Nearly every financial decision — a loan, a bond, a retirement account, a business valuation — comes down to the same question: what is money worth across time? Interest answers half of that question, and present value answers the other half. They're really the same formula pointed in opposite directions, and once that clicks, a surprising amount of finance stops looking like separate topics and starts looking like one idea applied repeatedly.
Simple interest vs. compound interest
Interest is the cost of borrowing money, charged as a percentage of the principal — the amount actually borrowed. There are two ways to charge it, and the difference between them compounds into real money faster than it looks like it should.
Simple interest only ever applies to the original principal. Borrow $1,000 at 10% for three years, and you owe 10% of that original $1,000 each year: $1,000 × (1 + 0.10 × 3) = $1,300. The formula is A = P(1 + rt) — amount equals principal times one plus rate times time.
Compound interest charges that same 10%, but on whatever you owe right now, not the original amount. Year one, you owe $100 on $1,000, same as before, bringing the balance to $1,100. Year two, the 10% applies to $1,100, not $1,000 — that's $110, not $100 — bringing the balance to $1,210. Year three, 10% of $1,210 is $121, for a final balance of $1,331. The formula swaps addition for an exponent: A = P(1 + r)^t, or $1,000 × (1.10)³ = $1,331.
Thirty-one dollars might not look dramatic over three years. Stretch the same math to thirty years and the gap between simple and compound stops being rounding error — it becomes the entire reason compound interest is treated as either a wealth-building engine or a debt trap, depending on which side of the loan you're standing on.
The dollar in your hand is worth more than the one you're owed
This is the same exponent, aimed the other direction. If a dollar today can be invested and grow, then a dollar promised to you later has to be worth less than a dollar today — the question is exactly how much less.
Finding out how much $100 today will be worth in three years at a 10% rate is just compound interest again, with "present value" standing in for principal: PV × (1 + r)^t = $100 × (1.10)³ = $133.10.
Running it backward is where present value actually earns its name. If someone offers you $100 three years from now, and the going rate is 10%, that promise is worth FV ÷ (1 + r)^t = $100 ÷ (1.10)³ = $75.13 today — not $100. That's the real comparison worth sitting with: $75.13 in your hand today, or $100 in your hand three years from now. They're worth exactly the same amount, by construction; the discount rate is what makes them equivalent.
A dollar owed to you in the future isn't a smaller dollar. It's the same dollar, minus everything it could have earned in the meantime.
Pricing anything with a future payoff is just this, stacked
Once you can discount a single future payment back to today, pricing something with multiple future payments — a bond, a business, a royalty stream — is just repeating the same calculation and adding the results together.
Say a bond promises $500 in five years and $650 in seven years, at a 5% discount rate, with both payments guaranteed. Discount each one separately: $500 ÷ (1.05)⁵ = $391.76, and $650 ÷ (1.05)⁷ = $461.94. Add them together and the bond is worth $853.70 today — not $1,150, which is what the two payments add up to with no accounting for time at all.
The rate does the real work in both directions — raise it, and future money shrinks faster and borrowed money grows faster. That's the entire mechanism behind why interest rate decisions move markets that have nothing to do with borrowing at all: change the rate, and you've just changed what every future dollar in the economy is worth today.