Present value is the future value formula read in reverse. Instead of asking how much money will grow, it asks how much a future amount is worth now, given what money could earn in the meantime. That single idea — discounting — powers bond prices, pension valuations, business appraisals and every loan payment you have ever made. This page explains the formula, where its pieces come from and how it extends to streams of payments, and the solver above shows each substitution with your numbers.
The formula, symbol by symbol
- PV — present value: what the future amount is worth today.
- FV — the future amount.
- i — the discount rate per period as a decimal; it represents the return you give up by waiting.
- n — the number of periods until the money arrives.
- 1 ÷ (1 + i)n — the discount factor (PVIF), always between 0 and 1 for a positive rate.
Where it comes from
If $3,122.99 is invested today at 4% per period, the future value formula says it grows to $3,122.99 × 1.0412 = $5,000 after 12 periods. So someone offering you $5,000 in 12 periods is offering you the equivalent of $3,122.99 now. Solving FV = PV(1 + i)n for PV gives the formula above. Nothing new is needed — just algebra.
Worked example
With the solver’s defaults — FV = $5,000, i = 4%, n = 12:
- 1.0412 = 1.601032
- Discount factor: 1 ÷ 1.601032 = 0.624597
- PV = $5,000 × 0.624597 = $3,122.99
Now add a $200 payment at the end of every period. The payments are worth $200 × (1 − 0.624597) ÷ 0.04 = $1,877.01, and the total present value is $3,122.99 + $1,877.01 = $5,000.00 exactly. That is no coincidence: $200 is 4% of $5,000, so the stream is identical to a $5,000 bond with a 4% coupon, and discounting it at 4% returns its face value.
The annuity version
For a level payment PMT at the end of each of n periods, discount every payment and add them. The sum is a geometric series with ratio 1 ÷ (1 + i):
The bracket divided by i is the present value annuity factor (9.385074 at 4% for 12 periods). For payments at the start of each period, multiply by (1 + i).
Versions you will meet
| Situation | Formula |
|---|---|
| Single future amount | FV ÷ (1 + i)n |
| Annual rate r, compounded m times a year for t years | FV ÷ (1 + r/m)mt |
| Continuous discounting | FV × e−rt |
| Ordinary annuity | PMT × [1 − (1 + i)−n] ÷ i |
| Perpetuity | PMT ÷ i |
| Growing perpetuity | PMT ÷ (i − g), with i > g |
Intuition checks
- Higher rate, lower value. Raising i shrinks the discount factor.
- Longer wait, lower value. Each extra period divides by another (1 + i).
- At 0%, nothing is discounted. PV equals the plain sum of the future amounts.
- Perpetuities are finite. Even infinite payments have a finite value, because distant payments shrink toward nothing.
To apply the formula with any compounding frequency, a lump sum and payments together, use the present value calculator. The growth side is on the future value formula page, and the present value table lists discount factors for many rates and periods.
This page is for education. Results are estimates and are not financial advice.
Frequently asked questions
What is the present value formula?
PV = FV ÷ (1 + i)^n for a single future amount, where i is the rate per period as a decimal and n the number of periods. For a level payment each period it becomes PV = PMT × [1 − (1 + i)^−n] ÷ i.
Why do we divide to find present value?
Present value is the future value formula run backward. If PV × (1 + i)^n = FV, then PV = FV ÷ (1 + i)^n. Dividing by the growth factor removes the interest that would have been earned in the meantime.
What is the discount factor?
1 ÷ (1 + i)^n, the present value of $1 received after n periods. At 4% for 12 periods it is 0.624597, so any amount due then is worth about 62.5% of its face value today.
What is the formula for a perpetuity?
PV = PMT ÷ i for a payment that continues forever, starting one period from now. It is the limit of the annuity formula as n grows without bound, because (1 + i)^−n shrinks toward zero.
Why is the example on this page worth exactly $5,000?
Because $200 is exactly 4% of $5,000. A stream that pays the full interest each period and returns the principal at the end is worth its principal when discounted at that same rate — the reason a bond trades at par when its coupon equals the market yield.