6
Time Value of Money Concepts
PowerPoint Authors:
Susan Coomer Galbreath, Ph.D., CPA
Charles W. Caldwell, D.B.A., CMA
Jon A. Booker, Ph.D., CPA, CIA
Cynthia J. Rooney, Ph.D., CPA
McGraw-Hill/Irwin
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.

6 - 2
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 - 3
Compound Interest
Assume we deposit $1,000 in a bank that earns
6% interest compounded annually.
What is the balance in our account at the end of three years?

Compound Interest
Original balance
First year interest ($1,000.00
× 6%)
Balance, end of year 1
Balance, beginning of year 2
Second year interest ($1,060.00 × 6%)
Balance, end of year 2
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Balance, beginning of year 3
Third year interest ($1,123.60 × 6%)
Balance, end of year 3
$ 1,000.00
60.00
$ 1,060.00
$ 1,060.00
63.60
$ 1,123.60
$ 1,123.60
67.42
$ 1,191.02

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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 deposit $1,000 for three years that earns 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

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Future Value of a Single Amount
Writing in a more efficient way, we can say . . . .
$1,191.02 = $1,000 × [1.06] 3
FV = PV (1 + i
) n
Future
Value
Amount
Invested at the
Beginning of the Period
Interest
Rate
Number of
Compounding
Periods

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Future Value of a Single Amount
Using the Future Value of $1 Table, we find the factor for 6% and 3 periods is 1.19102.
So, we can solve our problem like this. . .
FV = $1,000 × 1.19102
FV = $1,191.02

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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.

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Present Value of a Single Amount
Remember our equation?
FV = PV (1 + i) n
We can solve for PV and get . . . .
PV =
FV
(1 + i
) n

6 - 10
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?

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Present Value of a Single Amount i = .08, n = 5
Present Value Factor = .68058
$20,000 × .68058 = $13,611.60
Present Value of $1 Table
If you deposit $13,611.60 now, at
8% annual interest, you will have
$20,000 at the end of 5 years.

Solving for Other Values
FV = PV (1 + i
) n
Number of Compounding
Periods Future
Value
Present
Value
Interest
Rate
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There are four variables needed when determining the time value of money.
If you know any three of these, the fourth can be determined.

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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%
Present Value of $1 Table
$1,000 = $1,092 × ?
$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
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 .
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We impute an appropriate interest rate for noninterest-bearing notes.

6 - 15
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
Credit-Adjusted Risk-Free
Rate of Interest
Present Value

Basic Annuities
An annuity is a series of equal periodic payments.
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Period 1 Period 2 Period 3 Period 4
$10,000 $10,000 $10,000 $10,000

Ordinary Annuity
An annuity with payments at the end of the period is known as an ordinary annuity .
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Today 1 2 3 4
$10,000
End of year 1
$10,000
End of year 2
$10,000
End of year 3
$10,000
End of year 4

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Annuity Due
An annuity with payments at the beginning of the period is known as an annuity due .
Today 1
$10,000
2
$10,000
3
$10,000
4
$10,000
Beginning of year 1
Beginning of year 2
Beginning of year 3
Beginning of year 4

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Future Value of an Ordinary Annuity
To find the future value of an ordinary annuity , multiply the amount of the annuity by the future value of an ordinary annuity factor.

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 interest annually, on all invested funds.
What will be the fund balance at the end of 10 years?
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Am ount of annuity
Future value of ordinary annuity of $1
(i = 8%, n = 10)
Future value
×
$ 2,500.00
14.4866
$ 36,216.50

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Future Value of an Annuity Due
To find the future value of an annuity due , multiply the amount of the annuity by the future value of an annuity due factor.

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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.
Am ount of annuity
FV of annuity due of $1
(i=6%, n=4)
Future value
$ 10,000
× 4.63710
$ 46,371

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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?

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Present Value of an Ordinary Annuity
Today 1
$10,000
PV1
PV2
PV3
PV4
2
$10,000
3
$10,000
4
$10,000

Present Value of an Ordinary Annuity
PV1
PV2
PV3
PV4
Total
PV of $1
Annuity Factor
$ 10,000
10,000
10,000
10,000
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
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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.

Present Value of an Ordinary Annuity
PV1
PV2
PV3
PV4
Total
PV of $1
Annuity Factor
$ 10,000
10,000
10,000
10,000
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
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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

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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.
Am ount of annuity
PV of annuity due of $1
(i=6%, n=4)
Present value of annuity
$ 10,000
× 3.67301
$ 36,730

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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