6-1

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 inour account at theend of three years?

6-4

Compound Interest

6-5

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

6-6

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

Future Value of a Single Amount

6-7

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.

6-8

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 =

6-9

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 todaytoday in order to accumulate $20,000 in 5 years, if you can earn 8% interest compounded

annually?

6-10

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.

i = .08, n = 5Present Value Factor = .68058

$20,000 × .68058 = $13,611.60

Present Value of $1 Table

6-11

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

6-12Determining 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.

6-13

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

6-14

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

6-15

Annuity Due

An 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

6-16Present 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?

6-17

PV1PV2PV3PV4

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

1 2 3 4Today

Present Value of an Ordinary Annuity

6-18

Present Value of an Ordinary Annuity

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$

6-19

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

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

6-20

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.

6-21Solving for Unknown Values in

Present Value of Annuity Situations

In present value problems involvingannuities, there are four variables:

Present value of Present value of an ordinary an ordinary annuity or annuity or

present value of present value of an annuity duean annuity due

The amount of The amount of the annuity the annuity

paymentpayment

The number of The number of periodsperiods The interest rateThe interest rate

If you know any three of these,the fourth can be determined.

6-22Solving 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?

6-23Solving 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?

6-24

NOT COVERED ==

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 noninterest-

bearing notes.

Accounting Applications of Present Value Techniques—Single Cash AmountNOT COVERED

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 Value of Cash Flows

× Present Value Factor

= Present Value

Expected Cash Flow Approach

The present value factor is obtained using the company’s credit-adjusted risk-free rate of

interest.

6-27

Future Value of an Ordinary Annuity (NOT COVERED)

To find the future value of an

ordinary annuity, multiply the

amount of the annuity by the future value of

an ordinary annuity factor.

6-28

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?

6-29Future Value of an Annuity Due

(NOT COVERED)

To find the future value of an annuity due, multiply the

amount of the annuity by the

future value of an annuity due factor.

6-30

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.

6-31Present Value of a Deferred Annuity

(NOT COVERED)

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

date of the agreement.

6-32

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?

6-33

More Efficient Computation

1. Calculate the present value 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.

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

6-34

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?

6-35Accounting Applications of Present

Value Techniques—Annuities (NOT COVERED)

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 BondsCalculate 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 bond issue 926,405$

On June 30, 2013, Ebsen Electric issued 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 December 31, 2013. What was the price of the bond

issue?

6-37

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.

6-38

Valuation of Long-term LeasesOn 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).

6-39

Valuation 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.

6-40

Valuation of Pension Obligations

On January 1, 2013, Todd Furniture Company hired a new sales manager for the new showroom. The sales

manager is expected to work 30 years before retirement on December 31, 2042. Annual retirement benefits will be paid at the end of each year of retirement, a period

that is expected to be 25 years. The sales manager will earn $2,500 in annual retirement benefits for the first year worked, 2013. How much must Todd Furniture contribute to the company pension fund in 2013 to provide for $2,500 in annual pension benefits for 25 years that are expected to begin in 30 years? Todd

Furniture’s pension fund is expected to earn 5% interest.

6-41

Valuation of Pension ObligationsThis is a two-part calculation. The first part requires the computation of the present value of a 25-year ordinary annuity of $2,500 as of December 31, 2042. Next we calculate the present value of the December 31, 2042 amount. This second present value is the amount Todd Furniture will contribute in 2013 to fund the retirement

benefit earned by the sales manager in 2013.

6-42

End of Chapter 6