# Annuity

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{{Short description|Series of payments made at equal intervals}}
{{Other uses|Annuity (disambiguation)}}{{Personal finance}}{{more citations needed|date=April 2025}}
In [investment](/source/investment), an '''annuity''' is a series of payments of the same kind made at equal time intervals, usually over a finite term.<ref name="Broverman-MIC" /> Annuities are commonly issued by [life insurance](/source/life_insurance) companies, where an individual pays a lump sum or a series of premiums in return for regular income payments, often to provide [retirement](/source/retirement) or survivor benefits.<ref name="SEC-Annuities" />

Typical examples include regular deposits to a savings account, monthly home [mortgage](/source/mortgage) payments, monthly [insurance](/source/insurance) premiums and [pension](/source/pension) payments.<ref name="Broverman-MIC" /> The value of an annuity is usually expressed as a [present value](/source/present_value) or [future value](/source/future_value), calculated by discounting or accumulating the payments at a specified [interest rate](/source/interest_rate).

Annuities can be classified by the timing of payments, for example annuity-immediate and annuity-due, by whether the term is fixed or contingent on survival, and by whether the amounts are fixed, variable or linked to an index. Contracts may start paying immediately or after a deferral period, and a contract that continues indefinitely is a [perpetuity](/source/perpetuity).

== Types ==
Annuities may be classified in several ways.

=== Timing of payments ===
Payments in an annuity-immediate are made at the end of each payment period, so interest accrues during the period before each payment. By contrast, payments in an annuity-due are made at the beginning of each period, so each payment is made in advance.<ref name="Kent-BasicAnnuities" /><ref name="Broverman-MIC" />

Typical examples of annuity-immediate payment streams include home [mortgage](/source/mortgage_loan) and other loan repayments, where each installment covers interest that has accrued during the preceding period. Rent, leases and many [insurance](/source/insurance) premiums are usually paid in advance and are therefore examples of annuity-due payments.<ref name="Investopedia-AnnuityDue" /><ref name="MathFinance-Fundamentals" />

=== Contingency of payments ===
An annuity that pays over a fixed period, regardless of the survival of any individual, is an annuity certain. In this case the number of payments is known in advance and specified in the contract.<ref name="Khoruzhenko-Lecture" />

A [life annuity](/source/life_annuity) pays while one or more specified lives survive, so the number of payments is uncertain.<ref name="Khoruzhenko-Lecture" /><ref name="AMF-LifeAnnuityCertain" /> Pensions that pay a regular income for life are examples of life annuities.

A certain-and-life annuity, also called a life annuity with period certain, combines these features. Payments continue for at least a guaranteed minimum term and thereafter for as long as the annuitant is alive.<ref name="NC-DOI-LifeCertain" /><ref name="AnnuityOrg-LifeCertain" />

=== Variability of payments ===
* Fixed annuities provide payments determined using a fixed interest rate declared by the insurer, so the contract offers a guaranteed minimum rate of return on the account value.<ref name="NAIC-AnnuityTopics" /><ref name="FINRA-Annuities" />
* [Variable annuities](/source/Variable_annuity) invest premiums in underlying portfolios such as mutual funds, so the contract value and income payments vary with the performance of those investments.<ref name="SEC-Variable" /><ref name="FINRA-Variable" />
* [Equity-indexed annuities](/source/Equity-indexed_annuity) credit interest based partly on the performance of a specified market index, usually subject to a minimum guaranteed return and features such as caps or participation rates.<ref name="SEC-Indexed" /><ref name="FINRA-Indexed" />

=== Deferral of payments ===
A deferred annuity starts income payments after a deferral or accumulation period. During the deferral period the contract typically credits interest or investment returns to the account value.<ref name="NAIC-FixedDeferred" /><ref name="III-DeferredImmediate" /> A [immediate annuity](/source/Immediate_annuity) starts payments shortly after the contract is purchased, often within one year.<ref name="III-DeferredImmediate" /><ref name="Schwab-ImmediateDeferred" />

Fixed, variable and indexed annuities can each be written as immediate or deferred contracts.<ref name="NAIC-AnnuityTopics" /><ref name="FINRA-Annuities" />

==Valuation==
Valuation of an annuity treats the stream of payments as [cash flow](/source/cash_flow)s and summarises them by a [present value](/source/present_value) or a [future value](/source/future_value) at a given [interest rate](/source/interest_rate).<ref name="Broverman-MIC" /><ref name="Kellison-Interest" /> For a level annuity certain, the formulas depend on whether payments are made at the end or at the beginning of each period.
=== Annuity-certain ===
If the number of payments is known in advance, the contract is an annuity certain (also called a guaranteed annuity).<ref name="Broverman-MIC" /> Valuation uses the formulas below, which depend on the timing of payments.

====Annuity-immediate====
If payments are made at the end of each period, so interest accrues during the period before each payment, the annuity is an annuity-immediate (ordinary annuity).<ref name="Kellison-Interest" /> Mortgage payments are a typical example, since interest is charged between payments and then repaid at each due date.<ref name="MathFinance-Fundamentals" />
{| style="margin:1em auto;"
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Let <math>i</math> denote the effective interest rate per period and <math>n</math> the number of payments. The present value factor for a level annuity-immediate with unit payments is:
<math display="block">a_{\overline{n}|i} = \frac{1-(1+i)^{-n}}{i}</math>
and the present value of payments of amount <math>R</math> is:
:<math>\mathrm{PV}(i,n,R) = R\,a_{\overline{n}|i}.</math><ref name="Broverman-MIC" />

In practice, interest is often quoted as a nominal annual rate <math>J</math> convertible monthly or some other frequency. If payments are monthly and the nominal annual rate is <math>J</math>, then the rate per month is <math>i = J/12</math> and the number of payments over <math>t</math> years is <math>n = 12t</math>.<ref name="Kellison-Interest" />

The future value of a level annuity-immediate with unit payments is
<math display="block">s_{\overline{n}|i} = \frac{(1+i)^n-1}{i}</math>
and the accumulated value immediately after the last payment is:
:<math>\mathrm{FV}(i,n,R) = R\,s_{\overline{n}|i}.</math><ref name="PresentValueAnnuity" />

Example: The present value of a 5 year annuity with a nominal annual interest rate of 12% and monthly payments of $100 is
<math display="block">\mathrm{PV}\!\left( \frac{0.12}{12},5\times 12,100\right) = 100 \times a_{\overline{60}|0.01} \approx 4{,}495.50</math>
so the series of payments is equivalent to a single amount of about $4,496 at time zero.

Future and present values for an annuity-immediate are related by
<math display="block">s_{\overline{n}|i} = (1+i)^n\,a_{\overline{n}|i}</math>
and
:<math>\frac{1}{a_{\overline{n}|i}} - \frac{1}{s_{\overline{n}|i}} = i.</math><ref name="Kellison-Interest" />

===== Proof of annuity-immediate formula =====
To obtain the present value factor, consider a level annuity-immediate with unit payments. The payment at the end of period <math>k</math> is discounted by the factor <math>(1+i)^{-k}</math>, so the present value factor is
<math display="block">a_{\overline{n}|i} = \sum_{k=1}^{n} \frac{1}{(1+i)^k}.</math>
Let <math>v = (1+i)^{-1}</math> be the discount factor for one period. Then
<math display="block">a_{\overline{n}|i} = v + v^{2} + \cdots + v^{n} = v\sum_{k=0}^{n-1} v^{k}.</math>
Using the standard formula for the sum of a finite geometric series gives
:<math>a_{\overline{n}|i} = v\,\frac{1-v^{n}}{1-v} = \frac{1-v^{n}}{i} = \frac{1-(1+i)^{-n}}{i}.</math><ref name="Kellison-Interest" /><ref name="Broverman-MIC" />

====Annuity-due====
An annuity due is a series of equal payments made at the same interval at the beginning of each period.<ref name="MathFinance-Fundamentals" /> Periods can be monthly, quarterly, semi-annually, annually or any other defined period. Examples include rentals, leases and many insurance payments, which are made to cover services provided in the period following the payment.<ref name="Investopedia-AnnuityDue" />
{| style="margin:1em auto;"
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For an annuity-due with unit payments the present value factor is
<math display="block">\ddot{a}_{\overline{n}|i} = (1+i)\,a_{\overline{n}|i}</math>
and the future value factor is
:<math>\ddot{s}_{\overline{n}|i} = (1+i)\,s_{\overline{n}|i}.</math><ref name="Broverman-MIC" /><ref name="Kellison-Interest" />

The present and future values for an annuity-due satisfy
<math display="block">\ddot{s}_{\overline{n}|i} = (1+i)^n\,\ddot{a}_{\overline{n}|i}</math>
and
<math display="block">\frac{1}{\ddot{a}_{\overline{n}|i}} - \frac{1}{\ddot{s}_{\overline{n}|i}} = d,</math>
where <math>d = \frac{i}{1+i}</math> is the effective rate of discount.<ref name="PresentValueAnnuity" />

Example: The future value of a 7 year annuity-due with a nominal annual interest rate of 9% and monthly payments of $100 is
<math display="block">\mathrm{FV}_{\text{due}}\!\left(\frac{0.09}{12},7\times 12,100\right) = 100 \times \ddot{s}_{\overline{84}|0.0075} \approx 11{,}730.01.</math>

====Perpetuity====
A [perpetuity](/source/perpetuity) is an annuity for which the payments continue indefinitely.<ref name="Kellison-Interest" /> For a level perpetuity with payment <math>R</math> each period and per period interest rate <math>i</math>, the present value can be obtained as the limit of the level annuity-immediate present value as the term tends to infinity:
<math display="block">\lim_{n\to\infty} \mathrm{PV}(i,n,R) = \lim_{n\to\infty} R\,a_{\overline{n}|i} = \frac{R}{i}</math>
so the closed form is
<math display="block">\mathrm{PV}_{\text{perpetuity}} = \frac{R}{i}</math>
provided <math>i</math> is positive.<ref name="PresentValueAnnuity" /> In actuarial notation the present value factors for level perpetuities are
<math display="block">a_{\overline{\infty}|i} = \frac{1}{i}</math>
and
<math display="block">\ddot{a}_{\overline{\infty}|i} = \frac{1}{d},</math>
where <math>d = \frac{i}{1+i}</math> is the effective discount rate.<ref name="Broverman-MIC" />

=== Life annuities ===
Valuation of [life annuities](/source/Life_annuity) extends the level annuity formulas by taking into account mortality as well as interest. For a life aged <math>x</math> with annual payments of amount <math>R</math> payable while the life survives, the actuarial present value is the expected value of the discounted payment stream,
<math display="block">\mathrm{APV} = \sum_{t=1}^{\infty} R v^{t}\,{}_t p_x</math>
where <math>v = (1+i)^{-1}</math> is the discount factor per period and <math>{}_t p_x</math> is the probability that a life aged <math>x</math> survives at least <math>t</math> periods.<ref name="Bowers-LifeAnnuity" /><ref name="Goeters-LifeAnnuities" />

In actuarial notation the present value of a whole life annuity-immediate of 1 per year on a life aged <math>x</math> is written <math>a_x</math> and can be expressed as
<math display="block">a_x = \sum_{t=1}^{\infty} v^{t}\,{}_t p_x</math>
while the corresponding whole life annuity-due has present value factor
:<math>\ddot{a}_x = \sum_{t=0}^{\infty} v^{t}\,{}_t p_x.</math><ref name="Bowers-LifeAnnuity" /><ref name="Slud-LifeAnnuity" />

== Amortization calculations ==
If an annuity is used to repay a loan with level payments at the end of each period, the payment stream is an annuity-immediate. Let <math>P</math> be the initial loan principal, <math>R</math> the regular payment, <math>i</math> the effective interest rate per period and <math>N</math> the total number of payments. Then the present value of the payment stream is
<math display="block">P = R\,a_{\overline{N}|i} = R\,\frac{1-(1+i)^{-N}}{i},</math>
so the level payment that amortises the loan is
:<math>R = \frac{P}{a_{\overline{N}|i}} = P\,\frac{i}{1-(1+i)^{-N}}.</math><ref name="Kellison-Interest" /><ref name="Broverman-MIC" /><ref name="MathFinance-Fundamentals" />

The outstanding balance after <math>n</math> payments can be obtained in two equivalent ways. Under the retrospective method, the balance is the original principal accumulated with interest for <math>n</math> periods minus the accumulated value of the payments already made:
:<math>B_n = (1+i)^{n} P - R\,\frac{(1+i)^{n}-1}{i} = \frac{R}{i} - (1+i)^{n}\!\left(\frac{R}{i} - P\right).</math><ref name="Kellison-Interest" />

Under the prospective method, the outstanding balance is the present value of the remaining <math>N-n</math> payments:
:<math>B_n = R\,a_{\overline{N-n}|i} = R\,\frac{1 - (1+i)^{-(N-n)}}{i}.</math><ref name="Broverman-MIC" />

For an annuity due with payments at the beginning of each period, the same ideas apply but annuity-due factors are used. If <math>R</math> is the level payment and there are <math>N</math> payments in total, the outstanding balance after <math>n</math> payments is
:<math>B_n^{(\text{due})} = R\,\ddot{a}_{\overline{N-n}|i}, \qquad \ddot{a}_{\overline{m}|i} = (1+i)\,a_{\overline{m}|i}.</math><ref name="Kellison-Interest" />

Example. Let <math>P = 1{,}000</math>, <math>i = 0.10</math>, <math>N = 3</math>. Then
<math display="block">R = \frac{P\,i}{1-(1+i)^{-N}} = \frac{1{,}000\times 0.10}{1-(1.10)^{-3}} \approx 402.11.</math>
After one payment the retrospective and prospective balances coincide:
<math display="block">B_1 = 1{,}000\times 1.10 - 402.11\times\frac{1.10-1}{0.10} \approx 697.89,</math>
and
<math display="block">B_1 = R\,a_{\overline{2}|0.10} = 402.11\times\frac{1-(1.10)^{-2}}{0.10} \approx 697.89.</math>

See also [Fixed rate mortgage](/source/Fixed_rate_mortgage).

== Example calculations ==
This section gives worked examples for finding the periodic payment <math>R</math> for an annuity due from a given [present value](/source/present_value) or accumulated value. Throughout, <math>j</math> denotes a nominal annual interest rate convertible <math>m</math> times per year, <math>i = j/m</math> is the effective [interest rate](/source/interest_rate) per payment period and <math>n</math> is the total number of payments.

For an annuity-due with present value <math>A</math>, level payment <math>R</math> and <math>n</math> payments, the present value factor is
<math display="block">\ddot{a}_{\overline{n}|i} = \left(\frac{1-(1+i)^{-n}}{i}\right)(1+i)</math>
so the level payment is
:<math>R = \frac{A}{\ddot{a}_{\overline{n}|i}} = \frac{A}{\left(\frac{1-(1+i)^{-n}}{i}\right)(1+i)}.</math><ref name="Broverman-MIC" /><ref name="Kellison-Interest" /><ref name="MathFinance-Fundamentals" />

===Example 1: present value to payment (annuity-due)===
Suppose the present value of an annuity-due is <math>A = 70{,}000</math>, the effective interest rate per period is <math>i = 0.15</math> and there are <math>n = 3</math> annual payments. The annuity-due factor is
<math display="block">\ddot{a}_{\overline{3}|0.15} = \left(\frac{1-(1+0.15)^{-3}}{0.15}\right)(1+0.15) \approx 2.63</math>
so the level payment is
<math display="block">R = \frac{70{,}000}{2.63} \approx \$26{,}659.47.</math>

===Example 2: present value to payment (annuity-due)===
Suppose <math>250{,}700</math> is the present value of an annuity-due with quarterly payments for 8 years at a nominal annual interest rate of <math>j = 0.05</math> compounded quarterly. Then <math>i = j/m = 0.05/4 = 0.0125</math> and <math>n = 8\times 4 = 32</math>. The annuity-due factor is
<math display="block">\ddot{a}_{\overline{32}|0.0125} = \left(\frac{1-(1+0.0125)^{-32}}{0.0125}\right)(1+0.0125) \approx 26.57</math>
so the level payment is
<math display="block">R = \frac{250{,}700}{26.57} \approx \$9{,}435.71.</math>

For an annuity-due with accumulated value <math>S</math> at time <math>n</math>, level payment <math>R</math> and <math>n</math> payments, the accumulated value factor is
<math display="block">\ddot{s}_{\overline{n}|i} = (1+i)\,\frac{(1+i)^{n}-1}{i}</math>
so the level payment can be written as
:<math>R = \frac{S}{\ddot{s}_{\overline{n}|i}} = \frac{S\,i}{(1+i)\bigl((1+i)^{n}-1\bigr)}.</math><ref name="Kellison-Interest" /><ref name="Broverman-MIC" />

===Example 3: accumulated value to payment (annuity-due)===
Suppose the accumulated value of an annuity-due is <math>S = 55{,}000</math>, with monthly payments for 3 years at a nominal annual interest rate of <math>j = 0.15</math> compounded monthly. Then <math>i = j/m = 0.15/12 = 0.0125</math> and <math>n = 3\times 12 = 36</math>. The annuity-due accumulated value factor is
<math display="block">\ddot{s}_{\overline{36}|0.0125} = (1+0.0125)\,\frac{(1+0.0125)^{36}-1}{0.0125} \approx 45.68</math>
and the level payment is
<math display="block">R = \frac{55{,}000}{45.68} \approx \$1{,}204.04.</math>

==Legal regimes==
* [Annuities under American law](/source/Annuities_under_American_law)
* [Annuities under European law](/source/Annuities_under_European_law)
* [Annuities under Swiss law](/source/Annuities_under_Swiss_law)

==See also==
*[Amortization calculator](/source/Amortization_calculator)
*[Fixed rate mortgage](/source/Fixed_rate_mortgage)
*[Life annuity](/source/Life_annuity)
*[Perpetuity](/source/Perpetuity)
*[Time value of money](/source/Time_value_of_money)

==References==
{{Reflist|refs=
<ref name="Broverman-MIC">{{cite book |last=Broverman |first=Samuel A. |title=Mathematics of Investment and Credit |edition=5th |publisher=ACTEX Publications |isbn=978-1-56698-767-7}}</ref>

<ref name="SEC-Annuities">{{cite web |title=Annuities |url=https://www.investor.gov/introduction-investing/investing-basics/investment-products/insurance-products/annuities |website=US Securities and Exchange Commission |access-date=3 Oct 2025 |language=en}}</ref>

<ref name="Kent-BasicAnnuities">{{cite web |title=Basic Annuities |url=https://www.kent.ac.uk/learning/documents/slas-documents/basic-annuities.pdf |website=University of Kent |access-date=3 Oct 2025}}</ref>

<ref name="Investopedia-AnnuityDue">{{cite web |title=Annuity Due: Definition, Calculation, Formula, and Examples |url=https://www.investopedia.com/terms/a/annuitydue.asp |website=Investopedia |access-date=3 Oct 2025}}</ref>

<ref name="MathFinance-Fundamentals">{{cite web |title=Fundamentals of Annuities |url=https://ecampusontario.pressbooks.pub/mathematicsfinance/chapter/2-1-fundamentals-of-annuities/ |website=Mathematics of Finance |publisher=Fanshawe College |access-date=3 Oct 2025}}</ref>

<ref name="Khoruzhenko-Lecture">{{cite web |title=MAS200 Actuarial Statistics Lecture 5: Present Values of Annuities-Certain |url=https://webspace.maths.qmul.ac.uk/b.khoruzhenko/asnotes5.pdf |website=Queen Mary University of London |access-date=3 Oct 2025}}</ref>

<ref name="AMF-LifeAnnuityCertain">{{cite web |title=Life annuity and annuity certain |url=https://lautorite.qc.ca/en/general-public/investments/saving-plans/life-annuity-and-annuity-certain |website=Autorité des marchés financiers |access-date=3 Oct 2025}}</ref>

<ref name="NC-DOI-LifeCertain">{{cite web |title=Annuity options |url=https://www.ncdoi.gov/consumers/annuities/annuity-options |website=North Carolina Department of Insurance |access-date=3 Oct 2025}}</ref>

<ref name="AnnuityOrg-LifeCertain">{{cite web |title=Life Annuity With Period Certain |url=https://www.annuity.org/annuities/payout/life-annuity-period-certain/ |website=Annuity.org |access-date=3 Oct 2025}}</ref>

<ref name="NAIC-FixedDeferred">{{cite web |title=Buyer's Guide to Fixed Deferred Annuities |url=https://content.naic.org/sites/default/files/publication-anb-lp-consumer-annuities-fixed.pdf |website=National Association of Insurance Commissioners |access-date=3 Oct 2025}}</ref>

<ref name="III-DeferredImmediate">{{cite web |title=What are deferred and immediate annuities? |url=https://www.iii.org/article/what-are-deferred-and-immediate-annuities |website=Insurance Information Institute |access-date=3 Oct 2025}}</ref>

<ref name="Schwab-ImmediateDeferred">{{cite web |title=What is an annuity? Immediate vs. deferred |url=https://www.schwab.com/learn/story/4-questions-to-ask-before-buying-annuity |website=Charles Schwab |access-date=3 Oct 2025}}</ref>

<ref name="SEC-Variable">{{cite web |title=Variable Annuities: What You Should Know |url=https://www.sec.gov/investor/pubs/sec-guide-to-variable-annuities.pdf |website=US Securities and Exchange Commission |access-date=3 Oct 2025}}</ref>

<ref name="FINRA-Variable">{{cite web |title=Variable Annuities |url=https://www.finra.org/rules-guidance/key-topics/variable-annuities |website=FINRA |access-date=3 Oct 2025}}</ref>

<ref name="SEC-Indexed">{{cite web |title=Updated Investor Bulletin: Indexed Annuities |url=https://www.investor.gov/introduction-investing/general-resources/news-alerts/alerts-bulletins/investor-bulletins/updated-investor-bulletin-indexed-annuities |website=US Securities and Exchange Commission |access-date=3 Oct 2025}}</ref>

<ref name="NAIC-AnnuityTopics">{{cite web |title=Annuities |url=https://content.naic.org/insurance-topics/annuities |website=National Association of Insurance Commissioners |access-date=3 Oct 2025}}</ref>

<ref name="FINRA-Annuities">{{cite web |title=Annuities |url=https://www.finra.org/investors/investing/investment-products/annuities |website=FINRA |access-date=3 Oct 2025}}</ref>

<ref name="FINRA-Indexed">{{cite web |title=Equity-Indexed Annuities: A Complex Choice |url=https://www.maine.gov/pfr/insurance/themes/insurance/pdf/consumer-guides/pdf/equity_indexed_annuities_investor_alert.pdf |website=FINRA |access-date=3 Oct 2025}}</ref>

<ref name="Kellison-Interest">{{cite book |last=Kellison |first=Stephen G. |title=The Theory of Interest |edition=3rd |publisher=McGraw-Hill/Irwin |year=2008 |isbn=9780073382449}}</ref>

<ref name="PresentValueAnnuity">{{cite web |title=Present Value of Annuity |url=https://www.investopedia.com/terms/p/present-value-annuity.asp |website=Investopedia |access-date=3 October 2025 |language=en}}</ref>

<ref name="Bowers-LifeAnnuity">{{cite book |last=Bowers |first=Newton L. |title=Actuarial Mathematics |edition=2nd |publisher=Society of Actuaries |year=1997 |isbn=978-0-938959-46-5}}</ref>

<ref name="Goeters-LifeAnnuities">{{cite web |last=Goeters |first=Paul |title=Annuities, Insurance and Life |url=https://www.auburn.edu/~goetehp/acttex/acttex.pdf |website=Auburn University |access-date=3 October 2025 |language=en}}</ref>

<ref name="Slud-LifeAnnuity">{{cite web |last=Slud |first=Eric V. |title=Actuarial Mathematics and Life-Table Statistics, Chapter 4: Expected Present Values of Payments |url=https://www.math.umd.edu/~evs/s470/BookChaps/Chp45.pdf |website=University of Maryland |access-date=3 October 2025 |language=en}}</ref>

}}

==Other sources==
* {{cite book|title=Mathematics of Investment and Credit, 5th Edition|series=ACTEX Academic Series| year=2010|author = Samuel A. Broverman|publisher= ACTEX Publications|isbn=978-1-56698-767-7}}
* {{cite book|title=Theory of Interest, 3rd Edition|author=Stephen Kellison|publisher=McGraw-Hill/Irwin|year=2008|isbn= 978-0-07-338244-9}}
* {{cite EB9 |wstitle= Annuities |volume= II |last= Sprague |first= Thomas Bond |author-link= Thomas Bond Sprague |pages=72-89 |short=1}}

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