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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?
Illustration 6-10
Future
Value
?
End of End of End of
0 year 1 year 2 year 3
____________________________________________
$10,000 $10,000 $10,000
FV of $1 Future Value
Payment i=10% (at the end of year 3) n
First payment $10,000 x 1.331 = $13,310 3
T6-13
6-18 Intermediate Accounting, 8/e
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.
Illustration 6-11
Present Value
PV of $1 (at the beginning n
Payment i=10% of year 1)
First payment $10,000 x .90909 = $9,091 1
* Present value of an ordinary annuity of $1: n=3, i=10%
Present Future
Value Value
$24,868 $33,100
End of End of End of
0 year 1 year 2 year 3
____________________________________________
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.
What is the present value?
Illustration 6-13
Present
Value
?
End of End of End of
0 year 1 year 2 year 3
_____________________________________________
PVAD = $10,000 x 2.73554* = $27,355 (rounded)
annuity
amount
* Present value of an annuity due of $1: n=3, i=10%
Present Future
Value Value
$27,355 $36,410
End of End of End of
0 year 1 year 2 year 3
____________________________________________
$10,000 $10,000 $10,000
T6-15
6-20 Intermediate Accounting, 8/e
PRESENT VALUE OF A DEFERRED ANNUITY
At January 1, 2016, 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, 2018. The time value of money is 10%.
How much would you be willing to pay for this investment?
Illustration 6-14
Present
Value i = 10%
?
1/1/16 12/31/16 12/31/17 12/31/18 12/31/19 12/31/20
$10,000 $10,000 $10,000
Present Value
PV of $1 (at the beginning n
Payment i=10% of year 1)
First payment $10,000 x .75131 = $ 7,513 3
T6-16
PRESENT VALUE OF A DEFERRED ANNUITY
(continued)
or,
1. Calculate the PV of the annuity as of the beginning of the annuity
period.
PVA = $10,000 x 2.48685* = $24,868
2. Discount the single amount calculated in 1 to its present value as of
today.
PV = $24,868 x .82645* = $20,552
future
amount
Present Present Value
Value at beginning of i = 10%
the annuity period
$20,552 $24,868
1/1/16 12/31/16 12/31/17 12/31/18 12/31/19 12/31/20
$10,000 $10,000 $10,000
T6-16 (continued)
6-22 Intermediate Accounting, 8/e
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?
Illustration 6-15
Present
Value
$700 End of End of End of End of
0 year 1 year 2 year 3 year 4
____________________________________
________________________________________________
? ? ? ?
n = 4, i = 8%
$700 = 3.31213* x Annuity amount
present
value
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?
Illustration 6-16
Present
Value
$700 End of End of End of End of
0 year 1 year 2 year n-1 year n
________________ ________________
____________________________ ________________
n = ?, i = 7%
$700 = $100 x ? *
present annuity
value amount
When you consult the present value of an ordinary annuity 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.
T6-18
6-24 Intermediate Accounting, 8/e
VALUATION OF LONG-TERM BONDS
On June 30, 2016, Fumatsu Electric issued 10% stated rate bonds with
a face amount of $200 million. The bonds mature on June 30, 2036 (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, 2016. The interest payment is $10 million (5% X $200
million). What was the price of the bond issue? What amount of interest
expense will Fumatsu record on the bonds in 2016?
Illustration 6-19
T6-19
VALUATION OF LEASES
On January 1, 2016, the Stridewell Wholesale Shoe Company signed
a 25-year 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,
2016. Assuming that 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?
Illustration 6-20
PVAD = $10,000 x 9.98474* = $99,847
annuity
amount
T6-20
6-26 Intermediate Accounting, 8/e
VALUING A PENSION OBLIGATION
On January 1, 2016, the Stridewell Wholesale Shoe Company hired Sammy
Sossa. Sammy is expected to work for 25 years before retirement on December
31, 2040. 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, 2041. During 2016, 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 2016 to pay for pension benefits earned in 2016?
Illustration 6-21
Present
Value i = 6%
?
2016 2017 2040 2041 2042 2060
____________________ _____________________________________ ______
$2,000 $2,000 $2,000
n = 24 n = 20
PVA = $2,000 x 11.46992* = $22,940
annuity
amount
T6-21
SUMMARY OF TIME VALUE OF MONEY CONCEPTS
Concept
Summary
Formula
Table
Future value (FV) of $1
The amount of money that a
dollar will grow to at some
point in the future.
FV = $1(1 + i)n
1
Future value of an ordinary
annuity (FVA) of $1
The future value of a series
of equal-sized cash flows
with the first payment taking
place at the end of the first
compounding period.
FVA = (1+i)n-1
i
3
Future value of an annuity
due (FVAD) of $1
The future value of a series
of equal-sized cash flows
with the first payment taking
place at the beginning of the
annuity period.
FVAD = [(1+i)n-1
i] x (1+i)
5
T6-22
6-28 Intermediate Accounting, 8/e
Suggestions for Class Activities
1. Real World Scenario
An important accounting application of present value techniques illustrated in this chapter is the
valuation of bonds. Occasionally, corporations issue zero-coupon bonds or notes. These
instruments pay no stated interest over their life. For example, in 1997, Costco Wholesale
Corporation issued 20-year, zero coupons bonds with a maturity value of $900 million. The
company received $450 million upon issuance of these bonds.
Suggestions:
Have the class consider the Costco zero-coupon bonds. What is the approximate interest rate
implicit in these bonds when they were issued in 1997?
Points to note:
Students must solve for the unknown interest rate when PV and FV are known. In millions:
$450 = $900 x ? *
* Present value of $1: n =20 , i = ?
When you consult the present value table, Table 2, you search row 20 (n = 20) for this value. The
rate is approximately 3.5%. The company’s debt disclosure note reports that these notes were sold
to yield 3.5%.
2. Professional Skills Development Activities
The following are suggested assignments from the end-of-chapter material that will help your
students develop their communication, research, analysis, and judgment skills.
Communication Skills. In addition to Communication Case 6-3, Judgment Case 6-5 can be
adapted to ask students to choose one of the two alternatives and write a memo supporting their
Research Skills. In their careers, our graduates will be required to locate and extract relevant
information from available resource material to determine the correct accounting practice,
Analysis Skills. The “Broaden Your Perspective” section includes Analysis Cases that direct
students to gather, assemble, organize, process, or interpret data to provide options for making
Judgment Skills. The “Broaden Your Perspective” section includes Judgment Cases that require
6-30 Intermediate Accounting, 8/e
Assignment Chart
Learning Est. time
Questions Objective(s) Topic (min.)
6-1
1
Interest
5
6-2
1
Compound interest
5
6-3
1
Effective rate or yield
5
6-4
2
Future value
5
6-6
ó
Monetary versus nonmonetary assets and
liabilities
5
6-7
5
Annuity
5
6-8
5
Ordinary annuity versus an annuity due
5
6-9
3,5
Present value table relationships
5
6-10
5
Time diagram-ordinary annuity
5
6-12
7
Deferred annuity
5
6-13
8
Explain how to compute unknown annuity
payment
5
6-14
8
Compute unknown annuity payment
5
6-15
9
Long-term leases
5
Brief Learning Est. time
Exercises Objective(s) Topic (min.)
6-1
1
Simple versus compound interest
5
6-2
2
Future value; single amount
5
6-3
4
Future value; solving for unknown single amount
5
6-4
3
Present value; single amount
5
6-6
6
Future value; ordinary annuity
5
6-7
6
Future value; annuity due
5
6-8
7
Present value; ordinary annuity
5
6-10
7
Deferred annuity
10
6-11
8
Solve for unknown; annuity
5
6-12
9
Price of a bond
10
Learning Est. time
Exercises Objective(s) Topic (min.)
6-1
2
Future value; single amounts
10
6-2
2
Future value; single amounts
10
6-3
3
Present value; single amounts
10
6-4
3
Present value; multiple, unequal amounts
10
6-5
3
Noninterest-bearing note; single payment
10
6-7
6
Future value; annuities
20
6-8
7
Present value; annuities
10
6-9
8
Solving for unknowns; annuities
20
6-10
4,8
Future value; solving for annuities and single
amounts
15
6-11
3,6,7
Future and present value
20
6-13
8
Solving for unknown annuity payment
10
6-14
8
Solving for unknown interest rate
10
6-15
8
Solving for unknown annuity amount
10
6-16
7,8
Deferred annuities; solving for annuity amount
15
6-17
9
Price of a bond
10
6-18
9
Price of a bond; interest expense
15
6-19
9
Lease payments
10
6-21
1,2,3,5
Concepts; terminology
15
CPA/CMA Learning Est. time
Exam Questions Objective(s) Topic (min.)
CPA-1
3
Present value; single amount
3
CPA-3
7
Present value; annuities
3
CPA-4
7
Present value; annuities
3
CPA-5
7
Present value; annuities
3
CPA-6
9
Price of a bond
3
CPA-7
8
Solving for unknowns; annuities
3
CMA-2
5,9
Lease payments
3
6-32 Intermediate Accounting, 8/e
Learning Est. time
Problems Objective(s) Topic (min.)
6-1
3,7
Analysis of alternatives
20
6-3
3,7
Analysis of alternatives
15
6-4
3,7
Investment analysis
20
6-5
3,7
Investment decision; varying rates
25
6-6
3,8
Solving for unknowns
20
6-7
8
Solving for unknowns
20
6-9
7
Deferred annuities
15
6-10
3,7
Noninterest-bearing note; annuity and lump-sum
payment
10
6-11
8,9
Solving for unknown lease payment
20
6-13
3,7,9
Lease vs. buy alternatives
15
6-14
7,9
Deferred annuities; pension obligation
25
6-15
3,7,9
Bonds and leases; deferred annuities
30
Star Problems
Learning Est. time
Cases Objective(s) Topic (min.)
Ethics Case 6-1
1
Return on investment
20
Analysis Case 6-2
3,7
Bonus alternatives; present value analysis
15
Communication Case 6-3
7
Present value of annuities
60
Analysis Case 6-4
7
Present value of an annuity
15
