What present value means
A dollar due in ten years is worth less than a dollar in hand, because a dollar in hand can be invested and grow. Present value reverses that growth: it asks how much you would need today, invested at a given rate, to have the future amount when it falls due. PV = FV ÷ (1 + i)^n, where i is the rate per period and n the number of periods. $100,000 due in ten years, discounted at 6% a year, has a present value of $55,839: invest that at 6% and it grows to $100,000.
The rate is called the discount rate. It is the return available on alternatives of similar risk — a treasury yield for a near-certain payment, a higher rate for a risky one — and it drives everything: the higher the rate or the longer the wait, the less the future amount is worth today.
| Received in | At 3% | At 6% | At 8% | At 10% |
|---|---|---|---|---|
| 1 years | $97,087.38 | $94,339.62 | $92,592.59 | $90,909.09 |
| 5 years | $86,260.88 | $74,725.82 | $68,058.32 | $62,092.13 |
| 10 years | $74,409.39 | $55,839.48 | $46,319.35 | $38,554.33 |
| 20 years | $55,367.58 | $31,180.47 | $21,454.82 | $14,864.36 |
| 30 years | $41,198.68 | $17,411.01 | $9,937.73 | $5,730.86 |
Present value of a stream of payments
A series of equal payments — a pension, an annuity, rent, loan repayments — is valued by discounting each payment and adding them up, which collapses to PV = PMT × [1 − (1 + i)^−n] ÷ i. Payments at the start of each period (an annuity due) are worth one period's interest more: multiply by (1 + i). The table values $10,000 a year received at year-end.
| Discount rate | For 10 years | For 20 years | For 30 years |
|---|---|---|---|
| 2% | $89,825.85 | $163,514.33 | $223,964.56 |
| 4% | $81,108.96 | $135,903.26 | $172,920.33 |
| 6% | $73,600.87 | $114,699.21 | $137,648.31 |
| 8% | $67,100.81 | $98,181.47 | $112,577.83 |
| 10% | $61,445.67 | $85,135.64 | $94,269.14 |
Worked example: lump sum or payments?
Present value is the tool for choosing between a lump sum now and payments later — a lottery payout, a pension buyout, a structured settlement. Compare the lump sum to the present value of the payments at the return you could realistically earn. The break-even rate is the one at which they match; if you can beat it, take the lump sum.
| Option | Value | Verdict |
|---|---|---|
| Lump sum offered today | $600,000 | |
| 30 payments of $40,000 | $1,200,000 | Nominal total |
| PV of payments at 4% | $691,681 | Take the payments |
| PV of payments at 6% | $550,593 | Roughly even |
| PV of payments at 8% | $450,311 | Take the lump sum |
The discount factor
1 ÷ (1 + i)^n is the discount factor: multiply any future amount by it to get today's value. At 6% the ten-year factor is 0.5584, meaning a payment ten years away is worth about 56 cents on the dollar and 44% of its face value is 'lost' to waiting. The calculator reports the factor for whatever rate and period you enter.
| Years | Factor | Discount |
|---|---|---|
| 1 years | 0.9434 | 5.7% |
| 2 years | 0.8900 | 11.0% |
| 5 years | 0.7473 | 25.3% |
| 10 years | 0.5584 | 44.2% |
| 15 years | 0.4173 | 58.3% |
| 20 years | 0.3118 | 68.8% |
| 30 years | 0.1741 | 82.6% |
Compounding frequency
If the rate compounds more than once a year, the period rate is the annual rate divided by the number of periods and the period count is years times that number. Monthly compounding at 6% over ten years discounts by (1 + 0.005)^120 = 1.8194 instead of 1.06^10 = 1.7908, a slightly lower present value. Match the frequency to how the alternative investment actually compounds; for most planning questions annual is fine.
Where present value is used
Beyond lump-sum-or-payments decisions: valuing a bond (the PV of its coupons plus its face value), pricing a loan (the amount lent is the PV of the payments at the interest rate), judging a business project (net present value adds up the PVs of all future cash flows minus the cost today), and setting aside money for a known future expense such as tuition. The future value calculator runs the same math forward; the compound interest and investment calculators handle regular contributions.
- PV of a lump sum = FV ÷ (1 + i)^n.
- PV of level payments (end of period) = PMT × [1 − (1 + i)^−n] ÷ i.
- PV of payments at the beginning of each period = the above × (1 + i).
- A perpetuity — payments forever — has PV = PMT ÷ i.