annuities

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In investment, 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 A savings account is a bank account at a retail bank. Common features include a limited number of withdrawals, a lack of cheque and linked debit card facilities, limited transfer options and the inability to be overdrawn. Traditionally, transac ...
, 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 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. *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 in ...
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..

## 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.

### 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 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 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 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.

# 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 * Annuities under European law * Annuities under Swiss law