Math 325
Ch 3 (3.1 – 3.7) – Basic Annuities
NOTE 1: This set covers sections 3.1 - 3.7 of Chapter 3. We'll cover 3.8 and additional problems next
week.
NOTE 2: Don't worry about equations (3.21) and (3.22) on pg 94 of your book!
Annuity: An annuity is a sequence of payments made at equal intervals of time.
Annuity-certain: payments are certain to be made for a fixed period of time (the term).
Annuity-contingent: payments are not certain to be made. e.g. Life annuity.
For now, we will study annuities where the payment period and interest conversion period are equal
and coincide.
Annuity – Immediate: An annuity in which payments are made at the end of each of the n periods (n
is a positive integer). It is also known as an ordinary annuity.
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The present value (PV) of n payments of 1 made at the end of each period is denoted by _____.
The accumulated value (FV) at time n of n payments of 1 made at the end of each period is
denoted by _______.
Equation of value for PV
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Math 325
Ch 3 (3.1 – 3.7) – Basic Annuities
Accumulated value:
Relationship between PV and FV
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Math 325
Ch 3 (3.1 – 3.7) – Basic Annuities
Example 1. Bob has won an award in which he will receive payments of $25 at the end of each month
for 5 years. Assuming a nominal rate of 8% compounded monthly,
a) find the present value of this annuity.
b) how much will Bob have in his account at the end of 5 years if the account is for this annuity only
and he makes no other deposits or withdrawals?
Example 2. How much should be deposited at the end of each quarter so that at the end of 35 years, the
account balance is $1,000,000. Assume i(4) = 6%.
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Math 325
Ch 3 (3.1 – 3.7) – Basic Annuities
Annuity-Due: an annuity for which the payments are made at the beginning of the period.
Equations of value:
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Math 325
Ch 3 (3.1 – 3.7) – Basic Annuities
Example 3. Bob wishes to accumulate $250,000 in a fund for retirement at the end of 20 years. He
plans to make deposits at the end of each year with the final payment to be made one year prior to the
end of the investment period. How large should each deposit be, if the fund earns 9% effective?
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Math 325
Ch 3 (3.1 – 3.7) – Basic Annuities
Example 4. Suppose 6 payments of 1 are made at the end of the 5th through 10th periods, inclusive.
Find the values of this annuity at the
a) end of the 2nd period
b) the end of the 14th period
c) end of the 7th period
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Math 325
Ch 3 (3.1 – 3.7) – Basic Annuities
Perpetuities: an annuity whose payments continue forever.
Examples:
Equations of Value:
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Math 325
Ch 3 (3.1 – 3.7) – Basic Annuities
Example 5. Ted wishes to leave an inheritance to 3 charities. The total inheritance is a series of level
payments at the beginning of each year, continuing in perpetuity. He wants charities A and B to share
the payments equally for 10 years, and then all payments revert to charity C. If the shares received by
all 3 charities have the same value at the time bequest, find the annual effective interest rate i.
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Math 325
Ch 3 (3.1 – 3.7) – Basic Annuities
Example 6. Ted has $500,000 in his retirement account. He plans to withdraw $50,000 at the end of
each year for as long as possible. If i = 6%, find how many regular payments can be made, and find the
amount of the smaller payment
a) to be paid on the date of the last regular payment.
b) to be paid one year after the last regular payment
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Math 325
Ch 3 (3.1 – 3.7) – Basic Annuities
Example 7. A fund of $25,000 is to be accumulated by means of deposits of $1000 made at the end of
every year as long as necessary. If the fund earns an effective rate of interest of 8%, find how many
regular deposits will be necessary and the size of a final deposit to be made one year after the last
regular deposit.
Example 8. At what rate of interest convertible monthly is $12,500 the present value of $170 paid at
the beginning of every month for 9 years?
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