Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Time Value of Money Concepts 6.

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.   

Time Value of Money

Concepts

6

6-2

Time Value of Money

Interest is therent paid for the useof money over time.

That’s right! A dollartoday is more valuable

than a dollar to bereceived 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

Assume we will save $1,000 for three years and earn 6% interest compounded annually.

What is the balance inour account at the

end of three years?

Compound Interest

6-7

Compound Interest

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

Writing in a more efficient way, we can say . . . .

$1,000 × 1.06 × 1.06 × 1.06 = $1,191.02

or

$1,000 × [1.06]3 = $1,191.02

Future Value of a Single Amount

6-11

$1,000 × [1.06]3 = $1,191.02

We can generalize this as . . .

FV = PV (1 + i)n

FutureValue

FutureValue

Present Value

Present Value

InterestRate

InterestRate

Numberof

Compounding Periods

Numberof

Compounding Periods

Future Value of a Single Amount

6-12

Find the Future Value of $1 table in

your textbook.

Future Value of a Single Amount

Find the factor for 6% and 3 periods.

6-13

Find the factor for 6% and 3 periods.

Solve our problem like this. . .

FV = $1,000 × 1.19102

FV = $1,191.02

FV $1

Future Value of a Single Amount

6-14

Learning Objectives

Compute the present value of a single amount.

6-15

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.

Present Value of a Single Amount

6-16

Remember our equation?

FV = PV (1 + i) n

We can solve for PV and get . . . .

FV

(1 + i)nPV =

Present Value of a Single Amount

6-17

Find the Present Value of $1 table in

your textbook.

Hey, it looks familiar!

Present Value of a Single Amount

6-18

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 todaytoday in order to

accumulate $20,000 in 5 years, if you can earn 8% interest compounded annually?

Present Value of a Single Amount

6-19

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.

Present Value of a Single Amount

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

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

6-22

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%

Determining the Unknown Interest Rate

6-23

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%

Determining the Unknown Interest Rate

Present Value of $1 Table$1,000 = $1,092 × ?$1,000 ÷ $1,092 = .91575Search the PV of $1 table in row 2 (n=2) for this value.

6-24

Monetary assets and monetary liabilities are valued at the

present value of future cash flows.

Accounting Applications of Present Value Techniques—Single Cash Amount

Monetary Assets

Money and claims to receive money, the

amount which is fixed or determinable

Monetary Liabilities

Obligations to pay amounts of cash, the amount of which is

fixed or determinable

6-25

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.

No Explicit Interest

6-26

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 InterestPresent Value

Expected Cash Flow Approach

6-27

Learning Objectives

Explain the difference between an ordinary annuity and an annuity due.

6-28

An annuity is a series of equal periodic payments.

Basic Annuities

6-29

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

EndEnd EndEnd

Ordinary Annuity

6-30

An annuity with payments at the beginning of the period is known as an annuity due.

Beginning Beginning Beginning

Annuity Due

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

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?

Future Value of an Ordinary Annuity

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

Compute the future value of $10,000 invested at the beginning of each of the

next four years with interest at 6% compounded annually.

Future Value of an Annuity Due

6-36

Learning Objectives

Compute the present value of an ordinary annuity, an annuity due, and a deferred

annuity.

6-37

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?

Present Value of an Ordinary Annuity

6-38

PV1PV2PV3PV4

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

1 2 3 4Today

Present Value of an Ordinary Annuity

6-39

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.

PV of $1 PresentAnnuity Factor Value

PV1 10,000$ 0.90909 9,090.90$ PV2 10,000 0.82645 8,264.50 PV3 10,000 0.75131 7,513.10 PV4 10,000 0.68301 6,830.10 Total 3.16986 31,698.60$

Present Value of an Ordinary Annuity

6-40

PV of $1 PresentAnnuity Factor Value

PV1 10,000$ 0.90909 9,090.90$ PV2 10,000 0.82645 8,264.50 PV3 10,000 0.75131 7,513.10 PV4 10,000 0.68301 6,830.10 Total 3.16986 31,698.60$

Can you find this value in the Present Value of Ordinary Annuity of $1 table?

Present Value of an Ordinary Annuity

More Efficient Computation $10,000 × 3.16986 = $31,698.60

6-41

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. $153,981

b. $171,190

c. $167,324

d. $174,680

Present Value of an Ordinary Annuity

6-42

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. $153,981

b. $171,190

c. $167,324

d. $174,680

PV of Ordinary Annuity $1Payment $ 20,000.00PV Factor × 8.55948Amount $171,189.60

Present Value of an Ordinary Annuity

6-43

Compute the present value of $10,000 received at the beginning of each of the

next four years with interest at 6% compounded annually.

Present Value of an Annuity Due

6-44

In a deferred annuity, the first cash flow is expected to occur more than one

period after the date of the agreement.

Present Value of a Deferred Annuity

6-45

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?

1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10

Present Value? $12,500 $12,500

1 2 3 4

Present Value of a Deferred Annuity

6-46

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?

1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10

Present Value? $12,500 $12,500

1 2 3 4

Present Value of a Deferred Annuity

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

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?

1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10

Present Value? $12,500 $12,500

1 2 3 4

Present Value of a Deferred Annuity

6-48

Learning Objectives

Solve for unknown values in annuity situations involving present value.

6-49

In present value problems involving annuities, there are four variables:

Solving for Unknown Values in Present Value Situations

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.

6-50Solving 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?

Today End ofYear 1

Present Value $700

End ofYear 2

End ofYear 3

End ofYear 4

6-51Solving 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?

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.

6-53

Because financial instruments typically specify equal periodic

payments, these applications quite often involve annuity situations.

Accounting Applications of Present Value Techniques—Annuities

Long-term Bonds

Long-term Leases

Pension Obligations

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 Table Table Value Amount

Present Value

Face value of the bondPV of $1

n=10; i=6% 0.5584 1,000,000$ 558,400$

Interest (annuity)

PV of Ordinary

Annuity of $1n=10; i=6% 7.3601 50,000 368,005

Price of bonds 926,405$

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?

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