Annuity

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.

Typical examples of annuity-immediate payment streams include home mortgage and other loan repayments, where each installment covers interest that has accrued during the preceding period. Rent, leases and many insurance premiums are usually paid in advance and are therefore examples of annuity-due payments.

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.

A life annuity pays while one or more specified lives survive, so the number of payments is uncertain. 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.

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.
• 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.
• 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.

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. A immediate annuity starts payments shortly after the contract is purchased, often within one year.

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

Optional benefits and riders

Many annuities allow optional riders or benefits for an added cost. Death-benefit riders may provide a beneficiary with a minimum amount, a return of premium, or a stepped-up account value if the owner or annuitant dies before income begins. Living-benefit riders may provide guaranteed minimum income, withdrawal, or accumulation benefits while the owner remains alive.

Such features can change the distribution of risk between the owner and the insurer, but they may also add charges, restrict investment choices, limit withdrawal amounts, or reduce the value of guarantees after withdrawals. The economic effect of a rider therefore depends on both the stated guarantee and the contract conditions attached to using it.

Special-purpose forms

Annuities are used in legal, charitable, and pension arrangements. A structured settlement may use an annuity contract to fund periodic payments owed under a personal-injury or workers' compensation settlement. A charitable gift annuity is a transfer of property to a charitable organisation in return for annuity payments over one or two lives, with the charitable component depending on the relationship between the property transferred and the actuarial value of the payments. In pension risk transfer, a defined benefit pension scheme may use a bulk annuity, buy-in, or buy-out to transfer some or all pension-payment obligations to an insurer.

These special-purpose forms share the annuity feature of periodic or life-contingent payments, but their legal purpose and risk allocation differ from retail retirement annuities. Structured settlements focus on scheduled compensation payments, charitable gift annuities combine lifetime payments with a charitable transfer, and pension risk transfer annuities shift pension liabilities from a pension scheme or sponsor to an insurer.

Valuation

Valuation of an annuity treats the stream of payments as cash flows and summarises them by a present value or a future value at a given interest rate. 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). 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). Mortgage payments are a typical example, since interest is charged between payments and then repaid at each due date.

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

${\displaystyle \mathrm{P}\mathrm{V}(i,n,R)=R{a}_{\overline{n}|i}.}$

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

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

${\displaystyle \mathrm{F}\mathrm{V}(i,n,R)=R{s}_{\overline{n}|i}.}$

Example: The present value of a 5 year annuity with a nominal annual interest rate of 12% and monthly payments of $100 is $${\displaystyle \mathrm{P}\mathrm{V}\left(\frac{0.12}{12},5\times 12,100\right)=100\times a_{\overline{60}|0.01}\approx \mathrm{4,495.50}}$$ 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 by $${\displaystyle s_{\overline{n}|i}=(1+i{)}^n{a}_{\overline{n}|i}}$$ and

${\displaystyle \frac{1}{a_{\overline{n}|i}}-\frac{1}{s_{\overline{n}|i}}=i.}$

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 ${\displaystyle k}$ is discounted by the factor ${\displaystyle (1+i{)}^{-k}}$, so the present value factor is $${\displaystyle a_{\overline{n}|i}=\sum _{k=1}^n\frac{1}{(1+i{)}^k}.}$$ Let ${\displaystyle v=(1+i{)}^{-1}}$ be the discount factor for one period. Then $${\displaystyle a_{\overline{n}|i}=v+v^2+\cdots +v^n=v\sum _{k=0}^{n-1}{v}^k.}$$ Using the standard formula for the sum of a finite geometric series gives

${\displaystyle a_{\overline{n}|i}=v\frac{1-v^n}{1-v}=\frac{1-v^n}{i}=\frac{1-(1+i{)}^{-n}}{i}.}$

Annuity-due

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

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

${\displaystyle {\ddot{s}}_{\overline{n}|i}=(1+i)s_{\overline{n}|i}.}$

The present and future values for an annuity-due satisfy $${\displaystyle {\ddot{s}}_{\overline{n}|i}=(1+i{)}^n{\ddot{a}}_{\overline{n}|i}}$$ and $${\displaystyle \frac{1}{{\ddot{a}}_{\overline{n}|i}}-\frac{1}{{\ddot{s}}_{\overline{n}|i}}=d,}$$ where ${\displaystyle d=\frac{i}{1+i}}$ is the effective rate of discount.

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

Perpetuity

A perpetuity is an annuity for which the payments continue indefinitely. For a level perpetuity with payment ${\displaystyle R}$ each period and per period interest rate ${\displaystyle i}$, the present value can be obtained as the limit of the level annuity-immediate present value as the term tends to infinity: $${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\mathrm{P}\mathrm{V}(i,n,R)=\underset{n\to \mathrm{\infty }}{lim}R{a}_{\overline{n}|i}=\frac{R}{i}}$$ so the closed form is $${\displaystyle {\mathrm{P}\mathrm{V}}_{\text{perpetuity}}=\frac{R}{i}}$$ provided ${\displaystyle i}$ is positive. In actuarial notation the present value factors for level perpetuities are $${\displaystyle a_{\overline{\mathrm{\infty }}|i}=\frac{1}{i}}$$ and $${\displaystyle {\ddot{a}}_{\overline{\mathrm{\infty }}|i}=\frac{1}{d},}$$ where ${\displaystyle d=\frac{i}{1+i}}$ is the effective discount rate.

Life annuities

Valuation of life annuities extends the level annuity formulas by taking into account mortality as well as interest. For a life aged ${\displaystyle x}$ with annual payments of amount ${\displaystyle R}$ payable while the life survives, the actuarial present value is the expected value of the discounted payment stream, $${\displaystyle \mathrm{A}\mathrm{P}\mathrm{V}=\sum _{t=1}^{\mathrm{\infty }}R{v}^t{}_t{p}_x}$$ where ${\displaystyle v=(1+i{)}^{-1}}$ is the discount factor per period and ${\displaystyle {}_t{p}_x}$ is the probability that a life aged ${\displaystyle x}$ survives at least ${\displaystyle t}$ periods.

In actuarial notation the present value of a whole life annuity-immediate of 1 per year on a life aged ${\displaystyle x}$ is written ${\displaystyle a_x}$ and can be expressed as $${\displaystyle a_x=\sum _{t=1}^{\mathrm{\infty }}{v}^t{}_t{p}_x}$$ while the corresponding whole life annuity-due has present value factor

${\displaystyle {\ddot{a}}_x=\sum _{t=0}^{\mathrm{\infty }}{v}^t{}_t{p}_x.}$

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 ${\displaystyle P}$ be the initial loan principal, ${\displaystyle R}$ the regular payment, ${\displaystyle i}$ the effective interest rate per period and ${\displaystyle N}$ the total number of payments. Then the present value of the payment stream is $${\displaystyle P=R{a}_{\overline{N}|i}=R\frac{1-(1+i{)}^{-N}}{i},}$$ so the level payment that amortises the loan is

${\displaystyle R=\frac{P}{a_{\overline{N}|i}}=P\frac{i}{1-(1+i{)}^{-N}}.}$

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

${\displaystyle B_n=(1+i{)}^{n}P-R\frac{(1+i{)}^n-1}{i}=\frac{R}{i}-(1+i{)}^n\left(\frac{R}{i}-P\right).}$

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

${\displaystyle B_n=R{a}_{\overline{N-n}|i}=R\frac{1-(1+i{)}^{-(N-n)}}{i}.}$

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

${\displaystyle B_n^{(\text{due})}=R{\ddot{a}}_{\overline{N-n}|i},{\ddot{a}}_{\overline{m}|i}=(1+i)a_{\overline{m}|i}.}$

Example. Let ${\displaystyle P=\mathrm{1,000}}$, ${\displaystyle i=0.10}$, ${\displaystyle N=3}$. Then $${\displaystyle R=\frac{Pi}{1-(1+i{)}^{-N}}=\frac{\mathrm{1,000}\times 0.10}{1-(1.10{)}^{-3}}\approx \mathrm{402.11.}}$$ After one payment the retrospective and prospective balances coincide: $${\displaystyle B_1=\mathrm{1,000}\times 1.10-402.11\times \frac{1.10-1}{0.10}\approx 697.89,}$$ and $${\displaystyle B_1=R{a}_{\overline{2}|0.10}=402.11\times \frac{1-(1.10{)}^{-2}}{0.10}\approx \mathrm{697.89.}}$$

See also Fixed rate mortgage.

Example calculations

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

For an annuity-due with present value ${\displaystyle A}$, level payment ${\displaystyle R}$ and ${\displaystyle n}$ payments, the present value factor is $${\displaystyle {\ddot{a}}_{\overline{n}|i}=\left(\frac{1-(1+i{)}^{-n}}{i}\right)(1+i)}$$ so the level payment is

${\displaystyle R=\frac{A}{{\ddot{a}}_{\overline{n}|i}}=\frac{A}{\left(\frac{1-(1+i{)}^{-n}}{i}\right)(1+i)}.}$

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

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

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

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

For an annuity-due with accumulated value ${\displaystyle S}$ at time ${\displaystyle n}$, level payment ${\displaystyle R}$ and ${\displaystyle n}$ payments, the accumulated value factor is $${\displaystyle {\ddot{s}}_{\overline{n}|i}=(1+i)\frac{(1+i{)}^n-1}{i}}$$ so the level payment can be written as

${\displaystyle R=\frac{S}{{\ddot{s}}_{\overline{n}|i}}=\frac{Si}{(1+i)((1+i{)}^n-1)}.}$

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

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

Legal regimes

• Annuities under American law
• Annuities under European law
• Annuities under Swiss law

Other sources

• Samuel A. Broverman (2010). Mathematics of Investment and Credit, 5th Edition. ACTEX Academic Series. ACTEX Publications. ISBN 978-1-56698-767-7.
• Stephen Kellison (2008). Theory of Interest, 3rd Edition. McGraw-Hill/Irwin. ISBN 978-0-07-338244-9.
• Sprague, Thomas Bond (1878). "Annuities" . Encyclopædia Britannica. Vol. II (9th ed.). pp. 72–89.