Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.

PowerPoint Authors:Susan Coomer Galbreath, Ph.D., CPACharles W. Caldwell, D.B.A., CMAJon A. Booker, Ph.D., CPA, CIACynthia J. Rooney, Ph.D., CPA

Time Value of Money Concepts

Chapter 6

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)

Balance after 3 years is $1000 + $180 = $1,180

6-3

Compound Interest

Assume we deposit $1,000 in a bank that earns 6% interest compounded annually.

What is the balance inour account at theend of three years?

6-4

Compound Interest

Original balance 1,000.00$ First year interest ($1,000.00 × 6%) 60.00 Balance, end of year 1 1,060.00$

Balance, beginning of year 2 1,060.00$ Second year interest ($1,060.00 × 6%) 63.60 Balance, end of year 2 1,123.60$

Balance, beginning of year 3 1,123.60$ Third year interest ($1,123.60 × 6%) 67.42 Balance, end of year 3 1,191.02$

6-5

Future Value of a Single AmountThe 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

6-6

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

FutureValue

FutureValue

Amount Invested at

the Beginning

of the Period

Amount Invested at

the Beginning

of the Period

InterestRate

InterestRate

Numberof

Compounding

Periods

Numberof

Compounding

Periods

6-7

Excel Solution=FV(6%, 3, 0, 1000)

Future Value of a Single Amount

6-8

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.

6-9

Present Value of a Single Amount

Remember our equation?

FV = PV × (1 + i) n

We can solve for PV and get . . . .

FV

(1 + i)nPV =

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?

6-11

Present Value of a Single Amount

If you deposit $13,611.60 now, at8% annual interest, you will have

$20,000 at the end of 5 years.

Excel Solution:

=PV(8%, 5, 0, 20000)

6-12

FV = PV × (1 + i)n

FutureValue

FutureValue

PresentValue

PresentValue

InterestRate

InterestRate

Numberof Compounding

Periods

Numberof 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.

Solving for Other Values

6-13Determining 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%

Excel Solution

=RATE(2, 0, -1000, 1092)

6-14

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 noninterest-

bearing notes.

Accounting Applications of Present Value Techniques—Single Cash Amount

6-15

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 Approach

The present value of the asset or liability is obtained by

discounting the cash flow(s) using the company’s credit-

adjusted risk-free rate of interest.

6-16

Examples: Valuing a future asset/liability

ABC Company sells services to XYZ company in the amount of $100,000, with payment to be made five years after the date of sale. The appropriate discount rate for both companies is 10%.

How do ABC and XYZ record this transaction:

1. On the date of sale?

2. At the end of years 1-4?

3. At the end of year 5 when payment is due?

6-17

Basic Annuities

An annuity is a series

of equal periodic payments.

Period 1 Period 2 Period 3 Period 4

$10,000 $10,000 $10,000 $10,000

6-18

An annuity with payments at the end of the period is known as an ordinary

annuity.

Ordinary Annuity

End of year 1

$10,000 $10,000 $10,000 $10,000

1 2 3 4Today

End of year 2

End of year 3

End of year 4

6-19

Annuity DueAn annuity with payments at the

beginning of the period is known as an annuity due.

Beginning of year 1

$10,000 $10,000 $10,000 $10,000

1 2 3 4Today

Beginning of year 2

Beginning of year 3 Beginning

of year 4

6-20

Future Value of an Ordinary AnnuityWe 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?

Excel Solution: =FV(8%, 10, 2500) = $36,216

How would the solution/answer change if interest were compounded quarterly or

monthly?

6-21

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.

Excel Solution: =FV(6%, 4, 10000, 0, 1)

= $46,371

6-22

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?

6-23

PV1PV2PV3PV4

$10,000 $10,000 $10,000 $10,000

1 2 3 4Today

Present Value of an Ordinary Annuity

6-24

Present Value of an Ordinary Annuity

If you invest $31,699 today you will be able to withdraw $10,000 at the end of each of the next four

years.

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?

Excel Solution:

=PV(10%, 4, 10000)

= $31,699

6-25

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.

Excel Solution: =PV(6%, 4, 10000, 0, 1)

= $36,730

6-26

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.

6-27

Present Value of a Deferred Annuity

1/1/13 12/31/13 12/31/14 12/31/15 12/31/16 12/31/17

Present Value? $12,500 $12,500

1 2 3 4

On January 1, 2013, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2015. If

you require a 12% return on your investments, how much are you willing to pay

for this investment?

6-28

Correct Solution Process

1. Calculate the present value of the annuity as of the beginning of the annuity period. Excel: =PV(12%, 2, 12500) = $21,126

2. Discount the single value amount calculated in (1) to its present value as of today. Excel: =PV(12%, 2, 0, 21126) = $16,842

Present Value of a Deferred AnnuityOn January 1, 2013, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2015. If

you require a 12% return on your investments, how much are you willing to pay

for this investment?

1/1/13 12/31/13 12/31/14 12/31/15 12/31/16 12/31/17

Present Value? $12,500 $12,500

1 2 3 4

6-29Solving for Unknown Values in

Present Value of Annuity Situations

In present value problems involvingannuities, 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.

6-30Solving for Unknown Values

in Present Value Situations

Today End ofYear 1

Present Value $700

End ofYear 2

End ofYear 3

End ofYear 4

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?

6-31Solving 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?

Excel Solution: =PMT(8%, 4, 700)= $211.34

6-32Accounting 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

6-33

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.

6-34

Valuation of Long-term Leases

On January 1, 2013, Todd Furniture Company signed a 20-year lease for a new retail showroom. The lease agreement calls for annual payments of $25,000 for

20 years beginning on January 1, 2013. The appropriate rate of interest for this long-term lease is 8%. Calculate the value of the asset acquired and the

liability assumed by Todd (the present value of an annuity due at 8% for 20 years).

Excel Solution: =PV(8%, 20, 25000, 0, 1)

= $265,090

What journal entries are made?