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

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

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.

^{''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
A bank account is a financial account maintained by a bank or other financial institution in which the financial transaction
A financial transaction is an Contract, agreement, or communication, carried ...

, monthly home mortgage
A mortgage loan or simply mortgage () is a loan
In finance, a loan is the lending of money by one or more individuals, organizations, or other entities to other individuals, organizations etc. The recipient (i.e., the borrower) incurs a ...

payments, monthly insurance
Insurance is a means of protection from financial loss. It is a form of risk management
Risk management is the identification, evaluation, and prioritization of risk
In simple terms, risk is the possibility of something bad happening. ...

payments and pension
A pension (, from Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to be ...

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
Mathematics (from Greek: ) includes the study of such topics as quantity
Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", " ...

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
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 sav ...

.
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
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 sav ...

, 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 theSecurities and Exchange Commission
The U.S. Securities and Exchange Commission (SEC) is a large independent agency of the United States federal government, created in the aftermath of the Wall Street Crash of 1929
The Wall Street Crash of 1929, also known as the Great Cras ...

.
*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
Economics () is a social science
Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within those societies. ...

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
In and , interest is payment from a or deposit-taking financial institution to a or depositor of an amount above repayment of the (that is, the amount borrowed), at a particular rate. It is disti ...

, and future value
Future value is the value
Value or values may refer to:
* Value (ethics)
In ethics
Ethics or moral philosophy is a branch of philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about Metap ...

..
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.actuarial notation
Actuarial notation is a shorthand method to allow Actuary, actuaries to record mathematical formulas that deal with Interest, interest rates and life tables.
Traditional notation uses a halo system where symbols are placed as superscript or subscr ...

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 rateIn finance and economics, the nominal interest rate or nominal rate of interest is either of two distinct things:
# the rate of interest
In finance
Finance is the study of financial institutions, financial markets and how they operate within ...

, 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 $&= \frac\left(\frac\right)\quad\quad\text\\&= \frac\\&= \frac, \end which gives us the result as required. Similarly, we can prove the formula for the future value. The payment made at the end of the last year would accumulate no interest and the payment made at the end of the first year would accumulate interest for a total of (''n'' − 1) years. Therefore, :$s\_\; =\; 1\; +\; (1+i)\; +\; (1+i)^2\; +\; \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.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 aperpetuity
A perpetuity is an annuity
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 ...

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 theactuarial present valueThe actuarial present value (APV) is the expected value
In probability theory, the expected value of a random variable X, denoted \operatorname(X) or \operatorname is a generalization of the weighted average, and is intuitively the arithmetic me ...

of the future life contingent payments. Life table
In actuarial science
2003 US mortality ( life) table, Table 1, Page 1
Actuarial science is the discipline that applies mathematical
Mathematics (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( ...

s are used to calculate the probability
Probability is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

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 :$R\backslash left;\; href="/html/ALL/s/\backslash frac-\backslash frac\_\backslash right.html"\; ;"title="\backslash frac-\backslash frac\; \backslash right">\backslash frac-\backslash frac\; \backslash right$ See alsofixed rate mortgage
A fixed-rate mortgage (FRM) is a mortgage loan
A mortgage loan or simply mortgage () is a loan
In finance, a loan is the lending of money by one or more individuals, organizations, or other entities to other individuals, organizations et ...

.
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
In the United States, an annuity is a Structured product, structured (Insurance policy, insurance) product that each U.S. state, state approves and regulates. It is designed using a mortality table and mainly guaranteed by a life insurer. There ...

* Annuities under European law
Under European Union law
European Union law is a system of rules operating within the member states of the European Union. Since the founding of the European Coal and Steel Community following World War II, the EU has developed the aim to "pr ...

* Annuities under Swiss law
A Swiss annuity simply refers to a fixed or variable annuity (finance theory), 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 ...

See also

*Amortization calculator
An amortization calculator is used to determine the periodic payment amount due on a loan
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned wi ...

*Fixed rate mortgage
A fixed-rate mortgage (FRM) is a mortgage loan
A mortgage loan or simply mortgage () 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 purpose while p ...

*Life annuity
A life annuity is an annuity
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 sav ...

*Perpetuity
A perpetuity is an annuity
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 ...

*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