Compound Interest Explained: Deposits, Timing and Rates

What a compound interest projection answers

A compound interest projection asks what a starting balance and a regular saving policy would become if one rate applied consistently. It is useful for exploring a hypothetical plan, but the arithmetic cannot establish which rate you will actually receive. For investments with variable prices, a smooth growth assumption is especially different from a forecast.

This guide separates the three ingredients people often mix up: money already invested, new money added later, and interest earned on both. It also makes the deposit interval and beginning/end timing explicit, so two scenarios can be compared on a consistent basis.

Principal, deposits and the meaning of interest

Principal is money present at time zero. A recurring deposit is new money you add once per selected compounding period. The nominal annual percentage is divided by the number of equal periods per year. Complete years determine the number of deposits; the effective annual growth includes compounding and is shown separately.

Suppose you begin with 1,000 and add another 200 over time. A final 1,307.50 is not a profit of 307.50: you supplied 1,200 in total, so the modeled interest is 107.50. Keep this distinction when discussing savings progress. A target balance can be reached mostly by contributions, mostly by growth, or by both.

When this model is useful

Use it to explore a lump sum with reinvested interest, or a fixed recurring saving commitment over complete years. Set the deposit to explicit zero for a lump-sum-only comparison. Choosing monthly makes each entered deposit monthly and each interest period one month.

The calculator deliberately matches deposit and compounding intervals. It does not approximate monthly deposits into an annually credited account without saying so. That arrangement needs a separately specified accrual and crediting model. Withdrawals and changing rates also require a different model; do not enter a negative deposit to force them into this one.

Formula with deposits and timing

Let P be principal, C the deposit per period, R the nominal annual percentage, m periods per year, Y complete years, N=mY, and q=1+R/(100m). End-period balance is Pq^N+C times the sum of q^k for k=0 through N−1. Beginning timing multiplies only that deposit term by q. The engine uses stable binary composition of growth and geometric sums, not subtraction of nearly equal powers or division by a tiny rate.

The geometric sum is 1+q+…+q^(N−1). At a nonzero periodic rate it equals (q^N−1)/(q−1), but that quotient can lose precision when q is very close to one. The implementation composes growth factors and sums in logarithmic work instead. At zero rate the sum is N. When N is zero, the empty sum is zero for both timings.

q = 1 + R/(100m); N = mY
End: FV = Pq^N + CΣ(k=0…N−1)q^k
Beginning: FV = Pq^N + CqΣ(k=0…N−1)q^k
Contributed = P + CN; interest = FV − contributed
Effective annual growth (%) = 100(q^m − 1)

A verified two-year example

Start with 1,000, add 100 at the end of each year for two years at a nominal 5% compounded annually. Year one: 1,000×1.05+100=1,150. Year two: 1,150×1.05+100=1,307.50. Of this, 1,200 is contributed capital and 107.50 is interest. Beginning-year deposits instead produce 1,317.75, with the same contributed capital.

For beginning-year deposits, the first year is (1,000+100)×1.05=1,155 and the second is (1,155+100)×1.05=1,317.75. Deposited capital is still 1,200, making interest 117.75. The timing advantage is 10.25, not an extra deposit. At a negative rate, that extra exposure can instead reduce the final amount.

A separate lump-sum check is 2,000 at a nominal 5%, compounded monthly for two complete years with no recurring deposit: 2,000×(1+0.05/12)^24≈2,209.88. These are hypothetical examples independently checked by period-by-period recurrence, not current product rates.

1. Choose annual compounding, end timing, principal 1,000, recurring deposit 100, rate 5 and years 2.
2. After year one, multiply 1,000 by 1.05 and then add 100 to get 1,150.
3. Repeat once: 1,150×1.05+100=1,307.50.
4. Subtract all contributed capital: 1,307.50−(1,000+2×100)=107.50.
5. Switch only timing to beginning: final balance becomes 1,317.75, while total contributions remain 1,200.

How to use the calculator

Choose the display currency and matched compounding/deposit interval. Enter initial principal, recurring deposit per selected interval (explicit 0 if none), deposit timing, nominal annual rate and complete years. Re-enter the deposit after a frequency change. Compare contributed capital, interest and the optional selected-year milestones; use Clear values to reset.

The frequency selector is a combined convention, not merely a display preference. Changing it clears the deposit and asks you to re-enter an amount for the new interval. The market selector changes formatting alone and leaves all numeric values unchanged.

1. Confirm the rate you have is nominal; do not silently treat an effective APY/AER as nominal.
2. Choose an interval, then enter a deposit for that interval and explicit beginning/end timing.
3. Use complete integer years from 0 to 100. Zero years tests the time-zero balance.
4. Review contributions, interest, deposit count and effective annual growth together.
5. Open milestones for selected completed-year balances; clear or reload to discard the local scenario.

Interpreting a projection

Compare projected balance with total contributions to see which money came from you. Interest is balance minus contributed capital; it may be negative. Effective annual growth is (q^m−1)×100, a theoretical rate conversion rather than an investment performance observation or product guarantee.

For a positive nominal rate, more frequent compounding raises effective annual growth, holding the nominal rate fixed. But changing frequency also changes how often this model adds C. A higher ending balance could simply reflect more money contributed. Compare equal annual contribution totals before attributing a difference to compounding.

At zero interest, only your deposits increase the balance. At a negative rate, new deposits may keep the final balance rising while the interest component is negative. Neither the balance nor a theoretical annual growth rate measures investment risk, purchasing power or suitability.

Assumptions you should verify

Constant nominal rate; equal nonnegative deposits; matching deposit and compounding frequencies; complete integer years; initial principal separate from deposits. Daily means 365 equal periods every year, not calendar accrual. Growth is reinvested, with no contractual period-by-period cent rounding.

A beginning deposit occurs before that period grows; an end deposit occurs afterward. For positive interest this gives beginning deposits one extra earning period, while the last end deposit earns no interest before the horizon. At zero years there are no recurring deposits even with beginning timing.

No cent rounding occurs inside each period. The isolated calculation context uses 80 significant decimal digits, including guard precision for tiny rates and the subtraction needed to report interest. Currency display uses two places with half-even rounding; small nonzero amounts are marked instead of being silently presented as exact zero.

What this calculator intentionally leaves out

No withdrawals, variable rates, continuous compounding, fractional years, independently scheduled deposits, actual-date accrual, taxes, fees, inflation or FX. Currency is formatting only. Each amount is at most 10^15; rate greater than −100% through 1000%; years 0–100; projected balance at most 10^30. Decimal input uses a point, up to 12 fractional places and 48 characters. Unsupported results are refused, not clipped.

The result is future money, not an inflation-adjusted present value. The rate is constant, and neither product restrictions nor taxes, fees or market volatility are simulated. A balance outside the supported range is refused with an actionable error; it is not replaced by a capped number.

Selected milestones contain at most nine complete-year rows and are generated only on request. They are not a transaction ledger, a promise to match bank posting dates or an import/export facility. There are no actual date inputs, so no leap-day or timezone inference is made.

Common mistakes that change the answer

Entering an effective APY/AER as a nominal rate can apply compounding twice. Treating the entire projected balance as investment earnings counts your own deposits as profit. Changing a monthly deposit into a daily deposit without changing its amount radically changes contributions. A higher projection does not establish a better or safer investment.

Another error is applying the beginning-period multiplier to the initial principal as well as deposits. P is already at time zero in both modes, so it gets exactly N growth periods. Beginning timing changes only the deposit component. A zero-duration input likewise must not smuggle in an immediate recurring contribution.

Whole years and fixed equal periods are deliberate limits. Do not round a short or partial-year horizon up to one year and treat the projection as an exact accrual. Read your account terms or use a model with the necessary dates and day-count convention.

Projection versus historical return

CAGR describes the annualized change between two observed endpoints without external cash flows. This calculator instead starts with an assumed nominal rate and a contribution plan. A CAGR calculated on a balance inflated by new deposits is not a valid substitute for cash-flow-aware return measurement.

For actual irregular cash-flow dates, XIRR is a retrospective money-weighted rate equation with its own multiple-root and numerical limits. It does not predict savings growth. Choose the question first—future balance or historical return—rather than choosing whichever method reports the largest percentage.

Five-market considerations without invented tax rules

USA, UK, Canada, Australia and India use the same mathematics here. The market control provides USD, GBP, CAD, AUD and INR display conventions, including Indian digit grouping, but never converts currency. Keep the same currency basis for principal and every deposit.

In the UK, AER is an annual-equivalent comparison measure that includes compounding assumptions; a nominal input is a different basis. Similar effective-rate labels elsewhere must be read with their product definitions. This tool reports a mathematical effective annual growth rate, not a legally certified account APY or disclosure.

No ISA allowance, GIC condition, superannuation treatment, fixed-deposit rule, deposit protection or tax exemption is inferred from the selected country. These depend on the actual product and current rules. Check appropriate primary guidance or advice for a real decision; none of those changing rules is encoded in this projection.

Educational scope and sources

Hypothetical mathematical projections for education, not personalized financial or investment advice. Rates and future returns are uncertain. Verify actual product terms, cash-flow timing, fees, taxes and inflation before making a decision. No future balance or suitability is promised.

Prepared with AI assistance under the website operator’s editorial responsibility. No independent professional financial review, personalized recommendation or guaranteed return is claimed. Formula and recurrence evidence are engineering checks, not proof that an investment will behave like the model.

Frequently asked questions

Sources and review

Sources reviewed Nov 3, 2026. Consult current local rules before relying on time-sensitive information.

• OpenStax: compound interest
• OpenStax: savings and ordinary annuities
• OpenStax: annuities and beginning timing
• SEC Investor.gov: compound interest calculator
• ASIC Moneysmart: model assumptions
• Halifax: meaning of AER