Hypothetical worked example and primary educational references; not observed market data.
Annuity is a cash-flow shape
In financial mathematics, an annuity is a sequence of equal payments spaced at equal intervals. The word does not require an insurance product. Here it describes monthly savings deposits, with the amount and timing fixed by assumption. Investor.gov's savings-goal calculator uses contributions as a planning variable; the formula below explains one common contribution pattern. An ordinary annuity places each deposit at the end of its period. An annuity due places it at the beginning. Those names encode timing rather than quality. Before using a formula, establish which dates the deposits occupy and when the final balance is measured. A single period's shift changes the interest exposure of every payment in the series.
Supporting references: 1. SEC Investor.gov: Savings Goal Calculator
Build the total from separate deposits
Consider twenty-four end-of-month deposits and value the account just after the last deposit. The first deposit experiences twenty-three remaining monthly growth periods. The second experiences twenty-two, and the final deposit experiences none. Multiply each deposit by its own remaining growth factor and add the results. This direct construction explains why applying twenty-four months of growth to the entire contribution total overstates the ending value: most money was not present for the entire horizon. It also provides an independent way to check a closed-form formula. The worked example uses USD 200 deposits and an invented monthly effective rate of one quarter of one percent, alongside a zero-growth scenario.
Hypothetical worked example
End-of-month deposits under two invented growth assumptions
Model setup
Read these assumptions before comparing the scenarios.
- All monetary amounts in this example are hypothetical U.S. dollars (USD), not current prices, quoted account terms or observed market data. Percentages, counts, time periods and index levels retain their stated units.
- Deposit USD 200 at the end of each of 24 months.
- Start with zero; no withdrawals, fees, or taxes.
- Monthly effective growth is either zero or 0.25%; this is mathematics, not an insurance annuity.
Month-end balance (USD), USD 200 deposited at each month end
- Contributed principal / zero growth
- 0.25% monthly growth
Period-by-period model values (USD)
| Month | Contributed principal / zero growth | 0.25% monthly growth |
|---|---|---|
| 0 | 0.00 | 0.00 |
| 1 | 200.00 | 200.00 |
| 2 | 400.00 | 400.50 |
| 3 | 600.00 | 601.50 |
| 4 | 800.00 | 803.01 |
| 5 | 1,000.00 | 1,005.01 |
| 6 | 1,200.00 | 1,207.53 |
| 7 | 1,400.00 | 1,410.54 |
| 8 | 1,600.00 | 1,614.07 |
| 9 | 1,800.00 | 1,818.11 |
| 10 | 2,000.00 | 2,022.65 |
| 11 | 2,200.00 | 2,227.71 |
| 12 | 2,400.00 | 2,433.28 |
| 13 | 2,600.00 | 2,639.36 |
| 14 | 2,800.00 | 2,845.96 |
| 15 | 3,000.00 | 3,053.07 |
| 16 | 3,200.00 | 3,260.71 |
| 17 | 3,400.00 | 3,468.86 |
| 18 | 3,600.00 | 3,677.53 |
| 19 | 3,800.00 | 3,886.72 |
| 20 | 4,000.00 | 4,096.44 |
| 21 | 4,200.00 | 4,306.68 |
| 22 | 4,400.00 | 4,517.45 |
| 23 | 4,600.00 | 4,728.74 |
| 24 | 4,800.00 | 4,940.56 |
Intermediate points are independently calculated from the same fixed example assumptions; zero-based vertical scale. No changing rates, fees or taxes.
Hypothetical model, not observed market data or a forecast. Assumptions above; original values and calculations below.
| Hypothetical scenario | Calculated result (USD) |
|---|---|
| Zero monthly growth | 4,800.00 |
| 0.25% monthly effective growth | 4,940.56 |
Calculation details
- Zero monthly growth
- 4,800.00 USD
(200 * 24) - 0.25% monthly effective growth
- 4,940.56 USD
(200 * (((1.0024999999999999 ^ 24) - 1) / 0.0025000000000000001))
The geometric series gives a compact formula
When the periodic growth rate is i and there are n equal end-of-period deposits of C, future value is C multiplied by the quantity one plus i raised to n minus one, divided by i. The expression sums the different deposit exposures in a geometric series. The numerator and denominator must remain grouped together. Dividing only the final term or forgetting to subtract one produces a different expression. Use a monthly rate with monthly deposits and a monthly period count. A rate labeled annual cannot be substituted without conversion. This compact equation is convenient, but understanding the underlying individual deposits makes it easier to notice timing mistakes that a spreadsheet formula might otherwise conceal.
Zero growth is a separate evaluable case
The formula appears to divide by zero when i is zero, even though the savings question has a perfectly clear answer. With no growth, the final balance is simply deposit amount multiplied by deposit count. The zero-rate expression is also the limit of the positive-rate formula as the rate approaches zero. Software should handle that case deliberately rather than display an error or replace zero with an arbitrary small rate. The example includes a zero-growth row to show the contribution-only baseline. The difference between the two rows is modeled interest, while the entire baseline comes from the saver. Keeping those amounts conceptually separate prevents ordinary deposits from being mistaken for account performance.
A clean formula still needs realistic assumptions
Investor.gov's growth tool treats the interest rate as estimated, which is the appropriate status for a future illustration. Equal deposits might not fit variable income, and monthly account charges can change the series. Withdrawals break the simple accumulation pattern unless modeled as separate cash flows. The example intentionally excludes these features to teach the timing mechanism. It does not represent an insurance annuity, an offer, or a promised return. A practical savings forecast can begin with the ordinary-annuity model, then add irregular deposits and costs on their actual dates. Keep the assumptions visible so the closed-form convenience serves understanding rather than turning a simplified contribution pattern into a claim about a household's future balance.
Supporting references: 2. SEC Investor.gov: Compound Interest Calculator
Test another situationAudit the final deposit with a short series
A separate hypothetical case. Its inputs differ from the main worked example above.
Audit the final deposit with a short series
Consider hypothetical USD 100 deposited at the end of each of three equal periods. An invented 10% effective growth rate applies during each period before its deposit. At the endpoint immediately after the third deposit, the first contribution is worth USD 121, the second USD 110, and the last USD 100, totaling USD 331. Contributions supply USD 300 and modeled growth supplies USD 31. Applying all three periods to every deposit would produce USD 399.30 and overstate the balance by USD 68.30. The deliberately large rate exposes the timing error. Earlier money earns longer; later deposits preserve cash outside savings until their actual arrival dates.
Require the direct sum to match the formula
Investor.gov separates initial investment, contributions, and time. For an ordinary-annuity implementation, build a short direct-sum control before relying on a closed formula across a long horizon. The last deposit must receive no growth before the stated immediate valuation. Add a separate starting balance only if it actually exists, using its own full-horizon factor. A zero-rate input should reduce the series to deposited principal without a division error. When deposits vary, the direct dated sum remains usable but the equal-payment shortcut no longer describes the schedule. This distinguishes the mathematical series from any insurance product that happens to share the word annuity.
Supporting references: 2. SEC Investor.gov: Compound Interest Calculator
Check this case’s calculations
- Dated deposit sum
- 331.00 USD
(((100 * (1.1000000000000001 ^ 2)) + (100 * 1.1000000000000001)) + 100) - Contributed principal
- 300.00 USD
(100 * 3) - Modeled growth
- 31.00 USD
(331 - 300) - Overgrown contribution shortcut
- 399.30 USD
(300 * (1.1000000000000001 ^ 3)) - Shortcut error
- 68.30 USD
(399.30000000000001 - 331)
Limits of this additional case
- The 10% period rate is an exaggerated mathematical input, not a savings yield or contract promise.
Limits of this analysis
- Equal deposits and constant growth are simplifying assumptions.
- A quoted annual yield requires consistent conversion to a monthly rate.
- The term annuity here gives no insurance, longevity, or contractual guarantee.
Sources and scope
- SEC Investor.gov: Savings Goal Calculator
A savings target can be analyzed with initial funds, time, and estimated interest.
Scope: US planning tool; independently derived cash-flow examples.. Reference accessed 2026-10-06.
- SEC Investor.gov: Compound Interest Calculator
Growth illustrations distinguish time, contributions, estimated rate, and compounding frequency.
Scope: US educational calculator; assumptions are not forecasts.. Reference accessed 2026-10-06.
Illustrative calculations are not forecasts or personal recommendations. Editorial standards
