Accounting principles, 13th edition appxg

Accounting Principles
Thirteenth Edition
Weygandt Kimmel Kieso

Appendix G

Time Value of Money
Prepared by

Coby Harmon
University of California, Santa Barbara
Westmont College


Chapter Outline
Learning Objectives
LO 1 Compute interest and future values.
LO 2 Compute present values.
LO 3 Compute the present value in capital budgeting situations.
LO 4 Use a financial calculator to solve time value of money problems.

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2


Interest and Future Values
Nature of Interest
Payment for the use of money
Difference between amount borrowed or invested (principal) and amount repaid or
collected

Elements involved in financing transaction:

1.

Principal (p): Amount borrowed or invested.

2.

Interest Rate (i): An annual percentage.

3.

Time (n): Number of years or portion of a year that the principal is borrowed or
invested.

LO 1

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3


Nature of Interest
Simple Interest
Interest computed on principal only
Illustration: Assume you borrow $5,000 for 2 years at a simple interest rate of 12% annually. Calculate the
annual interest cost.
ILLUSTRATION G.1

Principal

Interest

LO 1

=

p

=

$5,000

=

$1,200

Rate
x

Time
x

i
x

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0.12

n

x

2

4


Nature of Interest
Compound Interest
Computes interest on

.

principal

.

interest earned that has not been paid or withdrawn

Most business situations use compound interest

LO 1

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5


Compound Interest
Illustration: Assume you deposit $1,000 in Bank Two, where it will earn simple interest of 9%

per year, and you deposit another $1,000 in Citizens Bank, where it will earn compound
interest of 9% per year compounded annually. Also assume that in both cases you will not
withdraw any cash until three years from the date of deposit.
Compute the interest to be received and the accumulated year-end balances for Citizens
Bank.

LO 1

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6


ILLUSTRATION G.2

Compound Interest

Simple versus compound interest

Bank Two
Simple Interest

Simple

Accumulated

Calculation

Interest

Year-End Balance

Year 1 $1,000 x 9%

$ 90.00

$1,090.00

Year 2 $1,000 x 9%

90.00

$1,180.00

Year 3 $1,000 x 9%

90.00

$1,270.00

$270.00

Citizens Bank
Compound

Compound

Accumulated

Interest Calculation

Interest

Year-End Balance

Year 1 $1,000 x 9%

$ 90.00

$1,090.00

Year 2 $1,090 x 9%

98.10

$1,188.10

Year 3 $1,188 x 9%

106.93

$1,295.03

$25.03

$295.03
LO 1

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7


Future Value of a Single Amount
Value at a future date of a given amount invested, assuming compound interest.

FV = p × (1 + i)

LO 1

n

FV

=

future value of a single amount

p

=

principal (or present value; the value today)

i

=

interest rate for one period

n

=

number of periods

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8


Future Value of a Single Amount
Illustration: The future value of a $1,000 investment earning 9% for three years is $1,295.03.
FV

=

p

n
× (1 + i)

3
= $1,000 × (1 + .09)
= $1,000 × 1.29503
= $1,295.03
ILLUSTRATION G.4
Time diagram

Present

Value (p)

0
$1,000

LO 1

Future

i = 9%

1

Value

2
n = 3 years

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3
$1,295.03

9


Future Value of a Single Amount
Another method to compute the future value of a single amount involves a compound interest table.

Table 1 Future Value of 1

(n) Periods

5%

6%

7%

8%

9%

10%

1

1.05000

1.06000

1.07000

1.08000

1.09000

1.10000

2

1.10250

1.12360

1.14490

1.16640

1.18810

1.21000

3

1.15763

1.19102

1.22504

1.25971

1.29503

1.33100

4

1.21551

1.26248

1.31080

1.36049

1.41158

1.46410

5

1.27628

1.33823

1.40255

1.46933

1.53862

1.61051

$1,000
Present Value
LO 1

x

1.29503
Factor
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=

$1,295.03
Future Value
10


Future Value of a Single Amount
Illustration: John and Mary Rich invested $20,000 in a savings account paying 6% interest at the time their
son, Mike, was born. The money is to be used by Mike for his college education. On his 18th birthday, Mike
withdraws the money from his savings account. How much did Mike withdraw from his account?

Present
Value (p)

0
$20,000

Future

i = 6%

1

2

3

4

5

6

7

8

9

Value = ?

10

11

12

13

14

15

16

17

18

n = 18 years
ILLUSTRATION G.5

LO 1

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11


Future Value of a Single Amount
Table 1 Future Value of 1
(n) Periods

5%

6%

7%

8%

9%

10%

1

1.05000

1.06000

1.07000

1.08000

1.09000

1.10000

2

1.10250

1.12360

1.14490

1.16640

1.18810

1.21000

16

2.18287

2.54035

2.95216

3.42594

3.97031

4.59497

17

2.29202

2.69277

3.15882

3.70002

4.32763

5.05447

18

2.40662

2.85434

3.37993

3.99602

4.71712

5.55992

19

2.52695

3.02560

3.61653

4.31570

5.14166

6.11591

20

2.65330

3.20714

3.86968

4.66096

5.60441

6.72750

$20,000
Present Value
LO 1

x

2.85434
Factor
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=

$57,086.80
Future Value
12


Future Value of an Annuity
Illustration: Assume you invest $2,000 at the end of each year for three years at 5% interest compounded
annually.
Future
i = 5%

Value = ?

Present
Value (p)

$2,000

$2,000

$2,000

0

1

2

3

n = 3 years
ILLUSTRATION G.6
Time diagram for a three-year annuity

LO 1

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13

Future Value of an Annuity
Table 1 Future Value of 1
(n) Periods

4%

5%

6%

7%

8%

9%

0

1.00000

1.00000

1.00000

1.00000

1.00000

1.00000

1

1.04000

1.05000

1.06000

1.07000

1.08000

1.09000

2

1.08160

1.10250

1.12360

1.14490

1.16640

1.18810

3

1.12486

1.15763

1.19102

1.22504

1.25971

1.29503

Invested

Number of

Future

at End

Compounding

Amount

of Year

Periods

Invested

1

2

$2,000

1.10250

$2,205

2

1

$2,000

1.05000

2,100

3

0

$2,000

1.00000

2,000

3.15250

$6,305

Value of 1
x

Factor at 5%

Future
=

Value

ILLUSTRATION G.7
Future value of periodic
LO 1

payment computation

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14


Future Value of an Annuity
When periodic payments (receipts) are the same in each period, the future value can be
computed by using a future value of an annuity of 1 table.

Table 2 Future Value of an Annuity of 1

LO 1

(n) Periods

5%

6%

7%

8%

9%

10%

1

1.00000

1.00000

1.00000

1.00000

1.00000

1.00000

2

2.05000

2.06000

2.07000

2.08000

2.09000

2.10000

3

3.15250

3.18360

3.21490

3.24640

3.27810

3.31000

4

4.31013

4.37462

4.43994

4.50611

4.57313

4.64100

5

5.52563

5.63709

5.75074

5.86660

5.98471

6.10510

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15


Future Value of an Annuity
Illustration: John and Char Lewis’s daughter, Debra, has just started high school. They decide to start a
college fund for her and will invest $2,500 in a savings account at the end of each year she is in high school
(4 payments total). The account will earn 6% interest compounded annually. How much will be in the
college fund at the time Debra graduates from high school?

Future
Value = ?

i = 6%

Present
Value (p)

$2,500

$2,500

$2,500

$2,500

0

1

2

3

4

ILLUSTRATION G.8

n = 4 years

Using Table 2 for FV of an annuity of 1
LO 1

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16


Future Value of an Annuity
Table 2 Future Value of an Annuity of 1
(n) Periods

5%

6%

7%

8%

9%

10%

1

1.00000

1.00000

1.00000

1.00000

1.00000

1.00000

2

2.05000

2.06000

2.07000

2.08000

2.09000

2.10000

3

3.15250

3.18360

3.21490

3.24640

3.27810

3.31000

4

4.31013

4.37462

4.43994

4.50611

4.57313

4.64100

5

5.52563

5.63709

5.75074

5.86660

5.98471

6.10510

What factor do we use?
$2,500
Payment
LO 1

x

4.37462
Factor

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=

$10,936.55
Future Value

17


Present Values
Present Value Variables
Present value is the value now of a given amount to be paid or received in the future,
assuming compound interest.
Present value variables:

LO 2

1.

Dollar amount to be received (future amount).

2.

Length of time until amount is received (number of periods).

3.

Interest rate (the discount rate).

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18


Present Value of a Single Amount
Present Value (PV) = Future Value (FV) ÷ (1 + i)

LO 2

p

=

principal (or present value)

i

=

interest rate for one period

n

=

number of periods

n

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19


Present Value of a Single Amount
Illustration: The computation of $1,000 discounted at 10% for one year is as follows.

PV

=

FV

n
÷ (1 + i)

= $1,000

1
÷ (1 + .10)

= $1,000

÷ 1.10

= $909.09

ILLUSTRATION G.10
Finding present value if
discounted for one period

Present
Value (?)

$909.09

LO 2

Future
i = 10%

n = 1 years

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Value

$1,000

20


Alternate Calculation

PV of a Single Amount
Table 3 Present Value of 1
(n) Periods

6%

7%

8%

9%

10%

11%

1

.94340

.93458

.92593

.91743

.90909

.90090

2

.89000

.87344

.85734

.84168

.82645

.81162

3

.83962

.81630

.79383

.77218

.75132

.73119

4

.79209

.76290

.73503

.70843

.68301

.65873

5

.74726

.71299

.68058

.64993

.62092

.59345

What factor do we use?
$1,000
Future Value
LO 2

x

.90909
Factor

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=

$909.09
Present Value

21


Present Value of a Single Amount
Illustration: If the single amount of $1,000 is to be received in two years and discounted at 10% [PV =
2
$1,000 ÷ (1 + .10) ], its present value is $826.45 [($1,000 ÷ 1.21).

Present

i = 10%

Value (?)

0

Value

1

$826.45
ILLUSTRATION G.11

Future

2
$1,000

n = 2 years

What table do we use?

LO 2

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22


Alternate Calculation

PV of a Single Amount
Table 3 Present Value of 1
(n) Periods

6%

7%

8%

9%

10%

11%

1

.94340

.93458

.92593

.91743

.90909

.90090

2

.89000

.87344

.85734

.84168

.82645

.81162

3

.83962

.81630

.79383

.77218

.75132

.73119

4

.79209

.76290

.73503

.70843

.68301

.65873

5

.74726

.71299

.68058

.64993

.62092

.59345

What factor do we use?
$1,000
Future Value
LO 2

x

.82645
Factor

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=

$826.45
Present Value

23


Present Value of a Single Amount
Illustration: Suppose you have a winning lottery ticket and the state gives you the option of taking $10,000
3 years from now or taking the present value of $10,000 now. The state uses an 8% rate in discounting.

How much will you receive if you accept your winnings now?

What table do we use?

LO 2

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24


PV of a Single Amount
Table 3 Present Value of 1
(n) Periods

6%

7%

8%

9%

10%

11%

1

.94340

.93458

.92593

.91743

.90909

.90090

2

.89000

.87344

.85734

.84168

.82645

.81162

3

.83962

.81630

.79383

.77218

.75132

.73119

4

.79209

.76290

.73503

.70843

.68301

.65873

5

.74726

.71299

.68058

.64993

.62092

.59345

What factor do we use?
$10,000
Future Value
LO 2

x

.79383
Factor

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=

$7,938.30
Present Value

25