mcgraw-hill /irwin© 2009 the mcgraw-hill companies, inc. time value of money concepts chapter 6

McGraw-Hill /Irwin © 2009 The McGraw-Hill Companies, Inc.

TIME VALUE OF MONEY TIME VALUE OF MONEY CONCEPTSCONCEPTS

Chapter 6

6-2

Simple InterestSimple 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 InterestCompound Interest

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

What is the balance inour account at the

end of three years?

6-4

Future Value of a Single AmountFuture 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

6-5

Future Value of a Single AmountFuture 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

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 × 1.19102FV = $1,191.02

6-6

Present Value of a Single AmountPresent 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.

6-7


Remember our equation?

FV = PV (1 + i) n

We can solve for PV and get . . . .

FV

(1 + i)nPV =

6-8


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?

6-9


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.

6-10

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 ValuesSolving for Other Values

6-11

Determining the Unknown Interest RateDetermining 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 = .91575Search the PV of $1 table in row 2 (n=2) for this value.

6-12

Monetary assets and monetary liabilities are valued at the present

value of future cash flows.

Accounting Applications of Present Value Accounting Applications of Present Value Techniques—Single Cash AmountTechniques—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

6-13

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 InterestNo Explicit Interest

6-14

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 InterestPresent Value

Expected Cash Flow ApproachExpected Cash Flow Approach

6-15

An annuity is a series of equal periodic payments.

Basic AnnuitiesBasic Annuities

6-16

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

Ordinary AnnuityOrdinary 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

6-17

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

Annuity DueAnnuity 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

6-18

Future Value of an Ordinary AnnuityFuture 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.

6-19

Future Value of an Ordinary AnnuityFuture 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?

6-20

Future Value of an Annuity DueFuture 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.

6-21

Future Value of an Annuity DueFuture 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.

6-22

Present Value of an Ordinary AnnuityPresent 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


6-24

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$


6-25

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?More Efficient Computation

$10,000 × 3.16986 = $31,698.60

6-26


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

6-27

Present Value of an Annuity DuePresent 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.

6-28

Present Value of a Deferred AnnuityPresent 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.

6-29

Present Value of a Deferred AnnuityPresent Value of a Deferred AnnuityOn January 1, 2009, you are considering an investment

that will pay $12,500 a year for 2 years beginning on December 31, 2011. If you require a 12% return on

your investments, how much are you willing to pay for this investment?

1/1/09 12/31/09 12/31/10 12/31/11 12/31/12 12/31/13

Present Value? $12,500 $12,500

1 2 3 4

6-30

Present Value of a Deferred AnnuityPresent 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.

On January 1, 2009, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2011. If you require a 12% return on


1/1/09 12/31/09 12/31/10 12/31/11 12/31/12 12/31/13


1 2 3 4

6-31

Present Value of a Deferred AnnuityPresent Value of a Deferred AnnuityOn January 1, 2009, you are considering an investment

that will pay $12,500 a year for 2 years beginning on December 31, 2011. If you require a 12% return on


1/1/09 12/31/09 12/31/10 12/31/11 12/31/12 12/31/13


1 2 3 4

6-32

Solving for Unknown Values in Present Value Solving for Unknown Values in Present Value SituationsSituations

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.

6-33


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

6-34


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?

6-35

Accounting Applications of Present Value Accounting Applications of Present Value Techniques—AnnuitiesTechniques—Annuities

Because financial instruments typically specify equal periodic

payments, these applications quite often involve annuity situations.

Long-term Bonds

Long-term Leases

Pension Obligations

6-36

Valuation of Long-term BondsValuation 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, 2009, 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, 2009.

What is the price of the bonds?

6-37

Valuation of Long-term LeasesValuation 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.

6-38

Valuation of Pension ObligationsValuation of Pension ObligationsSome 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.

mcgraw-hill /irwin© 2009 the mcgraw-hill companies, inc. time value of money concepts chapter 6

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