$100 now or $100 five years from now. Suppose the future payment is certain, and cash in hand can earn a steady 5% a year.
The cash now has five years to grow. Take the later payment, and you give up those five years.
How much would you need now to end up with the same $100 in five years? You can put a dollar amount on the cost of waiting.
Run compounding backward
With compound interest, you start with cash and find out what it can grow into. Here you start with the future payment and work backward.
Present value (PV) is the amount you would need now to match a future payment at a chosen rate of return. Finding it is called discounting.
A dollar has a date as well as an amount. The time value of money is the difference that date makes.
Waiting has an opportunity cost: what you give up by losing access to the cash. Here, it is the return you could earn. Even if prices never rise, that opportunity still has value.
Bring one payment back to today
Here, interest compounds annually and the payment arrives at year-end. The discount rate is the annual return used to bring future cash back to today's value.
r is the annual rate written as a decimal, and n is the number of whole years until payment.
For one year at 5%, divide $100 by 1.05 to get $95.24. The 1.05 is the growth factor: each $1 becomes $1.05 in a year.
Taking 5% off would give you $95 instead. But $95 × 1.05 = $99.75, short of the payment. To undo growth, divide by the growth factor; do not subtract the rate.
For five years, you undo five rounds of growth:
- Write 5% as 0.05, so 1 + r = 1.05.
- Raise 1.05 to the fifth power: 1.05⁵ = 1.2762815625.
- Divide $100 by that factor: $100 ÷ 1.2762815625 ≈ $78.35.
The superscript 5 means five factors of 1.05 multiplied together. Keep full precision until the final dollar answer, then round to cents.
Check it forward: $78.35 × 1.05⁵ ≈ $100.00. Discounting is compounding run backward.
The timeline strips away one year's growth at each step. Only the final $100 is paid; the earlier amounts show what that payment is worth at each date.
What the discount rate represents
The rate expresses the return you require for tying up money, given the wait and the risk. Choosing 10% in a calculator cannot make an investment earn 10%.
A company's expected payment can fall short or never arrive. Unlike the certain $100 in our opening example, it carries risk. The equity risk premium expresses the extra return investors seek for bearing stock-market risk; discounting shows how that demand affects value.
We use nominal dollars and rates here, without a separate inflation adjustment. The rule is to match the two: nominal forecasts with nominal rates, inflation-adjusted forecasts with real rates. Once both are in real terms, subtracting inflation again counts it twice.
Time changes the price of a promise
Keep the future payment at $100. The first two rows change only the wait; the last two change only the rate.
| Payment date | Rate | Future cash | Present value |
|---|---|---|---|
| Year 1 end | 5% | $100 | $95.24 |
| Year 5 end | 5% | $100 | $78.35 |
| Year 5 end | 10% | $100 | $62.09 |
Dates, rates and future cash are assumed; present values are calculated in US dollars.
At 10%, the five-year calculation is $100 ÷ 1.10⁵ ≈ $62.09. A longer wait or a higher rate lowers present value in these positive-rate examples.
A higher required return sounds better, yet it means a lower value now. You are solving for the starting amount: if each dollar grows faster, fewer are needed to reach the same $100. Requiring a higher return means paying less for the same future cash.
The answer to the opening puzzle is $78.35 now to match $100 in five years at 5%. That makes the $100-now offer worth $21.65 more today. You can compare the two offers because they now share a date.
In short
- Present value is the amount now equivalent to a future payment at a stated rate.
- Discounting runs compounding backward, one year at a time.
- At a positive rate, a longer wait or a higher required return lowers present value.
- A discount rate is a required return, not a promised result.