What is the present value of a future sum?

People type this into search every day: present value calculator.

A dollar you receive later is smaller, in today’s terms, if that dollar could have earned a return in the waiting. Investor.gov’s compound interest page is the forward version: interest on principal and on interest already earned. Their classroom example grows $100 at 5% a year. The compound interest post writes that growth as 100 × 1.05^10 = $162.89.

Present value runs the same relationship backward:

PV = future value ÷ (1 + annual rate)^years

The present value calculator compounds once a year, on one future sum. Extra deposits, fees, and taxes stay out. Investor.gov’s future value glossary names the forward idea: what an asset is worth on a later date. This page asks what you would set aside today to get there.

These are labeled examples. They are not a quote for an investment, and they are not personalized financial advice.

Example 1: $10,000 in 10 years at 5%

These are the calculator defaults.

Inputs: future amount $10,000, annual rate 5%, years 10.

Growth factor: (1.05)^10 = 1.62889..., shown as 1.63×

Present value: 10,000 ÷ 1.62889… = $6,139.13

Amount discounted: 10,000 − 6,139.13 = $3,860.87

Check the forward trip. 6,139.13 × 1.05^10 comes back to $10,000, within a cent of rounding. The $6,139.13 is the lump sum that grows to $10,000 in 10 years at 5%, compounded once a year.

The discounted piece is 3,860.87 ÷ 10,000 = 38.6% of the future sum. The present value is 61.4%.

Example 2: the classroom $162.89 comes back to $100

Investor.gov’s 5% classroom, run for 10 years with no extra deposits, lands on more than $162. The annual formula prints $162.89.

Inputs: future amount $162.89, rate 5%, years 10.

Present value: 162.89 ÷ (1.05)^10 = $100.00

Amount discounted: $62.89

Growth factor: 1.63×

This is Example 1’s formula on the SEC’s smaller number. $100 grows to $162.89. $162.89 discounts to $100. If you use the unrounded 100 × 1.05^10, which is 162.88946…, the present value is still $100.00 after the page rounds to cents.

Example 3: the same $10,000, 25 years out

The classroom page also looks 25 years ahead. Hold the future amount at $10,000 and the rate at 5%. Change the years to 25.

Growth factor: (1.05)^25 = 3.38635..., shown as 3.39×

Present value: 10,000 ÷ (1.05)^25 = $2,953.03

Amount discounted: 10,000 − 2,953.03 = $7,046.97

Ten years left $6,139.13. Twenty-five years leaves $2,953.03. The extra 15 years discount another 6,139.13 − 2,953.03 = $3,186.10.

Example 4: what if the rate is 2%

Future amount $10,000, years 10, rate 2%.

Growth factor: (1.02)^10 = 1.21899..., shown as 1.22×

Present value: $8,203.48

Amount discounted: $1,796.52

Against Example 1, a 3 percentage point drop in the rate raises today’s value by 8,203.48 − 6,139.13 = $2,064.35. You are saying the money could earn less while it waits, so you need more of it today to reach the same $10,000.

Example 5: what if the rate is 8%

Future amount $10,000, years 10, rate 8%.

Growth factor: (1.08)^10 = 2.15892..., shown as 2.16×

Present value: $4,631.93

Amount discounted: $5,368.07

A higher rate cuts today’s value. The same $10,000 in 10 years is worth $4,631.93 at 8%, against $6,139.13 at 5% and $8,203.48 at 2%.

Example 6: a short wait

Inputs: future amount $50,000, rate 4%, years 3.

Growth factor: (1.04)^3 = 1.12486..., shown as 1.12×

Present value: 50,000 ÷ (1.04)^3 = $44,449.82

Amount discounted: $5,550.18

Three years at 4% only pulls $5,550.18 off a $50,000 sum. That is 5,550.18 ÷ 50,000 = 11.1%. Discounting is quiet when the wait is short and the rate is modest. It gets loud when either one grows, which is the next example.

Example 7: a long wait at 6%

Inputs: future amount $100,000, rate 6%, years 30.

Growth factor: (1.06)^30 = 5.74349..., shown as 5.74×

Present value: 100,000 ÷ (1.06)^30 = $17,411.01

Amount discounted: $82,588.99

Most of the $100,000 is the discount. $17,411.01 is the lump sum that reaches $100,000 in 30 years at 6%, compounded annually, with nothing added along the way. If you will also deposit each month, this formula is the wrong one. Use the compound savings calculator.

Example 8: a 0% rate leaves the sum alone

Inputs: future amount $10,000, rate 0%, years 10.

Growth factor: (1.00)^10 = 1, shown as 1×

Present value: $10,000.00

Amount discounted: $0.00

With no return available in the waiting, a future dollar is a present dollar in this formula. The moment the rate is positive, the present value drops below the future amount. Example 4 is the mild version of that drop. Example 5 is the steeper one.

Example 9: dividing by (1 + rate × years) is a different formula

A common slip is simple interest in the denominator. For Example 1 that looks like:

10,000 ÷ (1 + 0.05 × 10) = 10,000 ÷ 1.5 = $6,666.67

The compound result is $6,139.13.

The simple version is higher by 6,666.67 − 6,139.13 = $527.54. You would set aside too much if the money really compounds once a year, or you would think the future sum is worth more today than this calculator says.

The gap gets wide on Example 7:

Simple: 100,000 ÷ (1 + 0.06 × 30) = 100,000 ÷ 2.8 = $35,714.29

Compound: $17,411.01

Difference: 35,714.29 − 17,411.01 = $18,303.28

Simple interest charges the rate on the original amount only. Compound interest charges it on the growing balance. The present-value formula on this page is the compound one. Use (1 + rate)^years in the denominator.

Example 10: the same algebra, an inflation question

The inflation calculator uses money today ÷ (1 + inflation)^years. The shape matches this page. The sentence does not.

Take $10,000, 5%, and 10 years on the inflation tool. It prints $6,139.13 of purchasing power, and $3,860.87 lost. Those are the same cents as Example 1.

Read as present value: a future $10,000 is worth $6,139.13 today if you can earn 5%.

Read as inflation: $10,000 you already hold buys what $6,139.13 buys today, after 10 years of 5% inflation.

At the inflation calculator’s own 3% default, the same $10,000 for 10 years is $7,440.94 of purchasing power, with $2,559.06 lost and a cumulative price increase of 34.4%. That 3% result is not a present value of a future bill. It is today’s cash, eroded.

When the question is prices, use the inflation post. When the question is a return that could turn today’s lump sum into a known future sum, stay on this page.

Example 11: what if the years change

Hold $10,000 and 5%. Move only the years.

5 years

• factor: 1.28×
• present value: $7,835.26
• discounted: $2,164.74

10 years: $6,139.13, from Example 1.

20 years

• factor: 2.65×
• present value: $3,768.89
• discounted: $6,231.11

30 years

• factor: 4.32×
• present value: $2,313.77
• discounted: $7,686.23

The points on that line, before the chart rounds to the dollar, are $10,000.00, $7,835.26, $6,139.13, $4,810.17, $3,768.89, $2,953.03, and $2,313.77. Year 15 is $4,810.17. Year 25 is the $2,953.03 from Example 3.

One more future bill, so the years are not the only thing moving. A $40,000 sum in 18 years at 5%:

Factor: 2.41×

Present value: 40,000 ÷ (1.05)^18 = $16,620.83

Amount discounted: $23,379.17

You would set aside $16,620.83 today, at 5% compounded annually, to have $40,000 in 18 years. No deposits in between. If the rate you can earn is different, Example 4 and Example 5 show how fast the answer moves.

The examples side by side

Example 9 is the simple-interest slip. Example 10 is the inflation calculator, which shares the algebra and answers a price question.

Open the present value calculator with one future amount, one annual rate, and the years until it arrives. If you want the years for a balance to double at that rate, the Rule of 72 is the shortcut, and the exact line next to it uses this same annual compounding.