annuity

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OR:

In
investment Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose of investing is ...
, an annuity is a series of payments made at equal intervals.Kellison, Stephen G. (1970). ''The Theory of Interest''. Homewood, Illinois: Richard D. Irwin, Inc. p. 45 Examples of annuities are regular deposits to a savings account, monthly home mortgage payments, monthly
insurance Insurance is a means of protection from financial loss in which, in exchange for a fee, a party agrees to compensate another party in the event of a certain loss, damage, or injury. It is a form of risk management, primarily used to hedge ...
payments and
pension A pension (, from Latin ''pensiō'', "payment") is a fund into which a sum of money is added during an employee's employment years and from which payments are drawn to support the person's retirement from work in the form of periodic payments ...
payments. Annuities can be classified by the frequency of payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other regular interval of time. Annuities may be calculated by
mathematical functions In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the func ...
known as "annuity functions". An annuity which provides for payments for the remainder of a person's lifetime is a
life annuity A life annuity is an annuity, or series of payments at fixed intervals, paid while the purchaser (or annuitant) is alive. The majority of life annuities are insurance products sold or issued by life insurance companies however substantial case ...
.

# Types

Annuities may be classified in several ways.

## Timing of payments

Payments of an ''annuity-immediate'' are made at the end of payment periods, so that interest accrues between the issue of the annuity and the first payment. Payments of an ''annuity-due'' are made at the beginning of payment periods, so a payment is made immediately on issueter.

## Contingency of payments

Annuities that provide payments that will be paid over a period known in advance are ''annuities certain'' or ''guaranteed annuities.'' Annuities paid only under certain circumstances are ''contingent annuities''. A common example is a
life annuity A life annuity is an annuity, or series of payments at fixed intervals, paid while the purchaser (or annuitant) is alive. The majority of life annuities are insurance products sold or issued by life insurance companies however substantial case ...
, which is paid over the remaining lifetime of the annuitant. ''Certain and life annuities'' are guaranteed to be paid for a number of years and then become contingent on the annuitant being alive.

## Variability of payments

*Fixed annuities – These are annuities with fixed payments. If provided by an insurance company, the company guarantees a fixed return on the initial investment. Fixed annuities are not regulated by the
Securities and Exchange Commission The U.S. Securities and Exchange Commission (SEC) is an independent agency of the United States federal government, created in the aftermath of the Wall Street Crash of 1929. The primary purpose of the SEC is to enforce the law against market ...
. *Variable annuities – Registered products that are regulated by the SEC in the United States of America. They allow direct investment into various funds that are specially created for Variable annuities. Typically, the insurance company guarantees a certain death benefit or lifetime withdrawal benefits. * Equity-indexed annuities – Annuities with payments linked to an index. Typically, the minimum payment will be 0% and the maximum will be predetermined. The performance of an index determines whether the minimum, the maximum or something in between is credited to the customer.

## Deferral of payments

An annuity that begins payments only after a period is a ''deferred annuity'' (usually after retirement). An annuity that begins payments as soon as the customer has paid, without a deferral period is an ''immediate annuity''.

# Valuation

Valuation of an annuity entails calculation of the
present value In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has int ...
of the future annuity payments. The valuation of an annuity entails concepts such as
time value of money The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference. The ...
,
interest rate An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
, and
future value Future value is the value of an asset at a specific date. It measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return; it ...
..

## Annuity-certain

If the number of payments is known in advance, the annuity is an ''annuity certain'' or ''guaranteed annuity''. Valuation of annuities certain may be calculated using formulas depending on the timing of payments.

### Annuity-immediate

If the payments are made at the end of the time periods, so that interest is accumulated before the payment, the annuity is called an ''annuity-immediate'', or ''ordinary annuity''. Mortgage payments are annuity-immediate, interest is earned before being paid. What is Annuity Due? Annuity due refers to 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 of annuity due payments include rentals, leases, and insurance payments, which are made to cover services provided in the period following the payment. The ''present value'' of an annuity is the value of a stream of payments, discounted by the interest rate to account for the fact that payments are being made at various moments in the future. The present value is given in
actuarial notation Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables. Traditional notation uses a halo system where symbols are placed as superscript or subscript before or af ...
by: :$a_ = \frac,$ where $n$ is the number of terms and $i$ is the per period interest rate. Present value is linear in the amount of payments, therefore the present value for payments, or ''rent'' $R$ is: :$\text\left(i,n,R\right) = R \times a_.$ In practice, often loans are stated per annum while interest is compounded and payments are made monthly. In this case, the interest $I$ is stated as a
nominal interest rate In finance and economics, the nominal interest rate or nominal rate of interest is the rate of interest stated on a loan or investment, without any adjustments or fees. Examples of adjustments or fees # An adjustment for inflation(in contrast with ...

### Perpetuity

A ''perpetuity'' is an annuity for which the payments continue forever. Observe that :$\lim_ \text\left(i,n,R\right) = \lim_ R \times a_ = \lim_ R \times \frac = \,\frac.$ Therefore a
perpetuity A perpetuity is an annuity that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence. For example, the United Kingdom (UK) government issued them in the past; these were known as cons ...
has a finite present value when there is a non-zero discount rate. The formulae for a perpetuity are :$a_ = \frac \text \ddot_ = \frac,$ where $i$ is the interest rate and $d=\frac$ is the effective discount rate.

## Life annuities

Valuation of life annuities may be performed by calculating the
actuarial present value The actuarial present value (APV) is the expected value of the present value of a contingent cash flow stream (i.e. a series of payments which may or may not be made). Actuarial present values are typically calculated for the benefit-payment or seri ...
of the future life contingent payments.
Life table In actuarial science and demography, a life table (also called a mortality table or actuarial table) is a table which shows, for each age, what the probability is that a person of that age will die before their next birthday ("probability of dea ...
s are used to calculate the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
that the annuitant lives to each future payment period. Valuation of life annuities also depends on the timing of payments just as with annuities certain, however life annuities may not be calculated with similar formulas because actuarial present value accounts for the probability of death at each age.

# Amortization calculations

If an annuity is for repaying a debt ''P'' with interest, the amount owed after ''n'' payments is :$\frac- \left(1+i\right)^n \left\left( \frac - P \right\right).$ Because the scheme is equivalent with borrowing the amount $\frac$ to create a perpetuity with coupon $R$, and putting $\frac-P$ of that borrowed amount in the bank to grow with interest $i$. Also, this can be thought of as the present value of the remaining payments : See also
fixed rate mortgage A fixed-rate mortgage (FRM) is a mortgage loan where the interest rate on the note remains the same through the term of the loan, as opposed to loans where the interest rate may adjust or "float". As a result, payment amounts and the duration of th ...
.

# Example calculations

Formula for finding the periodic payment ''R'', given ''A'': : $R = \frac A$ Examples: # Find the periodic payment of an annuity due of $70,000, payable annually for 3 years at 15% compounded annually. #* ''R'' = 70,000/(1+〖(1-(1+((.15)/1) )〗^(-(3-1))/((.15)/1)) #* R = 70,000/2.625708885 #* R =$26659.46724 Find PVOA factor as. 1) find ''r'' as, (1 ÷ 1.15)= 0.8695652174 2) find ''r'' × (''r''''n'' − 1) ÷ (''r'' − 1) 08695652174 × (−0.3424837676)÷ (−1304347826) = 2.2832251175 70000÷ 2.2832251175= $30658.3873 is the correct value # Find the periodic payment of an annuity due of$250,700, payable quarterly for 8 years at 5% compounded quarterly. #* R= 250,700/(1+〖(1-(1+((.05)/4) )〗^(-(32-1))/((.05)/4)) #* R = 250,700/26.5692901 #* R = $9,435.71 Finding the Periodic Payment(R), Given S: R = S\,/((〖((1+(j/m) )〗^(n+1)-1)/(j/m)-1) Examples: # Find the periodic payment of an accumulated value of$55,000, payable monthly for 3 years at 15% compounded monthly. #* R=55,000/((〖((1+((.15)/12) )〗^(36+1)-1)/((.15)/12)-1) #* R = 55,000/45.67944932 #* R = $1,204.04 # Find the periodic payment of an accumulated value of$1,600,000, payable annually for 3 years at 9% compounded annually. #* R=1,600,000/((〖((1+((.09)/1) )〗^(3+1)-1)/((.09)/1)-1) #* R = 1,600,000/3.573129 #* R = \$447,786.80

# Legal regimes

*
Annuities under American law In the United States, an annuity is a financial product which offers tax-deferred growth and which usually offers benefits such as an income for life. Typically these are offered as structured (insurance) products that each state approves and r ...
*
Annuities under European law Under European Union law, an annuity is a financial contract which provides an income stream in return for an initial payment with specific parameters. It is the opposite of a settlement funding. A Swiss annuity is not considered a European an ...
*
Annuities under Swiss law A Swiss annuity simply refers to a fixed or variable annuity marketed from Switzerland or issued by a Swiss based life insurance company but has no legal definition. Insurance brokers promoting annuity contracts issued by insurance companies domic ...

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Amortization calculator An amortization calculator is used to determine the periodic payment amount due on a loan (typically a mortgage), based on the amortization process. The amortization repayment model factors varying amounts of both interest and principal into ev ...
*
Fixed rate mortgage A fixed-rate mortgage (FRM) is a mortgage loan where the interest rate on the note remains the same through the term of the loan, as opposed to loans where the interest rate may adjust or "float". As a result, payment amounts and the duration of th ...
*
Life annuity A life annuity is an annuity, or series of payments at fixed intervals, paid while the purchaser (or annuitant) is alive. The majority of life annuities are insurance products sold or issued by life insurance companies however substantial case ...
*
Perpetuity A perpetuity is an annuity that has no end, or a stream of cash payments that continues forever. There are few actual perpetuities in existence. For example, the United Kingdom (UK) government issued them in the past; these were known as cons ...
*
Time value of money The time value of money is the widely accepted conjecture that there is greater benefit to receiving a sum of money now rather than an identical sum later. It may be seen as an implication of the later-developed concept of time preference. The ...

# References

* * {{DEFAULTSORT:Annuity (Finance Theory) Finance theories