Annuities
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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 A savings account is a bank account at a retail banking, 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. Tradit ...
, monthly
home mortgage A mortgage loan or simply mortgage (), in civil law jurisdicions known also as a hypothec loan, is a loan used either by purchasers of real property to raise funds to buy real estate, or by existing property owners to raise funds for any ...
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 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 l ...
.


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 l ...
, 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 agencies of the United States government, independent agency of the United States federal government, created in the aftermath of the Wall Street Crash of 1929. The primary pu ...
. *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 inte ...
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 Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, ...
,
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 In financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) that can be used to produce positive economic value. As ...
..


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_ = \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(i,n,R) = 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, 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_ = \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: :\text(i,n,R) = R \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: :\text\left( \frac,5\times 12,\$100\right) = \$100 \times a_ = \$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 \times a_ and :\frac - \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 \frac . Just considering ''R'' to be 1, then: :\begin a_ &= \sum_^n \frac = \frac\sum_^\left(\frac\right)^k \\ pt&= \frac\left(\frac\right)\quad\quad\text\\ pt&= \frac\\ pt&= \frac, \end 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 + \cdots + (1+i)^ = (1+i)^n a_ = \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. : \ddot_ = (1+i) \times a_ = \frac, :\ddot_ = (1+i) \times s_ = \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=\frac. The future and present values for annuities due are related since: :\ddot_ = (1+i)^n \times \ddot_, :\frac - \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: : \text_\left(\frac,7\times 12,\$100\right) = \$100 \times \ddot_ = \$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: : \ddot_=a_(1 + i)=a_+1. The value at the time of the first of ''n'' payments of 1. : \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 : \lim_ \text(i,n,R) = \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 The United Kingdom of Great Britain and Northern Ireland, ...
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 death ...
s are used to calculate the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), 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 ...
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- (1+i)^n \left( \frac - P \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 :R\left \frac-\frac \right= R \times a_. 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


See 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 l ...
*
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 The United Kingdom of Great Britain and Northern Ireland, ...
*
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 Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, ...


References

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