Time Value of Money

Money rarely arrives at the same moment you need to compare it. A job offer might pay a bonus today or a larger bonus in two years; a business might choose between a smaller payoff now and a bigger payoff later. You can't compare these fairly just by looking at the dollar amounts, because holding a dollar today comes with something a future dollar doesn't: the chance to put it to work right away. If you had that dollar now, you could save it, invest it, or spend it to avoid borrowing at interest later — earning or saving something in return. Economists call the value of the best thing you give up by waiting the opportunity cost of waiting, and it's the reason today's dollar and a same-sized future dollar aren't actually worth the same amount.

Time value of money supplies the missing piece: a way to translate amounts that arrive at different times onto a common basis, using an assumed rate of return (the discount rate). Moving money forward in time is compounding; moving money backward in time is discounting. Nearly every situation in finance where money changes hands at different times — a savings account, a loan, a business deciding whether a big purchase is worth it — is really just this same principle applied to a specific set of amounts and dates.

Time value of money is expressed in two directions, depending on which way you're moving in time:

FV = PV × (1 + r/n)^(n×t)   — moving forward (compounding)
PV = FV / (1 + r/n)^(n×t)   — moving backward (discounting)

• PV— present value, the amount in today's dollars
• FV — future value, the amount at some future date
• r — the discount/interest rate, as a decimal, per year
• n — number of compounding periods per year
• t — number of years between the two points in time

Both formulas describe the same relationship — they're just solved for different variables. Which one you use depends on whether you know the earlier amount and want to project it forward, or know the later amount and want to translate it back to today.

Suppose you're offered a choice: $1,000 today, or $1,050 in one year. Which is better depends entirely on what rate you could otherwise earn on money between now and then.

At 8% opportunity cost:
PV of $1,050 in 1 year = 1,050 / 1.08 ≈ $972.22 → take the $1,000 today

At 3% opportunity cost:
PV of $1,050 in 1 year = 1,050 / 1.03 ≈ $1,019.42 → take the $1,050 later

The $1,050 figure never changes — only the discount rate does. This is why time-value-of-money problems always require an assumed rate; without one, “$1,000 today vs. $1,050 in a year” has no right answer. Try the present value calculator to see how the crossover rate shifts with different amounts and time horizons.