Annuity

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Annuity (finance theory)
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For other uses, see Annuity (disambiguation).
An annuity is a series of payments made at fixed intervals of time. Examples of annuities are
regular deposits to a savings account, monthly home mortgage payments, monthly insurance
payments and pension 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.

Contents
[hide]


1 Valuation



2 Annuity-immediate



3 Annuity-due



4 Perpetuity



5 Proof of annuity formula



6 Amortization calculations



7 Example calculations



8 Other types



9 See also



10 References

Valuation[edit]
The valuation of an annuity entails concepts such as time value of money, interest rate, and
future value.[1]

Annuity-immediate[edit]
If the number of payments is known in advance, the annuity is an annuity-certain. 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.

———
0


———
1


———
2

...
———
...



n

payments
periods

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:

where is the number of terms and is the per period interest rate. Present value is linear in the
amount of payments, therefore the present value for payments, or rent is:

In practice, often loans are stated per annum while interest is compounded and payments are
made monthly. In this case, the interest is stated as a nominal interest rate, and

.

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:

where is the number of terms and is the per period interest rate. Future value is linear in the
amount of payments, therefore the future value for payments, or rent is:

Example: The present value of a 5 year annuity with nominal annual interest rate 12% and
monthly payments of $100 is:

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 as:

and

Annuity-due[edit]
An annuity-due is an annuity whose payments are made at the beginning of each period.[2]
Deposits in savings, rent or lease payments, and insurance premiums are examples of annuities
due.

———
0


———
1

...
———
...


———
n-1

payments

n

periods

Each annuity payment is allowed to compound for one extra period. Thus, the present and future
values of an annuity-due can be calculated through the formula:

and

where

are the number of terms, is the per term interest rate, and is the effective rate of

discount given by

.

Future and present values for annuities due are related as:

and

Example: The final value of a 7 year annuity-due with nominal annual interest rate 9% and
monthly payments of $100:

Note that 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:
(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)

Perpetuity[edit]
A perpetuity is an annuity for which the payments continue forever. Since:

even a perpetuity has a finite present value when there is a non-zero discount rate. The formula
for a perpetuity are:

where is the interest rate and

is the effective discount rate.

Proof of annuity formula[edit]
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 R/
(1+i)^k. Just considering R to be one, then:

which is the desired result.
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,

Amortization calculations[edit]
If an annuity is for repaying a debt P with interest, the amount owed after n payments is:

because the scheme is equivalent with borrowing the amount
coupon

, and putting

to create a perpetuity with

of that borrowed amount in the bank to grow with interest .

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

See also fixed rate mortgage.

Example calculations[edit]
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.
o R = 70000/(1+〖(1-(1+((.15)/1) )〗^(-(3-1))/((.15)/1))
o R = 70000/2.625708885
o R = $26659.46724
2. Find the periodic payment of an annuity due of $250700, payable quarterly for 8 years at
5% compounded quarterly.
o R= 250700/(1+〖(1-(1+((.05)/4) )〗^(-(32-1))/((.05)/4))
o R = 250700/26.5692901
o 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.
o R=55000/((〖((1+((.15)/12) )〗^(36+1)-1)/((.15)/12)-1)
o R = 55000/45.67944932
o R = $1204.04
2. Find the periodic payment of an accumulated value of $1600000, payable annually for 3
years at 9% compounded annually.
o R=1600000/((〖((1+((.09)/1) )〗^(3+1)-1)/((.09)/1)-1)
o R = 1600000/3.573129
o R = $447786.80

Other types[edit]



Fixed annuities – These are annuities with fixed payments. The insurance 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.

See also[edit]


Amortization calculator



Fixed rate mortgage



Life annuity



Perpetuity



Time value of money

References[edit]
1.

Jump up ^ Lasher, William (2008). Practical financial management. Mason, Ohio:
Thomson South-Western. p. 230. ISBN 0-324-42262-8..

2.

Jump up ^ Jordan, Bradford D.; Ross, Stephen David; Westerfield, Randolph (2000).
Fundamentals of corporate finance. Boston: Irwin/McGraw-Hill. p. 175. ISBN 0-07-231289-0.



Samuel A. Broverman (2010). Mathematics of Investment and Credit, 5th Edition.
ACTEX Academic Series. ACTEX Publications. ISBN 978-1-56698-767-7.



Stephen Kellison (2008). Theory of Interest, 3rd Edition. McGraw-Hill/Irwin. ISBN 9780-07-338244-9.

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