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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Chapter 11 | Other Types of Annuties
The formula for the present value of an ordinary perpetuity is derived using Formula 10.2(b),
1 - ^1 + ih- n
E
PV = PMT ;
i
As the number of payments, n, in an annuity gets larger and larger, the value of (1+i)-n (which is
1
a ratio of
) becomes smaller and smaller. Therefore, as n approaches infinity, as in the
^1 + ih- n
case of a perpetuity, the value of (1+i)-n approaches zero.
PV = PMT ;
1 - ^0 h
PMT
E=
i
i
Therefore, the formula for the present value of an ordinary perpetuity can be expressed as follows:
Formula 11.2(a)
Present Value of an Ordinary Perpetuity
PVPerp =
Similar to annuities, perpetuities can be classified as:
1. Ordinary perpetuity or perpetuity due.
2. Ordinary deferred perpetuity or deferred perpetuity due.
Each of the above can be simple or general.
If the periodic payment in a perpetuity is at the beginning of the payment period then it is a
perpetuity due. The factor (1 + i)-n in the PV formula of an annuity due approaches 0.
PVDue = PMT ;
PMT
1 - ^1 + ih- n
1 - ^0 h
E
E(1 + i) = PMT ;
E (1+ i) = ;
i (1 + i)
i
i
PMT
PVPerp Due = ; i E(1 + i)
PMT
i + PMT
PVPerp Due = PVPerp + PMT
=
Therefore, the present value of a perpetuity due can be expressed as follows:
Formula 11.2(b)
Present Value of a Perpetuity Due
PVPerp Due =
(1 + i) or PVPerp Due =
In the case of an ordinary general perpetuity (or a general perpetuity due), where the compounding
period is not equal to the payment period, we would carry out the same procedure by first finding
the equivalent periodic interest rate, 'i2', and substituting this in place of 'i' in the PVPerp formula.
Chapter 11 | Other Types of Annuties
A few examples of perpetuities are:
■■ Continuous interest payments from a lump sum of money invested at a fixed interest rate.
■■ A scholarship to students paid from an endowment fund on a perpetual basis.
■■ A charity fund established to provide regular payments indefinitely to needy children.
Example 11.2(a)
Calculating the Present Value of an Ordinary Simple Perpetuity
A college wants to provide students with a perpetual scholarship of $10,000 at the end of every three
months. How large should its endowment fund be if the money is growing at 8% compounded quarterly?
Solution
This is an ordinary simple perpetuity as:
■■ Payments are made at the end of each payment period (quarterly)
■■ Compounding period (quarterly) = payment period (quarterly)
Endowment Amount = PV =?
PMT
10, 000
=
= $500,000.00
PVPerp =
i
0.02
The endowment fund required is $500,000.00.
Example 11.2(b)
While solving perpetuity problems using
the financial calculator, we will have to
assume the number of periodic payments,
N, to be a very large number. However,
if N is too large for the computational
power of the financial calculator, an
error message will be displayed. In
the examples and exercises, we will
assume N = 10,000 payments, i.e., enter
N = 10,000 in the calculator.
In cases when an error message is
displayed, decrease the value of N but
keep it as large as possible.
Calculating the Present Value of a General Perpetuity Due
A wealthy donor sponsors an endowment fund that provides a hospital with $5000 at the beginning of
every month in perpetuity. What was the sponsorship amount if the fund is growing at 8% compounded
quarterly?
Solution
This is a general perpetuity due as:
■■ Payments are made at the beginning of each payment period (monthly)
■■ Compounding period (quarterly) ≠ payment period (monthly)
c=
Number of compounding periods per year
4
=
Number of payments per year
12
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Chapter 11 | Other Types of Annuties
Solution
continued
i2 = (1 + i)c - 1 = (1 + 0.02)(4/12) - 1 = 0.006622…
PMT
5000
(1+ i2) =
(1+ 0.006622…)
PVPerp Due =
i2
0.006622...
= $759,977.997240...
Therefore, the sponsorship amount is $759,978.00.
Example 11.2(c)
Calculating the Periodic Payment of an Ordinary General Perpetuity
A large telecommunications business purchases a perpetual bond at $500,000 that earns interest of 5.75%
compounded monthly. Calculate the end-of-quarter payments that will be received from the bond.
Solution
This is an ordinary general perpetuity as:
■■ Payments are made at the end of each payment period (quarterly).
■■ Compounding period (monthly) ≠ payment period (quarterly)
c=
Number of compounding periods per year
12
=
Number of payments per year
4
i2 = (1 + i)c -1 = (1.004791...)(12/4) -1 = 0.014443...­
Rearranging Formula 11.2(a), we obtain,
PMT = PV # i2 = 500,000.00 (0.014443...) = 7221.995113... = $7222.00
Therefore, the end-of-quarter payments will be $7222.00.
Example 11.2(d)
Calculating the Present Value of a Deferred Perpetuity Due
Green Splash Publishers invested a lump sum amount in a fund growing at 6% compounded
monthly. It plans to withdraw $2000 at the beginning of every month from the fund with
the first withdrawal in two years from now. How much should it invest in the fund?
Solution
This is a simple deferred perpetuity due as:
■■ Payments are made at the beginning of each payment period (monthly)
■■ Compounding period (monthly) = payment period (monthly):
Chapter 11 | Other Types of Annuties
Solution
continued
Step 1: Calculating PVPerp Due
PMT
PVPerp Due =
(1 + i) = 2000 (1+ 0.005)
i
0.005
= $402,000.00
Step 2: Calculating PV
PVDef = PVPerp Due (1 + i)-n
= 402,000.00 (1 + 0.005)-24
= 356,648.6389 = $356,648.64
Therefore, it should invest $356,648.64 in the fund.
11.2 |
Exercises Answers to the odd-numbered problems are available at the end of the textbook
1. A college wants to provide students with a perpetual scholarship of $5000 at the end of every three months. How
large should its endowment fund be if money can earn 12% compounded quarterly?
2. Ayes Inc., an e-learning company, wants to create a fund that pays $500 at the end of every month in perpetuity.
If money earns 6% compounded monthly, calculate the initial investment into this fund.
3. How much money is required to fund a perpetuity that pays $1000 at the end of every three months and that earns
6% compounded semi-annually?
4. Henry, a retired businessman, owns a perpetuity that pays him $200 at the end of every month and earns 8%
compounded quarterly. For how much did he purchase this perpetuity?
5. If Webcom Inc. pays dividends of $2.50 per share at the beginning of each quarter, calculate the market price per
share assuming that money earns 5% compounded quarterly.
6. A college was raising funds for a $5000 scholarship that it wanted to provide to its first semester students at the
beginning of every six months. If money is worth 3.5% compounded semi-annually, how much money should the
college raise for this endowment fund?
7. Alba was receiving rental payments of $3000 at the beginning of each month from the tenants of her commercial
property. What would be the value of her property on the market if she wants to sell it, assuming money earns 9%
compounded semi-annually?
8. A condominium was leased for $1200 per month with payments being made at the beginning of each month.
Calculate the market price of the condominium, assuming that money is worth 4% compounded quarterly.
9. What is the size of the scholarship payments that students at a college will receive at the end of every six months
if $700,000 is invested in an endowment fund that is earning 4.5% compounded semi-annually?
10. Roger runs a successful publishing business and invests his 2014 profits of $400,000 in a bank at 2% compounded
monthly. How much money can he withdraw from this account at the end of every month in perpetuity?
11. What is the nominal interest rate compounded quarterly that a fund of $110,540 is earning if it is providing
payments of $600 at the end of every month in perpetuity?
12. An endowment fund is providing scholarships of $7500 at the end of every year in perpetuity. If the investment in
the fund is $200,000, calculate the nominal interest rate compounded annually.
13. Anna purchased a perpetuity that was paying her $12,500 at the end of every six months and had an interest rate of
8% compounded semi-annually. By how much would this perpetuity be cheaper for Anna if it had an interest rate
of 8% compounded quarterly instead of 8% compounded semi-annually?
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Chapter 11 | Other Types of Annuties
14. A college endowment fund provides year-end scholarships of $7500 at 2.85% compounded semi-annually. If the fund
was set up to earn 2.85% compounded monthly instead, how much less money would be required to start the fund?
15. If money can earn 4.55% compounded monthly instead of 4.55% compounded semi-annually, how much less would
be the investment to start a perpetual scholarship that pays $11,500 at the beginning of every six months?
16. If money was invested at 6% compounded annually rather than 6% compounded quarterly, how much more of an
investment would be required to pay $1000 at the beginning of every three months in a perpetual scholarship?
17. Calculate the purchase price of a perpetuity that provides month-end payments of $1000, with the first payment
being received in three years. The interest on the perpetuity is 2.25% compounded quarterly.
18. Yara purchased a perpetuity that paid her $3000 at the end of every quarter, with the first payment received in five
years. If the perpetuity had an interest rate of 4% compounded quarterly, calculate its purchase price.
19. A technology company deposited $62,580 into an account for yearly scholarship amounts that will be awarded to
business students in Canada. The scholarship will be awarded at the beginning of every year forever with the first
scholarship being awarded four years after the amount gets deposited. How much scholarship can the company
provide each year if the interest rate is 6.6% compounded semi-annually?
20. The alumni of the Department of Engineering at a university collected $28,200 in an account for yearly scholarships
that will be awarded to first-year graduate students. The scholarship will be awarded at the beginning of every year
forever with the first scholarship to be awarded five years after the amount is deposited. How much scholarship can
the alumni provide each year if the interest rate is 7.8% compounded quarterly?
21. Toby invested $200,000 in a fund that will provide scholarship amounts in perpetuity. The fund earns 6%
compounded monthly. What is the monthly scholarship amount if:
a. the first payment starts at the end of the first month?
b. the first payment starts at the beginning of the first month?
c. the first payment starts five years from now?
d. the first payment starts at the end of the first month and the interest rate is 6% compounded semi-annually
instead of 6% compounded monthly?
22. A company invested $880,450 in an account that was providing monthly payments in perpetuity to the community
hospital. The account earns 4.75% compounded monthly. What is the monthly payment amount if:
a. the first payment starts at the end of the first month?
b. the first payment starts at the beginning of the first month?
c. the first payment starts three years from now?
d. the first payment starts at the end of the first month and the interest rate is 4.75% compounded quarterly
instead of 4.75% compounded monthly?
11.3 |
Constant-Growth Annuities
A constant-growth annuity is an annuity where the periodic payment amount (PMT) increases
by a constant rate (g%) over the preceding payment amount. A few examples of constant-growth
annuities may include retirement annuity contracts, insurance policies, leases, installment
purchases, and court-awarded payments.
Consider an example where you plan to deposit regular payments into your RRSP at the end of
every month for 3 years, starting with an initial investment of $100, and increasing the amount by
2% every month.