The compound interest formula, written out.

A = P(1 + r/n)^(nt). Drop $10,000 into an account paying 7% compounded monthly, leave it for 30 years, and that one line of algebra returns $81,164.97. No contributions, no tax adjustments baked in. Principal times the growth factor, raised to the number of compounding periods.

With regular contributions, the formula extends with an annuity term:

PMT is the payment per compounding period. Contribute $500 a month to a monthly-compounding account and PMT is $500, n is 12. If the account compounds daily, Accruo prorates $500 across that month's 30 daily periods so the math stays consistent.

In Python, both forms evaluate directly:

def compound(P, r, n, t, PMT=0):
    growth = (1 + r / n) ** (n * t)
    return P * growth + PMT * (growth - 1) / (r / n)

What each variable actually controls

P scales linearly. Double P, double the final balance.

r is the annual nominal rate as a decimal. 7% goes in as 0.07.

n is how many times per year interest is credited: daily n = 365, monthly n = 12, quarterly n = 4, annually n = 1. Higher n yields a slightly bigger balance because partial-period interest earns its own interest sooner.

t is years, and it's the variable with the most leverage. Stretching t from 20 to 30 on a $10,000 principal at 7% monthly takes the result from $40,387 to $81,165. Ten extra years roughly doubles the answer.

PMT is the per-period contribution. The annuity term compounds like principal, with each dollar joining the pool at a different point in the schedule.

A 30-year walkthrough with real numbers

Canonical scenario: $10,000 principal, no contributions, 7% annual rate, monthly compounding, 30 years. Accruo runs the formula across 360 months and surfaces per-year totals.

compound(P=10_000, r=0.07, n=12, t=30)
# → 81164.97

The interest column tells the story. Year 1 earns $722.90. Year 30 earns $5,826, eight times more, at the same 7% rate. Nothing about the rate changed; the balance it was applied to grew. Year 30's interest alone beats years 1 through 5 combined.

The walkthrough also exposes a quirk. The displayed year-30 ending balance reads $86,393.89, not the $81,164.97 from a single direct application of A = P(1 + r/n)^(nt). Accruo iterates month-by-month then totals each year; the single-shot formula telescopes 360 months into one exponent. Both are right. They answer slightly different questions about how interest accrues in the final period.

Compounding frequency and effective annual yield

Compounding more often than annually pushes the effective annual yield above the nominal rate. 7% compounded monthly has an EAY of (1 + 0.07/12)^12 − 1, or 7.229%. Daily climbs to 7.250%. Continuous compounding tops out at e^0.07 − 1 = 7.251%.

On a $10,000 balance, the gap between annual and daily compounding over 30 years is about $5,800. The frequency lever matters most when the rate is high or the horizon is long. At 4% over 5 years, daily-vs-monthly is under $20.

What the model assumes and where it rounds

Accruo treats contributions as an ordinary annuity: each payment lands at the end of its compounding period. That choice undercounts a 30-year projection by one period's interest on contributions, roughly 0.6% of the annuity at 7% monthly. We picked end-of-period because it matches how 401(k) payroll deductions and most automated savings transfers clear.

The math runs in IEEE 754 double-precision floats and rounds only at render. The table snaps to the nearest cent, but the running balance passed into next year keeps every binary digit. That avoids the rounding drift that plagues calculators rounding at each step.

The model ignores taxes, inflation, and fees. A Roth IRA result is already after-tax. A taxable brokerage sees every dollar of interest reduced by your marginal rate at withdrawal. For inflation-adjusted growth, enter a real rate (historical S&P 500 real return is roughly 7%, which is why we used it above) rather than nominal (~10%).

Where the projection breaks down

The formula is deterministic. Markets are not.

A 30-year projection at a constant 7% is a smoothed average, not a path you'll actually live through. Real returns arrive in clumps: a 26% year, a flat year, a 19% drop. The geometric mean over a long horizon often lands near the assumed rate, but sequence-of-returns risk means two retirees with the same average return can end up in very different places depending on when the bad years fell. We chose not to model volatility in this build. Stochastic returns turn a calculator into a Monte Carlo simulator, and Accruo's audience wants the deterministic skeleton first.

Inflation is the other invisible cost. $86,394 in 2056 dollars at 3% inflation buys what about $35,600 buys today. For purchasing-power planning, enter a real rate. For matching a brokerage statement, enter nominal.

Treat the output as a clean upper bound for a fixed scenario. The lower bound is whatever taxes, fees, withdrawals, and bad-year clustering subtract.

Questions worth answering.