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Lesson
5 Annuities |
Objective:
To develop an understanding
of the basic concepts and a proficiency with the mathematical
methods of annuities. Annuities are also an example of the basic
increase, or accumulation, relationship BI= B(1 +RI).
However, the accumulation factor, 1 + RI, is more
complex for annuities reflecting the particular characteristics
of annuities. The accumulation factor for annuities is called an
annuity factor.
Background:
As opposed to simple
interest and compound interest with only one payment at the
beginning of the time period for an investment, or one payment
at the end of time period for a loan, an annuity is more closely
related to the practice as illustrated in the U.S. Rule (or, partial
payments). That is, an annuity, whether as a loan or investment,
consists of a series of payments, not just a single payment.
(There are indeed single payment annuities, but are not
considered in this academic material.)
Annuities in modern applications are many and varied; and, are
much more complex and variable in modern applications than ever
before. The considerations herein are limited to fundamental
concepts and by no means are intended to be an exhaustive
treatment of the complex, challenging topic of annuities.
The purchase of an automobile and a house are two primary
financial transactions that most people will be involved in
during their lifetime. Both are examples of annuities in a loan
situation.
Saving for retirement as well as a provision for "rainy
day" savings on a regular periodic basis are examples of
annuities in an investment situation. Further, when one retires,
the periodic payment of retirement benefits is also an annuity.
From these few examples, it should
be evident that annuities are one of the most basic, useful and
beneficial financial considerations for most individuals.
Definitions and Development of
Concepts: For the purpose of this course, an annuity
is a financial instrument with regular periodic payments.
"Regular periodic" means same dollar amount and
the same frequency of payment (e.g. monthly payments of
$300 to pay off a car loan).
An annuity is a compound interest consideration. Therefore, a
compound interest description must be provided from which the
nominal rate, r, the frequency, k, and the representation of
time, t, is ascertained. From r, k, and t, the interest rate per
period, i, and the term measured in conversion periods, n, can
be determined. Recall these concepts/methods as developed in
compound interest and that
 ,
n= kt. Annuities give the additional consideration that
payments are made with the same frequency as interest is
compounded; e.g. monthly compounding of interest, monthly
payments, Therefore, n=kt not only gives the term, or
period of the financial transaction, in conversion periods but
also gives the number of payments for an annuity.
To
Summarize:
| r |
is
nominal interest rate from compound interest
description |
| k |
is
frequency of compounding as well as frequency of
annuity payments |
| t |
is
appropriate representation of time as previously
discussed for simple and compound interest |
 |
is
the interest rate per period |
| n
= kt |
is
the number of conversion periods as well as the
number of annuity payments for the term of the
financial transaction |
| P
= regular periodic
payment |
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An ordinary annuity
is an annuity with payments made at the end of the
period; e.g., mortgage payments are normally an ordinary
annuity.
An annuity due is an annuity with payments
made at the beginning of the period; e.g., contributions to a
retirement or savings program.
The accumulated value of an annuity, denoted S
for an ordinary annuity and
for an annuity due, represents the value of the series of
equal periodic payments (P) at the end of stated period
of time (n) at the stated rate per period (i).
The present value of an annuity, denoted A
for an ordinary annuity and
for an annuity due, represents the value of the series of
regular periodic payments at the beginning of the stated period
of time at the stated rate per period. Cost, purchase
price, amount paid are synonymous with present value.
Mathematically, the present value and accumulated value are
determined by multiplying P, regular periodic payment, by an
appropriate annuity factor:
= |
annuity
factor for accumulated value, ordinary annuity |
= |
annuity
factor for accumulated value, annuity due |
= |
annuity
factor for
present value, ordinary annuity |
= |
annuity
factor for
present value, annuity due. |
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These concepts/definitions are summarized in tabular format:
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The values of the annuity factors
and 
are obtained using the tables in the text. Additionally, the
following formulas can be used in calculation of the annuity
factors with a calculator:
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To find the values
for 
and 
use the relationships:
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Additional formulas
are also beneficial:
| Investment: |
Interest
(made) = S -nP (ordinary) |
Interest
(made) =
- nP (due) |
| Loan: |
Interest
(paid) = nP - A (ordinary)
Interest (paid) = nP -
 (due) |
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Copyright © Jim Pack,
2000. All Rights Reserved. Last modified
11/20/07
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in Table E to be 14.68033152
in Table F to be 9.78684805
- nP = 6840.17 - 12 * 500 = $840.17
= 12 * 500 - 5393.42 = $606.58 
using calculator.
using Calculator.
