Future value

Future value is the amount of money an investment will grow to over a specified period at a given interest rate (Brigham E.F., Houston J.F. 2019, p.152)^[1]. Put $1,000 in a savings account today earning 5% annually. Wait ten years. You'll have $1,628.89. That $628.89 difference? Future value thinking made it predictable before you deposited a dime.

The concept sits at the heart of finance. Every loan payment, retirement projection, and corporate investment decision uses future value calculations—or its mirror image, present value. Without understanding how money grows over time, financial planning becomes guesswork.

The time value of money

Here's the core insight: a dollar today is worth more than a dollar tomorrow.

Why? Three reasons. First, you can invest today's dollar and earn a return. Second, inflation erodes purchasing power over time. Third, future payments carry uncertainty—will you actually receive them?

This principle underpins all of finance. When someone offers you $1,000 today or $1,050 next year, you're not comparing equal amounts. You're comparing $1,000 now versus the present value of $1,050 received in twelve months. At 5% interest, those are identical. At 6% interest, take the cash today. At 4%, wait for next year's payment^[2].

Basic formula

For a single lump sum earning compound interest:

\[FV = PV \times (1 + r)^n\]

Where:

• FV = Future value
• PV = Present value (starting amount)
• r = Interest rate per period (as decimal)
• n = Number of periods

Invest $5,000 at 7% for 20 years: \[FV = 5{,}000 \times (1.07)^{20} = 5{,}000 \times 3.8697 = \$19{,}348.42\]

Your money nearly quadruples. Not from additional deposits—just from compound growth. Einstein reportedly called compound interest the eighth wonder of the world. Whether he actually said it is disputed. That it works is not (Ross S.A., Westerfield R.W., Jordan B.D. 2019, p.180)^[3].

Compounding frequency

Interest can compound annually, semi-annually, quarterly, monthly, daily—even continuously. More frequent compounding means higher future values.

When interest compounds m times per year:

\[FV = PV \times \left(1 + \frac{r}{m}\right)^{m \times n}\]

Example: $10,000 at 6% for 5 years with different compounding:

Compounding	Periods per year (m)	Future Value
Annual	1	$13,382.26
Semi-annual	2	$13,439.16
Quarterly	4	$13,468.55
Monthly	12	$13,488.50
Daily	365	$13,498.59

The differences seem small—$116 between annual and daily compounding. Over longer periods or higher rates, they grow substantial^[4].

Continuous compounding

At the mathematical limit—compounding infinitely often—the formula becomes:

\[FV = PV \times e^{r \times n}\]

Where e ≈ 2.71828 (Euler's number).

That same $10,000 at 6% for 5 years, compounded continuously: $13,498.59. Nearly identical to daily compounding. The formula matters more for theoretical finance (option pricing, for instance) than practical savings calculations.

Future value of annuities

Most savings aren't lump sums. People make regular contributions—monthly 401(k) deposits, annual IRA contributions. These payment streams are called annuities.

For an ordinary annuity (payments at period end):

\[FV = PMT \times \frac{(1 + r)^n - 1}{r}\]

Where PMT = payment amount per period.

Save $500 monthly for 30 years at 7% annual return (0.583% monthly):

\[FV = 500 \times \frac{(1.00583)^{360} - 1}{0.00583} = 500 \times 1,219.97 = \$609,985\]

You contributed $180,000 total. The account holds nearly $610,000. Compound growth added $430,000—more than double your deposits (Berk J., DeMarzo P. 2020, p.124)^[5].

Annuity due

When payments occur at the beginning of each period (annuity due), the formula adjusts:

\[FV_{due} = PMT \times \frac{(1 + r)^n - 1}{r} \times (1 + r)\]

Same example but with payments at month's start: \[FV = 609,985 \times 1.00583 = \$613,541\]

The extra $3,556 comes from each payment earning one additional period's interest. Rent payments, insurance premiums, and lease payments typically work this way.

The Rule of 72

Want a quick estimate of doubling time? Divide 72 by the interest rate.

At 6%, money doubles in about 12 years (72 ÷ 6 = 12). At 8%, roughly 9 years. At 4%, eighteen years.

The rule isn't exact but works surprisingly well for rates between 2% and 15%. It fails at extremes—at 72% interest, money doesn't double in one year^[6].

Real vs. nominal future value

Inflation complicates everything. $100,000 in 30 years won't buy what $100,000 buys today.

Real future value adjusts for expected inflation:

\[FV_{real} = PV \times \frac{(1 + r)^n}{(1 + i)^n} = PV \times \left(\frac{1 + r}{1 + i}\right)^n\]

Where i = inflation rate.

At 7% nominal return and 3% inflation: \[FV_{real} = 10,000 \times (1.0388)^{30} = \$31,346\]

Compare to nominal FV of $76,123. Inflation eats more than half the apparent growth. Financial planners should always think in real terms^[7].

Applications

Retirement planning. How much will your savings grow by age 65? Future value calculations answer this directly. Most retirement calculators are just future value formulas with user-friendly interfaces.

Loan analysis. What will you have paid over a mortgage's life? Future value of the payment stream shows total cost, revealing how much interest accumulates.

Investment comparison. Two investments with different compounding frequencies or rates can be compared by computing their future values. A 5.9% rate compounded monthly beats 6% compounded annually.

Capital budgeting. Companies projecting cash flows from projects use future value to understand cumulative benefits—or more commonly, present value to compare against today's investment cost.

Goal setting. Need $50,000 for a down payment in 5 years? Work backward from the future value formula to find required monthly savings.

Limitations

Rate uncertainty. Future value assumes a known, constant interest rate. Real-world returns fluctuate. The stock market returned 28.7% in 2019 and -18.1% in 2022. Planning with average rates obscures dramatic variability^[8].

Inflation unpredictability. Projecting real future values requires guessing future inflation. From 2010-2020, U.S. inflation averaged 1.8%. In 2022, it hit 8%. Thirty-year projections compound this uncertainty.

Ignores taxes. Future value formulas typically assume tax-free growth. In taxable accounts, annual taxes on interest or dividends reduce effective returns. A 6% nominal return might net 4% after taxes.

Assumes reinvestment. Compound interest calculations assume all earnings get reinvested at the same rate. Dividend-paying stocks or bonds with fluctuating reinvestment rates complicate the picture.

Single-rate limitation. Complex financial products—adjustable-rate mortgages, stepped CDs, variable annuities—don't fit the basic formula. More sophisticated models are required.

Future value vs. present value

The formulas are mirror images:

\[FV = PV \times (1 + r)^n\] \[PV = \frac{FV}{(1 + r)^n}\]

Use future value when you know today's amount and want tomorrow's. Use present value when you know a future amount and want today's equivalent.

The choice depends on the question being asked, not on which calculation is "better." Both express the same fundamental relationship between money and time (Brigham E.F., Houston J.F. 2019, p.168)^[9].

Sensitivity analysis

Small changes in assumptions create large outcome differences over long horizons.

$10,000 invested for 40 years:

• At 5%: $70,400
• At 6%: $102,857
• At 7%: $149,745

One percentage point difference? Over 40 years, it more than doubles the ending balance. This is why fee-conscious investors obsess over expense ratios. A fund charging 1.5% vs. 0.5% effectively reduces your return by a full percentage point—permanently^[10].

Future value — recommended articles

Investment — Present value — Cash flow — Financial planning — Compound interest

References

• Berk J., DeMarzo P. (2020), Corporate Finance, 5th Edition, Pearson, Boston.
• Brigham E.F., Houston J.F. (2019), Fundamentals of Financial Management, 15th Edition, Cengage Learning, Boston.
• Ross S.A., Westerfield R.W., Jordan B.D. (2019), Essentials of Corporate Finance, 10th Edition, McGraw-Hill, New York.
• Bodie Z., Kane A., Marcus A.J. (2021), Investments, 12th Edition, McGraw-Hill, New York.

Footnotes

1. ↑ Brigham E.F., Houston J.F. (2019), Fundamentals of Financial Management, p.152
2. ↑ Ross S.A., Westerfield R.W., Jordan B.D. (2019), Essentials of Corporate Finance, pp.145-160
3. ↑ Ross S.A., Westerfield R.W., Jordan B.D. (2019), Essentials of Corporate Finance, p.180
4. ↑ Brigham E.F., Houston J.F. (2019), Fundamentals of Financial Management, pp.172-178
5. ↑ Berk J., DeMarzo P. (2020), Corporate Finance, p.124
6. ↑ Bodie Z., Kane A., Marcus A.J. (2021), Investments, p.89
7. ↑ Ross S.A., Westerfield R.W., Jordan B.D. (2019), Essentials of Corporate Finance, pp.192-198
8. ↑ Bodie Z., Kane A., Marcus A.J. (2021), Investments, pp.134-156
9. ↑ Brigham E.F., Houston J.F. (2019), Fundamentals of Financial Management, p.168
10. ↑ Bodie Z., Kane A., Marcus A.J. (2021), Investments, pp.112-125

Author: Sławomir Wawak