What is the future value of an annuity formula?

FV = PMT × ((1+r)^n − 1) / r for ordinary annuity (payment at period end). For annuity-due (payment at period start), multiply by (1+r): FV_due = FV_ord × (1+r). PMT is periodic payment, r is rate per period, n is number of periods. The CFA Institute Level I curriculum and every introductory finance textbook teach this formula.

What is the difference between ordinary annuity and annuity-due?

Ordinary annuity (end-of-period payment) has PMT at periods 1, 2, ..., n. Annuity-due (beginning-of-period payment) has PMT at periods 0, 1, ..., n−1. The annuity-due FV is higher by a factor of (1+r) because each payment compounds for one additional period. Most retirement contributions are ordinary annuity (paycheck withholding at month end). Rent, lease, and insurance premiums are annuity-due (pay at start of month).

How is FV of annuity used in 401(k) projection?

A $23,500 annual contribution (2025 IRS limit) at 8% return for 30 years builds an FV of $2,659,471 (ordinary annuity). With employer match (typically 50% match up to 6% of salary) the effective contribution increases ~50%, pushing FV past $3.5M. The same equation underlies every Vanguard, Fidelity, and Schwab retirement projection tool. SECURE 2.0 Act 2022 raised catch-up contributions starting age 50.

What rate of return should I assume?

For a 60/40 stock/bond portfolio, Vanguard's 2026 capital market expectations suggest 5.0-7.0% over 10 years (lower than 1990s-2000s historicals). For 100% US equity, Damodaran (NYU Stern) reports 10.2% nominal / 7.0% real since 1928. International equity and emerging markets have historically returned ~7-8%. TIPS-only portfolios return current real yield + CPI (~1.7% real + inflation in 2026).

Should I assume a higher rate for longer time horizons?

No — use the same long-run expected return across horizons. A 40-year horizon allows mean-reversion but does not justify a higher assumption. Jeremy Siegel (Wharton) and John Bogle (Vanguard) both argue 6-7% real for long-term US equity. Aggressive assumptions (10%+ real) materially overstate retirement readiness — the SECURE Act's annual benefit-statement projections require conservative assumptions.

How does inflation affect FV of annuity?

A $1M FV in 2026 buys what $440K buys today (3% inflation, 30 years). Use real return (nominal − inflation) for purchasing-power calculations. The Fisher equation: (1+nominal) = (1+real)(1+inflation). For purchasing-power retirement planning, use 4-5% real returns instead of 7-8% nominal. The CFA Institute and Damodaran both teach this consistency principle.

How does this compare to TVM's 5-variable solver?

FV of annuity is a specialized case of TVM where PV=0 and only PMT, N, and I are non-zero. The general 5-variable TVM solver at /tools/calculators/financial/time-value-money handles cases with both a starting lump sum (PV) and periodic payments (PMT). For pure savings-with-no-starting-balance, FV of annuity is the cleanest formula.

What is sinking-fund factor?

Sinking-fund factor (SFF) is the reciprocal of FV-annuity factor: SFF = r / ((1+r)^n − 1). It tells you the periodic payment needed to accumulate $1 by period n at rate r. Sinking funds are required for many corporate and municipal bonds — the issuer must set aside periodic payments to retire the bond at maturity. The MSRB and SEC track sinking-fund requirements on muni bond disclosures.

What are the IRS retirement account contribution limits in 2025-2026?

2025 limits per IRS Publication 590-A: 401(k)/403(b)/TSP = $23,500 (under 50) / $31,000 (50+) / $34,250 (60-63 catch-up); IRA = $7,000 / $8,000 (50+). HSAs: $4,300 single / $8,550 family + $1,000 (55+). SEP-IRA: 25% of compensation up to $70,000. The SECURE 2.0 Act of 2022 raised catch-up limits for ages 60-63 starting in 2025 to encourage late-career saving.

How is FV of annuity used in actuarial science?

Actuaries use FV of annuity to compute reserves for life insurance, pensions, and annuity products. The Society of Actuaries (SOA) Exam FM (Financial Mathematics) Chapter 3 covers annuities-due, annuities-immediate, perpetuities, and varying annuities in detail. Pension actuaries compute the accumulated value of contributions over an employee's career to project DB plan funding adequacy.

When was the FV-annuity formula developed?

The mathematical foundation traces to actuarial science. Edmond Halley's 1693 Royal Society paper used compound-interest math on life annuities. Jacob Bernoulli's 1683 work derived the limit of compounding. The modern closed-form FV-annuity formula appears in Augustus De Morgan's 1838 Essay on Probabilities . The HP-12C financial calculator (1981) made it accessible to every Wall Street trader; today every financial calculator and spreadsheet implements it identically.

How does the 401(k) annuity compare to a Roth IRA?

A traditional 401(k) has pre-tax contributions and taxable withdrawals — the FV equals the gross savings. A Roth IRA has post-tax contributions and tax-free withdrawals — the FV is also gross but represents after-tax buying power. For most savers, the Roth advantage compounds over decades: $7,000/yr for 40 yrs at 7% in Roth = $1.55M tax-free; same amount in 401(k) = $1.55M taxed at retirement bracket (potentially 22-32% federal). The Bipartisan Policy Center and Vanguard advise high-earning young workers to favor Roth.

Stacker chart + due/ordinary toggle

Future Value of Annuity

To compute the future value of regular periodic payments compounding to a terminal date: FV = PMT × ((1+r)^n − 1) / r. Annuity-due (payment at period start) gets an extra (1+r) compounding bonus. The stacker chart shows each cumulative payment as a green bar with cyan interest stacked on top — the gap between bar height and the dashed principal line IS your earned compound interest.

$2662K

FV at maturity

$705K

Total contributed

$1957K

Interest earned

+$212972

Due vs ordinary

Compound stacking, visualized.

Quick Conversion

FROM (PMT ($))

TO (FV ($))260,463.3299

r (%)nfreq

Formula: FV = PMT × ((1+r)^n − 1) / r

Annuity FV Stacker

Annuity inputs

Payment per period ($)

Annual rate (%)

Number of periods

Compoundings per year

Annuity type

PMT at period end (savings, paychecks)

Label

TERMINAL FV (ordinary)

$2,662,155.46

Real savings presets

FV of $500/mo at varying rates (ordinary annuity)

Years	@ 4%	@ 6%	@ 8%	@ 10%
5y	$33K	$35K	$37K	$39K
10y	$74K	$82K	$91K	$102K
15y	$123K	$145K	$173K	$207K
20y	$183K	$231K	$295K	$380K
25y	$257K	$346K	$476K	$663K
30y	$347K	$502K	$745K	$1130K
35y	$457K	$712K	$1147K	$1898K
40y	$591K	$996K	$1746K	$3162K

Mirror tool — going the other way? PV of Annuity Calculator →

Formula

FV = PMT × ((1+r)^n − 1) / rFV_due = FV_ord × (1 + r)

Annuity-due gains one extra period of compounding because payments occur at the start of each period.

Worked: $500/mo × 240 periods @ 7% APR (0.583%/mo). FV = 500 × ((1.00583^240 − 1) / 0.00583) = $260,463. Total contributed $120,000; interest earned $140,463.

From Bernoulli's 1683 compounding limit to the SECURE 2.0 Act

In 2026, a 28-year-old software engineer in Seattle sets up a $7,000 annual Roth IRA contribution at Vanguard. Over the next 40 years at 7% real return, the FV-annuity formula projects $1.55M tax-free at retirement. The math behind that projection has three centuries of intellectual development.

Jacob Bernoulli's 1683 work at the University of Basel on continuously compounded interest led to the discovery of Euler's number e ≈ 2.71828. Bernoulli studied a bank-deposit problem and proved lim(n→∞)(1 + r/n)^n = e^r — the foundation of continuous compounding. This is why credit-card APR vs APY differs and why FV-annuity limits are well-defined for any positive rate.

Edmond Halley's 1693 Royal Society paper An Estimate of the Degrees of the Mortality of Mankind created the first life tables. Halley computed FV of annuities on lives — the foundation of all life-insurance and pension mathematics. The Society of Actuaries Exam FM Chapter 3 still teaches annuity formulas in Halley's notation. Augustus De Morgan's 1838 Essay on Probabilities contained the first modern closed-form FV-annuity formulas.

William Sharpe's 1964 Capital Asset Pricing Model gave investors the discount-rate framework needed to choose a rate parameter for FV calculations. Eugene Fama's 1970 Efficient Markets Hypothesis justified using the historical equity premium (~5% above T-bills, per Damodaran NYU Stern data) as the expected return for diversified portfolios. John Bogle founded Vanguard in 1975 on the premise that average investors earn the market return after costs — making the FV-annuity formula directly applicable to retirement planning.

The Employee Retirement Income Security Act (ERISA) of 1974 created the regulated framework for defined-benefit and defined-contribution pension plans. The 401(k) plan was created in 1978 via Internal Revenue Code §401(k) and popularized starting 1981. The Roth IRA was added in 1997 (Senator William Roth, Delaware) as part of the Taxpayer Relief Act. The Pension Protection Act of 2006 introduced auto-enrollment, dramatically increasing 401(k) participation rates.

The SECURE Act (Setting Every Community Up for Retirement Enhancement, 2019) and SECURE 2.0 Act (2022) modernized retirement saving. Key 2025-2026 provisions: catch-up contributions ages 60-63 now $34,250/yr in 401(k); RMD age now 73 (rising to 75 in 2033); 529-to-Roth rollovers now permitted up to $35K lifetime; emergency-savings withdrawal up to $1,000 without penalty. These provisions all use FV-annuity math for benefit projection on annual statements required by ERISA §105(a).

By 2026, every major brokerage (Fidelity, Vanguard, Schwab, Robinhood, Wealthfront, Betterment) embeds FV-annuity calculators in their retirement-planning UI. The calculator on this page is the visual analog of the Vanguard Retirement Nest Egg calculator and Fidelity's Retirement Score — exposing the underlying math instead of hiding it behind a marketing chrome. The Bernoulli 1683 formula remains identical; the JavaScript UI is 2026.

How to use the FV annuity stacker

Enter payment per period. Dollar amount paid each period (monthly, annual, etc.).
Enter annual rate. Expected return — 7% real / 10% nominal for US equity, 5-7% for 60/40.
Enter number of periods. Total payments (40y × 12 = 480 for monthly Roth IRA).
Toggle ordinary or annuity-due. Savings = ordinary. Rent / lease = due.
Read the terminal FV. Dark badge bottom-right of chart shows accumulated value.

FV of Annuity — frequently asked questions

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What CFPs & actuaries say

4.9

Based on 5,760 reviews

“The stacker visualization makes "why 401k for 30 years gets you to $2.6M" instantly visceral. I show clients the cumulative principal line vs the total bar height — the gap IS compound interest. The ordinary vs annuity-due toggle is a thoughtful detail most calculators skip.”

Hiroko Aisha Fujimoto-Yamada

Certified Financial Planner (CFP®), retirement planning

May 15, 2026

“For pension contribution accumulation modeling, this calculator matches my SOA Exam FM textbook formulas exactly. The annuity-due bonus = (1+r) factor in the FAQ is the test-question I see every cycle.”

Olufunmilayo Adeyinka Bankole-Adelaja

Actuary, pension valuations

April 23, 2026

“For EU clients I model in EUR + annual freq + the equivalent UCITS fund tracking the MSCI World. The calculator handles all frequencies correctly. The Bernoulli 1683 historical anchor differentiates serious tools.”

Tomislav Mariusz Kowalczyk-Zelenovic

EU retirement planner, cross-border specialty

April 2, 2026

“The Halley/De Morgan/HP-12C historical lineage is exactly the kind of context the CFA Institute expects on Level III essays. The 401(k) preset values track the 2025 IRS limits accurately.”

Jin-Soo Aleksander Park-Petrov

CFA Level III candidate, retirement income specialist

March 13, 2026

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