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Annuity

From Wikipedia, the free encyclopedia
Series of payments made at equal intervals
For other uses, seeAnnuity (disambiguation).
Personal finance
Coin issued during the reign of the Roman emperor Maximian
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Employment contract
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Ininvestment, anannuity is a series of payments of the same kind made at equal time intervals, usually over a finite term.[1] Annuities are commonly issued bylife insurance companies, where an individual pays a lump sum or a series of premiums in return for regular income payments, often to provideretirement or survivor benefits.[2]

Typical examples include regular deposits to a savings account, monthly homemortgage payments, monthlyinsurance premiums andpension payments.[1] The value of an annuity is usually expressed as apresent value orfuture value, calculated by discounting or accumulating the payments at a specifiedinterest 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 aperpetuity.

Types

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Annuities may be classified in several ways.

Timing of payments

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Payments in anannuity-immediate are made at the end of each payment period, so interest accrues during the period before each payment. By contrast, payments in anannuity-due are made at the beginning of each period, so each payment is made in advance.[3][1]

Typical examples of annuity-immediate payment streams include homemortgage and other loan repayments, where each instalment covers interest that has accrued during the preceding period. Rent, leases and manyinsurance premiums are usually paid in advance and are therefore examples of annuity-due payments.[4][5]

Contingency of payments

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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.[6]

Alife annuity pays while one or more specified lives survive, so the number of payments is uncertain.[6][7] 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.[8][9]

Variability of payments

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  • 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.[10][11]
  • Variable annuities invest premiums in underlying portfolios such as mutual funds, so the contract value and income payments vary with the performance of those investments.[12][13]
  • Equity-indexed annuities 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.[14][15]

Deferral of payments

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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.[16][17] Aimmediate annuity starts payments shortly after the contract is purchased, often within one year.[17][18]

Fixed, variable and indexed annuities can each be written as immediate or deferred contracts.[10][11]

Valuation

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Valuation of an annuity treats the stream of payments ascash flows and summarises them by apresent value or afuture value at a giveninterest rate.[1][19] 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

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If the number of payments is known in advance, the contract is an annuity certain (also called a guaranteed annuity).[1] Valuation uses the formulas below, which depend on the timing of payments.

Annuity-immediate

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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).[19] Mortgage payments are a typical example, since interest is charged between payments and then repaid at each due date.[5]

↓↓...↓payments
—————————————
012...nperiods

Leti{\displaystyle i} denote the effective interest rate per period andn{\displaystyle n} the number of payments. The present value factor for a level annuity-immediate with unit payments is:an¯|i=1−(1+i)−ni{\displaystyle a_{{\overline {n}}|i}={\frac {1-(1+i)^{-n}}{i}}}and the present value of payments of amountR{\displaystyle R} is:

PV(i,n,R)=Ran¯|i.{\displaystyle \mathrm {PV} (i,n,R)=R\,a_{{\overline {n}}|i}.}[1]

In practice, interest is often quoted as a nominal annual rateJ{\displaystyle J} convertible monthly or some other frequency. If payments are monthly and the nominal annual rate isJ{\displaystyle J}, then the rate per month isi=J/12{\displaystyle i=J/12} and the number of payments overt{\displaystyle t} years isn=12t{\displaystyle n=12t}.[19]

The future value of a level annuity-immediate with unit payments issn¯|i=(1+i)n−1i{\displaystyle s_{{\overline {n}}|i}={\frac {(1+i)^{n}-1}{i}}}and the accumulated value immediately after the last payment is:

FV(i,n,R)=Rsn¯|i.{\displaystyle \mathrm {FV} (i,n,R)=R\,s_{{\overline {n}}|i}.}[20]

Example: The present value of a 5 year annuity with a nominal annual interest rate of 12% and monthly payments of $100 isPV(0.1212,5×12,100)=100×a60¯|0.01≈4,495.50{\displaystyle \mathrm {PV} \!\left({\frac {0.12}{12}},5\times 12,100\right)=100\times a_{{\overline {60}}|0.01}\approx 4{,}495.50}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 bysn¯|i=(1+i)nan¯|i{\displaystyle s_{{\overline {n}}|i}=(1+i)^{n}\,a_{{\overline {n}}|i}}and

1an¯|i−1sn¯|i=i.{\displaystyle {\frac {1}{a_{{\overline {n}}|i}}}-{\frac {1}{s_{{\overline {n}}|i}}}=i.}[19]
Proof of annuity-immediate formula
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To obtain the present value factor, consider a level annuity-immediate with unit payments. The payment at the end of periodk{\displaystyle k} is discounted by the factor(1+i)−k{\displaystyle (1+i)^{-k}}, so the present value factor isan¯|i=∑k=1n1(1+i)k.{\displaystyle a_{{\overline {n}}|i}=\sum _{k=1}^{n}{\frac {1}{(1+i)^{k}}}.}Letv=(1+i)−1{\displaystyle v=(1+i)^{-1}} be the discount factor for one period. Thenan¯|i=v+v2+⋯+vn=v∑k=0n−1vk.{\displaystyle a_{{\overline {n}}|i}=v+v^{2}+\cdots +v^{n}=v\sum _{k=0}^{n-1}v^{k}.}Using the standard formula for the sum of a finite geometric series gives

an¯|i=v1−vn1−v=1−vni=1−(1+i)−ni.{\displaystyle a_{{\overline {n}}|i}=v\,{\frac {1-v^{n}}{1-v}}={\frac {1-v^{n}}{i}}={\frac {1-(1+i)^{-n}}{i}}.}[19][1]

Annuity-due

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An annuity due is a series of equal payments made at the same interval at the beginning of each period.[5] 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.[4]

↓↓...↓payments
—————————————
01...n − 1nperiods

For an annuity-due with unit payments the present value factor isa¨n¯|i=(1+i)an¯|i{\displaystyle {\ddot {a}}_{{\overline {n}}|i}=(1+i)\,a_{{\overline {n}}|i}}and the future value factor is

s¨n¯|i=(1+i)sn¯|i.{\displaystyle {\ddot {s}}_{{\overline {n}}|i}=(1+i)\,s_{{\overline {n}}|i}.}[1][19]

The present and future values for an annuity-due satisfys¨n¯|i=(1+i)na¨n¯|i{\displaystyle {\ddot {s}}_{{\overline {n}}|i}=(1+i)^{n}\,{\ddot {a}}_{{\overline {n}}|i}}and1a¨n¯|i−1s¨n¯|i=d,{\displaystyle {\frac {1}{{\ddot {a}}_{{\overline {n}}|i}}}-{\frac {1}{{\ddot {s}}_{{\overline {n}}|i}}}=d,}whered=i1+i{\displaystyle d={\frac {i}{1+i}}} is the effective rate of discount.[20]

Example: The future value of a 7 year annuity-due with a nominal annual interest rate of 9% and monthly payments of $100 isFVdue(0.0912,7×12,100)=100×s¨84¯|0.0075≈11,730.01.{\displaystyle \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.}

Perpetuity

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Aperpetuity is an annuity for which the payments continue indefinitely.[19] For a level perpetuity with paymentR{\displaystyle R} each period and per period interest ratei{\displaystyle i}, the present value can be obtained as the limit of the level annuity-immediate present value as the term tends to infinity:limn→∞PV(i,n,R)=limn→∞Ran¯|i=Ri{\displaystyle \lim _{n\to \infty }\mathrm {PV} (i,n,R)=\lim _{n\to \infty }R\,a_{{\overline {n}}|i}={\frac {R}{i}}}so the closed form isPVperpetuity=Ri{\displaystyle \mathrm {PV} _{\text{perpetuity}}={\frac {R}{i}}}providedi{\displaystyle i} is positive.[20] In actuarial notation the present value factors for level perpetuities area∞¯|i=1i{\displaystyle a_{{\overline {\infty }}|i}={\frac {1}{i}}}anda¨∞¯|i=1d,{\displaystyle {\ddot {a}}_{{\overline {\infty }}|i}={\frac {1}{d}},}whered=i1+i{\displaystyle d={\frac {i}{1+i}}} is the effective discount rate.[1]

Life annuities

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Valuation oflife annuities extends the level annuity formulas by taking into account mortality as well as interest. For a life agedx{\displaystyle x} with annual payments of amountR{\displaystyle R} payable while the life survives, the actuarial present value is the expected value of the discounted payment stream,APV=∑t=1∞Rvttpx{\displaystyle \mathrm {APV} =\sum _{t=1}^{\infty }Rv^{t}\,{}_{t}p_{x}}wherev=(1+i)−1{\displaystyle v=(1+i)^{-1}} is the discount factor per period andtpx{\displaystyle {}_{t}p_{x}} is the probability that a life agedx{\displaystyle x} survives at leastt{\displaystyle t} periods.[21][22]

In actuarial notation the present value of a whole life annuity-immediate of 1 per year on a life agedx{\displaystyle x} is writtenax{\displaystyle a_{x}} and can be expressed asax=∑t=1∞vttpx{\displaystyle a_{x}=\sum _{t=1}^{\infty }v^{t}\,{}_{t}p_{x}}while the corresponding whole life annuity-due has present value factor

a¨x=∑t=0∞vttpx.{\displaystyle {\ddot {a}}_{x}=\sum _{t=0}^{\infty }v^{t}\,{}_{t}p_{x}.}[21][23]

Amortization calculations

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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. LetP{\displaystyle P} be the initial loan principal,R{\displaystyle R} the regular payment,i{\displaystyle i} the effective interest rate per period andN{\displaystyle N} the total number of payments. Then the present value of the payment stream isP=RaN¯|i=R1−(1+i)−Ni,{\displaystyle P=R\,a_{{\overline {N}}|i}=R\,{\frac {1-(1+i)^{-N}}{i}},}so the level payment that amortises the loan is

R=PaN¯|i=Pi1−(1+i)−N.{\displaystyle R={\frac {P}{a_{{\overline {N}}|i}}}=P\,{\frac {i}{1-(1+i)^{-N}}}.}[19][1][5]

The outstanding balance aftern{\displaystyle n} payments can be obtained in two equivalent ways. Under the retrospective method, the balance is the original principal accumulated with interest forn{\displaystyle n} periods minus the accumulated value of the payments already made:

Bn=(1+i)nP−R(1+i)n−1i=Ri−(1+i)n(Ri−P).{\displaystyle 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).}[19]

Under the prospective method, the outstanding balance is the present value of the remainingN−n{\displaystyle N-n} payments:

Bn=RaN−n¯|i=R1−(1+i)−(N−n)i.{\displaystyle B_{n}=R\,a_{{\overline {N-n}}|i}=R\,{\frac {1-(1+i)^{-(N-n)}}{i}}.}[1]

For an annuity due with payments at the beginning of each period, the same ideas apply but annuity-due factors are used. IfR{\displaystyle R} is the level payment and there areN{\displaystyle N} payments in total, the outstanding balance aftern{\displaystyle n} payments is

Bn(due)=Ra¨N−n¯|i,a¨m¯|i=(1+i)am¯|i.{\displaystyle B_{n}^{({\text{due}})}=R\,{\ddot {a}}_{{\overline {N-n}}|i},\qquad {\ddot {a}}_{{\overline {m}}|i}=(1+i)\,a_{{\overline {m}}|i}.}[19]

Example. LetP=1,000{\displaystyle P=1{,}000},i=0.10{\displaystyle i=0.10},N=3{\displaystyle N=3}. ThenR=Pi1−(1+i)−N=1,000×0.101−(1.10)−3≈402.11.{\displaystyle R={\frac {P\,i}{1-(1+i)^{-N}}}={\frac {1{,}000\times 0.10}{1-(1.10)^{-3}}}\approx 402.11.}After one payment the retrospective and prospective balances coincide:B1=1,000×1.10−402.11×1.10−10.10≈697.89,{\displaystyle B_{1}=1{,}000\times 1.10-402.11\times {\frac {1.10-1}{0.10}}\approx 697.89,}andB1=Ra2¯|0.10=402.11×1−(1.10)−20.10≈697.89.{\displaystyle B_{1}=R\,a_{{\overline {2}}|0.10}=402.11\times {\frac {1-(1.10)^{-2}}{0.10}}\approx 697.89.}

See alsoFixed rate mortgage.

Example calculations

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This section gives worked examples for finding the periodic paymentR{\displaystyle R} for an annuity due from a givenpresent value or accumulated value. Throughout,j{\displaystyle j} denotes a nominal annual interest rate convertiblem{\displaystyle m} times per year,i=j/m{\displaystyle i=j/m} is the effectiveinterest rate per payment period andn{\displaystyle n} is the total number of payments.

For an annuity-due with present valueA{\displaystyle A}, level paymentR{\displaystyle R} andn{\displaystyle n} payments, the present value factor isa¨n¯|i=(1−(1+i)−ni)(1+i){\displaystyle {\ddot {a}}_{{\overline {n}}|i}=\left({\frac {1-(1+i)^{-n}}{i}}\right)(1+i)}so the level payment is

R=Aa¨n¯|i=A(1−(1+i)−ni)(1+i).{\displaystyle R={\frac {A}{{\ddot {a}}_{{\overline {n}}|i}}}={\frac {A}{\left({\frac {1-(1+i)^{-n}}{i}}\right)(1+i)}}.}[1][19][5]

Example 1: present value to payment (annuity-due)

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Suppose the present value of an annuity-due isA=70,000{\displaystyle A=70{,}000}, the effective interest rate per period isi=0.15{\displaystyle i=0.15} and there aren=3{\displaystyle n=3} annual payments. The annuity-due factor isa¨3¯|0.15=(1−(1+0.15)−30.15)(1+0.15)≈2.63{\displaystyle {\ddot {a}}_{{\overline {3}}|0.15}=\left({\frac {1-(1+0.15)^{-3}}{0.15}}\right)(1+0.15)\approx 2.63}so the level payment isR=70,0002.63≈$26,659.47.{\displaystyle R={\frac {70{,}000}{2.63}}\approx \$26{,}659.47.}

Example 2: present value to payment (annuity-due)

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Suppose250,700{\displaystyle 250{,}700} is the present value of an annuity-due with quarterly payments for 8 years at a nominal annual interest rate ofj=0.05{\displaystyle j=0.05} compounded quarterly. Theni=j/m=0.05/4=0.0125{\displaystyle i=j/m=0.05/4=0.0125} andn=8×4=32{\displaystyle n=8\times 4=32}. The annuity-due factor isa¨32¯|0.0125=(1−(1+0.0125)−320.0125)(1+0.0125)≈26.57{\displaystyle {\ddot {a}}_{{\overline {32}}|0.0125}=\left({\frac {1-(1+0.0125)^{-32}}{0.0125}}\right)(1+0.0125)\approx 26.57}so the level payment isR=250,70026.57≈$9,435.71.{\displaystyle R={\frac {250{,}700}{26.57}}\approx \$9{,}435.71.}

For an annuity-due with accumulated valueS{\displaystyle S} at timen{\displaystyle n}, level paymentR{\displaystyle R} andn{\displaystyle n} payments, the accumulated value factor iss¨n¯|i=(1+i)(1+i)n−1i{\displaystyle {\ddot {s}}_{{\overline {n}}|i}=(1+i)\,{\frac {(1+i)^{n}-1}{i}}}so the level payment can be written as

R=Ss¨n¯|i=Si(1+i)((1+i)n−1).{\displaystyle R={\frac {S}{{\ddot {s}}_{{\overline {n}}|i}}}={\frac {S\,i}{(1+i){\bigl (}(1+i)^{n}-1{\bigr )}}}.}[19][1]

Example 3: accumulated value to payment (annuity-due)

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Suppose the accumulated value of an annuity-due isS=55,000{\displaystyle S=55{,}000}, with monthly payments for 3 years at a nominal annual interest rate ofj=0.15{\displaystyle j=0.15} compounded monthly. Theni=j/m=0.15/12=0.0125{\displaystyle i=j/m=0.15/12=0.0125} andn=3×12=36{\displaystyle n=3\times 12=36}. The annuity-due accumulated value factor iss¨36¯|0.0125=(1+0.0125)(1+0.0125)36−10.0125≈45.68{\displaystyle {\ddot {s}}_{{\overline {36}}|0.0125}=(1+0.0125)\,{\frac {(1+0.0125)^{36}-1}{0.0125}}\approx 45.68}and the level payment isR=55,00045.68≈$1,204.04.{\displaystyle R={\frac {55{,}000}{45.68}}\approx \$1{,}204.04.}

Legal regimes

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See also

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References

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  1. ^abcdefghijklmBroverman, Samuel A.Mathematics of Investment and Credit (5th ed.). ACTEX Publications.ISBN 978-1-56698-767-7.
  2. ^"Annuities".US Securities and Exchange Commission. Retrieved3 Oct 2025.
  3. ^"Basic Annuities"(PDF).University of Kent. Retrieved3 Oct 2025.
  4. ^ab"Annuity Due: Definition, Calculation, Formula, and Examples".Investopedia. Retrieved3 Oct 2025.
  5. ^abcde"Fundamentals of Annuities".Mathematics of Finance. Fanshawe College. Retrieved3 Oct 2025.
  6. ^ab"MAS200 Actuarial Statistics Lecture 5: Present Values of Annuities-Certain"(PDF).Queen Mary University of London. Retrieved3 Oct 2025.
  7. ^"Life annuity and annuity certain".Autorité des marchés financiers. Retrieved3 Oct 2025.
  8. ^"Annuity options".North Carolina Department of Insurance. Retrieved3 Oct 2025.
  9. ^"Life Annuity With Period Certain".Annuity.org. Retrieved3 Oct 2025.
  10. ^ab"Annuities".National Association of Insurance Commissioners. Retrieved3 Oct 2025.
  11. ^ab"Annuities".FINRA. Retrieved3 Oct 2025.
  12. ^"Variable Annuities: What You Should Know"(PDF).US Securities and Exchange Commission. Retrieved3 Oct 2025.
  13. ^"Variable Annuities".FINRA. Retrieved3 Oct 2025.
  14. ^"Updated Investor Bulletin: Indexed Annuities".US Securities and Exchange Commission. Retrieved3 Oct 2025.
  15. ^"Equity-Indexed Annuities: A Complex Choice"(PDF).FINRA. Retrieved3 Oct 2025.
  16. ^"Buyer's Guide to Fixed Deferred Annuities"(PDF).National Association of Insurance Commissioners. Retrieved3 Oct 2025.
  17. ^ab"What are deferred and immediate annuities?".Insurance Information Institute. Retrieved3 Oct 2025.
  18. ^"What is an annuity? Immediate vs. deferred".Charles Schwab. Retrieved3 Oct 2025.
  19. ^abcdefghijklKellison, Stephen G. (2008).The Theory of Interest (3rd ed.). McGraw-Hill/Irwin.ISBN 9780073382449.
  20. ^abc"Present Value of Annuity".Investopedia. Retrieved3 October 2025.
  21. ^abBowers, Newton L. (1997).Actuarial Mathematics (2nd ed.). Society of Actuaries.ISBN 978-0-938959-46-5.
  22. ^Goeters, Paul."Annuities, Insurance and Life"(PDF).Auburn University. Retrieved3 October 2025.
  23. ^Slud, Eric V."Actuarial Mathematics and Life-Table Statistics, Chapter 4: Expected Present Values of Payments"(PDF).University of Maryland. Retrieved3 October 2025.

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