# Annuity

An annuity is a series of equal payments made at fixed intervals, valued by discounting each payment to the present at a stated interest rate. The term covers two distinct contracts: the annuity-certain of finance, whose payments run for a fixed term regardless of any person's survival, and the life annuity of insurance, whose payments are contingent on a named person's survival.

| Key fact | Detail |
|---|---|
| Present value, ordinary annuity | PV = PMT × (1 − (1+r)^(−N))/r; the annuity-due value is this times (1+r)<sup>[1](https://www.worldscientific.com/doi/pdf/10.1142/9789813224681_0002)</sup><sup> • </sup><sup>[2](https://openstax.org/books/principles-finance-2e/pages/8-2-annuities)</sup> |
| Future value | FV = PMT × ((1+r)^N − 1)/r, equal to the present value accumulated N periods<sup>[2](https://openstax.org/books/principles-finance-2e/pages/8-2-annuities)</sup><sup> • </sup><sup>[1](https://www.worldscientific.com/doi/pdf/10.1142/9789813224681_0002)</sup> |
| Perpetuity | PV = PMT/r (for r > 0); finite for a positive discount rate because discounting shrinks each added payment<sup>[1](https://www.worldscientific.com/doi/pdf/10.1142/9789813224681_0002)</sup><sup> • </sup><sup>[3](https://biz.libretexts.org/Courses/Aurora_University/Principles_of_Financial_Management/05%3A_Valuation_of_Financial_AssetsTime_Value_of_Money/5.03%3A_Annuities_and_Uneven_Cash_Flows)</sup> |
| Growing perpetuity (Gordon growth) | PV = CF₁/(i − g), valid when i > g<sup>[4](https://pthistle.faculty.unlv.edu/FIN301-London/Chapter06.pdf)</sup> |
| Life annuity value | Actuarial present value, the expectation of discounted payments over the curtate future lifetime K<sup>[5](https://users.math.msu.edu/users/valdezea/stt455f14/STT455Weeks9to11-F2014-annot.pdf)</sup> |
| U.S. market size, 2024 | Retail annuity sales a record $434.1 billion (LIMRA, 92% market coverage); annuity considerations $456 billion and reserves $4.5 trillion (ACLI, all annuities)<sup>[6](https://www.limra.com/en/newsroom/news-releases/2025/limra-2024-retail-annuity-sales-grow-12-to-a-record-$434.1-billion/)</sup><sup> • </sup><sup>[7](https://www.acli.com/-/media/public/pdf/news-and-analysis/publications-and-research/2025fb/8_annuities_acli_fact_book_2025.pdf)</sup> |
| Payout example, 2024 | $100,000 immediate annuity: $1,808 per month for 5 years certain, or $551 per month for a male age 60 single life<sup>[8](https://www.immediateannuities.com/comparative-annuity-reports/2024-08-comparative-annuity-reports.html)</sup> |

## What an annuity is

The defining choices are timing and contingency. An ordinary annuity (annuity-immediate) assumes a one-period lag before the first payment, as with a bond that pays interest at the end of every six months; an annuity due assumes payments begin immediately, as with rent due at the start of each month<sup>[2](https://openstax.org/books/principles-finance-2e/pages/8-2-annuities)</sup><sup> • </sup><sup>[9](https://www.investopedia.com/retirement/calculating-present-and-future-value-of-annuities/)</sup>. A life annuity adds a mortality contingency: payments are generally contingent on survival, though a guarantee period can continue payments after death, so its value depends on survival probabilities as well as interest rates.

## Valuation formulas

For an ordinary annuity of payment PMT for N periods at rate r per period,

\[ PV = \mathrm{PMT} \cdot \frac{1 - (1+r)^{-N}}{r} \]

The bracketed term is the annuity factor: it converts a level payment stream into a single present value. The future value is

\[ FV = \mathrm{PMT} \cdot \frac{(1+r)^{N} - 1}{r} \]

which is the present value accumulated N periods, sₙ = aₙ × (1+i)ⁿ in [actuarial notation](https://www.edgepedia.org/actuarial-notation)<sup>[2](https://openstax.org/books/principles-finance-2e/pages/8-2-annuities)</sup><sup> • </sup><sup>[1](https://www.worldscientific.com/doi/pdf/10.1142/9789813224681_0002)</sup>.

**The (1+r) shift.** Each payment of an annuity due is made one period ahead of the corresponding payment of an annuity immediate, so the present value of each due payment is (1+r) times the present value of the corresponding immediate payment; the whole stream's value is therefore multiplied by (1+r)<sup>[1](https://www.worldscientific.com/doi/pdf/10.1142/9789813224681_0002)</sup>. In practice one solves the problem as an ordinary annuity and multiplies by (1+r) once more<sup>[2](https://openstax.org/books/principles-finance-2e/pages/8-2-annuities)</sup>. For a positive rate, whether computing present or future value, the annuity-due result is larger than the ordinary-annuity result, all else constant<sup>[2](https://openstax.org/books/principles-finance-2e/pages/8-2-annuities)</sup>.

**Perpetuities.** For r > 0, letting the term grow without limit makes (1+r)^(−N) tend to zero and the annuity factor collapse to 1/r, giving a perpetuity value of PMT/r<sup>[1](https://www.worldscientific.com/doi/pdf/10.1142/9789813224681_0002)</sup>. Although a perpetuity has no ending date, its present value is finite for a positive discount rate because future payments are increasingly discounted, each additional payment contributing less than the one before<sup>[3](https://biz.libretexts.org/Courses/Aurora_University/Principles_of_Financial_Management/05%3A_Valuation_of_Financial_AssetsTime_Value_of_Money/5.03%3A_Annuities_and_Uneven_Cash_Flows)</sup>. A perpetuity has no single terminal future value, since its life is indefinite<sup>[10](https://oercollective.caul.edu.au/fundamental-finance/chapter/chapter-3-present-and-future-values-with-applications/)</sup>.

**Growing payments.** If payments grow at a constant rate g, the growing perpetuity is valued as PMT ÷ (i − g), the form underlying the [Gordon growth model](https://www.edgepedia.org/gordon-growth-model) of equity valuation<sup>[4](https://pthistle.faculty.unlv.edu/FIN301-London/Chapter06.pdf)</sup><sup> • </sup><sup>[10](https://oercollective.caul.edu.au/fundamental-finance/chapter/chapter-3-present-and-future-values-with-applications/)</sup>. Growth raises value sharply: a $90,000 payment stream discounted at 9% is worth $1,000,000 as a level perpetuity but $2,250,000 with 5% growth, since $90,000 ÷ 0.04 = $2,250,000<sup>[4](https://pthistle.faculty.unlv.edu/FIN301-London/Chapter06.pdf)</sup>.

**Continuous payments.** When payments are made continuously at unit rate, the accumulated value over n periods is s̄ₙ = ((1+i)ⁿ − 1)/ln(1+i), and there is no distinction between annuity-due and annuity-immediate<sup>[1](https://www.worldscientific.com/doi/pdf/10.1142/9789813224681_0002)</sup>.

## By the numbers

A structured-settlement example shows the factor at work: a $450,000 insurance settlement paid as $45,000 per year over 10 years, beginning one year from now, can be compared with a lump sum using the present value of an annuity formula at a 6% discount rate<sup>[2](https://openstax.org/books/principles-finance-2e/pages/8-2-annuities)</sup>. Market payout quotes from August 2024 put a $100,000 immediate annuity at $1,808 per month for a 5-year period certain and $551 per month for a male age 60 single life, reflecting higher interest rates<sup>[8](https://www.immediateannuities.com/comparative-annuity-reports/2024-08-comparative-annuity-reports.html)</sup>.

On market size, two measures differ by construction. LIMRA's U.S. Individual Annuity Sales Survey, covering 92% of the market, recorded record retail sales of $434.1 billion in 2024, up 13%<sup>[6](https://www.limra.com/en/newsroom/news-releases/2025/limra-2024-retail-annuity-sales-grow-12-to-a-record-$434.1-billion/)</sup>. The ACLI Fact Book, which counts all annuities including group contracts, reports considerations (payments into annuities) of $456 billion in 2024, up 18.8%, with annuity reserves of $4.5 trillion<sup>[7](https://www.acli.com/-/media/public/pdf/news-and-analysis/publications-and-research/2025fb/8_annuities_acli_fact_book_2025.pdf)</sup>. The two figures are not directly comparable: retail sales versus all considerations, and different survey coverage.

## How annuities compare with perpetuities and bonds

A perpetuity is the limiting case of an annuity whose equal payments continue forever<sup>[11](https://digitalcommons.georgefox.edu/cgi/viewcontent.cgi?article=1114&context=gfsb)</sup>. A bond decomposes into an annuity plus a lump sum: its cash flows are an ordinary annuity of equal coupon payments plus the par value repaid at maturity, so bond value is the sum of an annuity present value and a discounted lump sum<sup>[11](https://digitalcommons.georgefox.edu/cgi/viewcontent.cgi?article=1114&context=gfsb)</sup>. Perpetuity valuation also underpins preferred stock and dividend-paying equity<sup>[3](https://biz.libretexts.org/Courses/Aurora_University/Principles_of_Financial_Management/05%3A_Valuation_of_Financial_AssetsTime_Value_of_Money/5.03%3A_Annuities_and_Uneven_Cash_Flows)</sup>.

## Life annuities and mortality

A whole life annuity-due pays $1 at the beginning of each year the annuitant (x) survives. Its actuarial present value is äₓ = E[ä_{K+1}], where K is the curtate future lifetime, the number of complete years lived; the actuarial present value is the standard valuation quantity<sup>[5](https://users.math.msu.edu/users/valdezea/stt455f14/STT455Weeks9to11-F2014-annot.pdf)</sup>. Mortality enters through survival probabilities applied to each payment, and discounting through the interest rate; US expected-present-discounted-value calculations benchmark the purchase price against discounted monthly payments using the U.S. Treasury bond yield curve as the riskless discount rate series<sup>[12](https://adam-solomon.com/EPDVPaper.pdf)</sup>.

**A key difference from annuities-certain.** For fixed-term annuities, the annuity-immediate value is simply the annuity-due value discounted one period. For life annuities this shortcut fails: the payment streams of a life annuity-due deferred 1/m year and a life annuity-immediate end at the same time of death rather than with the same number of payments, so one cannot simply multiply the life annuity-due present value by the discount factor v^(1/m)<sup>[13](https://www.math.umd.edu/~slud/s470/BookChaps/Chp45.pdf)</sup>.

Regulated practice fixes the mortality inputs. UK pension illustrations must use the PMA92 and PFA92 mortality tables with the medium cohort year-of-birth projection from the Institute and Faculty of Actuaries<sup>[14](https://www.fca.org.uk/publication/policy/ps12-14.pdf)</sup>. US individual deferred variable annuity standards set the post-payout mortality factor from the Annuity 2000 Mortality Table, a modification with no lower life expectancy at any age, or an approved annuity mortality table<sup>[15](https://www.insurancecompact.org/sites/default/files/2024-08/20140915_core_individual_deferred_variable.pdf)</sup>.

**Measuring value for money.** The UK Financial Conduct Authority assesses annuity value with the Money's Worth ratio, the expected present value of annuity income divided by the annuity premium<sup>[16](https://www.fca.org.uk/publication/occasional-papers/occasional-paper-5.pdf)</sup>. Mortality differences show up in product features: adding a 20-year guarantee period reduces the annual payout for a 65-year-old man by 8.8% but by only 6.9% for a similar-aged woman, because a 65-year-old woman is more likely to survive to age 85<sup>[17](https://www.nber.org/system/files/working_papers/w28557/w28557.pdf)</sup>.

## Where annuity valuation is used

Loan amortisation is the ordinary annuity in reverse: a $20,000 car loan at 6% annual interest repaid monthly over 5 years, with the first installment one month after the loan is released, is priced with the level-installment annuity formula<sup>[1](https://www.worldscientific.com/doi/pdf/10.1142/9789813224681_0002)</sup>. Lenders issue amortization schedules showing the payments needed to pay off the loan and how the split between principal and interest shifts over time<sup>[9](https://www.investopedia.com/retirement/calculating-present-and-future-value-of-annuities/)</sup>. The ordinary-annuity present value is also the standard tool for evaluating leases and service contracts<sup>[3](https://biz.libretexts.org/Courses/Aurora_University/Principles_of_Financial_Management/05%3A_Valuation_of_Financial_AssetsTime_Value_of_Money/5.03%3A_Annuities_and_Uneven_Cash_Flows)</sup>, and structured settlements use the same factor to price periodic injury payouts<sup>[2](https://openstax.org/books/principles-finance-2e/pages/8-2-annuities)</sup>.

**Solving for the rate.** Given a price and a payment stream, the implied interest rate generally admits no analytic solution; numerical methods such as grid search, or the Excel RATE function, are used<sup>[1](https://www.worldscientific.com/doi/pdf/10.1142/9789813224681_0002)</sup>.

## Recent developments

U.S. annuity sales have set records in four consecutive years. After $434.1 billion in 2024<sup>[6](https://www.limra.com/en/newsroom/news-releases/2025/limra-2024-retail-annuity-sales-grow-12-to-a-record-$434.1-billion/)</sup>, sales rose 6% to $461.3 billion in 2025<sup>[18](https://www.limra.com/en/newsroom/news-releases/2026/limra-u.s.-retail-annuity-sales-top-$460-billion-in-2025-marking-fourth-year-of-record-sales/)</sup>.

**RILA growth.** Registered index-linked annuity (RILA) sales reached $65.6 billion in 2024, up 38% and the tenth consecutive record year, the first year RILA outsold traditional variable annuities, up from $3.7 billion in 2015<sup>[6](https://www.limra.com/en/newsroom/news-releases/2025/limra-2024-retail-annuity-sales-grow-12-to-a-record-$434.1-billion/)</sup>. In 2025 RILA sales rose a further 20% to $79.6 billion, roughly ten times the level of a decade earlier, and indexed products (RILA plus fixed indexed annuities) took 45% of total sales, up from 24% ten years before; LIMRA projects RILA sales above $85 billion in 2026<sup>[18](https://www.limra.com/en/newsroom/news-releases/2026/limra-u.s.-retail-annuity-sales-top-$460-billion-in-2025-marking-fourth-year-of-record-sales/)</sup>.

**Rate sensitivity.** Fixed-rate deferred (FRD) annuity sales track interest rates closely. They fell 7% to $153.2 billion in 2024, with fourth-quarter sales down 50% from the record fourth quarter of 2023 as falling rates dampened demand<sup>[6](https://www.limra.com/en/newsroom/news-releases/2025/limra-2024-retail-annuity-sales-grow-12-to-a-record-$434.1-billion/)</sup>, then recovered 5% to $160.6 billion in 2025<sup>[18](https://www.limra.com/en/newsroom/news-releases/2026/limra-u.s.-retail-annuity-sales-top-$460-billion-in-2025-marking-fourth-year-of-record-sales/)</sup>.

## References

1. [Financial Mathematics for Actuaries: Annuities, World Scientific](https://www.worldscientific.com/doi/pdf/10.1142/9789813224681_0002)
2. [Principles of Finance 2e, Section 8.2 Annuities, OpenStax](https://openstax.org/books/principles-finance-2e/pages/8-2-annuities)
3. [5.3: Annuities and Uneven Cash Flows, Business LibreTexts](https://biz.libretexts.org/Courses/Aurora_University/Principles_of_Financial_Management/05%3A_Valuation_of_Financial_AssetsTime_Value_of_Money/5.03%3A_Annuities_and_Uneven_Cash_Flows)
4. [Chapter 6: Interest Rates and Bond Valuation, UNLV](https://pthistle.faculty.unlv.edu/FIN301-London/Chapter06.pdf)
5. [STT 455 Actuarial Mathematics notes: Life Annuities, Michigan State University](https://users.math.msu.edu/users/valdezea/stt455f14/STT455Weeks9to11-F2014-annot.pdf)
6. [LIMRA: 2024 Retail Annuity Sales Grow to a Record $434.1 Billion](https://www.limra.com/en/newsroom/news-releases/2025/limra-2024-retail-annuity-sales-grow-12-to-a-record-$434.1-billion/)
7. [ACLI Fact Book 2025, Group and Individual Annuities](https://www.acli.com/-/media/public/pdf/news-and-analysis/publications-and-research/2025fb/8_annuities_acli_fact_book_2025.pdf)
8. [Comparative Annuity Reports, August 2024](https://www.immediateannuities.com/comparative-annuity-reports/2024-08-comparative-annuity-reports.html)
9. [Calculating the Present and Future Value of Annuities, Investopedia](https://www.investopedia.com/retirement/calculating-present-and-future-value-of-annuities/)
10. [Chapter 3: Present and Future Values with Applications, Fundamentals of Finance](https://oercollective.caul.edu.au/fundamental-finance/chapter/chapter-3-present-and-future-values-with-applications/)
11. [Helping Students Crack Annuity, Perpetuity, Bond, and Stock Valuation Formulas, George Fox Digital Commons](https://digitalcommons.georgefox.edu/cgi/viewcontent.cgi?article=1114&context=gfsb)
12. [Discount Rates, Mortality Projections, and Money's Worth Calculations for US Individual Annuities](https://adam-solomon.com/EPDVPaper.pdf)
13. [Actuarial Mathematics and Life-Table Statistics, University of Maryland, Ch. 4–5](https://www.math.umd.edu/~slud/s470/BookChaps/Chp45.pdf)
14. [FCA PS12/14: Mortality assumption for pension Key Features Illustrations](https://www.fca.org.uk/publication/policy/ps12-14.pdf)
15. [Core Standards for Individual Deferred Variable Annuity Contracts, Insurance Compact](https://www.insurancecompact.org/sites/default/files/2024-08/20140915_core_individual_deferred_variable.pdf)
16. [FCA Occasional Paper No.5: The value for money of annuities and other retirement income strategies in the UK](https://www.fca.org.uk/publication/occasional-papers/occasional-paper-5.pdf)
17. [NBER Working Paper w28557: annuity payout valuation](https://www.nber.org/system/files/working_papers/w28557/w28557.pdf)
18. [LIMRA: U.S. Retail Annuity Sales Top $460 Billion in 2025](https://www.limra.com/en/newsroom/news-releases/2026/limra-u.s.-retail-annuity-sales-top-$460-billion-in-2025-marking-fourth-year-of-record-sales/)

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*Topic: Encyclopedia › Society and history › Economics and business › Finance › Finance theory and quantitative methods › Valuation and corporate finance › Titles A to F*

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