Chapter 11 | Other Types of Annuties
11.1 |
Deferred Annuities
A deferred annuity is an annuity in which the first periodic payment is made after a certain
interval of time, known as the deferral period. The deferral period is the time interval from
'now' to the beginning of the annuity period.
A deferred annuity
is an annuity in which the
first periodic payment
is made after a certain
interval of time.
Ordinary Deferred Annuity
If the deferral period ends one payment interval before the first periodic payment, then it is an
ordinary deferred annuity.
Deferred Annuity Due
If the deferral period ends at the beginning of the first periodic payment, then it is a deferred
annuity due.
Deferred Annuity
Ordinary Deferred Annuity
Deferred Annuity Due
Deferral period ends one payment interval before
the first periodic payment.
Deferral period ends at the first periodic payment.
The ordinary deferred annuity or the deferred annuity due can be simple or general based on the
payment period and the compounding period.
A deferred annuity due can be modified to become an ordinary deferred annuity by
shortening the deferral period by one payment period. By doing this, the payments made at the
beginning of each payment period can be accommodated by making them into payments made
at the end of each payment period, as shown below:
Deferred Annuity Due
Ordinary Deferred Annuity
In the solved examples and exercise problems in this section, the type of annuity (ordinary
deferred annuity or deferred annuity due) is identified based on the periodic payment date (end
of the payment interval or beginning of the payment interval).
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Example 11.1(a)
Identifying the Deferral Period, Annuity Period, and the Number of Payments (n) in an
Annuity
Identify the deferral period, annuity period, and total number of payments for the following:
(i) Ordinary deferred annuity: Payments of $1000 at the end of each year for ten years with
the first payment made three years from now.
(ii) Deferred annuity due: Payments of $1000 at the beginning of each year for ten years with
the first payment made three years from now.
(iii)Ordinary deferred annuity: Payments of $5000 at the end of every 6 months for 15 years
with the first payment made 5 years from now.
(iv) Deferred annuity due: Payments of $100,000 at the beginning of every month for eight
years with the first payment made two years from now.
Solution
(i)
■■ Payments are made at the end of each
payment period (annually).
■■Payments start 3 years from now.
When considered as an ordinary deferred
annuity, the ordinary annuity term starts 2 years
from now; i.e., one payment interval before the
first periodic payment. Therefore, the deferral
period is 2 years.
Annuity period: 10 years
Total number of payments, n = 10
(ii)
■■Payments are made at the beginning of
each payment period (annually).
■■Payments start three years from now.
When considered as a deferred annuity due, the
annuity due term starts three years from now.
Therefore, the deferral period is 3 years.
Annuity period: 10 years
Total number of payments, n = 10
(iii)
■■Payments are made at the end of each
payment period (semi-annually).
■■Payments start 5 years from now.
When considered as an ordinary deferred
annuity, the ordinary annuity term starts
4 years and 6 months from now; i.e., one
payment interval before the first periodic
payment. Therefore, the deferral period is 4
years and 6 months.
Annuity period: 15 years
Total number of payments, n = 30
Chapter 11 | Other Types of Annuties
Solution
continued
(iv)
■■ Payments are made at the beginning of
each payment period (monthly).
■■Payments start two years from now.
When considered as a deferred annuity due,
the annuity due term starts two years from
now. Therefore, the deferral period is two
years.
Annuity period: 8 years
Total number of payments, n = 96
Note: Unless otherwise specified, problems involving deferred annuities are solved based on the
following criteria:
■■ If the periodic payments are made at the end of each payment period, it is considered as an
ordinary deferred annuity.
■■ If the periodic payments are made at the beginning of each payment period, it is considered as a
deferred annuity due.
Calculating Future Value and Present Value of a
Deferred Annuity
The future value of a deferred annuity (FVDef) is the accumulated value of the stream of
payments at the end of the annuity period. This is the same procedure as calculating the future value
of any annuity that you have learned in Chapter 10.
The present value of a deferred annuity (PVDef) is the discounted value of the stream of
payments at the beginning of the deferral period. This follows a two-step procedure and the
following examples will illustrate these calculations.
Example 11.1(b)
Calculating the Present Value of an Ordinary Simple Deferred Annuity
What amount should you invest now if you want to receive payments of $1000 at the end of
each year for ten years with the receipt of the first payment three years from now? Assume
that money earns 5% compounded annually.
Solution
■■ Payments are made at the end of each payment period (annually).
■■Compounding period (annually) = Payment period (annually)
■■Payments start three years from now.
When calculated as an ordinary simple deferred annuity, the ordinary annuity term starts two years
from now; i.e., one payment interval before the first periodic payment. Therefore, the deferral period
is two years.
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Solution
continued
The deferral period
is the region of
compound interest.
As you can see in the time line diagram, there is a deferral period of two years followed
by an annuity period for ten years. We are required to find the present value of the
deferred annuity (PVDef).
Step 1: Calculating the present value of the ordinary simple annuity (PVAnnuity )
Using Formula 10.2(b), PVAnnuity = PMT ;
= 1000 ;
1 - ^1 + ih- n
E
i
1 - ^1 + 0.05h- 10
E = $7721.734929…
0.05
Step 2: Calculating the present value of this amount at the beginning of the deferral
period (PVDef )
Using Formula 9.1(b),
PV = FV(1 + i)-n
PVDef = PVAnnuity (1 + i)-n
= 7721.734929...(1 + 0.05)-2 = $7003.841206...
Therefore, you should invest $7003.84 in the fund now.
Example 11.1(c)
Calculating the Present Value of a General Deferred Annuity Due
Calculate the amount of money an investment banker would have to deposit in an investment fund
that will provide him $1000 at the beginning of each month for 11 years. He receives his first
payment 2 years from now and the interest rate is 6% compounded semi-annually.
Solution
■■ Payments are made at the beginning of each payment period (monthly).
■■ Compounding period (semi-annually) ≠ Payment period (monthly).
■■ Payments start 2 years from now.
When calculated as a general deferred annuity due, the annuity due term starts 2 years from now.
Therefore, the deferral period is 2 years.
Chapter 11 | Other Types of Annuties
Solution
continued
We are required to calculate the present value of the deferral period (PVDef).
Step 1: Calculating the present value of the annuity (PVDue )
2
1
Number of compounding periods per year
=
=
12
6
Number of payments per year
c=
i2 = (1 + i)c - 1 = (1 + 0.03)(1/6) - 1 = 0.004938...
PVDue = PMT ;
1 - ^1 + i2h- n
E (1+i2)
i2
= 1000 ;
1 - ^1 + 0.004938...h- 132
E (1 + 0.004938...)
0.004938...
= $97,288.006312...
Step 2: Calculating the present value of this amount at the beginning
of the deferral period (PVDef)
PVDef = PVDue (1 + i)-n
= 97,288.006312… (1 + 0.03)-4 = $86,439.13353...
Or
PVDef = PVDue (1 + i2)-n
= 97,288.006312… (1 + 0.004938...)-24 = $86,439.13353...
Therefore, the investment banker would have to deposit $86,439.13 in the
investment fund.
Example 11.1(d)
Calculating the Periodic Payments of an Ordinary General Deferred Annuity
The owner of a business borrowed $7500 to purchase a new machine for his factory. The interest rate
charged on the loan is 4% compounded semi-annually and he is required to settle the loan by making equal
monthly payments, at the end of each month, for five years, with the first payment to be made 1 year and 1
month from now. Calculate the size of the monthly payments that are required to settle the loan.
Solution
■■ Payments are made at the end of each payment period (monthly).
■■ Compounding period (semi-annually) ≠ Payment period (monthly).
■■ Payments start 1 year and 1 month from now.
When calculated as an ordinary general deferred annuity, the ordinary annuity term starts one
year from now; i.e., one payment interval before the first periodic payment. Therefore, the
deferral period is one year.
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Solution
continued
Step 1: Calculating the future value of the investment at the end of
the deferral period (FVDef )
FVDef = PVDef (1 + i)n = 7500(1 + 0.02)2 = $7803.00
This amount becomes the present value for the ordinary general deferred
annuity (PVAnn).
Step 2: Calculating PMT of the general deferred annuity
2
1
Number of compounding periods per year
c=
=
=
12
6
Number of payments per year
c
(1/6)
i2 = (1 + i) - 1 = (1 + 0.02)
- 1 = 0.003305…
We use the PV formula for ordinary general annuity
to solve for PMT:
PV = PMT ;
1 - ^1 + i2h- n
E
i2
1 - ^1 + 0.003305...h- 60
E
7803.00 = PMT ;
0.003305...
PMT = $143.588188...
Therefore, the size of the monthly payments required to settle the loan are $143.60.
Example 11.1(e)
Calculating the Number of Payments and Term of a General Deferred Annuity Due
Neelima Glassware Corporation invested its annual net profits of $500,000 in a fixed deposit at
8% compounded quarterly. It wants to withdraw $90,000 at the beginning of every year, with the
first withdrawal made three years from now. Calculate the time period of the annuity. Round your
answer up to the nearest payment period.
Solution
■■ Withdrawals are made at the beginning of each payment period (annually).
■■ Compounding period (quarterly) ≠ Payment period (annually).
■■ Withdrawals start three years from now.
When calculated as a general deferred annuity due, the annuity due term starts three years from
now. Therefore, the deferral period is three years.
Chapter 11 | Other Types of Annuties
Solution
continued
Step 1: C
alculating the future value of the investment at the end of the deferral
period (FVDef )
FVDef = PVDef (1 + i)n
= 500,000 (1 + 0.02)12 = $634,120.897281…
This amount becomes the present value for the general annuity due (PVDue).
Step 2: Calculating ‘n’ for for the general annuity due
c =
4
Number of compounding periods per year
=
1
Number of payments per year
i2 = (1 + i)c - 1 = (1 + 0.02)4 - 1 = 0.082432...
We then use the PVDue formula for this general annuity due to solve for n:
n = -
i # PVdue
m
PMT(1 + i)
In^1 + ih
In c1 -
0.082432... # 634, 120.897281...
m
90, 000^1 + 0.082432...h
== 9.709541...
In^1 + 0.082432...h
= 10 payments (Rounded up to the nearest payment period)
n
t =
= 10
number of payments in a year
1
= 10 years
Therefore, the time period of the annuity is 10 years.
In c1 -
11.1 |
Exercises Answers to the odd-numbered problems are available at the end of the textbook
Identify the type of annuity (based on the periodic payment date), deferral period, annuity period, and number of payments
in Problems 1 and 2:
1. Investments:
a. $500 is deposited into a savings account at the end of each month for three years and the first deposit is
made five months from now.
b. $500 is deposited into a savings account at the beginning of every six months for 5 years and 6 months
and the first deposit is made 2 years from now.
2. Payments:
a. A certain amount is deposited into an RRSP that pays $2000 at the end of every month for ten years. The
first payment is received seven years from now.
b. A certain amount is deposited into a GIC today and payments of $10,000 are withdrawn at the beginning
of every quarter for five years. The first payment is received two years from now.
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3. How much would a business have to invest in a high-growth fund to receive $10,000 at the end of every quarter for five years?
The first payment is received two years from now and the investment has an interest rate of 12% compounded quarterly.
4. What amount would a company have to borrow from a bank to be able to repay $2800 at the end of every month
for ten years, with receipt of the first payment two years from now, if the bank charges an interest rate of 6%
compounded monthly?
5. Keira is planning to retire in seven years and would like to receive $3000 from her RRSP at the end of every month
for ten years during her retirement. She would like to receive her first periodic payment on the day she retires. How
much would she have to invest in an RRSP that has an interest rate of 8% compounded quarterly?
6. Tasty Pastries Inc. wants to purchase a deferred annuity that would provide the company annuity payments of
$20,000 at the end of every six months for seven years. Calculate the purchase price of the deferred annuity if the
deferral period is two years and the interest rate is 6% compounded monthly.
7. Calculate the amount of money an investment banker would have to deposit in an investment fund that would
provide him $1000 at the beginning of every month for 10 years if his payments were deferred by 3 years and 6
months and the interest rate is 6% compounded monthly.
8. A restaurant owner wants to invest in GICs that have an interest rate of 3.25% compounded semi-annually.
Calculate the amount he should invest in order to receive annuity payments of $2000 at the beginning of every 6
months for a period of 5 years. He wants his annuity to begin 2 years and 6 months from now.
9. On the day Cecilia was born, her grandmother made an investment in a fund that was growing at 6.75% compounded
quarterly. How much was invested in the fund to enable annual withdrawals of $10,000 for five years starting from
Cecilia's 18th birthday?
10. Jemi purchased a deferred annuity with his 2010 earnings. If it pays him $1200 at the beginning of every month
for five years and the payment period is deferred by one year, calculate the purchase price of the deferred annuity.
Assume an interest rate of 8% compounded semi-annually.
11. Yuan invested $10,000 in a fund earning 8% compounded monthly. He withdraws $800 from the fund at the end of every
quarter with the first withdrawal being made three years from now. How long will it take for the fund to be depleted?
12. How long will it take Ardiana to settle a $280,000 business loan if she makes equal month-end payments of $1500,
making her first payment six months from now? The interest rate on the mortgage is 3.45% compounded semiannually.
13. Russ Inc. invested its annual net profits of $25,000 into a fund at 8% compounded quarterly. The company wants
to withdraw $2500 at the beginning of every six months, with the first withdrawal being made two years from now.
For how long can withdrawals be made?
14. Samantha deposited her sales commission of $15,500 in an investment that was growing at 7% compounded
monthly. If she wanted to withdraw $2500 at the beginning of every quarter, with the first withdrawal being made
four years from now, how long will she be able to make withdrawals?
15. Jehona took an $8000 loan to purchase equipment for her hair salon. If the interest rate charged on the loan is 11%
compounded quarterly, what month-end payments of equal amounts will settle the loan in 5 years if she made her
first payment 1 year and 4 months from now?
16. A small business invested its profits of $20,850 in an annuity that pays equal amounts at the end of every quarter
for five years, receiving the first payment two years from now. Calculate the size of the payments if the annuity
has an interest rate of 8% compounded monthly.
Chapter 11 | Other Types of Annuties
17. A software company took a loan of $85,000 from a bank and was required to pay equal payment amounts at the
beginning of every month for ten years, with the first payment being made one year from now. The interest rate
charged is 8.5% compounded semi-annually.
a. What will be the size of the monthly payments?
b. What will be the total interest charged?
18. Amanda obtained a student loan of $55,000 for her two-year MBA program. The loan agreement states that she
would have to make equal payments at the beginning of each month to settle the loan over ten years after she
graduates two years from now. The interest rate on the loan is 6% compounded quarterly.
a. What will be the size of the monthly payments?
b. What will be the total interest charged?
19. Leigha invested $35,000 in a retirement fund and withdrew equal amounts at the end of each month for 20 years. She made
her first withdrawal 5 years after she made the initial investment. Calculate the size of the withdrawals if the fund was
earning 9.5% compounded quarterly during the deferral period and 8% compounded quarterly during the annuity period.
20. Five years ago, a bank offered an interest rate of 4% compounded semi-annually on an investment of $20,000.
Now, a month before the first withdrawal will be made, the rate will be changing to 4% compounded quarterly.
Calculate the size of the equal withdrawals at the end of each month that would ensure that the investment lasts
for ten years.
21. A company invested $380,000 in a fixed deposit at 5.75% compounded quarterly. After a deferral period, it wants
to withdraw $11,100.21 at the beginning of the month for four years. How long is the deferral period? Express
your answer in years rounded to two decimal places.
22. Beyonce received a student loan of $56,000 at 6.55% compounded semi-annually. She was required to settle the
loan by making payments of $1409.07 at the beginning of every month for a period of five years from the date of
graduation. How long is the deferral period? Express your answer in years rounded to two decimal places.
23. Ada is planning to retire in 15 years and would like to receive $2500 from her RRSP at the end of every month
for 20 years during her retirement. At the end of this 20-year period she would like to have $20,000 in the RRSP
after receiving her last payment. If she receives the first periodic payment one month after she retires, how much
would she have to invest in an RRSP that has an interest rate of 4.5% compounded semi-annually?
24. A company purchased a deferred annuity that provided it with annuity payments of $15,000 at the end of every
three months for ten years. At the end of this ten-year period, the account would have a balance of $10,000
after the last payment has been withdrawn. Calculate the purchase price of the deferred annuity if the period of
deferment is four years and the interest rate is 5.75% compounded monthly.
11.2 |
Perpetuities
A perpetuity is an annuity in which the periodic payments begin on a fixed date and continue
indefinitely. Therefore, it is not possible to calculate its future value. However, there is a definite
present value for a perpetuity.
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