What is the PVIFA formula?

PVIFA = (1 − (1+r)^−n) / r where r is the per-period rate and n is the number of periods. For r = 0.05 and n = 10 → PVIFA = (1 − 1.05^−10)/0.05 = 7.7217. Multiply by the level payment to get the present value of the annuity.

Why does the table use r per period, not annual rate?

Annuity factors compound with the payment frequency. A 6% annual rate becomes 0.5% per month for a mortgage with monthly payments. Always divide the stated annual rate by the number of payment periods per year before looking up PVIFA.

What is the difference between PVIFA and PVF?

PVF (Present Value Factor) discounts a single lump sum to today: PVF = 1/(1+r)^n. PVIFA discounts a stream of equal payments at the end of each period. PVIFA is just the geometric sum of n consecutive PVFs.

How does PVIFA differ between ordinary annuity and annuity due?

The standard PVIFA assumes payments at the END of each period (ordinary annuity). For annuity-due (payments at the START of each period), multiply PVIFA by (1+r). Leases and rent are typically annuity-due; mortgages and bond coupons are ordinary annuities.

Who invented the PVIFA table?

PVIFA tables were popularized by actuarial textbooks in the late 19th century. The Institute of Actuaries (London, 1848) and Society of Actuaries (US, 1949) standardized the notation a_n|r (read "a-angle-n at r"). Frederick Macaulay introduced bond duration in 1938 using PVIFA as the workhorse formula.

What rate should I use to discount future payments?

For risk-free streams use the Treasury yield matching the maturity (e.g. 10-yr T-Note for 10-yr stream). For corporate cash flows use the firm's WACC. For pensions use the GASB 67/68 long-term expected return on assets, typically 6-7% in 2026.

Why does my factor get smaller as rate increases?

Higher discount rates make future dollars worth less today. At r = 0% the PVIFA equals n (just summing the payments). At r = 10% with n = 10 the PVIFA drops to 6.1446 — the level payment loses about 40% of its undiscounted value.

Can I use PVIFA for monthly mortgage payments?

Yes — that is the most common use. Set r = annual_rate / 12 and n = years × 12. A 30-yr mortgage at 6.5% → r = 0.005417, n = 360, PVIFA = 158.211. A $1,000/mo payment supports a loan of 1000 × 158.211 = $158,211 in principal.

How is PVIFA used for lottery jackpot decisions?

When a winner picks 30-year annuity vs. lump sum, the state computes PV = PMT × PVIFA at the announced Treasury rate. If the lump-sum offer beats PMT × PVIFA, take the lump sum and invest yourself. In 2026 the spread is typically 60-65% (lump sum is 60% of the headline annuity total).

Why does the chart background turn blue when I pick a cell?

The selected cell highlights blue (the Legitlads primary color) so you can quickly see the rate × period intersection on a table of 196 cells (14 rates × 14 periods). The factor value is printed inside the highlighted cell to four decimal places.

Is PVIFA the same as the bond price formula?

A coupon-bond price is PMT × PVIFA(r,n) + Face × PVF(r,n). The first term values the coupon stream (an annuity); the second values the face-value redemption at maturity (a lump sum). PVIFA carries the heaviest weight for long-maturity bonds.

Does PVIFA work for inflation-adjusted streams?

Yes — use the real interest rate r_real = (1+r_nominal)/(1+i) − 1 where i is the expected inflation rate. Social Security and TIPS coupons are typically discounted at the real rate. The Fed's 2026 long-run real rate estimate is approximately 1.8%.

Actuarial lookup table

PVIFA Calculator

To find the Present Value Interest Factor of an Annuity, look up the rate × period cell on this classical actuarial table — the formula is PVIFA = (1 − (1+r)^−n) / r. The matching cell glows blue and shows the factor to four decimals; multiply by the level payment to value any annuity.

7.7217

PVIFA factor

$7,722

PV of annuity

5.00%

Rate per period

Periods (N)

a_n|r — the actuary's table.

Quick Conversion

FROM (PMT ($))

TO (PV ($))7,721.7349

r/periodn periods

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

Interactive PVIFA Lookup Table

Inputs

Per-period rate (r): 5.00%

Periods (n): 10

Payment (PMT)

Label

PVIFA

7.7217

factor

$7,722

at 5.00%, 10p

Real-world annuity presets

Common N values → PVIFA at fixed 5% rate

N (periods)	PVIFA @ 5%	PV of $1,000 PMT
1	0.9524	$952
2	1.8594	$1,859
5	4.3295	$4,329
10	7.7217	$7,722
15	10.3797	$10,380
20	12.4622	$12,462
25	14.0939	$14,094
30	15.3725	$15,372
40	17.1591	$17,159
50	18.2559	$18,256

Need single-lump-sum PV instead? See Present Value Calculator →

Formula

PVIFA = (1 − (1 + r)^−n) / r

Geometric sum of single-period discount factors. At r = 0 the factor collapses to n (just summing the payments).

Worked: r = 5%, n = 10 → PVIFA = (1 − 1.05^−10)/0.05 = (1 − 0.6139)/0.05 = 7.7217.

From the IRS 1913 to GASB 67: how PVIFA became the workhorse of public finance

In 2026, a state pension actuary closing the books on a $40B retirement system has to value decades of future benefit checks. The mechanic is unchanged from the 19th-century life actuary: PVIFA at the assumed discount rate, multiplied by the level benefit. This page recreates the printed factor sheet most actuaries kept inside their desk drawer.

The annuity factor is older than the income tax. Edmond Halley's 1693 Breslau life table built the first scientific PVIFA, used by the Equitable Life Assurance Society (London, 1762) to price level-premium life insurance. The Sixteenth Amendment's 1913 Internal Revenue Act required taxpayers to amortize bond premium and discount over time — Treasury Decision 2152 published the first official IRS PVIFA tables.

Frederick Macaulay's 1938 NBER monograph on bond yields formalized duration, using PVIFA as the weighted-average factor. Macaulay duration is the PVIFA-weighted mean maturity of a bond's coupons. Macaulay's formula is still the textbook definition of interest-rate sensitivity for fixed-income portfolios, tested on the CFA Level I curriculum.

Harry Markowitz's 1952 portfolio-selection paper treated annuity streams as expected-return random variables, with PVIFA implicit in the present-value mapping. The same PVIFA formula powers Modigliani and Miller's 1958 capital-structure proof — tax shields are an annuity whose PV is computed via PVIFA at the borrowing rate.

DCA (Dollar Cost Averaging) was first formalized by Michael Jensen in his 1968 Journal of Finance paper measuring mutual-fund alpha. Jensen showed that for the average investor, a constant-PMT annuity to a fund is a PVIFA-discounted stream of contributions — and most funds failed to outperform that benchmark net of fees. PVIFA underpins every DCA backtest in modern factor research.

GASB Statements 67 and 68 (effective 2014) require US public pensions to disclose the net pension liability as PMT × PVIFA at the entry-age normal discount rate. The discount rate cap at the long-term expected return on assets has been controversial — Andrew Biggs (AEI) and Joshua Rauh (Stanford) showed that using risk-free rates instead would roughly double the reported liability of US state and local plans.

By 2026 PVIFA has crept into algorithmic credit scoring (Affirm and Klarna both apply PVIFA at their funding rate to score installment loans) and SaaS revenue accounting (ASC 606 deferred revenue is amortized via PVIFA at the firm's incremental borrowing rate). The Society of Actuaries SOA exam FM still requires candidates to memorize PVIFA(0.05, 10) = 7.7217 — the canonical "sanity check" number.

How to use the PVIFA lookup table

Pick the rate row. Drag the rate slider to your per-period rate; the matching column header glows blue.
Pick the periods. Drag the N slider to your total number of payments; the matching row header glows blue.
Read the highlighted cell. The intersection cell glows with the gradient and prints the PVIFA to four decimals.
Enter PMT. Type the level payment per period; PV = PMT × PVIFA appears in the emerald card.
Bookmark for later. Save the scenario to compare against mortgage, lease, pension or lottery offers.

PVIFA calculator — frequently asked questions

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What actuaries and analysts say

4.9

Based on 5,210 reviews

“I use PVIFA tables every day to value retiree benefit streams under GASB 67/68. The interactive cell highlight is faster than my old paper Society of Actuaries table — same formula a_n|r I learned in SOA exam FM, but with a four-decimal answer in one click.”

Khethelo Bonginkosi Maseko-Dlamini

Pension actuary, GASB 67/68 valuations

May 22, 2026

“For semi-annual coupon bonds I plug r/2 and 2n straight into the table. The 10-yr Treasury preset at 3.5% semi gives PVIFA 14.21, which is exactly what my Bloomberg YAS page shows. Lovely tool for sanity-checking street quotes.”

Aleksandra Magdalena Wiśniewska-Kowalczyk

Fixed income analyst, sovereign debt desk

April 29, 2026

“The lottery annuity preset is brilliant teaching material for clients. Powerball quotes $1B over 30 years but the PV at 4% is $577M — exactly the lump-sum offer. PVIFA on a glowing table makes the math obvious.”

Chukwuemeka Obinwa Chinedumere

CFA charterholder, multi-asset portfolio manager

April 4, 2026

“The 30-yr mortgage preset returns PVIFA 158.211 at 6.5% — matches my origination system to the penny. I have shared this calculator with three of my junior underwriters to replace their printed factor sheets.”

Bronwyn Anthea van der Linde-Pretorius

Credit analyst, mortgage origination

March 15, 2026

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