Change a number. Watch the result.
Parameters
6 inputs| Year | Starting | Contributions | Interest | Ending |
|---|---|---|---|---|
| 1 | $10,000 | $6,000 | $955 | $16,955 |
| 2 | $16,955 | $6,000 | $1,458 | $24,413 |
| 3 | $24,413 | $6,000 | $1,997 | $32,411 |
| 4 | $32,411 | $6,000 | $2,575 | $40,986 |
| 5 | $40,986 | $6,000 | $3,195 | $50,182 |
| 6 | $50,182 | $6,000 | $3,860 | $60,042 |
| 7 | $60,042 | $6,000 | $4,573 | $70,614 |
| 8 | $70,614 | $6,000 | $5,337 | $81,952 |
| 9 | $81,952 | $6,000 | $6,157 | $94,108 |
| 10 | $94,108 | $6,000 | $7,036 | $107,144 |
| 11 | $107,144 | $6,000 | $7,978 | $121,122 |
| 12 | $121,122 | $6,000 | $8,988 | $136,110 |
| 13 | $136,110 | $6,000 | $10,072 | $152,182 |
| 14 | $152,182 | $6,000 | $11,234 | $169,416 |
| 15 | $169,416 | $6,000 | $12,480 | $187,895 |
| 16 | $187,895 | $6,000 | $13,815 | $207,710 |
| 17 | $207,710 | $6,000 | $15,248 | $228,958 |
| 18 | $228,958 | $6,000 | $16,784 | $251,742 |
| 19 | $251,742 | $6,000 | $18,431 | $276,173 |
| 20 | $276,173 | $6,000 | $20,197 | $302,370 |
| 21 | $302,370 | $6,000 | $22,091 | $330,461 |
| 22 | $330,461 | $6,000 | $24,121 | $360,582 |
| 23 | $360,582 | $6,000 | $26,299 | $392,881 |
| 24 | $392,881 | $6,000 | $28,634 | $427,515 |
| 25 | $427,515 | $6,000 | $31,138 | $464,653 |
| 26 | $464,653 | $6,000 | $33,822 | $504,475 |
| 27 | $504,475 | $6,000 | $36,701 | $547,176 |
| 28 | $547,176 | $6,000 | $39,788 | $592,964 |
| 29 | $592,964 | $6,000 | $43,098 | $642,062 |
| 30 | $642,062 | $6,000 | $46,647 | $694,709 |
Compound interest reference: rates, limits, and benchmarks
Accruo runs the formula. This reference covers the numbers around it: what real accounts pay in 2026, how often they compound, how much you can legally contribute, and how to map Accruo's inputs to the accounts you're actually opening.
Account types and where the rate comes from
Plug the rate from your account into Accruo's annual-rate slider. For market-linked accounts, use a long-run average rather than last quarter's return.
| Account | 2026 APY range | Compounding | Interest credited |
|---|---|---|---|
| Online HYSA | 4.00%–5.00% | Daily | Monthly |
| Brick-and-mortar savings | 0.01%–1.50% | Daily | Monthly |
| Money market account | 3.50%–4.75% | Daily | Monthly |
| CD (12 month) | 4.25%–5.25% | Daily | At maturity or monthly |
| 1-Year Treasury Bill | 4.10%–4.30% | Discount basis | At maturity |
| Brokerage MMF (SPAXX, VMFXX) | 4.00%–4.55% | Daily | Monthly |
| Series I Bond (Nov 2025 – Apr 2026 issue) | 3.11% composite | Semiannual | Semiannual (deferred) |
| S&P 500 index (long-run total return) | ~10% nominal, ~7% real | Continuous (price + dividends) | Variable |
Cash accounts compound daily and credit monthly because daily compounding lets the bank advertise a slightly higher APY than the nominal rate. The spread is small at 5% but real: set Accruo's compounding slider to "daily" and a 30-year balance rises by a few hundred dollars versus monthly compounding.
Yield lookup by nominal rate and compounding frequency
A 5% nominal rate compounds to slightly more than 5% per year once interest credits more than once. The exact yield is (1 + r/n)^n − 1.
| Nominal rate | Annual | Quarterly | Monthly | Daily |
|---|---|---|---|---|
| 1.00% | 1.000% | 1.004% | 1.005% | 1.005% |
| 3.00% | 3.000% | 3.034% | 3.042% | 3.045% |
| 4.50% | 4.500% | 4.577% | 4.594% | 4.602% |
| 5.00% | 5.000% | 5.094% | 5.116% | 5.127% |
| 7.00% | 7.000% | 7.186% | 7.229% | 7.250% |
| 9.00% | 9.000% | 9.308% | 9.381% | 9.416% |
| 12.00% | 12.000% | 12.551% | 12.683% | 12.747% |
The frequency lever does more work when the rate is high. At 1%, the spread from annual to daily is 5 basis points. At 12%, it's 75.
Doubling time at a fixed rate
Rule of 72 estimates years-to-double as 72 / rate. The exact answer is log(2) / log(1 + r). Rule of 72 lands closest to exact at around 8%.
| Annual rate | Rule of 72 | Exact (years) | Rule-of-72 error |
|---|---|---|---|
| 1% | 72.00 | 69.66 | +2.34 |
| 2% | 36.00 | 35.00 | +1.00 |
| 3% | 24.00 | 23.45 | +0.55 |
| 4% | 18.00 | 17.67 | +0.33 |
| 5% | 14.40 | 14.21 | +0.19 |
| 6% | 12.00 | 11.90 | +0.10 |
| 7% | 10.29 | 10.24 | +0.05 |
| 8% | 9.00 | 9.01 | -0.01 |
| 10% | 7.20 | 7.27 | -0.07 |
| 12% | 6.00 | 6.12 | -0.12 |
| 15% | 4.80 | 4.96 | -0.16 |
At 7%, a balance doubles in about 10 years and a quarter. It triples in roughly 16 years (log(3) / log(1.07)). It quadruples in just over 20.
2026 tax-advantaged account contribution limits
Annual federal limits. Sources: IRS COLA notices for 2026. Verify current figures at irs.gov before relying on them for filing decisions.
| Account | 2026 limit | Catch-up (age 50+) | Income phase-out (single filer) |
|---|---|---|---|
| Roth IRA | $7,000 | +$1,000 | $150,000–$165,000 MAGI |
| Traditional IRA | $7,000 | +$1,000 | $77,000–$87,000 (deduction, with workplace plan) |
| 401(k) employee deferral | $23,500 | +$7,500 (age 50+); +$11,250 (ages 60–63) | None |
| 401(k) combined (employee + employer) | $70,000 | — | None |
| HSA (self-only HDHP) | $4,300 | +$1,000 at 55+ | None |
| HSA (family HDHP) | $8,550 | +$1,000 at 55+ | None |
| 529 (federal gift-tax exclusion) | $19,000 per donor per beneficiary | 5-year front-load: $95,000 lump | None |
| SEP-IRA | 25% of compensation, up to $70,000 | — | None |
A 25-year-old maxing a Roth IRA at $7,000 a year is contributing about $583 per month. The annuity term in the compound interest formula treats that exactly the way Accruo's contribution slider does.
Real return after inflation
The shorthand real ≈ nominal − inflation is close enough for back-of-the-envelope work. The exact formula is (1 + nominal) / (1 + inflation) − 1.
| Nominal rate | At 2% inflation | At 3% inflation | At 4% inflation |
|---|---|---|---|
| 4% | 1.96% | 0.97% | 0.00% |
| 5% | 2.94% | 1.94% | 0.96% |
| 6% | 3.92% | 2.91% | 1.92% |
| 7% | 4.90% | 3.88% | 2.88% |
| 8% | 5.88% | 4.85% | 3.85% |
| 10% | 7.84% | 6.80% | 5.77% |
To project in today's purchasing power, enter the real rate into Accruo and read the ending balance as 2026 dollars. To match an account statement at the end of the horizon, enter the nominal rate.
Long-run real returns by asset class
Compound annual growth rates, US data, 1928 through end of 2025. Returns are pre-tax and pre-fee. Nominal includes inflation; real removes it.
| Asset | Nominal CAGR | Real CAGR | Worst 10-year real return |
|---|---|---|---|
| S&P 500 (total return) | 10.1% | 7.0% | -3.7% (1999–2008) |
| US 10-year Treasury | 4.9% | 1.9% | -3.5% (1970s) |
| US 3-month T-bills | 3.3% | 0.4% | -3.5% (1970s) |
| Aaa corporate bonds | 6.0% | 3.0% | -2.0% (1970s) |
| Gold (bullion) | 5.4% | 2.4% | -7.0% (1980s) |
| REITs (since 1972) | 10.4% | 7.4% | -4.0% (2007–2016) |
| US housing (Case-Shiller) | 4.4% | 1.4% | -5.0% (2007–2016) |
Source: Damodaran (NYU Stern) historical risk-premium dataset and Shiller online data. A 7% real assumption matches the S&P 500's century-scale average and is a defensible default for a long-horizon retirement projection.
Scenario presets for the sliders
Five starting points that map common life stages to Accruo's six inputs.
| Scenario | Initial | Contribution | Rate | Compounding | Years |
|---|---|---|---|---|---|
| First HYSA, building an emergency fund | $1,000 | $250/month | 4.50% | Daily | 5 |
| Roth IRA at age 25, target-date fund | $0 | $583/month | 7.00% | Monthly | 40 |
| 401(k) with full employer match | $5,000 | $1,000/month | 7.50% | Monthly | 30 |
| House down-payment fund (5-year horizon) | $15,000 | $1,500/month | 4.25% | Daily | 5 |
| Mid-career retirement catch-up | $80,000 | $1,500/month | 6.50% | Monthly | 20 |
The 25-year-old Roth IRA scenario ends near $1.5 million in 2026 dollars at the 7% real assumption, or roughly $3.3 million in nominal 2066 dollars at 10% nominal. The gap between those two numbers is inflation, not the formula.
Slider bounds, defaults, and rounding rules
What Accruo accepts as input, what it shows by default, and how it rounds output.
| Field | Minimum | Maximum | Default | Rounding |
|---|---|---|---|---|
| Initial balance | $0 | $10,000,000 | $10,000 | None at input; $0.01 in display |
| Contribution amount | $0 | $100,000 | $500 | None at input; $0.01 in display |
| Annual rate | 0.00% | 30.00% | 7.00% | 0.01% increments |
| Years | 1 | 50 | 30 | Whole years |
| Contribution frequency | Weekly | Annually | Monthly | 5 discrete options |
| Compounding frequency | Daily | Annually | Monthly | 4 discrete options |
Internal math runs in IEEE 754 double precision and rounds only at render. The running balance carried from year N into year N+1 keeps every binary digit.
Related concepts
- Sequence-of-returns risk. Two retirees with identical average returns can land in very different places depending on whether the bad years cluster early or late.
- Time-weighted vs money-weighted return. Brokerages report time-weighted; your personal experience is money-weighted. They diverge when you contribute or withdraw mid-period.
- Dollar-cost averaging. The contribution side of the formula, applied to assets with variable prices rather than fixed-rate accounts.
- Safe withdrawal rate. The Trinity Study's 4% rule, applied in reverse: a balance Accruo projects can sustain roughly 4% of itself per year in retirement spending.
- Geometric vs arithmetic mean. A 9% average return delivered as +30%, +30%, -30% is not a 9% compound return. Volatility drags geometric returns below arithmetic.
"Compound interest is the eighth wonder of the world. He who understands it, earns it; he who doesn't, pays it."
— often attributed to Albert Einstein