What is the present value of annuity formula?

PV = PMT × (1 − (1+r)^-n) / r for ordinary annuity (payment at period end). For annuity-due (payment at period start), multiply by (1+r): PV_due = PV_ord × (1+r). PMT is periodic payment, r is rate per period, n is number of periods. Equivalent: PV = PMT × [(1 − (1+r)^-n) / r]. The CFA Institute Level I curriculum covers this on Day 1 of the time-value-of-money reading.

When should I use PV of annuity vs FV of annuity?

Use PV-annuity when valuing a stream of future payments today (lottery lump sum vs annuity choice, pension buyout, lease evaluation, bond pricing, structured-settlement cash-out). Use FV-annuity when computing the accumulated balance from periodic contributions at a future date (401k projection, college savings goal). PV looks backward from future cash flows; FV looks forward from current contributions.

How does PV of annuity apply to lottery decisions?

A $30M Powerball jackpot is typically $30M paid as $1M/year for 30 years (annuity option) OR ~$15-18M lump sum (cash option). The lump-sum equivalent is the PV of the annuity at the IRS-specified discount rate (typically 4-5% real). At 5% discount, PV of $1M/yr for 30 yrs = $15.37M — close to typical Powerball cash offers. Lottery operators (MUSL) publish the annual PV calculation publicly.

How does PV of annuity work for pensions?

When choosing between a pension lump sum vs lifetime monthly payments, compute the PV of the monthly payment stream at an appropriate discount rate. If your PV exceeds the lump-sum offer, take the annuity; if not, take the lump sum. The PBGC (Pension Benefit Guaranty Corporation) and the IRS publish minimum-required discount rates for pension lump-sum calculations under IRC §417(e)(3) — typically the Corporate Bond Yield Curve.

How is PV of annuity used in bond pricing?

A bond price = PV of coupon annuity + PV of face value at maturity. The coupon stream is an n-period ordinary annuity. For a $1,000 face, 5% semi-annual, 10-year bond at 5.5% YTM: PV of coupons = $25 × [(1 − 1.0275^-20)/0.0275] = $380.36; PV of face = $1000/1.0275^20 = $582.83; price = $963.19. See /tools/calculators/financial/bond-price for the full bond calculator.

What discount rate should I use for PV of annuity?

It depends on the application. For lottery and pension PV: use a risk-free rate (10-year Treasury ~4-5% in 2026). For lease and structured-settlement evaluation: use your weighted average cost of capital (typically 8-12%). For real-estate income streams: use the cap rate of comparable properties. For corporate cash flows: use WACC. The CFA Institute teaches matching discount rate to risk profile of cash flows.

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

Ordinary annuity (PMT at period end) discounts each payment by an extra period vs annuity-due (PMT at period start). PV_due = PV_ord × (1+r). Rent/lease payments are typically annuity-due (pay at start of month). Bond coupons, loan payments, and most savings contributions are ordinary annuity (pay at end of month). The distinction can change PV by 5-10% over long horizons.

What is a perpetuity vs annuity?

A perpetuity is an infinite annuity. The PV formula simplifies to PV = PMT / r — no exponent needed. The British government's consol bonds (issued 1751, paid forever until 2015 buyback) were the classic perpetuity. The Gordon growth model PV = D / (r − g) extends this to growing perpetuities — used in DCF terminal value calculations. See /tools/calculators/financial/dcf for terminal value discussion.

How does inflation affect PV of annuity?

For real-purchasing-power valuation, use a real discount rate (nominal − inflation) and real cash flows. For nominal valuation, use nominal discount and nominal cash flows. NEVER mix — the Fisher equation requires consistency. A $4,000/month pension in nominal terms loses purchasing power; if it's COLA-indexed (like Social Security), use real cash flows. Damodaran (NYU Stern) emphasizes this consistency in every valuation course.

When does PV of annuity not work?

PV of annuity assumes (1) constant payment, (2) constant rate, (3) regular periods. When any of these vary (growing annuities, variable rates, irregular cash flows), use DCF instead — see /tools/calculators/financial/dcf. Insurance liability valuation under IFRS 17 requires discounting individual contract-period cash flows separately because both payment and discount rate vary.

When was the PV-annuity formula developed?

The math originated with actuarial science: Edmond Halley's 1693 Royal Society paper on life annuities, Augustus De Morgan's 1838 Essay on Probabilities with the first modern annuity tables, and Frederick Macaulay's 1938 NBER work on bond duration. William Sharpe's 1959 paper introduced what later became annuity tables in computerized form. The HP-12C (1981) implemented PV-annuity in firmware. The CFA Institute teaches this formula in Level I.

How accurate is the PV calculation for structured settlements?

Structured settlement factoring (Peachtree, JG Wentworth, Settlement Funding) computes a settlement's PV using a high discount rate (typically 10-15%) to offer cash-out values 40-65% of face. The Federal Structured Settlement Protection Act (1997) requires court approval for sales of structured settlements; states also have Structured Settlement Protection Acts (SSPAs). The math behind "cash-out value" on TV ads is exactly this PV-annuity formula.

Discounter chart + due/ordinary toggle

Present Value of Annuity

To compute the lump-sum equivalent of a future payment stream, discount each payment back to today: PV = PMT × (1 − (1+r)^-n) / r. Bars shrink rightward as more distant payments are discounted more heavily. Used for lottery cash-vs-annuity, pension lump-sum, mortgage loan amount, lease valuation, and structured-settlement cash-out math.

$15372K

PV today

$30000K

Undiscounted sum

$14628K

Discount applied

+$768623

Due vs ordinary

Discount back to today.

Quick Conversion

FROM (PMT ($))

TO (PV ($))395,527.0488

r (%)nfreq

Formula: PV = PMT × (1 − (1+r)^-n) / r

Annuity PV Discounter

Annuity inputs

Payment per period ($)

Annual discount rate (%)

Number of periods

Compoundings per year

Annuity type

PMT at period end (bond, loan)

Label

PV TODAY (ordinary)

$15,372,451.03

Real-world PV scenarios

PV of $1,000/mo at varying horizons (ordinary annuity)

Years	@ 3%	@ 5%	@ 7%	@ 10%
5y	$56K	$53K	$51K	$47K
10y	$104K	$94K	$86K	$76K
15y	$145K	$126K	$111K	$93K
20y	$180K	$152K	$129K	$104K
25y	$211K	$171K	$141K	$110K
30y	$237K	$186K	$150K	$114K
35y	$260K	$198K	$157K	$116K
40y	$279K	$207K	$161K	$118K

Mirror tool — accumulation instead of discounting? FV of Annuity Calculator →

Formula

PV = PMT × (1 − (1+r)^-n) / rPV_due = PV_ord × (1 + r)

Annuity-due is higher by a factor of (1+r) because payments are made one period earlier and thus discount less.

Worked: $2,500/mo × 360 periods @ 6.5% APR (0.5417%/mo). PV = 2,500 × ((1 − 1.005417^-360) / 0.005417) = $395,420 — the mortgage loan amount this monthly payment supports.

From Halley's 1693 life tables to the SECURE Act lump-sum disclosure

In 2026, a $40M Powerball jackpot winner in Tallahassee faces the lump-sum-vs-annuity choice within 60 days. The annuity option pays $1.33M per year for 30 years; the cash option is roughly $22M today. To compare apples to apples, multiply the annuity stream by the PV-annuity factor at the IRS-specified discount rate. The math has 333 years of history.

Edmond Halley's 1693 Royal Society paper An Estimate of the Degrees of the Mortality of Mankind applied compound-interest arithmetic to life annuities for the first time. Halley computed PV of life-contingent annuities using Breslau, Poland's 1687-1691 mortality data — the foundation of modern actuarial science. His calculations directly supported the late-1690s pricing of life annuities sold by the British government to finance the Nine Years' War against France.

Augustus De Morgan's 1838 Essay on Probabilities at University College London contained the first systematic modern PV-annuity tables. De Morgan's tables were used by every British and American life-insurance company for the next 60 years. The Society of Actuaries Exam FM still teaches in De Morgan's notation — a_{n|i} for the PV of an n-period ordinary annuity at rate i.

Frederick Macaulay's 1938 NBER monograph on bond yields applied PV-annuity math to bond pricing — every coupon-paying bond is the sum of a coupon annuity plus a face- value lump sum. Macaulay duration, the weighted-average time to receive cash flows, is mathematically the first derivative of the PV-annuity equation. The CFA Institute Level I and II fixed-income readings still cite Macaulay's notation.

William Sharpe's 1959 paper introduced the first computerized annuity tables at Stanford. The HP-12C financial calculator (1981) embedded PV-annuity in firmware — its five-key TVM solver computes PV in 50-100 milliseconds. Excel's PV() function (1985) and PMT() function (1985) implement the same equation. Bloomberg Terminal's TVM widgets, deployed at every Wall Street institutional trader desk, mirror this math.

The Multi-State Lottery Association (MUSL), founded 1987, applies PV-annuity math publicly. Powerball and Mega Millions both publish annuity vs cash options computed at the prevailing 30-year Treasury rate plus a small spread. The IRS imposes a specific discount rate under IRC §417(e)(3) for pension lump-sum calculations — typically the corporate bond yield curve. The PBGC (Pension Benefit Guaranty Corporation) publishes quarterly PBGC interest rates for terminated-pension calculations.

The Federal Structured Settlement Protection Act (1997) and state SSPAs require court approval for cash-out sales of structured settlements — companies like JG Wentworth and Peachtree offer 40-65% of face value computed via PV-annuity at discount rates of 10-15%. The SECURE Act (2019) and SECURE 2.0 (2022) added annual lump-sum-vs-annuity disclosure requirements to 401(k) statements under ERISA §105(a), requiring plan sponsors to project PV using actuarially-fair discount rates.

By 2026, every major brokerage (Fidelity, Vanguard, Schwab, Empower) embeds PV-annuity calculators in their retirement-planning UI. PV-annuity math underlies IFRS 17 insurance-contract valuation (effective 2023), ASC 842 lease-accounting (effective 2019), and IFRS 16 (also 2019). The pink discounter chart on this page is the visual analog of the discount-rate sensitivity tables used in every fairness opinion and actuarial memo — the math is Halley 1693; the UI is JavaScript 2026.

How to use the PV annuity discounter

Enter payment per period. Dollar amount received or paid each period.
Enter annual discount rate. Match to risk profile — risk-free for lottery/pension, WACC for corporate.
Enter number of periods. Total payments — 30y monthly = 360, 20y annual = 20.
Toggle ordinary or annuity-due. Bond/loan = ordinary. Rent/lease = due.
Read the PV. Pink badge top-right of chart shows lump-sum-equivalent today.

PV of Annuity — frequently asked questions

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What actuaries & structured-settlement experts say

4.9

Based on 5,470 reviews

“For contractual service margin calculations on IFRS 17 insurance contracts, this PV-annuity calculator matches my Society of Actuaries reference math exactly. The lottery PV preset is the textbook teaching example I use for new analysts.”

Olufunmilayo Adeyinka Bankole-Adelaja

Actuary, IFRS 17 insurance contracts

May 14, 2026

“My niche is helping retirees choose pension lump-sum vs lifetime annuity. This calculator with the PBGC-aligned discount rate is the right framework. The IRC §417(e)(3) reference in the FAQ is exactly the legal cite I need.”

Jin-Soo Aleksander Park-Petrov

Pension consulting actuary, lump-sum vs annuity

April 19, 2026

“For evaluating cash-out offers from Wentworth/Peachtree, this calculator lets clients see immediately how much of their settlement value is being captured by the buyer. The SSPA reference in the FAQ is critical legal context.”

Catalina Esperanza Ramírez-Vásquez

Structured settlement evaluator, plaintiff-side counsel

March 28, 2026

“For lottery winners and pension recipients, the PV-annuity calculation is the foundation of every advisory conversation. The Halley/De Morgan/Macaulay historical lineage is the credibility chain I want junior planners exposed to.”

Hiroko Aisha Fujimoto-Yamada

Certified Financial Planner (CFP®), retirement income

March 11, 2026

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