# Annuity (finance theory)

﻿
Annuity (finance theory)

The term "annuity" is used in finance theory to refer to any terminating stream of fixed payments over a specified period of time. This usage is most commonly seen in academic discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money concepts such as interest rate and future value. [cite web |url=http://www.college-cram.com/study/finance/presentations/1128|title=Calculate Annuity Payment: Funding an Annuity|accessdate=2008-07-10 ]

Examples of annuities are regular deposits to a savings account, monthly home mortgage payments and monthly insurance payments. Annuities are classified by payment dates. The payments (deposits) may be made weekly, monthly, quarterly, yearly, or at any other interval of time.

Ordinary annuity

An ordinary annuity (also referred as annuity-immediate) is an annuity whose payments are made at the end of each period (e.g. a month, a year). The values of an ordinary annuity can be calculated through the following [Finite Mathematics, Eighth Edition, by Margaret L. Lial, Raymond N. Greenwell, and Nathan P. Ritchey. Published by Addison Wesley. ISBN 032122826X] :

Formulae

Let::$r$ = the yearly nominal interest rate.:$t$ = the number of years.:$m$ = the number of periods per year.:$i$ = the interest rate per period.:$n$ = the number of periods.

Note::$i=frac\left\{r\right\}\left\{m\right\}$:$n=tm$

Also let::$P$ = the principal (or present value).:$S$ = the future value of an annuity.:$R$ = the periodic payment in an annuity (the amortized payment).

:$S ,=,Rleft \left[frac\left\{left\left(1+i ight\right)^n-1\right\}\left\{i\right\} ight\right] ,=,Rcdot s_\left\{overline\left\{n\right\}|i\right\}$ (annuity notation)

Also:

:$P ,=,Rleft \left[frac\left\{1-frac\left\{1\right\}\left\{left\left(1+i ight\right)^n\left\{i\right\} ight\right] = Rcdot a_\left\{overline\left\{n\right\}|i\right\}$

Clearly, in the limit as $n$ increases,

$lim_\left\{n, ightarrow,infty\right\},P,=,frac\left\{R\right\}\left\{i\right\}$

Thus even an infinite series of finite payments (perpetuity) with a non-zero discount rate has a finite present value.

Proof

The next payment is to be paid in one period. Thus, the present value is computed to be:

:$P , = , frac\left\{R\right\}\left\{1+i\right\} + frac\left\{R\right\}\left\{\left(1+i\right)^2\right\} + dots + frac\left\{R\right\}\left\{\left(1+i\right)^n\right\} = frac\left\{R\right\}\left\{1+i\right\} left \left[ 1 + frac\left\{1\right\}\left\{1+i\right\} + frac\left\{1\right\}\left\{\left(1+i\right)^2\right\} + dots + frac\left\{1\right\}\left\{\left(1+i\right)^\left\{n-1 ight\right] .$

We notice that the second term is a geometric progression of scale factor $1$ and of common ratio $frac\left\{1\right\}\left\{1+i\right\}$. We can write

:$P , = , frac\left\{R\right\}\left\{1+i\right\} imes frac\left\{1 - frac\left\{1\right\}\left\{\left(1+i\right)^n\left\{1-frac\left\{1\right\}\left\{1+i.$

Finally, after simplifications, we obtain

:$P , = , frac\left\{R\right\}\left\{i\right\} left \left[1 - frac\left\{1\right\}\left\{\left(1+i\right)^n\right\} ight\right] = frac\left\{Rm\right\}\left\{r\right\} left \left[1 - frac\left\{1\right\}\left\{\left(1+frac\left\{r\right\}\left\{m\right\}\right)^\left\{\left(tm\right) ight\right] .$

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,

:$S , = , R + R\left(1+i\right) + R\left(1+i\right)^2 + dots + R\left(1+i\right)^\left\{n-1\right\} = R left \left[ 1 + \left(1+i\right) + \left(1+i\right)^2 + dots + \left(1+i\right)^\left\{n-1\right\} ight\right] .$

Hence:

:$S , = , R left \left[ frac\left\{\left(1+i\right)^n-1\right\}\left\{i\right\} ight\right] .$

If an annuity is for repaying a debt "P" with interest, the amount owed after "n" payments is: :$frac\left\{S\right\}\left\{i\right\}-\left(1+i\right)^n\left(frac\left\{S\right\}\left\{i\right\}-P\right)$because the scheme is equivalent with lending an amount $frac\left\{S\right\}\left\{i\right\}$ and putting part of that, an amount $frac\left\{S\right\}\left\{i\right\}-P$, in the bank to grow due to interest. See also fixed rate mortgage.

Annuity-due

An annuity-due is an annuity whose payments are made at the beginning of each period. [cite web |url=http://www.college-cram.com/study/finance/presentations/1129|title=Future Value of an Annuity Due|accessdate=2008-07-10 ] Deposits in savings, rent payments, and insurance premiums are examples of annuities due.

Because each annuity payment is allowed to compound for one extra period, the value of an annuity-due is equal to the value of the corresponding ordinary annuity multiplied by (1+i). Thus, the future value of an annuity-due can be calculated through the formula (variables named as above) [ibid Lial.] :

:$S , = , R left \left[ \left\{ \left(1+i\right)^\left\{n+1\right\} - 1 over i \right\} ight\right] - R,=,Rcdot ddot\left\{s\right\}_\left\{overline\left\{ni\right\}$ (annuity notation)It can also be written as

:$S ,=,Rleft \left[frac\left\{left\left(1+i ight\right)^n-1\right\}\left\{i\right\} ight\right] \left(1 + i \right) ,=,Rcdot s_\left\{overline\left\{n\right\}|r\right\}$$\left(1 + i\right)$

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 with one payment more, minus the last payment.

Thus we have: :$ddot\left\{a\right\}_\left\{overline\left\{ni\right\}=a_\left\{overline\left\{n\right\}|i\right\}\left(1 + i\right)=a_\left\{overline\left\{n-1i\right\}+1$ (value at the time of the first of "n" payments of 1):$ddot\left\{s\right\}_\left\{overline\left\{ni\right\}=s_\left\{overline\left\{n\right\}|i\right\}\left(1 + i\right)=s_\left\{overline\left\{n+1i\right\}-1$ (value one period after the time of the last of "n" payments of 1)

Other types of annuities

*Fixed annuities - These are annuities with fixed payments. They are primarily used for low risk investments like government securities or corporate bonds. Fixed annuities offer a fixed rate up to ten years but are not regulated by the Securities and Exchange Commission.

*Variable annuities - Unlike fixed annuities, these are regulated by the SEC. They allow you to invest in portions of money markets.

*Equity-indexed annuities - Lump sum payments are made to an insurance company.

Annuity due is useful for lease payment calculations

References

ee also

*Annuity (financial contracts)
*Perpetuity
*Life annuity
*Fixed rate mortgage

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Annuity — See also: Pension Annuity may refer to: Annuity (finance theory): any terminating stream of fixed payments over a specified period of time Life annuity: a financial contract providing payments for a person s lifetime Annuity (US financial… …   Wikipedia

• Annuity (European financial arrangements) — An annuity can be defined as a contract which provides an income stream in return for an initial payment.Immediate annuityAn immediate annuity is an annuity for which the income stream begins at a time after the initial payment which is less than …   Wikipedia

• Corporate finance — Corporate finance …   Wikipedia

• Outline of finance — The following outline is provided as an overview of and topical guide to finance: Finance – addresses the ways in which individuals, businesses and organizations raise, allocate and use monetary resources over time, taking into account the risks… …   Wikipedia

• List of finance topics — Topics in finance include:Fundamental financial concepts* Finance an overview ** Arbitrage ** Capital (economics) ** Capital asset pricing model ** Cash flow ** Cash flow matching ** Debt *** Default *** Consumer debt *** Debt consolidation ***… …   Wikipedia

• Life annuity — The life annuity is a financial contract according to which a seller (issuer) typically a financial institution such as a life insurance company makes a series of payments in the future to the buyer (annuitant) in exchange for the immediate… …   Wikipedia

• Institute of Business and Finance — Type Financial Services Education Provider Industry Financial Services, Finance, Education Headquarters San Diego [United States, US] …   Wikipedia

• Dedicated Portfolio Theory — Dedicated Portfolio Theory, in finance, deals with the characteristics and features of a portfolio built to generate a predictable stream of future cash inflows. This is achieved by purchasing bonds and/or other fixed income securities (such as… …   Wikipedia

• Amortization (business) — For other uses of Amortization, see the Amortization disambiguation page. Amortization is the distribution of a single lump sum cash flow into many smaller cash flow installments, as determined by an amortization schedule. Unlike other repayment… …   Wikipedia

• Time value of money — The time value of money is the value of money figuring in a given amount of interest earned over a given amount of time. The time value of money is the central concept in finance theory. For example, \$100 of today s money invested for one year… …   Wikipedia