Time Value of
Money
Concepts
6
Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
6-2
Time Value of Money
Interest is the
rent paid for the use
of money over time.
That’s right! A dollar
today is more valuable
than a dollar to be
received in one year.
6-3
Learning Objectives
Explain the difference between simple and
compound interest.
6-4
Simple Interest
Interest amount = P × i × n
Assume you invest $1,000 at 6% simple interest
for 3 years.
You would earn $180 interest.
($1,000 × .06 × 3 = $180)
(or $60 each year for 3 years)
6-5
Compound Interest
Compound interest includes interest not only on
the initial investment but also on the
accumulated interest in previous periods.
Principal
Interest
6-6
Compound Interest
Assume we will save $1,000 for three years and
earn 6% interest compounded annually.
What is the balance in
our account at the
end of three years?
6-7
Compound Interest
Original balance
First year interest
Balance, end of year 1
$ 1,000.00
60.00
$ 1,060.00
Balance, beginning of year 2
Second year interest
Balance, end of year 2
$ 1,060.00
63.60
$ 1,123.60
Balance, beginning of year 3
Third year interest
Balance, end of year 3
$ 1,123.60
67.42
$ 1,191.02
6-8
Learning Objectives
Compute the future value of a single amount.
6-9
Future Value of a Single Amount
The future value of a single amount is the amount of
money that a dollar will grow to at some point in
the future.
Assume we will save $1,000 for three years and
earn 6% interest compounded annually.
$1,000.00 × 1.06 = $1,060.00
and
$1,060.00 × 1.06 = $1,123.60
and
$1,123.60 × 1.06 = $1,191.02
6-10
Future Value of a Single Amount
Writing in a more efficient way, we can say . . . .
$1,000 × 1.06 × 1.06 × 1.06 = $1,191.02
or
3
$1,000 × [1.06] = $1,191.02
6-11
Future Value of a Single Amount
3
$1,000 × [1.06] = $1,191.02
We can generalize this as . . .
FV = PV (1 +
Future
Value
Present
Value
n
i)
Interest
Rate
Number
of
Compounding
Periods
6-12
Future Value of a Single Amount
Find the
Future Value
of $1 table in
your
textbook.
Find the factor for 6% and 3
periods.
6-13
Future Value of a Single Amount
Find the factor for 6% and 3 periods.
Solve our problem like this. . .
FV = $1,000 × 1.19102
FV = $1,191.02
FV $1
6-14
Learning Objectives
Compute the present value of a single amount.
6-15
Present Value of a Single Amount
Instead of asking what is the future value of a
current amount, we might want to know what
amount we must invest today to accumulate a
known future amount.
This is a present value question.
Present value of a single amount is today’s
equivalent to a particular amount in the future.
6-16
Present Value of a Single Amount
Remember our equation?
FV = PV (1 + i)
n
We can solve for PV and get . . . .
PV =
FV
n
(1 + i)
6-17
Present Value of a Single Amount
Find the
Present Value
of $1 table in
your textbook.
Hey, it looks
familiar!
6-18
Present Value of a Single Amount
Assume you plan to buy a new car in 5
years and you think it will cost $20,000 at
that time.
What amount must you invest today in order to
accumulate $20,000 in 5 years, if you can
earn 8% interest compounded annually?
6-19
Present Value of a Single Amount
i = .08, n = 5
Present Value Factor = .68058
$20,000 × .68058 = $13,611.60
If you deposit $13,611.60 now, at 8% annual
interest, you will have $20,000 at the end of 5
years.
6-20
Learning Objectives
Solving for either the interest rate or the
number of compounding periods when present
value and future value of a single amount are
known.
6-21
Solving for Other Values
FV = PV (1 +
Future
Value
Present
Value
n
i)
Interest
Rate
Number
of Compounding
Periods
There are four variables needed when
determining the time value of money.
If you know any three of these, the fourth
can be determined.
6-22
Determining the Unknown Interest Rate
Suppose a friend wants to borrow $1,000 today
and promises to repay you $1,092 two years
from now. What is the annual interest rate you
would be agreeing to?
a.
3.5%
b.
4.0%
c.
4.5%
d.
5.0%
6-23
Determining the Unknown Interest Rate
Suppose a friend wants to borrow $1,000 today
and promises to repay you $1,092 two years
from now. What is the annual interest rate you
would be agreeing to?
a.
3.5%
b.
4.0%
Present Value of $1 Table
c.
4.5%
$1,000 = $1,092 × ?
d.
5.0%
$1,000 ÷ $1,092 = .91575
Search the PV of $1 table
in row 2 (n=2) for this value.
Accounting Applications of Present Value
Techniques—Single Cash Amount
Monetary assets and monetary
liabilities are valued at the
present value of future cash
flows.
Monetary
Assets
Monetary
Liabilities
Money and claims to
receive money, the
amount which is
fixed or determinable
Obligations to pay
amounts of cash, the
amount of which is
fixed or determinable
6-24
6-25
No Explicit Interest
Some notes do not include a stated
interest rate. We call these notes
noninterest-bearing notes.
Even though the agreement states it
is a noninterest-bearing note, the
note does, in fact, include interest.
We impute an appropriate interest
rate for a loan of this type to use
as the interest rate.
6-26
Expected Cash Flow Approach
Statement of Financial Accounting Concepts No. 7
“Using Cash Flow Information and Present Value in
Accounting Measurements”
The objective of
valuing an asset or
liability using
present value is to
approximate the fair
value of that asset
or liability.
Expected Cash Flow
× Risk-Free Rate of Interest
Present Value
6-27
Learning Objectives
Explain the difference between an ordinary
annuity and an annuity due.
6-28
Basic Annuities
An annuity is a series of equal periodic
payments.
6-29
Ordinary Annuity
An annuity with payments at the end of the
period is known as an ordinary annuity.
End
End
6-30
Annuity Due
An annuity with payments at the beginning of
the period is known as an annuity due.
Beginning
Beginning
Beginning
6-31
Learning Objectives
Compute the future value of both an ordinary
annuity and an annuity due.
6-32
Future Value of an Ordinary Annuity
To find the future
value of an
ordinary annuity,
multiply the
amount of a single
payment or receipt
by the future value
of an ordinary
annuity factor.
6-33
Future Value of an Ordinary Annuity
We plan to invest $2,500 at the end of each of the
next 10 years. We can earn 8%, compounded
annually, on all invested funds.
What will be the fund balance at the end of 10
years?
Am ount of annuity
$
2,500.00
Future value of ordinary annuity of $1
(i = 8%, n = 10)
Future value
×
14.4866
$
36,216.50
6-34
Future Value of an Annuity Due
To find the future
value of an annuity
due, multiply the
amount of a single
payment or receipt
by the future value
of an ordinary
annuity factor.
6-35
Future Value of an Annuity Due
Compute the future value of $10,000
invested at the beginning of each of the
next four years with interest at 6%
compounded annually.
Amount of annuity
$ 10,000
FV of annuity due of $1
(i=6%, n=4)
Future value
× 4.63710
$ 46,371
6-36
Learning Objectives
Compute the present value of an ordinary
annuity, an annuity due, and a deferred
annuity.
6-37
Present Value of an Ordinary Annuity
You wish to withdraw $10,000 at the end
of each of the next 4 years from a
bank account that pays 10% interest
compounded annually.
How much do you need to invest today to
meet this goal?
6-38
Present Value of an Ordinary Annuity
Today
PV1
PV2
PV3
PV4
1
2
3
4
$10,000
$10,000
$10,000
$10,000
6-39
Present Value of an Ordinary Annuity
PV1
PV2
PV3
PV4
Total
Annuity
$ 10,000
10,000
10,000
10,000
PV of $1
Factor
0.90909
0.82645
0.75131
0.68301
3.16986
Present
Value
$ 9,090.90
8,264.50
7,513.10
6,830.10
$ 31,698.60
If you invest $31,698.60 today you will be
able to withdraw $10,000 at the end of
each of the next four years.
6-40
Present Value of an Ordinary Annuity
PV1
PV2
PV3
PV4
Total
Annuity
$ 10,000
10,000
10,000
10,000
PV of $1
Factor
0.90909
0.82645
0.75131
0.68301
3.16986
Present
Value
$ 9,090.90
8,264.50
7,513.10
6,830.10
$ 31,698.60
Can you find this value in the Present Value of
Ordinary Annuity of $1 table?
More Efficient Computation
$10,000 × 3.16986 = $31,698.60
6-41
Present Value of an Ordinary Annuity
How much must a person 65 years old invest
today at 8% interest compounded annually to
provide for an annuity of $20,000 at the end
of each of the next 15 years?
a.
b.
c.
d.
$153,981
$171,190
$167,324
$174,680
6-42
Present Value of an Ordinary Annuity
How much must a person 65 years old invest
today at 8% interest compounded annually to
provide for an annuity of $20,000 at the end
of each of the next 15 years?
a.
b.
c.
d.
$153,981
$171,190
$167,324
$174,680
PV of Ordinary Annuity $1
Payment
$ 20,000.00
PV Factor
× 8.55948
Amount
$171,189.60
6-43
Present Value of an Annuity Due
Compute the present value of $10,000
received at the beginning of each of the
next four years with interest at 6%
compounded annually.
Amount of annuity
$ 10,000
PV of annuity due of $1
(i=6%, n=4)
Present value of annuity
× 3.67301
$ 36,730
6-44
Present Value of a Deferred Annuity
In a deferred annuity, the first cash flow
is expected to occur more than one
period after the date of the
agreement.
6-45
Present Value of a Deferred Annuity
On January 1, 2006, you are considering an investment
that will pay $12,500 a year for 2 years beginning on
December 31, 2008. If you require a 12% return on
your investments, how much are you willing to pay for
this investment?
Present
Value?
1/1/06
1
2
12/31/06
1
Payment
$
12,500
12,500
12/31/07
2
$12,500
$12,500
12/31/08
3
12/31/09
4
PV of $1
i = 12%
0.71178
0.63552
$
$
12/31/10
PV
8,897
7,944
16,841
n
3
4
6-46
Present Value of a Deferred Annuity
On January 1, 2006, you are considering an investment
that will pay $12,500 a year for 2 years beginning on
December 31, 2008. If you require a 12% return on
your investments, how much are you willing to pay for
this investment?
Present
Value?
1/1/06
12/31/06
1
12/31/07
2
$12,500
$12,500
12/31/08
3
12/31/09
4
12/31/10
More Efficient Computation
1.
Calculate the PV of the annuity as of the beginning of the annuity
period.
2.
Discount the single value amount calculated in (1) to its present
value as of today.
6-47
Present Value of a Deferred Annuity
On January 1, 2006, you are considering an investment
that will pay $12,500 a year for 2 years beginning on
December 31, 2008. If you require a 12% return on
your investments, how much are you willing to pay for
this investment?
Present
Value?
1/1/06
Payment
$
12,500
12/31/06
1
PV of
Ordinary
Annuity of $1
n=2, i = 12%
1.69005
$
12/31/07
2
PV
21,126
$12,500
$12,500
12/31/08
3
12/31/09
4
Future Value
$
21,126
12/31/10
PV of $1
n=2, i = 12%
0.79719
$
PV
16,841
6-48
Learning Objectives
Solve for unknown values in annuity situations
involving present value.
Solving for Unknown Values in Present
Value Situations
In present value problems involving annuities,
there are four variables:
Present value of an
ordinary annuity or
Present value of an
annuity due
The amount of the
annuity payment
The number of
periods
The interest rate
If you know any three of these, the fourth can be
determined.
6-49
Solving for Unknown Values in Present
Value Situations
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?
Present
Value
$700
Today
End of
Year 1
End of
Year 2
End of
Year 3
End of
Year 4
6-50
Solving for Unknown Values in Present
Value Situations
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?
Present value
$ 700.00
PV of ordinary annuity of $1
(i=8%, n=4)
Annuity amount
÷ 3.31213
$ 211.34
6-51
6-52
Learning Objectives
Briefly describe how the concept of the time
value of money is incorporated into the
valuation of bonds, long-term leases, and
pension obligations.
Accounting Applications of Present Value
Techniques—Annuities
Because financial instruments
typically specify equal periodic
payments, these applications quite
often involve annuity situations.
Long-term
Bonds
Long-term
Leases
Pension
Obligations
6-53
6-54
Valuation of Long-term Bonds
Calculate the Present Value
of the Lump-sum Maturity
Payment (Face Value)
Calculate the Present Value
of the Annuity Payments
(Interest)
Cash Flow
Face value of the bond
Interest (annuity)
Price of bonds
On January 1, 2006, Fumatsu Electric
issues 10% stated rate bonds with a
face value of $1 million. The bonds
mature in 5 years. The market rate of
interest for similar issues was 12%.
Interest is paid semiannually beginning
on June 30, 2006. What is the price of
the bonds?
Table
PV of $1
n=10; i=6%
PV of
Ordinary
Annuity of $1
n=10; i=6%
Table
Value
Amount
0.5584 $ 1,000,000
7.3601
Present
Value
$
558,400
$
368,005
926,405
50,000
6-55
Valuation of Long-term Leases
Certain long-term
leases require the
recording of an asset
and corresponding
liability at the present
value of future lease
payments.
6-56
Valuation of Pension Obligations
Some pension plans
create obligations during
employees’ service periods
that must be paid during
their retirement periods.
The amounts contributed
during the employment
period are determined
using present value
computations of the
estimate of the future
amount to be paid during
retirement.
6-57
End of Chapter 6