Annuity (finance theory)

# Annuity (finance theory)

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Encyclopedia
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 discussions of finance, usually in connection with the valuation of the stream of payments, taking into account time value of money
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....

concepts such as interest rate
Interest rate
An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...

and future value
Future value
Future value is the value of an asset at a specific date. It measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate, or more generally, rate of return; it is the present value multiplied by the accumulation...

.

Examples of annuities are regular deposits to a savings account, monthly home mortgage payments and monthly insurance payments. Annuities are classified by the frequency of 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:

Let: = the yearly nominal interest rate. = the number of years. = the number of periods per year. = the interest rate per period. = the number of periods.

Note:

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

Also:

Clearly, in the limit as increases,

Thus, even an infinite series of finite payments (perpetuity
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...

) 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:
.

We notice that the second factor is a geometric progression
Geometric progression
In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression...

of scale factor and of common ratio . We can write
.

Finally, after simplifications
Elementary algebra
Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. It is typically taught in secondary school under the term algebra. The major difference between algebra and...

, we obtain
.

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,
.

Hence:
.

If an annuity is for repaying a debt P with interest, the amount owed after n payments is:
because the scheme is equivalent with lending an amount and putting part of that, an amount , in the bank to grow due to interest. See also fixed rate mortgage
Fixed rate mortgage
A fixed-rate mortgage is a mortgage loan first developed by the Federal Housing Administration where the interest rate on the note remains the same through the term of the loan, as opposed to loans where the interest rate may adjust or "float." Other forms of mortgage loan include interest only...

.

Also, this can be thought of as the present value of the remaining payments:

## 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.

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):
(annuity notation)
It can also be written as

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: (value at the time of the first of n payments of 1) (value one period after the time of the last of n payments of 1)

Formula for Finding the Periodic payment(R), Given A:

R = A/(1+〖(1-(1+(j/m) )〗^(-(n-1))/(j/m))

Examples:
1. Find the periodic payment of an annuity due of \$70000, payable annually for 3 years at 15% compounded annually.
R= 70000/(1+〖(1-(1+((.15)/1) )〗^(-(3-1))/((.15)/1))
R = 70000/2.625708885
R = \$26659.46724

2. Find the periodic payment of an annuity due of \$250700, payable quarterly for 8 years at 5% compounded quarterly.
R= 250700/(1+〖(1-(1+((.05)/4) )〗^(-(32-1))/((.05)/4))
R = 250700/26.5692901
R = \$9435.71

Finding the Periodic Payment(R), Given S:

R = S\,/((〖((1+(j/m) )〗^(n+1)-1)/(j/m)-1)

Examples:
1. Find the periodic payment of an accumulated value of \$55000, payable monthly for 3 years at 15% compounded monthly.
R=55000/((〖((1+((.15)/12) )〗^(36+1)-1)/((.15)/12)-1)
R = 55000/45.67944932
R = \$1204.04

2. Find the periodic payment of an accumulated value of \$1600000, payable annually for 3 years at 9% compounded annually.
R=1600000/((〖((1+((.09)/1) )〗^(3+1)-1)/((.09)/1)-1)
R = 1600000/3.573129
R = \$447786.80

## Other types

• 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 but are not regulated by the Securities and Exchange Commission. This type can be adversely affected by high inflation.

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

• Equity-indexed annuities
Equity-indexed annuity
An index annuity in the United States is a type of tax-deferred annuity whose credited interest is linked to an equity index — typically the S&P 500 or international index...

– Lump sum payments are made to an insurance company. Can be implemented with a Call option
Call option
A call option, often simply labeled a "call", is a financial contract between two parties, the buyer and the seller of this type of option. The buyer of the call option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument from the seller...

.

• Annuity (financial contracts)
• Perpetuity
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...

• Life annuity
Life annuity
A life annuity is a financial contract in the form of an insurance product according to which a seller — typically a financial institution such as a life insurance company — makes a series of future payments to a buyer in exchange for the immediate payment of a lump sum or a series...

• Fixed rate mortgage
Fixed rate mortgage
A fixed-rate mortgage is a mortgage loan first developed by the Federal Housing Administration where the interest rate on the note remains the same through the term of the loan, as opposed to loans where the interest rate may adjust or "float." Other forms of mortgage loan include interest only...

• Amortization calculator
Amortization calculator
An amortization calculator is used to determine the periodic payment amount due on a loan , based on the amortization process....