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Opener
Page Chapter 1 of 3
Intermediate Accounting
eBook
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Content
Chapter6: Time Value of Money Concepts
Chapter Opener
/ / / OVERVIEW
Time value of money concepts, specifically future value and present value, are essential in a
variety of accounting situations. These concepts and the related computational procedures are
the subjects of this chapter. Present values and future values of single amounts and present
values and future values of annuities (series of equal periodic payments) are described
separately but shown to be interrelated.
/ / / LEARNING OBJECTIVES
After studying this chapter, you should be able to:
LO1
Explain the difference between simple and compound interest. (page 300)
LO2
Compute the future value of a single amount. (page 301)
LO3
Compute the present value of a single amount. (page 302)
LO4
Solve for either the interest rate or the number of compounding periods when present
value and future value of a single amount are known. (page 304)
LO5
Explain the difference between an ordinary annuity and an annuity due situation. (page
309)
LO6
Compute the future value of both an ordinary annuity and an annuity due. (page 310)
LO7
Compute the present value of an ordinary annuity, an annuity due, and a deferred
annuity. (page 312)
LO8
Solve for unknown values in annuity situations involving present value. (page 316)
LO9
Briefly describe how the concept of the time value of money is incorporated into the
valuation of bonds, long-term leases, and pension obligations. (page 320)
FINANCIAL REPORTING CASE
The Winning Ticket
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p. 298
299
Al Castellano had been buying California State lottery tickets for 15 years at his
neighborhood grocery store. On Sunday, June 24, 2010, his world changed. When he
awoke, opened the local newspaper, and compared his lottery ticket numbers with
Saturday night's winning numbers, he couldn't believe his eyes. All of the numbers on his
ticket matched the winning numbers. He went outside for a walk, came back into the
kitchen and checked the numbers again. He woke his wife Carmen, told her what had
happened, and they danced through their apartment. Al, a 66-year-old retired supermarket
clerk, and Carmen, a 62-year-old semiretired secretary, had won the richest lottery in
California's history, $141 million!
On Monday when Al and Carmen claimed their prize, their ecstasy waned slightly when
they were informed that they would soon be receiving a check for approximately $43
million. When the Castellanos purchased the lottery ticket, they indicated that they would
like to receive any lottery winnings in one lump payment rather than in 26 equal annual
installments beginning now. They knew beforehand that the State of California is required
to withhold 31% of lottery winnings for federal income tax purposes, but this reduction was
way more than 31%.
Source: This case is adapted from an actual situation.
QUESTIONS / / /
By the time you finish this chapter, you should be able to respond appropriately
to the questions posed in this case. Compare your response to the solution
provided at the end of the chapter.
1. Why were the Castellanos to receive $43 million rather than the $141 million lottery
prize? (page 312)
2. What interest rate did the State of California use to calculate the $43 million lumpsum payment? (page 318)
3. What are some of the accounting applications that incorporate the time value of
money into valuation? (page 320)
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Chapter Opener
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Part A: Basic Concepts
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Intermediate Accounting
eBook
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Content
Chapter6: Time Value of Money Concepts
Part A: Basic Concepts
Time Value of Money
The key to solving the problem described in the financial reporting case is an understanding of the
concept commonly referred to as the time value of money This concept means that money invested
today will grow to a larger dollar amount in the future. For example, $100 invested in a savings account at
your local bank yielding 6% annually will grow to $106 in one year. The difference between the $100
invested nowthe present value of the investmentand its $106 future value represents the time value
of money.
The time value of money means
that money can be invested
today to earn interest and grow
to a larger dollar amount in the
future.
This concept has nothing to do with the worth or buying power of those dollars. Prices in our economy can change. If the inflation rate were
higher than 6%, then the $106 you would have in the savings account actually would be worth less than the $100 you had a year earlier. The time
value of money concept concerns only the growth in the dollar amounts of money.
Time value of money concepts
are useful in valuing several
assets and liabilities.
The concepts you will learn in this chapter are useful in solving many business decisions such as, for
example, the determination of the lottery award presented in the financial reporting case at the beginning of
this chapter. More important, the concepts also are necessary when valuing assets and liabilities for
financial reporting purposes. As you will see in this and subsequent chapters, most accounting applications
that incorporate the time value of money involve the concept of present value. The valuation of leases,
bonds, pension obligations, and certain notes receivable and payable are a few prominent examples. It is
important that you master the concepts and tools we review here. This knowledge is essential to the
remainder of your accounting education.
Simple versus Compound Interest
Interest is the rent paid for the use of money for some period of time. In dollar terms, it is the
amount of money paid or received in excess of the amount of money borrowed or lent. If you lent the
bank $100 today and received $106 a year from now, your interest earned would be $6. Interest
also can be expressed as a rate at which money will grow. In this case, that rate is 6%. It is this
interest that gives money its time value.
LO1
Interest is the amount of money paid
or received in excess of the amount
borrowed or lent.
Simple interest is computed by multiplying an initial investment times both the applicable interest rate and the period of time for which the
money is used. For example, simple interest earned each year on a $1,000 investment paying 10% is $100 ($1,000 10%).
Compound interest results in increasingly larger interest amounts for each period of the investment.
The reason is that interest is now being earned not only on the initial investment amount but also on the
accumulated interest earned in previous periods.
For example, Cindy Johnson invested $1,000 in a savings account paying 10% interest compounded
annually. How much interest will she earn each year, and what will be her investment balance after three
years?
Compound interest includes
interest not only on the initial
investment but also on the
accumulated interest in previous
periods.
With compound interest at 10% annually, the $1,000 investment would grow to $1,331at the end of the three-year period. Of course, if Cindy
withdrew the interest earned each year, she would earn only $100 in interest each year, that is, the amount of simple interest. If the investment
period had been 20 years, 20 individual calculations would be needed. However, calculators, computer programs, and compound interest tables
make these calculations much easier.
Most banks compound interest more frequently than once a year. Daily compounding is common for
savings accounts. More rapid compounding has the effect of increasing the actual rate, which is called
the effective rate, at which money grows per year. It is important to note that interest is typically stated
Interest rates are typically stated
as annual rates.
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Part A: Basic Concepts
p. 300
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as an annual rate regardless of the length of the compounding period involved. In situations when the
compounding period is less than a year, the interest rate per compounding period is determined by
dividing the annual rate by the number of periods. Assuming an annual rate of 12%:
p. 301
As an example, now let's assume Cindy Johnson invested $1,000 in a savings account paying 10% interest compounded twice a year. There
are two six-month periods paying interest at 5% (the annual rate divided by two periods). How much interest will she earn the first year, and what
will be her investment balance at the end of the year?
The $1,000 would grow by $102.50, the interest earned, to $1,102.50, $2.50 more than if interest were
compounded only once a year. The effective annual interest rate, often referred to as the annual yield, is
10.25% ($102.50 $1,000).
The effective interest rate is the
rate at which money actually will
grow during a full year.
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Valuing a Single Cash Flow Amount
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Intermediate Accounting
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Content
Chapter6: Time Value of Money Concepts
Valuing a Single Cash Flow Amount
Future Value of a Single Amount
In the first Cindy example, in which $1,000 was invested for three years at 10% compounded annually, the $1,331 is referred to
as the future value (FV). A time diagram is a useful way to visualize this relationship, with 0 indicating the date of the initial
investment.
LO2
Future
value of a
single
amount.
The future value after one year can be calculated as $1,000 1.10 (1.00 + .10) = $1,100. After three years, the future value is $1,000 1.10
1.10 1.10 = $1,331. In fact, the future value of any invested amount can be determined as follows:
where: FV=Future value of the invested amount
I=Amount invested at the beginning of the period
i =Interest rate
The future value of a single
amount is the amount of money
that a dollar will grow to at some
point in the future.
n =Number of compounding periods
The future value can be determined by using Table 1, Future Value of $1, located at the end of this textbook. The table contains the future value
of $1 invested for various periods of time, n, and at various rates, i.
With this table, it's easy to determine the future value of any invested amount simply by multiplying it by the table value at the intersection of the
column for the desired rate and the row for the number of compounding periods. Graphic 6-1 contains an excerpt from Table 1.
GRAPHIC 6-1
Future Value of $1 (excerpt from Table 1 located at the end of this textbook)
The table shows various values of (1 + i)n for different combinations of i and n. From the table you can find the future value factor for three
periods at 10% to be 1.331. This means that $1 invested at 10% compounded annually will grow to approximately $1.33 in three years. So, the
future value of $1,000 invested for three years at 10% is $1,331:
The future value function in financial calculators or in computer spreadsheet programs calculates future values in the same way. Determining
future values (and present values) electronically avoids the need for tables such as those in the chapter appendix. It's important to remember that
the n in the future value formula refers to the number of compounding periods, not necessarily the number of years. For example, suppose you
wanted to know the future value two years from today of $1,000 invested at 12% with quarterly compounding. The number of periods is therefore
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Valuing a Single Cash Flow Amount
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eight and the compounding rate is 3% (12% annual rate divided by four, the number of quarters in a year). The future value factor from Table 1 is
1.26677, so the future value is $1,266.77 ($1,000 1.26677).1
Present Value of a Single Amount
The example used to illustrate future value reveals that $1,000 invested today is equivalent to $1,100
received after one year, $1,210 after two years, or $1,331 after three years, assuming 10% interest
compounded annually. Thus, the $1,000 investment (I) is the present value (PV) of the single sum
of $1,331 to be received at the end of three years. It is also the present value of $1,210 to be
received in two years and $1,100 in one year.
LO3
The present value of a single
amount is todays equivalent to a
particular amount in the future.
Remember that the future value of a present amount is the present amount times (1 + i)n. Logically, then, that computation can be reversed to
find the present value of a future amount to be the future amount divided by (1 + i)n. We substitute PV for I (invested amount) in the future value
formula above.
In our example,
Of course, dividing by (1 + i)n is the same as multiplying by its reciprocal, 1/(1 + i)n.
As with future value, these computations are simplified by using calculators, computer programs, or present value tables. Table 2, Present
Value of $1, located at the end of this textbook provides the solutions of 1/(1 + i)n for various interest rates (i) and compounding periods (n).
These amounts represent the present value of $1 to be received at the end of the different periods. The table can be used to find the present
value of any single amount to be received in the future by multiplying that amount by the value in the table that lies at the intersection of the
column for the appropriate rate and the row for the number of compounding periods.2Graphic 6-2 contains an excerpt from Table 2.
GRAPHIC 6-2
Present Value of $1 (excerpt from Table 2 located at the end of this textbook)
Notice that the farther into the future the $1 is to be received, the less valuable it is now. This is the essence of the concept of the time value of
money. Given a choice between $1,000 now and $1,000 three years from now, you would choose to have the money now. If you have it now, you
could put it to use. But the choice between, say, $740 now and $1,000 three years from now would depend on your time value of money. If your
time value of money is 10%, you would choose the $1,000 in three years, because the $740 invested at 10% for three years would grow to only
$984.94 [$740 1.331 (FV of $1, i = 10%, n = 3)]. On the other hand, if your time value of money is 11% or higher, you would prefer the $740
now. Presumably, you would invest the $740 now and have it grow to $1,012.05 ($740 1.36763) in three years.
Using the present value table above, the present value of $1,000 to be received in three years assuming a time value of money of 10% is
$751.31 [$1,000 .75131 (PV of $1, i = 10% and n = 3)]. Because the present value of the future amount, $1,000, is higher than $740 we could
have today, we again determine that with a time value of money of 10%, the $1,000 in three years is preferred to the $740 now.
In our earlier example, $1,000 now is equivalent to $1,331 in three years, assuming the time value of money is 10%. Graphically, the relation
between the present value and the future value can be viewed this way:
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Valuing a Single Cash Flow Amount
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While the calculation of future value of a single sum invested today requires the inclusion of compound
interest, present value problems require the removal of compound interest. The process of computing
present value removes the $331 of interest earned over the three-year period from the future value of
$1,331, just as the process of computing future value adds $331 of interest to the present value of $1,000
to arrive at the future value of $1,331.
The calculation of future value
requires the addition of interest,
while the calculation of present
value requires the removal of
interest.
As we demonstrate later in this chapter and in subsequent chapters, present value calculations are
incorporated into accounting valuation much more frequently than future value.
Accountants use PV calculations
much more frequently than FV.
Solving for Other Values When FV and PV are Known
There are four variables in the process of adjusting single cash flow amounts for the time value of money: the present value (PV),
the future value (FV), the number of compounding periods (n), and the interest rate (i). If you know any three of these, the fourth
can be determined. Illustration 6-1 solves for an unknown interest rate and Illustration 6-2 determines an unknown number of
periods.
LO4
DETERMINING THE UNKNOWN INTEREST RATE
ILLUSTRATION 6-1
Determining i When PV, FV, and n are Known
Suppose a friend asks to borrow $500 today and promises to repay you $605 two years from
now. What is the annual interest rate you would be agreeing to?
The following time diagram illustrates the situation:
The interest rate is the discount rate that will provide a present value of $500 when discounting (determining present value) the $605 to be
received in two years:
Rearranging algebraically, we find that the present value table factor is .82645.
The unknown variable is the
interest rate.
When you consult the present value table, Table 2, you search row two (n = 2) for this value and find it in the 10% column. So the effective
interest rate is 10%. Notice that the computed factor value exactly equals the table factor value.3
DETERMINING THE UNKNOWN NUMBER OF PERIODS
ILLUSTRATION 6-2
Determining n When PV, FV, and i are Known
You want to invest $10,000 today to accumulate $16,000 for graduate school. If you can
invest at an interest rate of 10% compounded annually, how many years will it take to
accumulate the required amount?
The following time diagram illustrates the situation:
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Valuing a Single Cash Flow Amount
The number of years is the value of n that will provide a present value of $10,000 when discounting
$16,000 at a rate of 10%:
Page 4 of 6
The unknown variable is the
number of periods.
Rearranging algebraically, we find that the present value table factor is .625.
When you consult the present value table, Table 2, you search the 10% column (i = 10%) for this value and find .62092 in row five. So it would
take approximately five years to accumulate $16,000 in the situation described.
ADDITIONAL CONSIDERATION
Solving for the unknown factor in either of these examples could just as easily be done
using the future value tables. The number of years is the value of n that will provide a
present value of $10,000 when discounting $16,000 at a discount rate of 10%.
Rearranging algebraically, the future value table factor is 1.6.
When you consult the future value table, Table 1, you search the 10% column (i =
10%) for this value and find 1.61051 in row five. So it would take approximately five
years to accumulate $16,000 in the situation described.
CONCEPT REVIEW EXERCISE
VALUING A SINGLE CASH FLOW AMOUNT
Using the appropriate table, answer each of the following independent questions.
1. What is the future value of $5,000 at the end of six periods at 8% compound
interest?
2. What is the present value of $8,000 to be received eight periods from today
assuming a compound interest rate of 12%?
3. What is the present value of $10,000 to be received two years from today assuming
an annual interest rate of 24% and monthly compounding?
4. If an investment of $2,000 grew to $2,520 in three periods, what is the interest rate
at which the investment grew? Solve using both present and future value tables.
5. Approximately how many years would it take for an investment of $5,250 to
accumulate to $15,000, assuming interest is compounded at 10% annually? Solve
using both present and future value tables.
SOLUTION
1. FV = $5,000 1.58687* = $7,934
*
Future value of $1: n = 6, i = 8% (from Table 1)
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p.p. 306
305
304
303
302
2. FV = $8,000 .40388* = $3,231
*
Present value of $1: n = 8, i = 12% (from Table 2)
3. FV = $10,000 .62172* = $6,217
*
4.
Present value of $1: n = 24, i = 2% (from Table 2)
Using present value table,
*
Present value of $1: n = 3, i = ? (from Table 2, i approximately 8%)
Using future value table,
*
Future value of $1: n = 3, i = ? (from Table 1, i approximately 8%)
5. Using present value table,
*
Present value of $1: n = ?, i = 10% (from Table 2, n approximately 11 years)
Using future value table,
*
Future value of $1: n = ?. i = 10% (from Table 1, n approximately 11 years)
1
When interest is compounded more frequently than once a year, the effective annual interest rate, or yield, can be determined using the
following equation:
with i being the annual interest rate and p the number of compounding periods per year. In this example, the annual yield would be 12.55%,
calculated as follows:
Determining the yield is useful when comparing returns on investment instruments with different compounding period length.
2
The factors in Table 2 are the reciprocals of those in Table 1. For example, the future value factor for 10%, three periods is 1.331, while the
present value factor is .75131. $1 1.331 = $.75131, and $1 .75131 = $1.331.
3
If the calculated factor lies between two table factors, interpolation is useful in finding the unknown value. For example, if the future value in our
example is $600, instead of $605, the calculated PV factor is .83333 ($500 600). This factor lies between the 9% factor of .84168 and the 10%
factor of .82645. The total difference between these factors is .01523 (.84168 .82645). The difference between the calculated factor of .83333
and the 10% factor of .82645 is .00688. This is 45% of the difference between the 9% and 10% factors:
Therefore, the interpolated interest rate is 9.55% (10 .45).
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Preview of Accounting Applications of Present Value TechniquesSingle Cash Amount
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Intermediate Accounting
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Content
Chapter6: Time Value of Money Concepts
Preview of Accounting Applications of Present Value TechniquesSingle Cash Amount
Kile Petersen switched off his television set immediately after watching the Super Bowl game and swore to himself that this would be the last year
he would watch the game on his 10-year-old 20-inch TV set. Next year, a big screen TV, he promised himself. Soon after, he saw an
advertisement in the local newspaper from Slim Jim's TV and Appliance offering a Philips 60-inch large screen television on sale for $1,800. And
the best part of the deal was that Kile could take delivery immediately but would not have to pay the $1,800 for one whole year! In a year, I can
easily save the $1,800, he thought.
In the above scenario, the seller, Slim Jim's TV and Appliance, records a sale when the TV is delivered to Kile. How should the company value
its receivable and corresponding sales revenue? We provide a solution to this question at the end of this section on page 308. The following
discussion will help you to understand that solution.
Many assets and most liabilities are monetary in nature. Monetary assets include money and claims to
receive money, the amount of which is fixed or determinable. Examples include cash and most receivables.
Monetary liabilities are obligations to pay amounts of cash, the amount of which is fixed or determinable. Most
liabilities are monetary. For example, if you borrow money from a bank and sign a note payable, the amount of
cash to be repaid to the bank is fixed. Monetary receivables and payables are valued based on the fixed amount
of cash to be received or paid in the future with proper reflection of the time value of money. In other words, we
value most receivables and payables at the present value of future cash flows, reflecting an appropriate time
Most monetary assets and
monetary liabilities are
valued at the present value
of future cash flows.
value of money.4
The example in Illustration 6-3 demonstrates this concept.
ILLUSTRATION 6-3
Valuing a Note: One Payment, Explicit Interest
Explicit Interest
The Stridewell Wholesale Shoe Company manufactures athletic shoes for sale to retailers.
The company recently sold a large order of shoes to Harmon Sporting Goods for $50,000.
Stridewell agreed to accept a note in payment for the shoes requiring payment of $50,000 in
one year plus interest at 10%.
How should Stridewell value the note receivable and corresponding sales revenue earned? How should Harmon value the note payable and
corresponding inventory purchased? As long as the interest rate explicitly stated in the agreement properly reflects the time value of money, the
answer is $50,000, the face value of the note. It's important to realize that this amount also equals the present value of future cash flows at 10%.
Future cash flows equal $55,000, $50,000 in note principal plus $5,000 in interest ($50,000 10%). Using a time diagram:
In equation form, we can solve for present value as follows:
By calculating the present value of $55,000 to be received in one year, the interest of $5,000 is removed from the future value, resulting in a
proper note receivable/sales revenue value of $50,000 for Stridewell and a $50,000 note payable/inventory value for Harmon.
While most notes, loans, and mortgages explicitly state an interest rate that will properly reflect the time value of money, there can be
exceptions. Consider the example in Illustration 6-4.
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Preview of Accounting Applications of Present Value TechniquesSingle Cash Amount
p. 307
Page 2 of 2
ILLUSTRATION 6-4
Valuing a Note: One Payment, No Explicit Interest
No Explicit Interest
The Stridewell Wholesale Shoe Company recently sold a large order of shoes to Harmon
Sporting Goods. Terms of the sale require Harmon to sign a noninterest-bearing note of $60,500
with payment due in two years.
How should Stridewell and Harmon value the note receivable/payable and corresponding sales revenue/inventory? Even though the agreement
states a noninterest-bearing note, the $60,500 does, in fact, include interest for the two-year period of the loan. We need to remove the interest
portion of the $60,500 to determine the portion that represents the sales price of the shoes. We do this by computing the present value. The
following time diagram illustrates the situation assuming that a rate of 10% reflects the appropriate interest rate for a loan of this type:
p. 308
Again, using the present value of $1 table,
Both the note receivable for Stridewell and the note payable for Harmon initially will be valued at $50,000. The difference of $10,500 ($60,500
50,000) represents interest revenue/expense to be recognized over the life of the note. The appropriate journal entries are illustrated in later
chapters.
Now can you answer the question posed in the scenario at the beginning of this section? Assuming that a rate of 10% reflects the appropriate
interest rate in this situation, Slim Jim's TV and Appliance records a receivable and sales revenue of $1,636 which is the present value of the
$1,800 to be received from Kile Petersen one year from the date of sale.
$1,800 (future value) .90909* = $1,636 (present value)
*
Present value of $1: n = 1, i = 10% (from Table 2)
4
FASB ASC 83530: InterestImputation of Interest (previously Interest on Receivables and Payables, Accounting Principles Board Opinion
No. 21 (New York: AICPA, 1971)). As you will learn in Chapter 7, normal trade accounts receivable and accounts payable are valued at the
amount expected to be received or paid, not the present value of those amounts. The difference between the amount expected to be received or
paid and present values often is immaterial.
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Expected Cash Flow Approach
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Intermediate Accounting
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Content
Chapter6: Time Value of Money Concepts
Expected Cash Flow Approach
Present value measurement has long been integrated with accounting valuation and is specifically addressed in
several accounting standards. Because of its increased importance, the FASB in 2000 issued Statement of
Financial Accounting Concepts No. 7, Using Cash Flow Information and Present Value in Accounting
Measurements.5 This statement provides a framework for using future cash flows as the basis for accounting
measurement and asserts that the objective in valuing an asset or liability using present value is to approximate the
fair value of that asset or liability. Key to that objective is determining the present value of future cash flows
associated with the asset or liability, taking into account any uncertainty concerning the amounts and timing of the
cash flows. Although future cash flows in many instances are contractual and certain, the amounts and timing of
cash flows are less certain in other situations.
For example, lease payments are provided in the contract between lessor and lessee. On
the other hand, the future cash flows to be paid to settle a pending lawsuit may be highly
uncertain. Traditionally, the way uncertainty has been considered in present value
calculations has been by discounting the best estimate of future cash flows applying a
discount rate that has been adjusted to reflect the uncertainty or risk of those cash flows.
With the approach described by SFAC No. 7, though, the adjustment for uncertainty or risk
of cash flows is applied to the cash flows, not the discount rate. This new expected cash flow
approach incorporates specific probabilities of cash flows into the analysis. Consider
Illustration 6-5.
SFAC No. 7
SFAC No. 7 provides a
framework for using
future cash flows in
accounting
measurements.
SFAC NO. 7
While many accountants do not routinely
use the expected cash flow approach,
expected cash flows are inherent in the
techniques used in some accounting
measurements, like pensions, other
postretirement benefits, and some insurance
obligations.6
Compare the approach described in Illustration 6-5 to the traditional approach that uses the
present value of the most likely estimate of $200 million and ignores information about cash
flow probabilities.
The discount rate used to determine present value when applying the expected cash flow
approach should be the company's credit-adjusted risk-free rate of interest. Other elements of
uncertainty are incorporated into the determination of the probability-weighted expected cash
flows. In the traditional approach, elements of uncertainty are incorporated into a riskadjusted discount rate.
The companys creditadjusted risk-free rate
of interest is used when applying the
expected cash flow approach to the
calculation of present value.
The FASB expects that the traditional approach to calculating present value will continue to be used in many situations, particularly those where
future cash flows are contractual. The Board also believes that the expected cash flow approach is more appropriate in more complex situations.
In fact, the board has incorporated the concepts developed in SFAC No. 7 into recent standards on asset retirement obligations, impairment
losses, and business combinations. In Chapter 10 we illustrate the use of the expected cash flow approach as it would be applied to the
measurement of an asset retirement obligation. In Chapter 13, we use the approach to measure the liability associated with a loss contingency.
ILLUSTRATION 6-5
Expected Cash Flow Approach
LDD Corporation faces the likelihood of having to pay an uncertain amount in five years in
connection with an environmental cleanup. The future cash flow estimate is in the range of $100
million to $300 million with the following estimated probabilities:
The expected cash flow, then, is $220 million:
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Expected Cash Flow Approach
p. 309
Page 2 of 2
If the company's credit-adjusted risk-free rate of interest is 5%, LDD will report a liability of
$172,376,600, the present value of the expected cash outflow:
*
Present value of $1, n = 5, i = 5% (from Table 2)
5
Using Cash Flow Information and Present Value in Accounting Measurements, Statement of Financial Accounting Concepts No. 7 (Norwalk,
Conn.: FASB, 2000). Recall that Concept Statements do not directly prescribe GAAP, but instead provide structure and direction to financial
accounting.
6
Ibid., para. 48.
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Part B: Basic Annuities
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Chapter6: Time Value of Money Concepts
Part B: Basic Annuities
The previous examples involved the receipt or payment of a single future amount. Financial instruments frequently involve multiple
receipts or payments of cash. If the same amount is to be received or paid each period, the series of cash flows is referred to as an
annuity. A common annuity encountered in practice is a loan on which periodic interest is paid in equal amounts. For example,
bonds typically pay interest semiannually in an amount determined by multiplying a stated rate by a fixed principal amount. Some
loans and most leases are paid in equal installments during a specified period of time.
An agreement that creates an annuity can produce either an ordinary annuity or an annuity due
(sometimes referred to as an annuity in advance) situation. The first cash flow (receipt or payment) of an
ordinary annuity is made one compounding period after the date on which the agreement begins. The
final cash flow takes place on the last day covered by the agreement. For example, an installment note
payable dated December 31, 2011, might require the debtor to make three equal annual payments, with
the first payment due on December 31, 2012, and the last one on December 31, 2014. The following time
diagram illustrates an ordinary annuity:
LO5
In an ordinary annuity cash flows
occur at the end of each period.
Ordinary annuity.
The first payment of an annuity due is made on the first day of the agreement, and the last payment is
made one period before the end of the agreement. For example, a three-year lease of a building that
begins on December 31, 2011, and ends on December 31, 2014, may require the first year's lease
payment in advance on December 31, 2011. The third and last payment would take place on December
31, 2013, the beginning of the third year of the lease. The following time diagram illustrates this
situation:
In an annuity due cash flows
occur at the beginning of each
period.
p. 310
Annuity due.
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Future Value of an Annuity
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Content
Chapter6: Time Value of Money Concepts
Future Value of an Annuity
Future Value of an Ordinary Annuity
Let's first consider the future value of an ordinary annuity in Illustration 6-6.
ILLUSTRATION 6-6
Future Value of an Ordinary Annuity
Sally Rogers wants to accumulate a sum of money to pay for graduate school. Rather
than investing a single amount today that will grow to a future value, she decides to invest
$10,000 a year over the next three years in a savings account paying 10% interest
compounded annually. She decides to make the first payment to the bank one year from
today.
The following time diagram illustrates this ordinary annuity situation. Time 0 is the start of the first period.
Using the FV of $1 factors from Table 1, we can calculate the future value of this annuity by calculating the future value of each of the individual
payments as follows:
From the time diagram, we can see that the first payment has two compounding periods to
earn interest. The factor used, 1.21, is the FV of $1 invested for two periods at 10%. The
second payment has one compounding period and the last payment does not earn any
interest because it is invested on the last day of the three-year annuity period. Therefore, the
factor used is 1.00.
In the future value of an ordinary annuity,
the last cash payment will not earn any
interest.
This illustration shows that it's possible to calculate the future value of the annuity by separately calculating the FV of each
payment and then adding these amounts together. Fortunately, that's not necessary. Table 3, Future Value of an Ordinary Annuity
of $1, located at the end of this textbook simplifies the computation by summing the individual FV of $1 factors for various factors
of n and i.Graphic 6-3 contains an excerpt from Table 3.
LO6
GRAPHIC 6-3
Future Value of an Ordinary Annuity of $1 (excerpt from Table 3 located at the end of this textbook)
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Future Value of an Annuity
p. 311
Page 2 of 2
The future value of $1 at the end of each of three periods invested at 10% is shown in Table 3 to be $3.31. We can simply multiply this factor by
$10,000 to derive the FV of our ordinary annuity (FVA):
Future Value of an Annuity Due
Let's modify the previous illustration to create an annuity due in Illustration 6-7.
ILLUSTRATION 6-7
Future Value of an Annuity Due
Sally Rogers wants to accumulate a sum of money to pay for graduate school. Rather than
investing a single amount today that will grow to a future value, she decides to invest $10,000 a
year over the next three years in a savings account paying 10% interest compounded annually.
She decides to make the first payment to the bank immediately. How much will Sally have
available in her account at the end of three years?
The following time diagram depicts the situation. Again, note that 0 is the start of the first period.
The future value can be found by separately calculating the FV of each of the three payments and then summing those individual future values:
In the future value of an annuity
due, the last cash payment will
earn interest.
p. 312
And, again, this same future value can be found by using the future value of an annuity due (FVAD) factor from Table 5, Future Value of an
Annuity Due of $1, located at the end of this textbook, as follows:
Of course, if unequal amounts are invested each year, we can't solve the problem by using the annuity tables. The future value of each
payment would have to be calculated separately.
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Present Value of an Annuity
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Content
Chapter6: Time Value of Money Concepts
Present Value of an Annuity
Present Value of an Ordinary Annuity
You will learn in later chapters that liabilities and receivables, with the exception of certain trade receivables and
payables, are reported in financial statements at their present values. Most of these financial instruments specify
equal periodic interest payments or installment payments. As a result, the most common accounting applications
of the time value of money involve determining present value of annuities. As in the future value applications we
discussed above, an annuity can be either an ordinary annuity or an annuity due. Let's look at an ordinary
annuity first.
In Illustration 6-6 on page 310, we determined that Sally Rogers could accumulate $33,100 for graduate
school by investing $10,000 at the end of each of three years at 10%. The $33,100 is the future value of the
ordinary annuity described. Another alternative is to invest one single amount at the beginning of the three-year
period. (See Illustration 6-8.) This single amount will equal the present value at the beginning of the three-year
period of the $33,100 future value. It will also equal the present value of the $10,000 three-year annuity.
LO7
FINANCIAL
Reporting Case
Q1, p.299
ILLUSTRATION 6-8
Present Value of an Ordinary Annuity
Sally Rogers wants to accumulate a sum of money to pay for graduate school. She wants
to invest a single amount today in a savings account earning 10% interest compounded
annually that is equivalent to investing $10,000 at the end of each of the next three years.
The present value can be found by separately calculating the PV of each of the three payments and then summing those individual present
values:
A more efficient method of calculating present value is to use Table 4, Present Value of an Ordinary Annuity of $1, located at the end of this
textbook. Graphic 6-4 contains an excerpt from Table 4.
GRAPHIC 6-4
Present Value of an Ordinary Annuity of $1 (excerpt from Table 4 located at the end of this textbook)
Using Table 4, we calculate the PV of the ordinary annuity (PVA) as follows:
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Present Value of an Annuity
The relationship between the present value and the future value of the annuity can be depicted
graphically as follows:
Page 2 of 5
Relationship between present
value and future valueordinary
annuity.
This can be interpreted in several ways:
1. $10,000 invested at 10% at the end of each of the next three years will accumulate to $33,100 at the end of the third year.
2. $24,868 invested at 10% now will grow to $33,100 after three years.
3. Someone whose time value of money is 10% would be willing to pay $24,868 now to receive $10,000 at the end of each of the next three
years.
4. If your time value of money is 10%, you should be indifferent with respect to paying/receiving (a) $24,868 now, (b) $33,100 three years from
now, or (c) $10,000 at the end of each of the next three years.
ADDITIONAL CONSIDERATION
We also can verify that these are the present value and future value of the same
annuity by calculating the present value of a single cash amount of $33,100 three years
hence:
Present Value of an Annuity Due
ILLUSTRATION 6-9
Present Value of an Annuity Due
In the previous illustration, suppose that the three equal payments of $10,000 are to be
made at the beginning of each of the three years. Recall from Illustration 6-7 on page 311
that the future value of this annuity is $36,410. What is the present value?
The following time diagram depicts this situation:
Present value of an annuity due.
Once again, using individual PV factors of $1 from Table 2, the PV of the annuity due can be calculated as follows:
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Present Value of an Annuity
Page 3 of 5
The first payment does not contain any interest since it is made on the first day of the three-year
annuity period. Therefore, the factor used is 1.00. The second payment has one compounding period and
the factor used of .90909 is the PV factor of $1 for one period and 10%, and we need to remove two
compounding periods of interest from the third payment. The factor used of .82645 is the PV factor of $1
for two periods and 10%.
In the present value of an annuity
due, no interest needs to be
removed from the first cash
payment.
The relationship between the present value and the future value of the annuity can be depicted
graphically as follows:
Relationship between present
value and future valueannuity
due.
Using Table 6, Present Value of an Annuity Due, located at the end of this book, we can more efficiently calculate the PV of the annuity due
(PVAD):
To better understand the relationship between Tables 4 and 6, notice that the PVAD factor for three periods, 10%, from Table 6 is 2.73554. This
is simply the PVA factor for two periods, 10%, of 1.73554, plus 1.0. The addition of 1.0 reflects the fact that the first payment does not require the
removal of any interest.
Present Value of a Deferred Annuity
Accounting valuations often involve the present value of annuities in which the first cash flow is expected
to occur more than one time period after the date of the agreement. As the inception of the annuity is
deferred beyond a single period, this type of annuity is referred to as a deferred annuity.7
A deferred annuity exists when
the first cash flow occurs more
than one period after the date the
agreement begins.
ILLUSTRATION 6-10
Deferred Annuity
At January 1, 2011, you are considering acquiring an investment that will provide three
equal payments of $10,000 each to be received at the end of three consecutive years.
However, the first payment is not expected until December 31, 2013. The time value of
money is 10%. How much would you be willing to pay for this investment?
The following time diagram depicts this situation:
Cash flows for a deferred
annuity.
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Present Value of an Annuity
Page 4 of 5
The present value of the deferred annuity can be calculated by summing the present values of the three individual cash flows, each discounted
to today's PV:
A more efficient way of calculating the present value of a deferred annuity involves a two-step process:
1. Calculate the PV of the annuity as of the beginning of the annuity period.
2. Discount the single amount calculated in (1) to its present value as of today.
In this case, we compute the present value of the annuity as of December 31, 2012, by multiplying the annuity amount by the three-period
ordinary annuity factor:
This is the present value as of December 31, 2012. This single amount is then reduced to present value as of January 1, 2011, by making the
following calculation:
The following time diagram illustrates this two-step process:
Present value of a deferred
annuity two-step process.
If you recall the concepts you learned in this chapter, you might think of other ways the present value of a deferred annuity can be determined.
Among them:
1. Calculate the PV of an annuity due, rather than an ordinary annuity, and then discount that amount three periods rather than two:
This is the present value as of December 31, 2013. This single amount is then reduced to present value as of January 1, 2011 by making the
following calculation:
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Present Value of an Annuity
p. 313
316
315
314
Page 5 of 5
2. From Table 4, subtract the two-period PVA factor (1.73554) from the five-period PVA factor (3.79079) and multiply the difference (2.05525) by
$10,000 to get $20,552.
7
The future value of a deferred annuity is the same as the future amount of an annuity not deferred. That is because there are no interest
compounding periods prior to the beginning of the annuity period.
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Financial Calculators and Excel
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Content
Chapter6: Time Value of Money Concepts
Financial Calculators and Excel
As previously mentioned, financial calculators can be used to solve future and present value problems. For example, a Texas Instruments model
BA-35 has the following pertinent keys:
These keys are defined as follows:
N=number of periods
%I=interest rate
PV=present value
FV=future value
PMT=annuity payments
CPT=compute button
To illustrate its use, assume that you need to determine the present value of a 10-period ordinary
annuity of $200 using a 10% interest rate. You would enter
10,
10,
-200, then press
and
to obtain the answer of $1,229.
Many professionals choose to use spreadsheet software, such as Excel, to solve time value of money
problems. These spreadsheets can be used in a variety of ways. A template can be created using the
formulas shown in Graphic 6-5 on page 323. An alternative is to use the software's built-in financial
functions. For example, Excel has a function called PV that calculates the present value of an ordinary
annuity. To use the function, you would select the pull-down menu for Insert, click on Function and
choose the category called Financial. Scroll down to PV and double-click. You will then be asked to
input the necessary variablesinterest rate, the number of periods, and the payment amount.
In subsequent chapters we illustrate the use of both a calculator and Excel in addition to present value tables to solve present value calculations
for selected examples and illustrations.
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Solving for Unknown Values in Present Value Situations
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Chapter6: Time Value of Money Concepts
Solving for Unknown Values in Present Value Situations
In present value problems involving annuities, there are four variables: (1) present value of an ordinary annuity (PVA) or present
value of an annuity due (PVAD), (2) the amount of each annuity payment, (3) the number of periods, n, and (4) the interest rate, i.
If you know any three of these, the fourth can be determined.
LO8
ILLUSTRATION 6-11
Determining the Annuity Amount When Other Variables Are Known
Assume that you borrow $700 from a friend and intend to repay the amount in four equal
annual installments beginning one year from today. Your friend wishes to be reimbursed
for the time value of money at an 8% annual rate. What is the required annual payment
that must be made (the annuity amount), to repay the loan in four years?
The following time diagram illustrates the situation:
Determining the unknown
annuity amountordinary
annuity.
The required payment is the annuity amount that will provide a present value of $700 when discounting
that amount at a discount rate of 8%:
The unknown variable is the
annuity amount.
Rearranging algebraically, we find that the annuity amount is $211.34.
You would have to make four annual payments of $211.34 to repay the loan. Total payments of $845.36 (4 $211.34) would include $145.36 in
interest ($845.36 700.00).
ILLUSTRATION 6-12
Determining n When Other Variables Are Known
Assume that you borrow $700 from a friend and intend to repay the amount in equal
installments of $100 per year over a period of years. The payments will be made at the
end of each year beginning one year from now. Your friend wishes to be reimbursed for
the time value of money at a 7% annual rate. How many years would it take before you
repaid the loan?
Once again, this is an ordinary annuity situation because the first payment takes place one year from
now. The following time diagram illustrates the situation:
Determining the unknown
number of periodsordinary
annuity.
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Solving for Unknown Values in Present Value Situations
The number of years is the value of n that will provide a present value of $700 when discounting $100
at a discount rate of 7%:
Page 2 of 5
The unknown variable is the
number of periods.
Rearranging algebraically, we find that the PVA table factor is 7.0.
When you consult the PVA table, Table 4, you search the 7% column (i = 7%) for this value and find 7.02358 in row 10. So it would take
approximately 10 years to repay the loan in the situation described.
ILLUSTRATION 6-13
Determining i When Other Variables Are Known
Suppose that a friend asked to borrow $331 today (present value) and promised to repay
you $100 (the annuity amount) at the end of each of the next four years. What is the
annual interest rate implicit in this agreement?
First of all, we are dealing with an ordinary annuity situation as the payments are at the end of each period.
The following time diagram illustrates the situation:
FINANCIAL
Reporting Case
Q2, p.299
Determining the unknown
interest rateordinary annuity.
The interest rate is the discount rate that will provide a present value of $331 when discounting the
$100 four-year ordinary annuity:
The unknown variable is the
interest rate.
Rearranging algebraically, we find that the PVA table factor is 3.31.
When you consult the PVA table, Table 4, you search row four (n = 4) for this value and find it in the 8% column. So the effective interest rate is
8%.
ILLUSTRATION 6-14
Determining i When Other Variables Are KnownUnequal Cash
Flows
Suppose that you borrowed $400 from a friend and promised to repay the loan by making
three annual payments of $100 at the end of each of the next three years plus a final
payment of $200 at the end of year four. What is the interest rate implicit in this
agreement?
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Solving for Unknown Values in Present Value Situations
Page 3 of 5
The following time diagram illustrates the situation:
Determining the unknown
interest rateunequal cash flow.
The interest rate is the discount rate that will provide a present value of $400 when discounting the
$100 three-year ordinary annuity plus the $200 to be received in four years:
The unknown variable is the
interest rate.
This equation involves two unknowns and is not as easily solved as the two previous examples. One way to solve the problem is to trial-anderror the answer. For example, if we assumed i to be 9%, the total PV of the payments would be calculated as follows:
Because the present value computed is less than the $400 borrowed, using 9% removes too much interest. Recalculating PV with i = 8%
results in a PV of $405. This indicates that the interest rate implicit in the agreement is between 8% and 9%.
CONCEPT REVIEW EXERCISE
ANNUITIES
Using the appropriate table, answer each of the following independent questions.
1. What is the future value of an annuity of $2,000 invested at the end of each of the
next six periods at 8% interest?
2. What is the future value of an annuity of $2,000 invested at the beginning of each of
the next six periods at 8% interest?
3. What is the present value of an annuity of $6,000 to be received at the end of each
of the next eight periods assuming an interest rate of 10%?
4. What is the present value of an annuity of $6,000 to be received at the beginning of
each of the next eight periods assuming an interest rate of 10%?
5. Jane bought a $3,000 audio system and agreed to pay for the purchase in 10 equal
annual installments of $408 beginning one year from today. What is the interest rate
implicit in this agreement?
6. Jane bought a $3,000 audio system and agreed to pay for the purchase in 10 equal
annual installments beginning one year from today. The interest rate is 12%. What
is the amount of the annual installment?
7. Jane bought a $3,000 audio system and agreed to pay for the purchase by making
nine equal annual installments beginning one year from today plus a lump-sum
payment of $1,000 at the end of 10 periods. The interest rate is 10%. What is the
required annual installment?
8. Jane bought an audio system and agreed to pay for the purchase by making four
equal annual installments of $800 beginning one year from today plus a lump-sum
payment of $1,000 at the end of five years. The interest rate is 12%. What was the
cost of the audio system? (Hint: What is the present value of the cash payments?)
9. Jane bought an audio system and agreed to pay for the purchase by making five
equal annual installments of $1,100 beginning four years from today. The interest
rate is 12%. What was the cost of the audio system? (Hint: What is the present
value of the cash payments?)
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Solving for Unknown Values in Present Value Situations
p. 317
319
318
Page 4 of 5
SOLUTION
1. FVA = $2,000 7.3359* = $14,672
*
Future value of an ordinary annuity of $1: n = 6, i = 8% (from Table 3)
2. FVAD = $2,000 7.9228* = $15,846
*
Future value of an annuity due of $1: n = 6, i = 8% (from Table 5)
3. PVA = $6,000 5.33493* = $32,010
*
Present value of ordinary annuity of $1: n = 8, i = 10% (from Table 4)
4. PVAD = $6,000 5.86842* = $35,211
*
Present value of an annuity due of $1: n = 8, i = 10% (from Table 6)
5.
*
Present value of an ordinary annuity of $1: n = 10, i = ? (from Table 4, i
approximately 6%)
6.
*
Present value of an ordinary annuity of $1: n = 10, i = 12% (from Table 4)
*
Present value of an ordinary annuity of $1: n = 9, i = 10% (from Table 4)
Present value of $1: n = 10, i = 10% (from Table 2)
7.
8. PV = ($800 3.03735*) + ($1,000 .56743 = $2,997
*
Present value of an ordinary annuity of $1: n = 4, i = 12% (from Table 4)
Present value of $1: n = 5, i = 12% (from Table 2)
9. PVA = $1,100 3.60478* = $3,965
*
Present value of ordinary an annuity of $1: n = 5, i = 12% (from Table 4)
This is the present value three years from today (the beginning of the five-year
ordinary annuity). This single amount is then reduced to present value as of today
by making the following calculation:
PV = $3,965 .71178 = $2,822
Present value of $1: n = 3, i = 12%, (from Table 2)
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Solving for Unknown Values in Present Value Situations
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Preview of Accounting Applications of Present Value TechniquesAnnuities
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Content
Chapter6: Time Value of Money Concepts
Preview of Accounting Applications of Present Value TechniquesAnnuities
The time value of money has many applications in accounting. Most of these applications involve the concept of present value.
Because financial instruments typically specify equal periodic payments, these applications quite often involve annuity situations.
For example, let's consider one accounting situation using both an ordinary annuity and the present value of a single amount
(long-term bonds), one using an annuity due (long-term leases), and a third using a deferred annuity (pension obligations).
LO9
Valuation of Long-Term Bonds
You will learn in Chapter 14 that a long-term bond usually requires the issuing (borrowing) company to repay a
specified amount at maturity and make periodic stated interest payments over the life of the bond. The stated
interest payments are equal to the contractual stated rate multiplied by the face value of the bonds. At the date
the bonds are issued (sold), the marketplace will determine the price of the bonds based on the market rate of
interest for investments with similar characteristics. The market rate at date of issuance may not equal the
bonds' stated rate in which case the price of the bonds (the amount the issuing company actually is borrowing)
will not equal the bonds' face value. Bonds issued at more than face value are said to be issued at a premium,
while bonds issued at less than face value are said to be issued at a discount. Consider the example in
Illustration 6-15.
FINANCIAL
Reporting Case
Q3, p.299
ILLUSTRATION 6-15
Valuing a Long-term Bond Liability
On June 30, 2011, Fumatsu Electric issued 10% stated rate bonds with a face amount of
$200 million. The bonds mature on June 30, 2031 (20 years). The market rate of interest
for similar issues was 12%. Interest is paid semiannually (5%) on June 30 and December
31, beginning December 31, 2011. The interest payment is $10 million (5% $200
million). What was the price of the bond issue? What amount of interest expense will
Fumatsu record for the bonds in 2011?
To determine the price of the bonds, we calculate the present value of the 40-period annuity (40 semiannual interest payments of $10 million)
and the lump-sum payment of $200 million paid at maturity using the semiannual market rate of interest of 6%. In equation form,
The bonds will sell for $169,907,000, which represents a discount of $30,093,000 ($200,000,000 169,907,000). The discount results from the
difference between the semiannual stated rate of 5% and the market rate of 6%. Fumatsu records a $169,907,000 increase in cash and a
corresponding liability for bonds payable.
Interest expense for the first six months is determined by multiplying the carrying value (book value) of the bonds ($169,907,000) by the
semiannual effective rate (6%) as follows:
The difference between interest expense ($10,194,420) and interest paid ($10,000,000) increases the carrying value of the bond liability.
Interest for the second six months of the bond's life is determined by multiplying the new carrying value by the 6% semiannual effective rate.
We discuss the specific accounts used to record these transactions in Chapter 14.
Valuation of Long-Term Leases
Companies frequently acquire the use of assets by leasing rather than purchasing them. Leases usually require the payment of fixed amounts at
regular intervals over the life of the lease. You will learn in Chapter 15 that certain long-term, noncancelable leases are treated in a manner
similar to an installment sale by the lessor and an installment purchase by the lessee. In other words, the lessor records a receivable and the
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lessee records a liability for the several installment payments. For the lessee, this requires that the leased asset and corresponding lease liability
be valued at the present value of the lease payments. Consider the example in Illustration 6-16.
ILLUSTRATION 6-16
Valuing a Long-Term Lease Liability
On January 1, 2011, the Stridewell Wholesale Shoe Company signed a 25-year noncancelable lease agreement for an office building. Terms of the lease call for Stridewell to
make annual lease payments of $10,000 at the beginning of each year, with the first
payment due on January 1, 2011. Assuming an interest rate of 10% properly reflects the
time value of money in this situation, how should Stridewell value the asset acquired and
the corresponding lease liability if it is to be treated in a manner similar to an installment
purchase?
Once again, by computing the present value of the lease payments, we remove the portion of the
payments that represents interest, leaving the portion that represents payment for the asset itself.
Because the first payment is due immediately, as is common for leases, this is an annuity due situation.
In equation form:
Certain long-term leases require
the recording of an asset and
corresponding liability at the
present value of future lease
payments.
Stridewell initially will value the leased asset and corresponding lease liability at $99,847.
Journal entry at the inception of a
lease.
The difference between this amount and total future cash payments of $250,000 ($10,000 25) represents the interest that is implicit in this
agreement. That difference is recorded as interest over the life of the lease.
Valuation of Pension Obligations
Pension plans are important compensation vehicles used by many U.S. companies. These plans are essentially forms of deferred compensation
as the pension benefits are paid to employees after they retire. You will learn in Chapter 17 that some pension plans create obligations during
employees' service periods that must be paid during their retirement periods. These obligations are funded during the employment period. This
means companies contribute cash to pension funds annually with the intention of accumulating sufficient funds to pay employees the retirement
benefits they have earned. The amounts contributed are determined using estimates of retirement benefits. The actual amounts paid to
employees during retirement depend on many factors including future compensation levels and length of life. Consider Illustration 6-17.
ILLUSTRATION 6-17
Valuing a Pension Obligation
On January 1, 2011, the Stridewell Wholesale Shoe Company hired Sammy Sossa.
Sammy is expected to work for 25 years before retirement on December 31, 2035.
Annual retirement payments will be paid at the end of each year during his retirement
period, expected to be 20 years. The first payment will be on December 31, 2036. During
2011 Sammy earned an annual retirement benefit estimated to be $2,000 per year. The
company plans to contribute cash to a pension fund that will accumulate to an amount
sufficient to pay Sammy this benefit. Assuming that Stridewell anticipates earning 6% on
all funds invested in the pension plan, how much would the company have to contribute at
the end of 2011 to pay for pension benefits earned in 2011?
To determine the required contribution, we calculate the present value on December 31, 2011, of the deferred annuity of $2,000 that begins on
December 31, 2036, and is expected to end on December 31, 2055.
The following time diagram depicts this situation:
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p. 320
322
321
We can calculate the present value of the annuity using a two-step process. The first step computes the present value of the annuity as of
December 31, 2035, by multiplying the annuity amount by the 20-period ordinary annuity factor:
This is the present value as of December 31, 2035. This single amount is then reduced to present value as of December 31, 2011, by a second
calculation:
Stridewell would have to contribute $5,666 at the end of 2011 to fund the estimated pension benefits earned by its employee in 2011. Viewed in
reverse, $5,666 invested now at 6% will accumulate a fund balance of $22,940 at December 31, 2035. If the fund balance remains invested at
6%, $2,000 can be withdrawn each year for 20 years before the fund is depleted.
Among the other situations you'll encounter using present value techniques are valuing notes (Chapters 10 and 14) and other postretirement
benefits (Chapter 17).
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Summary of Time Value of Money Concepts
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Chapter6: Time Value of Money Concepts
Summary of Time Value of Money Concepts
Graphic 6-5 summarizes the time value of money concepts discussed in this chapter.
p. 323
GRAPHIC 6-5
Summary of Time Value of Money Concepts
FINANCIAL REPORTING CASE SOLUTION
1. Why were the Castellanos to receive $43 million rather than the $141 million lottery prize?(p.
312) The Castellanos chose to receive their lottery winnings in one lump payment immediately
rather than in 26 equal annual installments beginning immediately. The state calculates the present
value of the 26 equal payments, withholds the necessary federal income tax, and pays the
Castellanos the remainder.
2. What interest rate did the State of California use to calculate the $43 million lump-sum payment?(p. 318) The equal payment is
determined by dividing $141 million by 26 periods:
Since the first payment is made immediately, this is an annuity due situation. We must find the interest rate that provides a present value of
$43 million. There is no 26 period row in Table 6. We can subtract the first payment from the $43 million since it is paid immediately and
solve using the 25-period ordinary annuity table (that is, the 25 remaining annual payments beginning in one year):
So, the interest rate used by the state was approximately 8%.
p. 324
3.
What are some of the accounting applications that incorporate the time value of money into valuation?(p. 320) Accounting
applications that incorporate the time value of money techniques into valuation include the valuation of long-term notes receivable and
various long-term liabilities that include bonds, notes, leases, pension obligations, and postretirement benefits other than pensions. We
study these in detail in later chapters.
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The Bottom Line
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Content
Chapter6: Time Value of Money Concepts
The Bottom Line
p. 324
LO1
A dollar today is worth more than a dollar to be received in the future. The difference between the present value of cash flows and
their future value represents the time value of money. Interest is the rent paid for the use of money over time. (p. 300)
LO2
The future value of a single amount is the amount of money that a dollar will grow to at some point in the future. It is computed by
multiplying the single amount by (1 + i)n, where i is the interest rate and n the number of compounding periods. The Future Value
of $1 table allows for the calculation of future value for any single amount by providing the factors for various combinations of i
and n. (p. 301)
LO3
The present value of a single amount is the amount of money today that is equivalent to a given amount to be received or paid in
the future. It is computed by dividing the future amount by (1 + i)n. The Present Value of $1 table simplifies the calculation of the
present value of any future amount. (p. 302)
LO4
There are four variables in the process of adjusting single cash flow amounts for the time value of money: present value (PV),
future value (FV), i and n. If you know any three of these, the fourth can be computed easily. (p. 304)
LO5
An annuity is a series of equal-sized cash flows occurring over equal intervals of time. An ordinary annuity exists when the cash
flows occur at the end of each period. An annuity due exists when the cash flows occur at the beginning of each period. (p. 309)
LO6
The future value of an ordinary annuity (FVA) is the future value of a series of equalsized cash flows with the first payment taking
place at the end of the first compounding period. The last payment will not earn any interest since it is made at the end of the
annuity period. The future value of an annuity due (FVAD) is the future value of a series of equal-sized cash flows with the first
payment taking place at the beginning of the annuity period (the beginning of the first compounding period). (p. 310)
LO7
The present value of an ordinary annuity (PVA) is the present value of a series of equal-sized cash flows with the first payment
taking place at the end of the first compounding period. The present value of an annuity due (PVAD) is the present value of a
series of equal-sized cash flows with the first payment taking place at the beginning of the annuity period. The present value of a
deferred annuity is the present value of a series of equal-sized cash flows with the first payment taking place more than one time
period after the date of the agreement. (p. 312)
LO8
In present value problems involving annuities, there are four variables: PVA or PVAD, the annuity amount, the number of
compounding periods (n) and the interest rate (i). If you know any three of these, you can determine the fourth. (p. 316)
LO9
Most accounting applications of the time value of money involve the present values of annuities. The initial valuation of long-term
bonds is determined by calculating the present value of the periodic stated interest payments and the present value of the lumpsum payment made at maturity. Certain long-term leases require the lessee to compute the present value of future lease
payments to value the leased asset and corresponding lease obligation. Also, pension plans require the payment of deferred
annuities to retirees. (p. 320)
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Questions for Review of Key Topics
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Intermediate Accounting
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Content
Chapter6: Time Value of Money Concepts
Questions for Review of Key Topics
Q 6-1 Define interest.
Q 6-2 Explain compound interest.
Q 6-3 What would cause the annual interest rate to be different from the annual effective rate or yield?
Q 6-4 Identify the three items of information necessary to calculate the future value of a single amount.
Q 6-5 Define the present value of a single amount.
p. 325
Q 6-6 Explain the difference between monetary and nonmonetary assets and liabilities.
Q 6-7 What is an annuity?
Q 6-8 Explain the difference between an ordinary annuity and an annuity due.
Q 6-9 Explain the relationship between Table 2, Present Value of $1, and Table 4, Present Value of an Ordinary Annuity of $1.
Q 6-10 Prepare a time diagram for the present value of a four-year ordinary annuity of $200. Assume an interest rate of 10% per year.
Q 6-11 Prepare a time diagram for the present value of a four-year annuity due of $200. Assume an interest rate of 10% per year.
Q 6-12 What is a deferred annuity?
Q 6-13 Assume that you borrowed $500 from a friend and promised to repay the loan in five equal annual installments beginning one
year from today. Your friend wants to be reimbursed for the time value of money at an 8% annual rate. Explain how you would
compute the required annual payment.
Q 6-14 Compute the required annual payment in Question 6-13.
Q 6-15 Explain how the time value of money concept is incorporated into the valuation of long-term leases.
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Brief Exercises
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Chapter6: Time Value of Money Concepts
Brief Exercises
BE 6-1
Simple versus compound interest
LO1
Fran Smith has two investment opportunities. The interest rate for both investments is 8%. Interest on the first investment will compound
annually while interest on the second will compound quarterly. Which investment opportunity should Fran choose? Why?
BE 6-2
Future value; single amount
LO2
Bill O'Brien would like to take his wife, Mary, on a trip three years from now to Europe to celebrate their 40th anniversary. He has just received a
$20,000 inheritance from an uncle and intends to invest it for the trip. Bill estimates the trip will cost $23,500 and he believes he can earn 5%
interest, compounded annually, on his investment. Will he be able to pay for the trip with the accumulated investment amount?
BE 6-3
Future value; solving for unknown; single amount
LO4
Refer to the situation described in BE 6-2. Assume that the trip will cost $26,600. What interest rate, compounded annually, must Bill earn to
accumulate enough to pay for the trip?
BE 6-4
Present value; single amount
LO3
John has an investment opportunity that promises to pay him $16,000 in four years. He could earn a 6% annual return investing his money
elsewhere. What is the maximum amount he would be willing to invest in this opportunity?
BE 6-5
Present value; solving for unknown; single amount
LO4
Refer to the situation described in BE 6-4. Suppose the opportunity requires John to invest $13,200 today. What is the interest rate John would
earn on this investment?
BE 6-6
Future value; ordinary annuity
LO6
Leslie McCormack is in the spring quarter of her freshman year of college. She and her friends already are planning a trip to Europe after
graduation in a little over three years. Mary would like to contribute to a savings account over the next three years in order to accumulate
enough money to take the trip. Assuming an interest rate of 4%, compounded quarterly, how much will she accumulate in three years by
depositing $500 at the end of each of the next 12 quarters, beginning three months from now?
BE 6-7
Future value; annuity due
LO6
Refer to the situation described in BE 6-6. How much will Leslie accumulate in three years by depositing $500 at the beginning of each of the
next 12 quarters?
BE 6-8
Present value; ordinary annuity
LO7
Canliss Mining Company borrowed money from a local bank. The note the company signed requires five annual installment payments of
$10,000 beginning one year from today. The interest rate on the note is 7%. What amount did Canliss borrow?
BE 6-9
Present value; annuity due
LO7
Refer to the situation described in BE 6-8. What amount did Canliss borrow assuming that the first $10,000 payment was due immediately?
BE 6-10 Deferred annuity
LO7
Refer to the situation described in BE 6-8. What amount did Canliss borrow assuming that the first of the five annual $10,000 payments was not
due for three years?
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Brief Exercises
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BE 6-11 Solve for unknown; annuity
LO8
Kingsley Toyota borrowed $100,000 from a local bank. The loan requires Kingsley to pay 10 equal annual installments beginning one year from
today. Assuming an interest rate of 8%, what is the amount of each annual installment payment?
BE 6-12 Price of a bond
LO9
On December 31, 2011, Interlink Communications issued 6% stated rate bonds with a face amount of $100 million. The bonds mature on
December 31, 2041. Interest is payable annually on each December 31, beginning in 2012. Determine the price of the bonds on December 31,
2011, assuming that the market rate of interest for similar bonds was 7%.
BE 6-13 Lease payment
LO9
On September 30, 2011, Ferguson Imports leased a warehouse. Terms of the lease require Ferguson to make 10 annual lease payments of
$55,000 with the first payment due immediately. Accounting standards require the company to record a lease liability when recording this type of
lease. Assuming an 8% interest rate, at what amount should Ferguson record the lease liability on September 30, 2011, before the first payment
is made?
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Exercises
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Chapter6: Time Value of Money Concepts
Exercises
An alternate exercise and problem set is available on the text website: www.mhhe.com/spiceland6e
E 6-1
Future value; single amount
LO2
Determine the future value of the following single amounts:
E 6-2
Future value; single amounts
LO2
Determine the future value of $10,000 under each of the following sets of assumptions:
E 6-3
Present value; single amount
LO3
Determine the present value of the following single amounts:
E 6-4
Present value; multiple, unequal amounts
LO3
Determine the combined present value as of December 31, 2011, of the following four payments to be received at the end of each of the
designated years, assuming an annual interest rate of 8%.
E 6-5
Noninterest-bearing note; single payment
LO3
The Field Detergent Company sold merchandise to the Abel Company on June 30, 2011. Payment was made in the form of a noninterestbearing note requiring Abel to pay $85,000 on June 30, 2013. Assume that a 10% interest rate properly reflects the time value of money in this
situation.
Required:
Calculate the amount at which Field should record the note receivable and corresponding sales revenue on June 30, 2011.
E 6-6
Solving for unknowns; single amounts
LO4
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For each of the following situations involving single amounts, solve for the unknown (?). Assume that interest is compounded annually. (i =
interest rate, and n = number of years)
E 6-7
LO6
Future value; annuities
Wiseman Video plans to make four annual deposits of $2,000 each to a special building fund. The fund's assets will be invested in mortgage
instruments expected to pay interest at 12% on the fund's balance. Using the appropriate annuity table, determine how much will be
accumulated in the fund on December 31, 2014, under each of the following situations:
1. The first deposit is made on December 31, 2011, and interest is compounded annually.
2. The first deposit is made on December 31, 2010, and interest is compounded annually.
3. The first deposit is made on December 31, 2010, and interest is compounded quarterly.
4. The first deposit is made on December 31, 2010, interest is compounded annually, and interest earned is withdrawn at the end of each year.
E 6-8
Present value; annuities
LO7
Using the appropriate present value table and assuming a 12% annual interest rate, determine the present value on December 31, 2011, of a
five-period annual annuity of $5,000 under each of the following situations:
1. The first payment is received on December 31, 2012, and interest is compounded annually.
2. The first payment is received on December 31, 2011, and interest is compounded annually.
3. The first payment is received on December 31, 2012, and interest is compounded quarterly.
E 6-9
Solving for unknowns; annuities
LO8
For each of the following situations involving annuities, solve for the unknown (?). Assume that interest is compounded annually and that all
annuity amounts are received at the end of each period. (i = interest rate, and n = number of years)
E 6-10
Future value; solving for annuities and single amount
LO4 LO8
John Rider wants to accumulate $100,000 to be used for his daughter's college education. He would like to have the amount available on
December 31, 2016. Assume that the funds will accumulate in a certificate of deposit paying 8% interest compounded annually.
Required:
Answer each of the following independent questions.
1. If John were to deposit a single amount, how much would he have to invest on December 31, 2011?
2. If John were to make five equal deposits on each December 31, beginning on December 31, 2012, what is the required deposit?
3. If John were to make five equal deposits on each December 31, beginning on December 31, 2011, what is the required deposit?
E 6-11
Future and present value
LO3 LO6 LO7
Answer each of the following independent questions.
1. Alex Meir recently won a lottery and has the option of receiving one of the following three prizes: (1) $64,000 cash immediately, (2) $20,000
cash immediately and a six-period annuity of $8,000 beginning one year from today, or (3) a six-period annuity of $13,000 beginning one
year from today. Assuming an interest rate of 6%, which option should Alex choose?
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2. The Weimer Corporation wants to accumulate a sum of money to repay certain debts due on December 31, 2020. Weimer will make annual
deposits of $100,000 into a special bank account at the end of each of 10 years beginning December 31, 2011. Assuming that the bank
account pays 7% interest compounded annually, what will be the fund balance after the last payment is made on December 31, 2020?
E 6-12
Deferred annuities
LO7
Lincoln Company purchased merchandise from Grandville Corp. on September 30, 2011. Payment was made in the form of a noninterestbearing note requiring Lincoln to make six annual payments of $5,000 on each September 30, beginning on September 30, 2014.
Required:
Calculate the amount at which Lincoln should record the note payable and corresponding purchases on September 30, 2011, assuming that an
interest rate of 10% properly reflects the time value of money in this situation.
E 6-13
Solving for unknown annuity payment
LO8
Don James purchased a new automobile for $20,000. Don made a cash down payment of $5,000 and agreed to pay the remaining balance in
30 monthly installments, beginning one month from the date of purchase. Financing is available at a 24% annual interest rate.
Required:
Calculate the amount of the required monthly payment.
E 6-14
Solving for unknown interest rate
LO8
Lang Warehouses borrowed $100,000 from a bank and signed a note requiring 20 annual payments of $13,388 beginning one year from the
date of the agreement.
Required:
Determine the interest rate implicit in this agreement.
E 6-15
Solving for unknown annuity amount
LO8
Sandy Kupchack just graduated from State University with a bachelors degree in history. During her four years at the U, Sandy accumulated
$12,000 in student loans. She asks for your help in determining the amount of the quarterly loan payment. She tells you that the loan must be
paid back in five years and that the annual interest rate is 8%. Payments begin in three months.
Required:
Determine Sandy's quarterly loan payment.
E 6-16
Deferred annuities; solving for annuity amount
LO7 LO8
On April 1, 2011, John Vaughn purchased appliances from the Acme Appliance Company for $1,200. In order to increase sales, Acme allows
customers to pay in installments and will defer any payments for six months. John will make 18 equal monthly payments, beginning October 1,
2011. The annual interest rate implicit in this agreement is 24%.
Required:
Calculate the monthly payment necessary for John to pay for his purchases.
E 6-17
Price of a bond
LO9
On September 30, 2011, the San Fillipo Corporation issued 8% stated rate bonds with a face amount of $300 million. The bonds mature on
September 30, 2031 (20 years). The market rate of interest for similar bonds was 10%. Interest is paid semiannually on March 31 and
September 30.
Required:
Determine the price of the bonds on September 30, 2011.
E 6-18
Price of a bond; interest expense
LO9
On June 30, 2011, Singleton Computers issued 6% stated rate bonds with a face amount of $200 million. The bonds mature on June 30, 2026
(15 years). The market rate of interest for similar bond issues was 5% (2.5% semiannual rate). Interest is paid semiannually (3%) on June 30
and December 31, beginning on December 31, 2011.
Required:
1. Determine the price of the bonds on June 30, 2011.
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2. Calculate the interest expense Singleton reports in 2011 for these bonds.
E 6-19
Lease payments
LO9
On June 30, 2011, Fly-By-Night Airlines leased a jumbo jet from Boeing Corporation. The terms of the lease require Fly-By-Night to make 20
annual payments of $400,000 on each June 30. Generally accepted accounting principles require this lease to be recorded as a liability for the
present value of scheduled payments. Assume that a 7% interest rate properly reflects the time value of money in this situation.
Required:
1. At what amount should Fly-By-Night record the lease liability on June 30, 2011, assuming that the first payment will be made on June 30,
2012?
2. At what amount should Fly-By-Night record the lease liability on June 30, 2011, before any payments are made, assuming that the first
payment will be made on June 30, 2011?
E 6-20
Lease payments; solve for unknown interest rate
LO8 LO9
On March 31, 2011, Southwest Gas leased equipment from a supplier and agreed to pay $200,000 annually for 20 years beginning March 31,
2012. Generally accepted accounting principles require that a liability be recorded for this lease agreement for the present value of scheduled
payments. Accordingly, at inception of the lease, Southwest recorded a $2,293,984 lease liability.
Required:
Determine the interest rate implicit in the lease agreement.
E 6-21
Concepts; terminology
LO1 through LO3 LO5
Listed below are several terms and phrases associated with concepts discussed in the chapter. Pair each item from List A with the item from List
B (by letter) that is most appropriately associated with it.
List A
List B
_____ 1. Interest
a. First cash flow occurs one period after agreement
begins.
_____ 2. Monetary asset
b. The rate at which money will actually grow during
a year.
_____ 3. Compound interest
c. First cash flow occurs on the first day of the
agreement.
_____ 4. Simple interest
d. The amount of money that a dollar will grow to.
_____ 5. Annuity
e. Amount of money paid/received in excess of
amount borrowed/lent.
_____ 6. Present value of a single f. Obligation to pay a sum of cash, the amount of
amount
which is fixed.
_____ 7. Annuity due
g. Money can be invested today and grow to a larger
amount.
_____ 8. Future value of a single
amount
h. No fixed dollar amount attached.
_____ 9. Ordinary annuity
i. Computed by multiplying an invested amount by
the interest rate.
_____ 10. Effective rate or yield
j. Interest calculated on invested amount plus
accumulated interest.
_____ 11. Nonmonetary asset
k. A series of equal-sized cash flows.
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_____ 12. Time value of money
l. Amount of money required today that is equivalent
to a given future amount.
_____ 13. Monetary liability
m. Claim to receive a fixed amount of money.
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CPA and CMA Review Questions
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Chapter6: Time Value of Money Concepts
CPA and CMA Review Questions
CPA Exam Questions
The following questions are used in the Kaplan CPA Review Course to study the time value of money while preparing for the
CPA examination. Determine the response that best completes the statements or questions.
1.
An investment product promises to pay $25,458 at the end of nine years. If an investor feels this investment should
produce a rate of return of 14 percent, compounded annually, what's the most the investor should be willing to pay for
the investment?
LO3
a. $6,866
b. $7,828
c. $8,926
d. $9,426
2.
On January 1, 2011, Ott Company sold goods to Fox Company. Fox signed a noninterest-bearing note requiring
payment of $60,000 annually for seven years. The first payment was made on January 1, 2011. The prevailing rate of
interest for this type of note at date of issuance was 10%. Information on present value factors is as follows:
LO7
Ott should record sales revenue in January 2011 of
a. $214,200
b. $261,600
c. $292,600
d. $321,600
3.
An annuity will pay eight annual payments of $100, with the first payment to be received one year from now. If the
interest rate is 12 percent per year, what is the present value of this annuity? Use the appropriate table located at the
end of the textbook to solve this problem.
LO7
a. $497
b. $556
c. $801
d. $897
4.
An annuity will pay four annual payments of $100, with the first payment to be received three years from now. If the
interest rate is 12 percent per year, what is the present value of this annuity? Use the appropriate table located at the
end of the textbook to solve this problem.
LO7
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CPA and CMA Review Questions
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a. $181
b. $242
c. $304
d. $400
5.
Justin Banks just won the lottery and is trying to decide between the annual cash flow payment option of $100,000 per
year for 15 years beginning today, or the lump-sum option. Justin can earn 8 percent investing his money. At what
lump-sum payment amount would he be indifferent between the two alternatives? Use the appropriate table located at
the end of the textbook to solve this problem.
LO7
a. $824,424
b. $855,948
c. $890,378
d. $924,424
6.
An investor purchases a 10-year, $1,000 par value bond that pays annual interest of $100. If the market rate of interest
is 12 percent, what is the current market value of the bond?
LO3 LO7 LO9
a. $ 887
b. $ 950
c. $1,000
d. $1,100
7.
You borrow $15,000 to buy a car. The loan is to be paid off in monthly installments over five years at 12 percent
interest annually. The first payment is due one month from today. If the present value of an ordinary annuity of $1 for 5
years @ 12% with monthly compounding is $44.955, what is the amount of each monthly payment?
LO8
a. $334
b. $456
c. $546
d. $680
CMA Exam Questions
The following questions dealing with the time value of money are adapted from questions that previously appeared on Certified
Management Accountant (CMA) examinations. The CMA designation sponsored by the Institute of Management Accountants
(www.imanet.org) provides members with an objective measure of knowledge and competence in the field of management
accounting. Determine the response that best completes the statements or questions.
1.
Janet Taylor Casual Wear has $75,000 in a bank account as of December 31, 2009. If the company plans on
depositing $4,000 in the account at the end of each of the next 3 years (2010, 2011, and 2012) and all amounts in the
account earn 8% per year, what will the account balance be at December 31, 2012? Ignore the effect of income taxes.
LO2
a. $ 87,000
b. $ 88,000
c. $ 96,070
d. $107,500
2.
Essex Corporation is evaluating a lease that takes effect on March 1. The company must make eight equal payments,
with the first payment due on March 1. The concept most relevant to the evaluation of the lease is
LO7 LO9
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a. The present value of an annuity due.
b. The present value of an ordinary annuity.
c. The future value of an annuity due.
d. The future value of an ordinary annuity.
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Problems
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P 6-1
Analysis of alternatives
LO3 LO7
Esquire Company needs to acquire a molding machine to be used in its manufacturing process. Two types of machines that would be
appropriate are presently on the market. The company has determined the following:
Machine A could be purchased for $48,000. It will last 10 years with annual maintenance costs of $1,000 per year. After 10 years the machine
can be sold for $5,000.
Machine B could be purchased for $40,000. It also will last 10 years and will require maintenance costs of $4,000 in year three, $5,000 in year
six, and $6,000 in year eight. After 10 years, the machine will have no salvage value.
Required:
Determine which machine Esquire should purchase. Assume an interest rate of 8% properly reflects the time value of money in this situation and
that maintenance costs are paid at the end of each year. Ignore income tax considerations.
P 6-2
Present and future value
LO6 LO7 LO9
Johnstone Company is facing several decisions regarding investing and financing activities. Address each decision independently.
1. On June 30, 2011, the Johnstone Company purchased equipment from Genovese Corp. Johnstone agreed to pay Genovese $10,000 on the
purchase date and the balance in five annual installments of $8,000 on each June 30 beginning June 30, 2012. Assuming that an interest
rate of 10% properly reflects the time value of money in this situation, at what amount should Johnstone value the equipment?
2. Johnstone needs to accumulate sufficient funds to pay a $400,000 debt that comes due on December 31, 2016. The company will
accumulate the funds by making five equal annual deposits to an account paying 6% interest compounded annually. Determine the required
annual deposit if the first deposit is made on December 31, 2011.
3. On January 1, 2011, Johnstone leased an office building. Terms of the lease require Johnstone to make 20 annual lease payments of
$120,000 beginning on January 1, 2011. A 10% interest rate is implicit in the lease agreement. At what amount should Johnstone record the
lease liability on January 1, 2011, before any lease payments are made?
P 6-3
Analysis of alternatives
LO3 LO7
Harding Company is in the process of purchasing several large pieces of equipment from Danning Machine Corporation. Several financing
alternatives have been offered by Danning:
1. Pay $1,000,000 in cash immediately.
2. Pay $420,000 immediately and the remainder in 10 annual installments of $80,000, with the first installment due in one year.
3. Make 10 annual installments of $135,000 with the first payment due immediately.
4. Make one lump-sum payment of $1,500,000 five years from date of purchase.
Required:
Determine the best alternative for Harding, assuming that Harding can borrow funds at an 8% interest rate.
P 6-4
Investment analysis
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LO3 LO7
John Wiggins is contemplating the purchase of a small restaurant. The purchase price listed by the seller is $800,000. John has used past
financial information to estimate that the net cash flows (cash inflows less cash outflows) generated by the restaurant would be as follows:
If purchased, the restaurant would be held for 10 years and then sold for an estimated $700,000.
Required:
Assuming that John desires a 10% rate of return on this investment, should the restaurant be purchased? (Assume that all cash flows occur at
the end of the year.)
P 6-5
Investment decision; varying rates
LO3 LO7
John and Sally Claussen are contemplating the purchase of a hardware store from John Duggan. The Claussens anticipate that
the store will generate cash flows of $70,000 per year for 20 years. At the end of 20 years, they intend to sell the store for an
estimated $400,000. The Claussens will finance the investment with a variable rate mortgage. Interest rates will increase twice
during the 20-year life of the mortgage. Accordingly, the Claussens' desired rate of return on this investment varies as follows:
Required:
What is the maximum amount the Claussens should pay John Duggan for the hardware store? (Assume that all cash flows occur at the end of
the year.)
P 6-6
Solving for unknowns
LO8
The following situations should be considered independently.
1. John Jamison wants to accumulate $60,000 for a down payment on a small business. He will invest $30,000 today in a bank account paying
8% interest compounded annually. Approximately how long will it take John to reach his goal?
2. The Jasmine Tea Company purchased merchandise from a supplier for $28,700. Payment was a noninterest-bearing note requiring Jasmine
to make five annual payments of $7,000 beginning one year from the date of purchase. What is the interest rate implicit in this agreement?
3. Sam Robinson borrowed $10,000 from a friend and promised to pay the loan in 10 equal annual installments beginning one year from the
date of the loan. Sam's friend would like to be reimbursed for the time value of money at a 9% annual rate. What is the annual payment Sam
must make to pay back his friend?
P 6-7
Solving for unknowns
LO8
Lowlife Company defaulted on a $250,000 loan that was due on December 31, 2011. The bank has agreed to allow Lowlife to
repay the $250,000 by making a series of equal annual payments beginning on December 31, 2012.
Required:
1. Calculate the required annual payment if the bank's interest rate is 10% and four payments are to be made.
2. Calculate the required annual payment if the bank's interest rate is 8% and five payments are to be made.
3. If the bank's interest rate is 10%, how many annual payments of $51,351 would be required to repay the debt?
4. If three payments of $104,087 are to be made, what interest rate is the bank charging Lowlife?
P 6-8
Deferred annuities
LO7
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Problems
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On January 1, 2011, the Montgomery company agreed to purchase a building by making six payments. The first three are to be $25,000 each,
and will be paid on December 31, 2011, 2012, and 2013. The last three are to be $40,000 each and will be paid on December 31, 2014, 2015,
and 2016. Montgomery borrowed other money at a 10% annual rate.
Required:
1. At what amount should Montgomery record the note payable and corresponding cost of the building on January 1, 2011?
2. How much interest expense on this note will Montgomery recognize in 2011?
P 6-9
Deferred annuities
LO7
John Roberts is 55 years old and has been asked to accept early retirement from his company. The company has offered John three alternative
compensation packages to induce John to retire:
1. $180,000 cash payment to be paid immediately.
2. A 20-year annuity of $16,000 beginning immediately.
3. A 10-year annuity of $50,000 beginning at age 65.
Required:
Which alternative should John choose assuming that he is able to invest funds at a 7% rate?
P 6-10
Noninterest-bearing note; annuity and lump-sum payment
LO3 LO7
On January 1, 2011, The Barrett Company purchased merchandise from a supplier. Payment was a noninterest-bearing note requiring five
annual payments of $20,000 on each December 31 beginning on December 31, 2011, and a lump-sum payment of $100,000 on December 31,
2015. A 10% interest rate properly reflects the time value of money in this situation.
Required:
Calculate the amount at which Barrett should record the note payable and corresponding merchandise purchased on January 1, 2011.
P 6-11
Solving for unknown lease payment
LO8 LO9
Benning Manufacturing Company is negotiating with a customer for the lease of a large machine manufactured by Benning. The machine has a
cash price of $800,000. Benning wants to be reimbursed for financing the machine at an 8% annual interest rate.
Required:
1. Determine the required lease payment if the lease agreement calls for 10 equal annual payments beginning immediately.
2. Determine the required lease payment if the first of 10 annual payments will be made one year from the date of the agreement.
3. Determine the required lease payment if the first of 10 annual payments will be made immediately and Benning will be able to sell the
machine to another customer for $50,000 at the end of the 10-year lease.
P 6-12
Solving for unknown lease payment; compounding periods of
varying length
LO9
(This is a variation of the previous problem focusing on compounding periods of varying length.)
Benning Manufacturing Company is negotiating with a customer for the lease of a large machine manufactured by Benning. The machine has
a cash price of $800,000. Benning wants to be reimbursed for financing the machine at a 12% annual interest rate over the five-year lease term.
Required:
1. Determine the required lease payment if the lease agreement calls for 10 equal semiannual payments beginning six months from the date of
the agreement.
2. Determine the required lease payment if the lease agreement calls for 20 equal quarterly payments beginning immediately.
3. Determine the required lease payment if the lease agreement calls for 60 equal monthly payments beginning one month from the date of the
agreement. The present value of an ordinary annuity factor for n = 60 and i = 1% is 44.9550.
P 6-13
Lease vs. buy alternatives
LO3 LO7 LO9
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Kiddy Toy Corporation needs to acquire the use of a machine to be used in its manufacturing process. The machine needed is manufactured by
Lollie Corp. The machine can be used for 10 years and then sold for $10,000 at the end of its useful life. Lollie has presented Kiddy with the
following options:
1. Buy machine. The machine could be purchased for $160,000 in cash. All maintenance and insurance costs, which approximate $5,000 per
year, would be paid by Kiddy.
2. Lease machine. The machine could be leased for a 10-year period for an annual lease payment of $25,000 with the first payment due
immediately. All maintenance and insurance costs will be paid for by the Lollie Corp. and the machine will revert back to Lollie at the end of
the 10-year period.
Required:
Assuming that a 12% interest rate properly reflects the time value of money in this situation and that all maintenance and insurance costs are
paid at the end of each year, determine which option Kiddy should choose. Ignore income tax considerations.
P 6-14
Deferred annuities; pension obligation
LO7 LO9
Three employees of the Horizon Distributing Company will receive annual pension payments from the company when they
retire. The employees will receive their annual payments for as long as they live. Life expectancy for each employee is 15 years
beyond retirement. Their names, the amount of their annual pension payments, and the date they will receive their first payment
are shown below:
Required:
1. Compute the present value of the pension obligation to these three employees as of December 31, 2011. Assume an 11% interest rate.
2. The company wants to have enough cash invested at December 31, 2014, to provide for all three employees. To accumulate enough cash,
they will make three equal annual contributions to a fund that will earn 11% interest compounded annually. The first contribution will be made
on December 31, 2011. Compute the amount of this required annual contribution.
P 6-15
Bonds and leases; deferred annuities
LO3 LO7 LO9
On the last day of its fiscal year ending December 31, 2011, the Sedgwick & Reams (S&R) Glass Company completed two
financing arrangements. The funds provided by these initiatives will allow the company to expand its operations.
1. S&R issued 8% stated rate bonds with a face amount of $100 million. The bonds mature on December 31, 2031 (20 years). The market rate
of interest for similar bond issues was 9% (4.5% semiannual rate). Interest is paid semiannually (4%) on June 30 and December 31,
beginning on June 30, 2012.
2. The company leased two manufacturing facilities. Lease A requires 20 annual lease payments of $200,000 beginning on January 1, 2012.
Lease B also is for 20 years, beginning January 1, 2012. Terms of the lease require 17 annual lease payments of $220,000 beginning on
January 1, 2015. Generally accepted accounting principles require both leases to be recorded as liabilities for the present value of the
scheduled payments. Assume that a 10% interest rate properly reflects the time value of money for the lease obligations.
Required:
What amounts will appear in S&R's December 31, 2011, balance sheet for the bonds and for the leases?
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Chapter6: Time Value of Money Concepts
Broaden Your Perspective
Apply your critical-thinking ability to the knowledge you've gained. These cases will provide you an
opportunity to develop your research, analysis, judgment, and communication skills. You also will
work with other students, integrate what you've learned, apply it in real world situations, and
consider its global and ethical ramifications. This practice will broaden your knowledge and further
develop your decision-making abilities.
Ethics Case 6-1
Rate of return
LO1
The Damon Investment Company manages a mutual fund composed mostly of speculative stocks. You recently saw an ad claiming that
investments in the funds have been earning a rate of return of 21%. This rate seemed quite high so you called a friend who works for one of
Damon's competitors. The friend told you that the 21% return figure was determined by dividing the two-year appreciation on investments in the
fund by the average investment. In other words, $100 invested in the fund two years ago would have grown to $121 ($21 $100 = 21%).
Required:
Discuss the ethics of the 21% return claim made by the Damon Investment Company.
Analysis Case 6-2
Bonus alternatives; present value analysis
LO3 LO7
Sally Hamilton has performed well as the chief financial officer of the Maxtech Computer Company and has earned a bonus. She has a choice
among the following three bonus plans:
1. A $50,000 cash bonus paid now.
2. A $10,000 annual cash bonus to be paid each year over the next six years, with the first $10,000 paid now.
3. A three-year $22,000 annual cash bonus with the first payment due three years from now.
Required:
Evaluate the three alternative bonus plans. Sally can earn a 6% annual return on her investments.
Communication Case 6-3 Present value of annuities
LO7
Harvey Alexander, an all-league professional football player, has just declared free agency. Two teams, the San Francisco 49ers and the Dallas
Cowboys, have made Harvey the following offers to obtain his services:
49ers: $1 million signing bonus payable immediately and an annual salary of $1.5
million for the five-year term of the contract.
Cowboys: $2.5 million signing bonus payable immediately and an annual salary of $1
million for the five-year term of the contract.
With both contracts, the annual salary will be paid in one lump sum at the end of the football season.
Required:
You have been hired as a consultant to Harvey's agent, Phil Marks, to evaluate the two contracts. Write a short letter to Phil with your
recommendation including the method you used to reach your conclusion. Assume that Harvey has no preference between the two teams and
that the decision will be based entirely on monetary considerations. Also assume that Harvey can invest his money and earn an 8% annual
return.
Analysis Case 6-4
Present value of an annuity
LO7
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On a rainy afternoon two years ago, John Smiley left work early to attend a family birthday party. Eleven minutes later, a careening truck
slammed into his SUV on the freeway causing John to spend two months in a coma. Now he can't hold a job or make everyday decisions and is
in need of constant care. Last week, the 40-year-old Smiley won an out-of-court settlement from the truck driver's company. He was awarded
payment for all medical costs and attorney fees, plus a lump-sum settlement of $2,330,716. At the time of the accident, John was president of
his family's business and earned approximately $200,000 per year. He had anticipated working 25 more years before retirement.8
John's sister, an acquaintance of yours from college, has asked you to explain to her how the attorneys came up with the settlement amount.
They said it was based on his lost future income and a 7% rate of some kind, she explained. But it was all legal-speak to me.
Required:
How was the amount of the lump-sum settlement determined? Create a calculation that might help John's sister understand.
Replacement decision
Judgment Case 6-5
LO3 LO7
Hughes Corporation is considering replacing a machine used in the manufacturing process with a new, more efficient model. The purchase price
of the new machine is $150,000 and the old machine can be sold for $100,000. Output for the two machines is identical; they will both be used
to produce the same amount of product for five years. However, the annual operating costs of the old machine are $18,000 compared to
$10,000 for the new machine. Also, the new machine has a salvage value of $25,000, but the old machine will be worthless at the end of the five
years.
Required:
Should the company sell the old machine and purchase the new model? Assume that an 8% rate properly reflects the time value of money in
this situation and that all operating costs are paid at the end of the year. Ignore the effect of the decision on income taxes.
p. 335
Real World Case 6-6
Zero-coupon bonds; Johnson & Johnson
LO3 LO9
Johnson & Johnson is one of the world's largest manufacturers of health care products. The company's 2009 financial
statements included the following information in the long-term debt disclosure note:
Real World
Financials
The disclosure note stated that the debenture bonds were issued early in 2000 and have a maturity value of $272.5 million. The maturity value
indicates the amount that Johnson & Johnson will pay bondholders in 2020. Each individual bond has a maturity value (face amount) of $1,000.
Zero-coupon bonds pay no cash interest during the term to maturity. The company is accreting (gradually increasing) the issue price to
maturity value using the bonds' effective interest rate computed on a semiannual basis.
Required:
1. Determine the effective interest rate on the bonds.
2. Determine the issue price in early 2000 of a single, $1,000 maturity-value bond.
Real World Case 6-7
Leases; Southwest Airlines
LO3 LO9
Southwest Airlines provides scheduled air transportation services in the United States. Like many airlines, Southwest leases
many of its planes from Boeing Company. In its long-term debt disclosure note included in the financial statements for the
year ended December 31, 2008, the company listed $39 million in lease obligations. The note also disclosed that existing
leases had a three-year remaining life and that future lease payments averaged approximately $14 million per year.
Real World
Financials
Required:
1. Determine the effective interest rate the company used to determine the lease liability assuming that lease payments are made at the end of
each fiscal year.
2. Repeat requirement 1 assuming that lease payments are made at the beginning of each fiscal year.
8
This case is based on actual events.
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