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

^{''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

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 alife 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 thepresent 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.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 by: :$a\_\; =\; \backslash frac,$ 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: :$\backslash text(i,n,R)\; =\; R\; \backslash times\; a\_.$ 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, and $i\; =\; I/12$. The ''future value'' of an annuity is the accumulated amount, including payments and interest, of a stream of payments made to an interest-bearing account. For an annuity-immediate, it is the value immediately after the n-th payment. The future value is given by: :$s\_\; =\; \backslash frac,$ where $n$ is the number of terms and $i$ is the per period interest rate. Future value is linear in the amount of payments, therefore the future value for payments, or ''rent'' $R$ is: :$\backslash text(i,n,R)\; =\; R\; \backslash times\; s\_$ Example: The present value of a 5-year annuity with a nominal annual interest rate of 12% and monthly payments of $100 is: :$\backslash text\backslash left(\; \backslash frac,5\backslash times\; 12,\backslash \$100\backslash right)\; =\; \backslash \$100\; \backslash times\; a\_\; =\; \backslash \$4,495.50$ The rent is understood as either the amount paid at the end of each period in return for an amount PV borrowed at time zero, the ''principal'' of the loan, or the amount paid out by an interest-bearing account at the end of each period when the amount PV is invested at time zero, and the account becomes zero with the n-th withdrawal. Future and present values are related since: :$s\_\; =\; (1+i)^n\; \backslash times\; a\_$ and :$\backslash frac\; -\; \backslash frac\; =\; i$= Proof of annuity-immediate formula

= To calculate present value, the ''k''-th payment must be discounted to the present by dividing by the interest, compounded by ''k'' terms. Hence the contribution of the ''k''-th payment ''R'' would be $\backslash frac$. Just considering ''R'' to be 1, then: :$\backslash begin\; a\_\; \&=\; \backslash sum\_^n\; \backslash frac\; =\; \backslash frac\backslash sum\_^\backslash left(\backslash frac\backslash right)^k\; \backslash \backslash ;\; href="/html/ALL/l/pt.html"\; ;"title="pt">pt$ which gives us the result as required. Similarly, we can prove the formula for the future value. The payment made at the end of the last year would accumulate no interest and the payment made at the end of the first year would accumulate interest for a total of (''n'' − 1) years. Therefore, :$s\_\; =\; 1\; +\; (1+i)\; +\; (1+i)^2\; +\; \backslash cdots\; +\; (1+i)^\; =\; (1+i)^n\; a\_\; =\; \backslash frac.$Annuity-due

An ''annuity-due'' is an annuity whose payments are made at the beginning of each period. Deposits in savings, rent or lease payments, and insurance premiums are examples of annuities due. Each annuity payment is allowed to compound for one extra period. Thus, the present and future values of an annuity-due can be calculated. :$\backslash ddot\_\; =\; (1+i)\; \backslash times\; a\_\; =\; \backslash frac,$ :$\backslash ddot\_\; =\; (1+i)\; \backslash times\; s\_\; =\; \backslash frac,$ where $n$ is the number of terms, $i$ is the per-term interest rate, and $d$ is the effective rate of discount given by $d=\backslash frac$. The future and present values for annuities due are related since: :$\backslash ddot\_\; =\; (1+i)^n\; \backslash times\; \backslash ddot\_,$ :$\backslash frac\; -\; \backslash frac\; =\; d.$ Example: The final value of a 7-year annuity-due with a nominal annual interest rate of 9% and monthly payments of $100 can be calculated by: : $\backslash text\_\backslash left(\backslash frac,7\backslash times\; 12,\backslash \$100\backslash right)\; =\; \backslash \$100\; \backslash times\; \backslash ddot\_\; =\; \backslash \$11,730.01.$ In Excel, the PV and FV functions take on optional fifth argument which selects from annuity-immediate or annuity-due. An annuity-due with ''n'' payments is the sum of one annuity payment now and an ordinary annuity with one payment less, and also equal, with a time shift, to an ordinary annuity. Thus we have: :$\backslash ddot\_=a\_(1\; +\; i)=a\_+1$. The value at the time of the first of ''n'' payments of 1. :$\backslash ddot\_=s\_(1\; +\; i)=s\_-1$. The value one period after the time of the last of ''n'' payments of 1.Perpetuity

A ''perpetuity'' is an annuity for which the payments continue forever. Observe that :$\backslash lim\_\; \backslash text(i,n,R)\; =\; \backslash lim\_\; R\; \backslash times\; a\_\; =\; \backslash lim\_\; R\; \backslash times\; \backslash frac\; =\; \backslash ,\backslash frac.$ Therefore a perpetuity has a finite present value when there is a non-zero discount rate. The formulae for a perpetuity are :$a\_\; =\; \backslash frac\; \backslash text\; \backslash ddot\_\; =\; \backslash frac,$ where $i$ is the interest rate and $d=\backslash frac$ 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 :$\backslash frac-\; (1+i)^n\; \backslash left(\; \backslash frac\; -\; P\; \backslash right).$ Because the scheme is equivalent with borrowing the amount $\backslash frac$ to create a perpetuity with coupon $R$, and putting $\backslash frac-P$ 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 :$R\backslash left;\; href="/html/ALL/l/\backslash frac-\backslash frac\_\backslash right.html"\; ;"title="\backslash frac-\backslash frac\; \backslash right">\backslash frac-\backslash frac\; \backslash right$ See also fixed rate mortgage.Example calculations

Formula for finding the periodic payment ''R'', given ''A'': : $R\; =\; \backslash frac\; A$ 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''Legal regimes

* Annuities under American law * Annuities under European law * Annuities under Swiss lawSee also

* Amortization calculator * Fixed rate mortgage *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
*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