FINANCE · FOUNDATIONS
Time Value of Money: Present Value, Future Value, and Worked Examples
The time value of money is the idea that a dollar available today and a dollar received later are not directly equivalent. Money available now can be put to work, while future receipts involve waiting and uncertainty. To compare amounts at different dates, move them to one common date using an appropriate rate.
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The four inputs
| Input | Meaning | Example |
|---|---|---|
| PV | Present value at the starting date | $10,000 today |
| FV | Future value at the ending date | Value after three years |
| r | Rate per period, expressed as a decimal | 8% annually = 0.08 per year |
| n | Number of matching periods | Three annual periods = 3 |
Future value: move money forward
FV = PV × (1 + r)n
Assume $10,000 earns a hypothetical 8% effective annual return for three years, with no fees, tax, withdrawals, or extra deposits. Interest remains invested. Future value is $10,000 × 1.08³ = $12,597.12.
| Year | Opening value | 8% growth | Closing value |
|---|---|---|---|
| 1 | $10,000.00 | $800.00 | $10,800.00 |
| 2 | $10,800.00 | $864.00 | $11,664.00 |
| 3 | $11,664.00 | $933.12 | $12,597.12 |
Under simple interest, interest would be calculated only on the original principal: $10,000 × (1 + 0.08 × 3) = $12,400. Compounding produces another $197.12 because earlier interest also earns interest. This is an illustration, not a promised investment return.
Present value: discount a future receipt
PV = FV ÷ (1 + r)n
Suppose a customer offers $15,000 exactly three years from now. At an illustrative 8% annual discount rate, its present value is $15,000 ÷ 1.08³ = $11,907.48. The discount rate must reflect the relevant opportunity cost and risk; 8% is an assumption for this example.
If the alternative is $12,000 today, that immediate amount has the higher present value by $92.52 at this rate. That narrow numerical comparison does not settle credit risk, contractual rights, taxes, or an immediate need for cash.
Match the rate to the period
A quoted nominal annual rate compounded monthly is different from an effective annual rate. For a nominal annual rate j compounded m times a year:
Periodic rate = j ÷ m
Number of periods = years × m
Effective annual rate = (1 + j ÷ m)m − 1
At a nominal annual rate of 12% compounded monthly, the monthly rate is 1%. $10,000 becomes $10,000 × 1.01¹² = $11,268.25 after one year. The effective annual rate is approximately 12.6825%. If 12% is instead an effective annual rate, the matching monthly rate is 1.121/12 − 1, approximately 0.9489%; do not divide that effective rate by 12.
Several payments: value each cash flow
For uneven cash flows, discount each amount using its own date and add the results. For equal end-of-period payments, the ordinary-annuity shortcut is:
PV = Payment × [1 − (1 + r)−n] ÷ r
Three $5,000 payments received at the end of years 1, 2, and 3 have a present value of $12,885.48 at 8%. A stream starting immediately has different timing. For equal beginning-of-period payments with the same count, multiply the ordinary-annuity value by (1 + r). When r is zero, the value is simply the sum of the payments.
The NPV guide applies this logic to a project’s inflows and outflows. IRR works in the opposite direction by finding a rate that makes NPV zero.
Try it in a spreadsheet
| Question | Excel-style formula | Result |
|---|---|---|
| Grow $10,000 for three years at 8% | =FV(8%,3,0,-10000) | $12,597.12 |
| Value $15,000 received in year 3 | =PV(8%,3,0,-15000) | $11,907.48 |
| Value three year-end $5,000 receipts | =PV(8%,3,-5000,0,0) | $12,885.48 |
Financial functions use opposite signs for cash paid and cash received. A negative input above produces a positive value output. Treat the signs consistently from one person’s or business’s perspective. These formulas assume regular periods; irregular dates need a date-aware calculation.
Common mistakes
- Mixing years and months. A monthly rate needs monthly periods.
- Comparing future money with today’s money directly. First bring both to the same date.
- Ignoring payment timing. A receipt today has a different value from one at year-end.
- Mixing real and nominal assumptions. Cash flows that include inflation need a consistent nominal discount rate; purchasing-power cash flows need a consistent real rate.
- Using one rate for every risk. A customer receivable and a speculative project need different analysis.
- Rounding every step. Retain precision internally and round displayed results.
Frequently asked questions
Is time value just inflation?
No. Inflation affects purchasing power, but time value also concerns the opportunity to earn a return and the risk of waiting. Be explicit about what the chosen rate represents.
Does a positive present value mean I can afford it?
No. Value and liquidity answer different questions. Check actual payment timing with a cash-flow forecast before committing resources.
What should I learn next?
Use NPV to compare project value and the payback period to assess recovery timing. They answer related but different questions.
Sources and learning notes
Further reading: OpenStax, Time Value of Money Basics. All figures, comparisons, and calculations on this page are original hypothetical teaching examples. Rates are illustrative; they are not current market quotes or individualized advice.