take a guess
Guess before reading: the main reason a dollar today is worth more than a dollar next year is…
A dollar today is worth more than a dollar tomorrow for one concrete reason: today’s dollar can go earn something in the meantime. Lend it, deposit it, invest it — by next year it has grown. So comparing money across different dates is like comparing prices in different currencies: you can’t just line up the face values, you have to convert them to the same point in time first.
That conversion runs in two directions, and they’re mirror images of each other:
| Direction | Name | Question it answers |
|---|---|---|
| Push money forward in time | Future value | What will today’s money grow into? |
| Pull money back to today | Present value | What is tomorrow’s money worth right now? |
Three reasons time has a price
Waiting for money costs you because of (1) opportunity cost — the return you could have earned meanwhile; (2) inflation — tomorrow’s dollar buys a little less; and (3) risk — a promise of future money might not be kept. The first one is the engine of the math below; the other two make the price of time even higher.
Future Value: Money Growing Forward
Future value (FV) answers: if I invest a sum today and let it compound, what will it be worth later? Each year the balance earns the rate r, and crucially, next year’s interest is earned on this year’s interest too. That’s compounding — interest on interest — and over time it does the heavy lifting.
where PV is the amount today, r is the rate per period, and t is the number of periods.
Worked example
You put $1,000 in an account earning 8% a year and leave it for 10 years:
Notice you didn’t just earn 8% × 10 = 80% (which would give $1,800). You earned
$1,159, because each year’s 8% was charged on a bigger and bigger base. That extra
$359 over the naive straight-line guess is the compounding.
Drag the rate and the horizon below and watch the compound curve peel away from the flat “simple growth” line — the gap between them is interest-on-interest:
- Final value
- $4,661
- CAGR
- 8%
Simple growth adds the same amount each year. Compound growth earns interest on past interest — so it curves upward and leaves the straight line behind.
The Rule of 72
Want a fast mental estimate of how long money takes to double? Divide 72 by the
percentage rate. At 8%, that’s 72 ÷ 8 = 9 years. Check it: 1.08⁹ ≈ 1.999 — almost
exactly double. The Rule of 72 is a back-of-the-napkin shortcut for compounding, no
calculator required.
An account earns 6% a year. Roughly how long until the balance doubles?
Present Value: Tomorrow’s Money, Today’s Price
Now flip the arrow. Present value (PV) answers the question that actually shows up
in real decisions: someone promises me a sum in the future — what is that promise
worth to me right now? Since money grows forward by multiplying by (1 + r), pulling
it back to today means dividing by (1 + r) for each year you have to wait. That’s
discounting — compounding played in reverse.
Worked example
A bond will pay you $1,000 in 10 years. If you can earn 8% elsewhere, what is that payment worth today?
So a $1,000 promise a decade out is worth only about $463 today — less than half its face value. You’d be indifferent between $463 now and $1,000 in ten years, because $463 invested at 8% grows right back to $1,000. Present value and future value are the same trip, walked in opposite directions.
Drag the discount rate and the years-to-payment and watch a fixed $1,000 promise shrink the longer you wait and the higher the rate:
- Worth today
- $215
- Cents on the dollar
- 21¢
A promised payment loses value the longer you wait and the higher the discount rate. The curve is compounding played backwards — each year divides by another (1 + r).
Before you read — take a guess
Guess: holding everything else fixed, what happens to the present value of a future payment as the discount rate rises?
The Discount Rate: The Dial That Sets the Price of Time
The rate r does double duty: going forward it’s the growth rate, going backward it’s the discount rate. It’s the single most important — and most argued-about — number in finance, because it sets how harshly the future is penalized. Watch what a $1,000 payment 30 years out is worth today at different rates:
| Discount rate | PV of $1,000 in 30 years | Cents on the dollar |
|---|---|---|
| 3% | ≈ $412 | 41¢ |
| 6% | ≈ $174 | 17¢ |
| 10% | ≈ $57 | 6¢ |
| 15% | ≈ $15 | 2¢ |
At a 15% discount rate, a $1,000 promise three decades away is worth a measly $15
today. Two forces are multiplying here: time (more years = more divisions by
1 + r) and rate (a bigger divisor each year). Stack them and far-future money
all but vanishes.
Where does the discount rate come from?
There’s no single “correct” rate — it’s the return you could earn on a comparable alternative (your opportunity cost), bumped up for risk. A safe government bond might justify 3–4%; a risky startup’s promised payout might demand 20%+. Because PV is so sensitive to this number, a valuation can swing wildly just from the rate someone chooses — which is why arguments about “the right discount rate” are really arguments about value.
Pick the right word for each blank.
Pick the right option for each blank, then check.
Pushing money forward in time is called , which by (1 + r) each period. Pulling money back to today is called , which by (1 + r). A higher rate makes present value .
Compounding Frequency: The Rate Isn’t the Whole Story
One subtlety trips people up: how often interest compounds matters, not just the headline rate. The more frequently interest is added, the sooner it starts earning its own interest. Take 12% a year on $1,000:
| Compounding | Periods/year | Growth factor | Value after 1 year |
|---|---|---|---|
| Annual | 1 | $1,120.00 | |
| Quarterly | 4 | $1,125.51 | |
| Monthly | 12 | $1,126.83 | |
| Daily | 365 | $1,127.47 |
Same 12% on the label, but monthly compounding actually earns 12.68% once you account for interest-on-interest within the year. That “true” yearly rate has a name — the effective annual rate — and untangling it from the nominal rate is exactly what the interest & yield lesson digs into next.
Connect every concept on the left to its correct description.
Pick a term, then click its definition.
Putting It Together
Two values, one engine. Future value pushes today’s money forward by multiplying; present value pulls tomorrow’s money back by dividing. The rate r is the hinge between them. Chunk it into one picture:
Big picture
Money's clock
- Time value of money
- Future value — forward
- FV = PV × (1 + r)^t
- Compounding: interest on interest
- Rule of 72: doubling time ≈ 72 ÷ rate
- Present value — backward
- PV = FV ÷ (1 + r)^t
- Discounting: compounding in reverse
- Higher rate or longer wait → smaller PV
- The rate r
- Growth rate going forward
- Discount rate going backward
- Set by opportunity cost + risk
- Future value — forward
Sort each task by which direction in time it travels.
Place each item in the right group.
- Price a promise of money you'll receive later
- Divide by (1 + r) for each year
- Find what a $1,000 bond payment in 20 years is worth today
- Multiply by (1 + r) for each year
- Estimate doubling time with the Rule of 72
- Find what $1,000 saved today is worth in 20 years
A mixed recap — it pulls from everything above:
What is the present value of $1,000 to be received in 10 years, discounted at 8%?
Check your answer to continue.
Key Takeaways
What to remember
- A dollar today beats a dollar tomorrow because today’s dollar can be put to work and grow. Money has a clock; comparing sums across dates means converting them to the same point in time first.
- Future value pushes money forward:
FV = PV × (1 + r)^t. Compounding (interest on interest) makes it grow faster than simple straight-line interest. Rule of 72: doubling time ≈ 72 ÷ rate. - Present value pulls money back:
PV = FV ÷ (1 + r)^t. Discounting is compounding in reverse — divide by(1 + r)for every year you wait. - The rate r is the hinge — a growth rate going forward, a discount rate going backward. A higher rate or a longer wait both shrink present value, and far-future money can be worth almost nothing today.
- Compounding frequency matters too: the same nominal rate is worth more the more often it compounds — the bridge to nominal vs. effective rates in the next topic.