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Summary
8.1 Perpetuities
A perpetuity is an investment that is intended to provide an expected return indefinitely, either remaining
constant or growing by an incremental amount. Preferred stock is a common example with a preestablished
dividend formula. An indefinite stream of payments cannot be compounded into a future value, but it can be
discounted to a present value, providing an opportunity to determine the amount an investor should be
willing to pay for a share of that stock.
8.2 Annuities
An annuity is a stream of fixed periodic payments that is expected to be paid or received. Calculations of future
value or present value are commonly performed on these payment streams for a wide number of reasons in
business and personal financial analysis, as seen in the chapter focusing on single amounts, particularly in
loan repayment. Annuities may be ordinary annuities, in which the first cash flow of a series occurs at the end
of the first period, or annuities due, if the first cash flow occurs at the beginning point of the first period.
8.3 Loan Amortization
Loans are contracts between a lender and a borrower. Failure to observe the rules of that contract, such as
payment of interest or repayment of the amount owed, can subject the borrower to substantial penalties as
well as damage to their credit. Loan agreements bearing a fixed rate of interest have a scheduled
amortization, or rate and time of repayments with interest. Several types of business and personal loans were
described.
8.4 Stated versus Effective Rates
For a borrower to understand the true cost of financing, they must be familiar with interannual compounding,
which can cause a stated interest rate that appears to be annual to actually be higher. The effective rate of
interest was demonstrated to understand that true cost.
8.5 Equal Payments with a Financial Calculator and Excel
The use of two tools for managing and understanding the time value of money and its many applications was
discussed: a professional financial calculator and the popular Microsoft Office Suite spreadsheet application
Excel.
Key Terms
annuity a stream of regular, periodic payments to be received or paid
annuity due a stream of periodic payments in which the payment or receipt occurs at the beginning of each
period
constant perpetuity a stream of periodic payments that is expected to continue indefinitely with no change
in the amount paid or received
discount rate an interest rate used in time value of money calculations to determine present value; may
derive from several sources, such as stated contract rates, costs to borrow, or expected rates of return on
investments
effective interest rate the interest rate that results when compounding occurs multiple times within a year;
the true cost of borrowing
growing perpetuity a stream of periodic payments that is expected to continue indefinitely with growth of
the amount paid or received in the future, usually by a fixed percentage
loan amortization the scheduling of periodic repayment of a debt, typically involving regular payments or
receipts of amounts that include both interest payment and repayment of the principal of the amount owed
lump sum a single cash payment made in lieu of a series of future payments, such as a lottery payout or a
8 • Summary 257
legal settlement
ordinary annuity a stream of periodic payments in which the payment or receipt occurs at the end of each
period
perpetuity a stream of periodic payments that is expected to continue indefinitely
preferred stock shares of ownership in a corporation that typically entitle the holder to a fixed dividend per
share, if declared by the corporation, with priority over holders of that corporation’s common stock
required rate of return the minimum amount of return that an investor will accept on an investment given
the level of risk involved
retirement planning the process of determining one’s objectives for retirement, including one’s finances,
and developing strategies and tactics to achieve them
structured settlements monetary legal settlements that are paid out in installments, such as an annuity,
rather than a lump sum cash amount
CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session
(https://openstax.org/r/CFA_Level_I_Study_Session2). Reference with permission of CFA Institute.
Multiple Choice
1. The best example of a constant perpetuity would most likely be ________.
a. an annuity due
b. dividends from common stock
c. preferred stock
d. an ordinary annuity
2. You wish to endow a university chair of accounting for a salary of $100,000 per year to the recipient. The
university will withdraw $100,000 each year for the recipient’s salary. The amount of your gift will remain
untouched indefinitely, in perpetuity. The university can lock in a fixed rate for your investment of 2.8% per
year. In order to achieve this, what is the approximate amount of the gift you would have to make now?
a. $3,103,569
b. $3,571,429
c. $4,101,218
d. $4,227,827
3. Preferred stock in Blue Agate Inc. is issued for dividends of $3.00 per share. The dividends will increase
each year at 0.178%, a growing perpetuity. The required rate of return on a stock such as this is 2.5%. At
what approximate price will this preferred stock most likely sell today?
a. $117.21
b. $119.87
c. $120.00
d. $129.20
4. Julio’s attorney negotiates a structured settlement after an injury, consisting of seven equal payments to
Barry of $150,000 each. The first payment is due today, and the remaining payments will be received in
annual amounts, with the second payment occurring one year from now. What is the approximate value of
this settlement in today’s dollars if Barry uses a discount rate of 5%?
a. $911,352
b. $867,960
c. $746,235
258 8 • CFA Institute
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d. $1,050,000
5. What is the approximate present value of an ordinary annuity (beginning one year from now) of a stream
of 12 annual payments of $87,000 if you use a discount rate of 6%?
a. $773,154.04
b. $747,278.92
c. $729,394.95
d. $718,974.58
6. If Maria invests $2,700 at the end of each six-month period for six years at an annual rate of 4%, what is
the approximate future value of her ordinary annuity? Review Chapter 7 for the techniques of interannual
compounding.
a. $17,909.10
b. $20,248.23
c. $31,755.54
d. $36,212.67
7. Assume all of the same facts as in exercise 6 above, except that Maria begins immediately and makes each
of her payments at the beginning of each 6-month period instead of the end. What is the approximate
future value of her annuity due at the end of the six years?
a. $17,909.10
b. $36,936.92
c. $32,707.24
d. $22,997.88
8. Rather than spending her $48,000 in casino winnings, Christy places the money in an investment that will
earn her 5% per year, compounded annually. She will withdraw the money in four equal annual
installments beginning one year from today. What must the approximate amount of each annual
withdrawal be for this investment to be fully depleted in four years?
a. $11,136.38
b. $12,892.56
c. $12,243.47
d. $13,536.61
9. Your friend Jamal borrows $5,000 from you, agreeing to pay you back with 8% annual interest, with the
first payment due to you one year from today. You ask that you be fully repaid over the next four years.
However, to lower his annual payment, Jamal asks you to extend the period over five full years instead.
What will be the approximate difference in his total payments to you, including interest and principal, if
the debt is amortized over five years rather than four?
a. Jamal will pay $544 less.
b. Jamal will pay $544 more.
c. Jamal will pay $223 less.
d. Jamal will pay $223 more.
10. Your new El Supremo credit card arrangement indicates that you will owe interest on unpaid balances at a
nominal (stated) rate of 1.2% per month. If the interest rate is compoundedmonthly, what is the
approximate effective annual rate of interest?
a. 15.39%
b. 12.00%
c. 14.02%
8 • Multiple Choice 259
d. 14.40%
Problems
Use four decimal places on time value of money factors unless otherwise specified. Approximations and minor
differences because of rounding are acceptable. Ignore the effect of taxes. Assume that all percentages are
annual rates and that compounding occurs annually unless indicated otherwise.
1. Steve purchases preferred stock in Berklee Corporation, with each share paying a $2.50 dividend. This
dividend will remain constant. If the public’s required rate of return for Berklee stock is 8%, at what price
should this company’s stock sell?
2. Donna enters into an investment contract that will guarantee her 4% per year if she deposits $3,500 each
year for the next 10 years. She must make the first deposit one year from today, the day she signs the
agreement. How much will she have when she makes her last payment 10 years from now?
3. Assume the same facts as in problem 2 above, except that Donna negotiates the chance to make her first
payment now and continue to pay at the beginning of each year for the 10-year period. How much will she
have accumulated?
4. Bill will receive a royalty payment of $18,000 per year for the next 25 years, beginning one year from now,
as a result of a book he has written. If a discount rate of 10 percent is applied, should he be willing to sell
out his future rights now for $160,000? How about $162,500? $165,000?
5. Debbie won the $60 million lottery. She is to receive $1 million a year for the next 50 years beginning one
year from now, plus an additional lump sum payment of $10 million after 50 years. The discount rate is 10
percent. How much cash would she need to be offered today to tempt her to take a lump-sum cash offer
instead, all things equal?
6. Kim started a paper route on January 1, 2016. Every three months, she deposited $300 in her new bank
account, which earned 4 percent annually but was compounded quarterly. On December 31, 2019, she
placed the entire balance in her bank account in an investment that earned 5 percent annually. How much
will she have on December 31, 2022?
7. You hire Thomas to work for you for five years, and you agree to put away enough money as a lump sum
now to fund an annuity for him. At the end of those five years, he will retire and may begin drawing out
$20,000 per year for five years, starting on the last day of each year (in this case, the end of year 6, from
when this arrangement began, through year 10). How much must you invest today if your guaranteed
interest rate is 3% compounded annually for all 10 years?
8. Your new boss doesn’t have a pension or 401(k) plan for your retirement, but she agrees to place aside
$12,000 every year once a year for four years. She gives you the option of either starting immediately on
your first day of work or starting one year from now. That makes this the difference between an ordinary
annuity and an annuity due. If the plan earns 5% per year, compounded annually, what will be the
difference between the two approaches after the four years / four payments?
9. Jada is borrowing $40,000 from you today. She agrees to pay you back in annual installments beginning a
year from now over eight years, with interest at 3%. What would her annual payment amount be, including
both interest and principal?
10. You agree to finance your new SUV with an auto loan of $38,000. This loan will be repaid over three years
with monthly payments (and compounding) at a 4% annual interest rate (0.33% per month). What will
your monthly loan payment be?
260 8 • Problems
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Video Activity
Future Value of Ordinary Annuities
Click to view content (https://openstax.org/r/Future_Value_of_an_Annuity)
1. What is the primary difference between this demonstration and our chapter examples, keeping the
chapter "Time Value of Money I" in mind?
2. Explain the significance of Dr. van Biezen’s comment at 4:32 regarding a difference when payments are
made at the beginning of each pay period rather than the end.
Practical Example of Annuities
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3. What is the primary difference between a fixed annuity and a variable annuity?
4. Annuities are often recommended to retirees and seniors. Why would a fixed annuity be more attractive to
such a person than a variable annuity?
8 • Video Activity 261
https://openstax.org/r/Future_Value_of_an_Annuity
https://openstax.org/r/What_is_an_annuity?
	Chapter 8 Time Value of Money II: Equal Multiple Payments
	Summary
	Key Terms
	CFA Institute
	Multiple Choice
	Problems
	Video Activity