Copyright © Cengage Learning. All rights reserved.
5
Finance
1
Copyright © Cengage Learning. All rights reserved.
5.3
Annuities
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Objectives
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Annuities
An annuity is simply a sequence of equal, regular payments into an account in which each payment receives compound interest.
Because most annuities involve relatively small periodic payments, they are affordable for the average person.
Over longer periods of time, the payments themselves start to amount to a significant sum, but it is really the power of compound interest that makes annuities so amazing.
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Annuities
If you pay $50 a month into an annuity for the next forty years, then you will have about $300,000 in your account, even though your total payment is
The interest portion is
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Annuities as Compound Interest, Repeated
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Example 1 – Understanding Annuities
On September 12, Patty Leitner started an annuity. She arranged to have $200 taken out of each of her end-of-the
month paychecks. The money would earn interest
compounded monthly.
Find the future value of the �account on November 30 �by applying the Compound�Interest Formula to the �September payment, the �October payment, and the �November payment.
Figure 5.11
Patty’s payment schedule and interest earning periods.
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Example 1 – Solution
A timeline for Patty’s payments is shown in Figure 5.11.
We are given P = 200 and r =
To find the future value of the first payment (made on September 30), use n = 2, because the payment will receive interest during October and November.
FV = P (1 + i) n
=
= 202.9273
the Compound Interest Formula
substituting
rounding
≈ $202.93
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Example 1 – Solution
To calculate the future value of the second payment (made on October 31), use n = 1. This payment will receive interest during November.
FV = P (1 + i) n
=
= 201.45833
≈ $201.46
cont’d
the Compound Interest Formula
substituting
rounding
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Example 1 – Solution
We are to find the future value of the account on November 30. The third payment is made on November 30, so it earns no interest (until after November 30, which is beyond the scope of the example). Therefore,
FV = $200
The future value of Patty’s annuity is:
FV = $200 + $201.46 + $203.93 = $604.39
Patty made three payments of $200 each, so her total contribution was $600. She earned $4.39 interest on these payments.
cont’d
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The Annuity Formula
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The Annuity Formula
The procedure followed in Examples 1 reflects what actually happens with annuities, and it works fine for a small number of payments.��However, most annuities are long-term, and the procedure would be impractical if we were computing the future value after forty years.
Because of this, long-term annuities are calculated with their own formula. To get that formula, we’re going to generalize on the work we did in Example 1. That is, we’ll use letters rather than numbers.
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The Annuity Formula
For an annuity with payment pymt, a periodic rate i, and a term of n = 3 payments, the first payment receives interest for n – 1 = 2 periods, just as Patty’s first payment did in Example 1. It is made at the end of the first period, so it received no interest for that one period. Its future value is:
The second payment receives interest for 1 period (1 less period than the first payment):
in Example 1
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The Annuity Formula
The third payment receives interest for 0 periods (1 less period than the second payment):
The future value of the three payments is:
To get the annuity formula, we’re going to multiply each side by (1 + i)
multiplying by (1 + i)
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The Annuity Formula
Now we’re going to subtract:
subtracting
since �1 + i – 1 = i
canceling
factoring
dividing by i
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The Annuity Formula
This is the formula for the future value of an annuity with �n = 3 payments.
The formula for an annuity with n payments is:
replacing 3 with n
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Two Types of Annuities
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Two Types of Annuities
There are two types of annuities. The type we’ve been talking about so far has payments due at the end of each period.
For example, Patty’s annuity in Example 1 had payments due at the end of September, the end of October, and the end of November. This type of annuity is called an ordinary annuity.
The other type of annuity has payments due at the beginning of each period; this type of annuity is called an annuity due.
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Two Types of Annuities
If in Example 1 Patty had an annuity payment due at the beginning of the month, the payments would be due at the beginning of September, the beginning of October, and the beginning of November. Each payment is due one month earlier, so each payment receives interest for one more month. This means that the future value of an annuity due is the future value of the ordinary annuity, plus one more period’s interest.
to get one period’s
interest, multiply by i
factoring
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Two Types of Annuities
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Tax-Deferred Annuities
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Tax-Deferred Annuities
A tax-deferred annuity (TDA) is an annuity that is set up to save for retirement. Money is automatically deducted from the participant’s paychecks until retirement, and the federal (and perhaps state) tax deduction is computed after the annuity payment has been deducted, resulting in significant tax savings.
In some cases, the employer also makes a regular contribution to the annuity.�
The next Example involves a long-term annuity. Usually, the interest rate of a long-term annuity varies somewhat from year to year. In this case, calculations must be viewed as predictions, not guarantees.
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Example 2 – Using an Annuity to Save for Retirement
Tom and Betty decided that they should start saving for retirement, so they set up a tax-deferred annuity. They arranged to have $200 taken out of each of Tom’s monthly checks, which will earn interest.��Because of the tax-deferring effect of the TDA, Tom’s �take-home pay went down by only $115. Tom just had his thirtieth birthday, and his ordinary annuity will come to term when he is 65.
a. Find the future value of the annuity.
b. Find Tom’s contribution and the interest portion.
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Example 2(a) – Solution
This is an ordinary annuity with pymt = 200,�
i = , and�
n = 35 years
= 35 years ● 12 months/year
= 420 monthly payments.
≈ $552,539.96
the Ordinary Annuity Formula
substituting
rounding
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Example 2(b) – Solution
The principal part of this $552,539.96 is Tom’s contribution, and the rest is interest.
• Tom’s contribution is 420 payments of $200 each
� = 420 ● $200
= $84,000.�
• The interest portion is then �
$552,539.96 – $84,000 = $468,539.96.
cont’d
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Sinking Funds
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Sinking Funds
A sinking fund is an annuity in which the future value is a specific amount of money that will be used for a certain purpose, such as a child’s education or the down payment on a home.
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Example 3 – Using an Annuity to Save a Specific Amount
Tom and Betty have a new baby. They agreed that they would need $30,000 in eighteen years for the baby’s college education. They decided to set up a sinking fund and have money deducted from each of Betty’s biweekly paychecks. That money will earn interest in Betty’s ordinary annuity. Find their biweekly payment.
Solution:
This is an ordinary annuity, with i = = 0.0925 / 26, �and n = 18 years ● 26 periods/year = 468 periods, and�FV = $30,000.
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Example 3 – Solution
FV (ord) = pymt
$30,000 = pymt
To find pymt, we must divide 30,000 by the fraction on the right side of the equation.
Because the fraction is so complicated, it is best to first calculate the fraction and then multiply its reciprocal by 30,000.
the Ordinary Annuity Formula
substituting
cont’d
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Example 3 – Solution
This gives pymt = 24.995038…. Betty would need to have only $25 taken out of each of her biweekly paychecks to save $30,000 in eighteen years.
Notice that she will not have exactly $30,000 saved, because she cannot have exactly $24.995048… deducted from each paycheck.
cont’d
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Present Value of an Annuity
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Present Value of an Annuity
The present value of an annuity is the lump sum that can be deposited at the beginning of the annuity’s term, at the same interest rate and with the same compounding period, that would yield the same amount as the annuity.��This value can help the saver to understand his or her options; it refers to an alternative way of saving the same amount of money in the same time.��It is called the present value because it refers to the single action that the saver can take in the present (i.e., at the beginning of the annuity’s term) that would have the same effect as would the annuity.
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Example 4 – Finding the Present Value
Find the present value of Tom and Betty’s annuity.
Solution:
FV = P(1 + i)n�
30,005.95588 =
P =
= 5,693.6451 . . .
the Compound �Interest Formula
substituting
solving for P
rounding
≈ $5693.65
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Example 4 – Solution
This means that Tom and Betty would have to deposit $5,693.55 as a lump sum to save as much money as the annuity would yield.
They chose an annuity over a lump sum deposit because they could not afford to tie up $5,700 for eighteen years, but they could afford to deduct $25 out of each paycheck.
cont’d
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Present Value of an Annuity
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