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Annuity Future Value Calculator (Uniform Series)

Future value equals annual amount times the quantity one plus i to the n minus one, divided by i

Solution

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How It Works

The uniform-series compound-amount factor (F/A) converts a stream of equal periodic deposits A into a single future value F. The formula F = A × [(1 + i)^n − 1] / i sums n equal payments, each compounded for the remaining periods at rate i. This is the ordinary-annuity future value — payments occur at the END of each period. Use it for retirement-savings projections, IRA accumulations, and any goal where you deposit a fixed amount on a regular schedule.

Example Problem

You deposit $1,000 at the end of every year for 10 years into an account earning 10% per year. What is the balance immediately after the tenth deposit?

  1. Choose Solve For = Future Value (F). The formula is F = A × [(1 + i)^n − 1] / i.
  2. Substitute A = 1,000, i = 0.10, n = 10.
  3. Compute (1.10)^10 = 2.593742.
  4. Numerator: 2.593742 − 1 = 1.593742. Divide by i: 1.593742 / 0.10 = 15.93742.
  5. Multiply: F = 1,000 × 15.93742 = 15,937.42.

Total deposits were 10 × 1,000 = 10,000; interest earned was 5,937.42 over the 10-year horizon.

Key Concepts

An ordinary annuity has level cash flows at the end of each period — the most common annuity convention in engineering economics. The factor F/A = [(1 + i)^n − 1] / i depends only on rate and periods, not the dollar amount, so doubling the deposit doubles the future value at the same terms. If deposits occur at the START of each period (annuity due), multiply the ordinary-annuity FV by (1 + i) for the additional period of compounding on each payment.

Applications

  • Retirement savings projections — IRA, 401(k), or pension accumulations from regular contributions
  • College savings plans (529) — periodic deposits compounding to a target tuition fund
  • Sinking funds — accumulating money to retire a bond or replace equipment
  • Goal-based saving — how much your emergency fund will be worth after N years of regular deposits
  • Insurance premium reserves — how a stream of premium payments accumulates with investment income

Common Mistakes

  • Assuming deposits compound from period 0 — the first deposit of an ordinary annuity earns no interest in period 1 (it's deposited at the end)
  • Using an annual rate with monthly deposits — convert to a monthly rate and use total months
  • Confusing ordinary annuity (end-of-period) with annuity due (start-of-period) — multiply by (1 + i) to convert
  • Entering interest rate as a percent instead of a decimal — use 0.10 for 10%, not 10
  • Forgetting that this gives the future value of deposits only — a starting lump sum needs a separate future-value compound calculation added on

Frequently Asked Questions

How do you calculate the future value of an annuity?

Use F = A × [(1 + i)^n − 1] / i where A is the periodic deposit, i is the periodic rate, and n is the number of deposits. Example: $1,000 per year at 10% for 10 years gives F = 1,000 × [(1.10)^10 − 1] / 0.10 = 1,000 × 15.937 = $15,937.42.

What is the formula for ordinary annuity future value?

F = A × [(1 + i)^n − 1] / i. The bracketed factor [(1 + i)^n − 1] / i is called the uniform-series compound-amount factor (F/A). Multiply it by the periodic deposit A to get the accumulated future value F.

What is the difference between ordinary annuity and annuity due?

Ordinary annuity payments occur at the END of each period; annuity-due payments occur at the START. Annuity due has one extra period of compounding on every payment, so its future value is (1 + i) × the ordinary-annuity value. Engineering economics defaults to ordinary annuity.

Can I use this for monthly retirement contributions?

Yes — convert the annual rate to a monthly rate (annual ÷ 12) and use the total number of months. Depositing $500 per month for 30 years at 7% annual gives i = 0.005833, n = 360, and F = $500 × 1,219.971 = $609,985.

How is annuity future value different from single-payment compound amount?

Single-payment compound amount F = P(1 + i)^n grows ONE lump sum over time. Annuity future value F = A × [(1 + i)^n − 1] / i grows a STREAM of equal deposits. If you start with a lump sum and also make periodic deposits, add the two future values together.

Why does the first deposit not compound for the full term?

In an ordinary annuity, the first deposit is made at the end of period 1 — so it compounds for n − 1 periods, not n. The last deposit is made at the end of period n and earns no compounding before the final balance is measured. The formula already accounts for this.

Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.

Worked Examples

401(k) Retirement

What is the future value of $500 per month for 30 years at 7% annual interest?

  • Convert to monthly: i = 0.07 / 12 ≈ 0.005833, n = 360
  • F = A × [(1 + i)^n − 1] / i
  • F = 500 × [(1.005833)^360 − 1] / 0.005833
  • F ≈ $609,985

College Fund

If you deposit $3,000 per year for 18 years at 6%, what will the college fund be worth?

  • F = A × [(1 + i)^n − 1] / i
  • F = 3,000 × [(1.06)^18 − 1] / 0.06
  • F = 3,000 × [2.8543 − 1] / 0.06
  • F ≈ $92,713

Reverse: Required Deposit

What annual deposit accumulates to $1,000,000 in 25 years at 8%?

  • A = F × i / [(1 + i)^n − 1]
  • A = 1,000,000 × 0.08 / [(1.08)^25 − 1]
  • A = 1,000,000 × 0.08 / 5.8485
  • A ≈ $13,678.78 per year

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