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Chapter 6

Time Value of Money Concepts

Time Value of Money

• Means money can be invested today to earn interest and grow to a larger dollar amount in the future

Example:

• Useful in valuing a variety of assets and liabilities

$100 $1066%

Invested in bank

Annual yield

Future value

Simple versus Compound Interest

Interest• Amount of money paid or received in excess of the

amount of money borrowed or lent Simple Interest

Compound Interest• Includes interest not only on the initial investment

but also on the accumulated interest in previous periods

LO6-1

Initial investment

Interest rate

Period of time

× ×

Simple Interest

Example:What is the simple interest earned each year on a $1,000 investment paying 10% interest?

LO6-1

Investment Interest rate Time period× × = Simple interest$1,000 10% 1 Year× × $100=

Compound Interest

Example:Cindy Johnson invested $1,000 in a savings account paying 10% interest compounded annually. How much interest will she earn in each of the next three years, and what will be her investment balance after three years?

Date Interest Balance(Interest rate × Outstanding Balance)

Initial deposit$1,000 End of year 1 10% × $1,000 = $100 $1,100 End of year 2 10% × $1,100 = $110 $1,210 End of year 3 10% × $1,210 = $121 $1,331

LO6-1

Compound Interest

Effective rate• Actual rate at which money grows per yearExample:Assuming an annual rate of 12%:

LO6-1

Interest Rate PerCompounded Compounding PeriodSemiannually 12% ÷ 2 = 6%Quarterly 12% ÷ 4 = 3%Monthly 12% ÷ 12 = 1%

Compound InterestExample: Cindy Johnson invested $1,000 in a savings account paying 10% interest compounded twice a year. What will be her investment balance at the end of the year? What is the effective annual interest rate?

LO6-1

Interest

Date (Interest rate ×

Outstanding balance) BalanceInitial deposit $1,000.00 After six months 5% × $1,000 = $50.00 $1,050 .00End of year 1 5% × $1,050 = $52.50 $1,102.50

Effective annual interest rate = $102.50 ÷ $1,000 = 10.25%

$1,102.50 – $1,000

10% ÷ 2 = 5%

Valuing a Single Cash Flow Amount

Future Value of a Single Amount • The amount of money that a dollar will grow to at

some point in the future

LO6-2

FV = I (1 + i)ⁿ

I = Amount invested at the beginning of the periodi = Interest rate

n = Number of compounding periods

Future Value of a Single Amount

Cindy Johnson invested $1,000 in a savings account for three years paying 10% interest compounded annually.

LO6-2

FV = $1,000 × 1.331

FV = I × FV Factor

FV = $1,331

Future value$1,331End ofyear 3

End ofyear 2

End ofyear 10

$1,000

Example:

Present Value of a Single Amount

• Today’s equivalent to a particular amount in the future

Example:

LO6-3

PV $1,331(1 + .10)3

$1,3311.331

$1,000= = =

FV = PV (1 + i) n PV FV(1 + i) n

=

Present Value of a Single Amount

Example:The present value of $1,331 received at the end of three years:

LO6-3

PV = FV × PV Factor

PV = $1,331 × .75131

PV = $1,000

• Future value entails the addition of interest• Present value entails the removal of interest• Accountants use PV calculations much more

frequently than FV

Relation between the Present Value and the Future Value

LO6-3

$100 $110 $121

End ofyear 2

End ofyear 10

$1,000 $1,331PV FV

End ofyear 3

Concept Check √

The Versa Tile Company purchased a delivery truck on February 1, 2016. The agreement required Versa Tile to pay the purchase price of $44,000 on February 1, 2017. Assuming an 8% rate of interest, to calculate the price of the truck Versa Tile would multiply $44,000 by the:a. Future value of an ordinary annuity of $1.b. Present value of $1.c. Present value of an ordinary annuity of $1.d. Future value of $1.

The calculation is for the present value today of the $44,000 to be received one year from now.

Solving for Other Values When FV and PV Are Known

LO6-4

Determining an Unknown Interest RateSuppose a friend asks to borrow $500 today and promises to repay you $605 two years from now. What is the annual interest rate you would be agreeing to?

Futurevalue$605

End ofyear 2

End ofyear 10

Presentvalue$500

n = 2, i = ?

Determining an Unknown Interest Rate (continued)

LO6-4

$500 (present value) = $605 (future value) × PVF* *Present value of $1; n = 2, i = ?

$500 (present value) ÷ $605 (future value) = 0.82645**Present value of $1; n = 2,

i = 10%

.

.75131.75131

i = ?

LO6-4

End ofyear 2

End ofyear 10

$10,000 $16,000

Presentvalue

End ofyear n-1

End ofyear n

Futurevalue

n = ?, i = 10%

Determining an Unknown Number of PeriodsYou want to invest $10,000 today to accumulate $16,000 for graduate school. If you can invest at an interest rate of 10% compounded annually, how many years will it take to accumulate the required amount?

Solving for Other Values When FV and PV Are Known

LO6-4

$10,000 (present value) = $16,000 (future value) × PVF**Present value of $1; i = 10%, n = ?

$10,000 (present value) ÷ $16,000 (future value) = 0.625**Present value of $1; i = 10%,

n = 5

.

Determining an Unknown Number of Periods (continued)

n = ?

.

.75131.75131

Preview of Accounting Applications of Present Value Techniques — Single Cash AmountMost monetary assets and monetary liabilities are valued at the present value of future cash flowsMonetary assets• Include money and claims to receive money in the

future, the amount of which is fixed or determinableExamples: Cash and most receivables

Monetary liabilities• Obligations to pay amounts of cash in the future, the

amount of which is fixed or determinableExample: Notes payable

LO6-4

Valuing a Note: One Payment, Explicit InterestExample:The Stridewell Wholesale Shoe Company manufactures athletic shoes for sale to retailers. The company recently sold a large order of shoes to Harmon Sporting Goods for $50,000. Stridewell agreed to accept a note in payment for the shoes requiring payment of $50,000 after one year plus interest at 10%.

Present value

?

Future value

$55,000 End of year 1

0

n = 1, i = 10%

LO6-4

LO6-4

$55,000 (future value) × 0.90909* = $50,000 (present value)*Present value of $1; n = 1, i = 10%

Valuing a Note: One Payment, Explicit Interest

Valuing a Note: One Payment, No Interest StatedExample:The Stridewell Wholesale Shoe Company recently sold a large order of shoes to Harmon Sporting Goods. Terms of the sale require Harmon to sign a noninterest-bearing note of $60,500 with payment due in two years.

Present value

?

Future value $60,500

End of year 2

n = 2, i = 10%

LO6-4

To find the PV of the note (price of the shoes), we need to know either the cash price of the shoes or the appropriate interest rate for a transaction like this one. Let’s say the market rate is 10%.

LO6-4

$60,500 (future value) × .82645* = $50,000 (present value)*Present value of $1; n = 2, i = 10%

Valuing a Note: One Payment, No Explicit Interest

Concept Check √

Turp and Tyne Distillery is considering investing in a two-year project. The company’s required rate of return is 10%. The present value of $1 for one period at 10% is .909 and .826 for two periods at 10%. The project is expected to create cash flows, net of taxes, of $240,000 in the first year, and $300,000 in the second year. The distillery should invest in the project if the project's cost is less than or equal to:a. $540,000b. $490,860c. $465,960d. $446,040

$218,160 ($240,000 x 0.909) 247,800 ($300,000 x 0.826)$465,960

Expected Cash Flow Approach

Statement of Financial Accounting Concepts No. 7 (SFAC No. 7)• Provides a framework for using future cash flows in

accounting measurement when uncertainty is present. • The objective in valuing an asset or liability using

present value is to approximate fair value of that asset or liabilityo Key to that objective is determining the present value

of future cash flows, taking into account any uncertainty concerning the amounts and timing of the cash flows

LO6-4

Illustration: Expected Cash Flow ApproachLO6-4

LDD Corporation faces the likelihood of having to pay an uncertain amount in five years in connection with an environmental cleanup. Calculate the expected cash flow. Also calculate the present value of the expected cash flow if the company’s credit-adjusted risk-free rate of interest is 5%. The future cash flow estimate is in the range of $100 million to $300 million with the following estimated probabilities:

Loss Amount Probability$100 million 10%

$200 million 60%

$300 million 30%

Present value of expected cash flows:

$220,000,000 X .78353 = $172,376,600

Present value of $1: n = 5, i = 5%

The expected cash flow:

$100 X 10% = $ 10 million 200 X 60% = 120 million 300 X 30% = 90 million

$220 million

Concept Check √

Willie Winn Track Shoes used the expected cash flow approach to determine the present of a future obligation to be paid to Betty Will Company in four years. Estimated future payment possibilities were as follows:

Possible payment Probability$100 million 20% 140 million 40% 180 million 40%

The risk-free interest rate is 5%. The present value of $1 in 4 periods at 5% is 0.82270. What is the estimated present value of the future obligation?a. $115 million.b. $122 million.c. $140 million.d. $148 million.

$100 million x 0.20 = $ 20 million $140 million x 0.40 = 56 million $180 million x 0.40 = 72 million

$148 million x .82270$121.76 million

Basic Annuities

Annuity• Series of cash flows of same amount received or

paid each periodExample: A loan on which periodic interest is paid in equal amounts

LO6-5

Ordinary Annuity

• Cash flows occur at the end of each period

Annuity Due

• Cash flows occur at the beginning of each period

Example:An installment note payable dated December 31, 2016, might require the debtor to make three equal annual payments, with the first payment due on December 31, 2017, and the last one on December 31, 2019.

Ordinary Annuity

LO6-5

Example:A three-year lease of a building that begins on December 31, 2016, and ends on December 31, 2019, may require the first year’s lease payment in advance on December 31, 2016. The third and last payment would take place on December 31, 2018, the beginning of the third year of the lease.

Annuity Due

LO6-5

Concept Check √

Justin Investor wants to calculate how much money he needs to deposit today into a savings account that earns 4% in order to be able to withdraw $6,000 at the end of each of the next 5 years. He should use which present value concept?a. Present value of $1 for 5 periods.b. Present value of an annuity due of $1 for 5 periods.c. Present value of an ordinary annuity of $1 for 5 periods.d. Future value of $1 for 5 periods.

The calculation is how much needs to be deposited today, the present value, so that equal amounts can be withdrawn over the next six years at the end of the year (ordinary annuity).

Concept Check √

The Knotworth Gedding Consulting Company purchased a machine for $15,000 down and $500 a month payable at the end of each of the next 36 months. How would the company calculate the cash price of the machine, assuming the annual interest rate is known?a. $15,000 plus the present value of $18,000 ($500 x 36).b. $15,000 plus the present value of an annuity due of $500 for 36

periods.c. $33,000.d. $15,000 plus the present value of an ordinary annuity of $500 for 36

periods.

The cash price is equal to the present value of the future cash outflows. This includes the $15,000 today plus the value today, present value, of the $500 payments made at the end of each month (ordinary annuity).

Concept Check √

If you have a set of present value tables, an annual interest rate, the dollar amount of equal payments made, and the number of semiannual payments, what other information do you need to calculate the present value of the series of payments?a. The rate of inflation.b. The timing of the payments (whether they are at the beginning or end

of the period).c. The future value of the annuity.d. No other information is needed.

If the payments are made at the end of each period, it is an ordinary annuity. If the payments are made at the beginning of each period, it is an annuity due.

LO6-6

Future Value of an Ordinary Annuity

First payment $10,000 × 1.21 = $12,100

Second payment 10,000 × 1.10 = 11,000

Third payment 10,000 × 1.00 = 10,000

Total 3.31 $33,100

210

PaymentFV of $1 i = 10%

Future value (at the end of year 3) n

Sally Rogers wants to accumulate a sum of money to pay for graduate school. Rather than investing a single amount today that will grow to a future value, she decides to invest $10,000 a year over the next three years in a savings account paying 10% interest compounded annually. She decides to make the first payment to the bank one year from today.

LO6-6

FVA = $10,000 (annuity amount) × 3.31* = $33,100*Future value of an ordinary annuity of $1: n = 3, i =10%

Using the FVA Table to Calculate the Future Value

Future Value of an Annuity DueLO6-6

Easier way:FVA = $10,000 (annuity amount) × 3.641* = $36,410 *Future value of an ordinary annuity of $1: n = 3, i =10%

First payment $10,000 × 1.331 = $13,310

Second payment 10,000 × 1.210 = 12,100

Third payment 10,000 × 1.100 = 11,000

Total 3.641 $36,410

3

21

PaymentFuture value

(at the end of year 3) nFV of $1 i = 10%

LO6-7

Present Value of an Ordinary Annuity

First payment $10,000 × .90909 = $9,091

Second payment 10,000 × .82645 = 8,264

Third payment 10,000 × .75131 = 7,513

Total 2.48685 $24,868

1

23

PaymentPV of $1 i = 10%

Present value (at the beginning of the year 1) n

Sally wants to accumulate a sum of money to pay for graduate school. She wants to invest a single amount today in a savings account earning 10% interest compounded annually that is equivalent to investing $10,000 at the end of each of the next three years.

LO6-7

Using the PVA Table to Calculate the Present Value

PVA = $10,000 (annuity amount) × 2.48685* = $24,868

Present Value of an Annuity Due

LO6-7

First payment $10,000 × 1.00000 = $10,000

Second payment 10,000 × .90909 = 9,091

Third payment 10,000 × .82645 = 8,264

Total 2.73554 $27,355

0

12

PaymentPV of $1 i = 10%

Present value (at the beginning of the year 1) n

In the previous illustration, suppose that the three equal payments of $10,000 are tobe made at the beginning of each of the three years. What is the present value of this annuity?

Using the PVAD Table to Calculate the Present Value

LO6-7

PVA= $10,000 (annuity amount) × 2.73554* = $27,355*Present value of an annuity due of $1: n = 3, i = 10%

From Table 6

Concept Check √

The Stinch Fertilizer Corporation wants to accumulate $8,000,000 for plant expansion. The funds are needed on January 1, 2021. Stinch intends to make five equal annual deposits in a fund that will earn interest at 7% compounded annually. The first deposit is to be made on January 1, 2016. Present value and future value facts are as follows:

Future value of an ordinary annuity of $1 at 7% for 5 periods 5.75Future value of an annuity due of $1 at 7% for 5 periods 6.15Present value of $1 at 7% for 5 periods .713Present value of an ordinary annuity of $1 at 7% for 5 periods 4.10

What is the amount of the required annual deposit?a. $1,300,813b. $1,391,304c. $1,951,220d. $1,704,000

$8,000,000 6.15 * = $1,300,813

*Future value of an annuity due of $1 at 7% for 5 periods)

Concept Check √

I. R. Wright plans to make quarterly deposits of $200 for 5 years into a savings account. The first deposit will be made immediately. The savings account pays interest at an annual rate of 8%, compounded quarterly. How much will Wong have accumulated in the savings account at the end of the five-year period?

Future value of an ordinary annuity of $1 at 8% for 5 periods 6.3359Future value of an annuity due of $1 at 8% for 5 periods 5.8666Future value of an ordinary annuity of $1 at 2% for 20 periods

26.1833Future value of an annuity due of $1 at 2% for 20 periods 24.2974

a. $2,672b. $4,000c. $4,860d. $5,237

$200 x 24.2974* = $4,860 *future value of an annuity due for 20 periods at 2%

Concept Check √

U. B. Wong plans to make quarterly deposits of $200 for 5 years into a savings account. The deposits will be made at the end of each quarter. The savings account pays interest at an annual rate of 8%, compounded quarterly. How much will Wong have accumulated in the savings account at the end of the five-year period?

Future value of an ordinary annuity of $1 at 8% for 5 periods 6.3359Future value of an annuity due of $1 at 8% for 5 periods 5.8666Future value of an ordinary annuity of $1 at 2% for 20 periods 26.1833Future value of an annuity due of $1 at 2% for 20 periods

24.2974 a. $2,672b. $4,000c. $4,860d. $5,237

$200 x 26.1833* = $5,237 *future value of an ordinary annuity for 20 periods at 2%

Present Value of a Deferred AnnuityDeferred annuity:• Exists when the first cash flow occurs more than one

period after the date the agreement begins

LO6-7

At January 1, 2016, you are considering acquiring an investment that will provide three equal payments of $10,000 each to be received at the end of three consecutive years. However, the first payment is not expected until December 31, 2018. The time value of money is 10%. How much would you be willing to pay for this investment?

Present Value of a Deferred Annuity (continued)

LO6-7

First payment $10,000 × .75131 = $7, 513

Second payment 10,000 × .68301 = 6,830

Third payment 10,000 × .62092 = 6,209

Total $20,552

3

45

PaymentPV of $1 i = 10% Present value n

At January 1, 2016, you are considering acquiring an investment that will provide three equal payments of $10,000 each to be received at the end of three consecutive years. However, the first payment is not expected until December 31, 2018. The time value of money is 10%. How much would you be willing to pay for this investment?

Present Value of a Deferred AnnuityAlternative: Two-Step Process

1. Calculate the PV of the annuity as of the beginning of the annuity period.

2. Reduce the single amount calculated in (1) to its present value as of today.

Illustration:

LO6-7

Present Value of a Deferred Annuity—Two-Step Process (continued)

LO6-7

Step 1:PVA= $10,000 (annuity amount) × 2.48685* = $24,868

Step 2: PV = $24,868 (future amount) × .82645* = $20,552

*Present value of an ordinary annuity of $1: n = 3, i = 10%*

*Present value of $1: n = 2, i = 10%*

Concept Check √

Harry Byrd’s Chicken Shack agrees to pay an employee $50,000 a year for six years beginning two years from today and decides to fund the payments by depositing one lump sum in a savings account today. The company should use which present value concept to determine the required deposit?a. Future value of $1.b. Future value of a deferred annuity.c. Present value of a deferred annuity.d. None of the above.

The calculation is the amount to be deposited today, the present value, of six equal payments (an annuity), that doesn’t start for two years (deferred annuity).

Financial Calculators

LO6-7

Texas Instruments model BA-35 has:

Example: • Assume you need to determine the present value of a 10-

period ordinary annuity of $200 using a 10% interest rate.• Enter N to be 10, %I to be 10, and PMT to be −200,

then press CPT and PV to obtain the answer of $1,229.

Excel

LO6-7

• We can use spreadsheet software, such as Excel, to solve time value of money problems.

• Excel has a function called PV that calculates the present value of an ordinary annuity.

• To use the function, you would select the pull-down menu for “Insert,” click on “Function” and choose the category called “Financial.” Scroll down to PV and double-click.

• Then input the necessary variables—interest rate, the number of periods, and the payment amount.

Example: • Assume you need to determine the present value of a 10-

period ordinary annuity of $200 using a 10% interest rate.• Enter N to be 10, %I to be 10, and PMT to be −200,

then press CPT and PV to obtain the answer of $1,229.

Solving for Unknown Values in Present Value Situations

LO6-8

Determining the Annuity Amount when Other Variables are Known

$700 (present value) = 3.31213* × annuity amount

$700 (present value) ÷ 3.31213* = $211.34 (annuity amount) * Present value of an ordinary annuity of $1: n = 4, i = 8%

Solving for Unknown Values in Present Value SituationsDetermining the Unknown Number of Periods—Ordinary Annuity

LO6-8

$700 (present value) = $100 (annuity amount) × ?*

$700 (present value) ÷ $100 (annuity amount) = 7.0000*

In the PVA table (Table 4), search the 7% column ( i = 7%) for this value and find 7.02358 in row 10. So it would take about 10 years to repay the loan.

* Present value of an ordinary annuity of $1: n = ?, i = 7%

* Present value of an ordinary annuity of $1: n = ?, i = 7%

PVA table factor

Solving for Unknown Values in Present Value SituationsDetermining i When Other Variables Are Known

LO6-8

$331(present value) = $100 (annuity amount) × ?*

$331 (present value) ÷ $100 (annuity amount) = 3.31*

In the PVA table (Table 4), search row four (n = 4) for 3.31. We find it in the 8% column. So the effective interest rate is 8%.

* Present value of an ordinary annuity of $1: n = 4, i = ?

* Present value of an ordinary annuity of $1: n = 4, i = ?

PVA table factor

Determining i When Other Variables Are Known— Unequal Cash Flows

LO6-8

$400 (present value) = $100 (annuity amount) × PVA* + $200 (single payment) × PV†

Using i = 9% PV = $100 (2.53129)* + $200 (.70843)† = $395Using i = 8% PV = $100 (2.57710)* + $200 (.73503)† = $405

* Present value of an ordinary annuity (PVA) of $1: n = 3, i = ?† Present value (PV) of $1: n = 4, i = ?

This indicates that the interest rate implicit in the agreement is about 8.5%.

$400

Concept Check √

The Omagosh Company purchased office furniture for $25,800 and agreed to pay for the purchase by making five annual installment payments beginning one year from today. The installment payments include interest at 8%. The present value of an ordinary annuity for 5 periods at 8% is 3.99271. The present value of an annuity due for 5 periods at 8% is 4.31213. What is the required annual installment payment? a. $5,160b. $6,462c. $5,982d. $4,398

$25,800 3.99271* = $ 6,462 *present value of an ordinary annuity for 5 periods at 8%

Valuing a Long-Term Bond LiabilityLO6-9

On June 30, 2016, Fumatsu Electric issued 10% stated rate bonds with a face amount of $200 million. The bonds mature on June 30, 2036 (20 years). The market rate of interest for similar issues was 12%. Interest is paid semiannually (5%) on June 30 and December 31, beginning December 31, 2016. The interest payment is $10 million (5% X $200 million).What was the price of the bond issue? What amount of interest expense will Fumatsu record for the bonds in 2016?

PVA = $10 million (annuity amount) X 15.04630PV = $200 million (lump-sum) X .09722

$150,463,00019,444,000

Price of the bond issue

==

= $169,907,000

Interest expense $169,907,000 × 6% = $10,194,420=

Present value of an ordinary annuity of $1: n = 40, i = 6%

Present value of $1: n = 40, i = 6%

Valuing a Long-Term Lease LiabilityLO6-9

On January 1, 2016, the Stridewell Wholesale Shoe Company signed a 25-year lease agreement for an office building. Terms of the lease call for Stridewell to make annual lease payments of $10,000 at the beginning of each year, with the first payment due on January 1, 2016. Assuming an interest rate of 10% properly reflects the time value of money in this situation, how should Stridewell value the asset acquired and the corresponding lease liability?

PVAD = $10,000 (annuity amount) × 9.98474 = $99,847

Journal Entry CreditDebitLeased office building 99,847

99,847Lease payable

Present value of an annuity due of $1: n = 25, i =10%

Concept Check √

On March 31, 2016, the Gusto Beer Company leased a machine from B. A. Lush, Inc. The lease agreement requires Gusto to pay 8 annual payments of $16,000 on each May 31, with the first payment due on May 31, 2016. Assuming an interest rate of 6% and that this lease is treated as an installment sale (capital lease), Gusto will initially value the machine by multiplying $16,000 by which of the following?a. Present value of $1 at 10% for 6 periods.b. Present value of an ordinary annuity of $1 at 10% for 6 periods.c. Present value of an annuity due of $1 at 10% for 6 periods.d. Future value of an annuity due of $1 at 10% for 6 periods.

Present value of an annuity due of $1 at 10% for 6 periods. The calculation is how much is recorded today, the present value of equal payments that start today (annuity due).

Valuing a Pension ObligationLO6-9

On January 1, 2016, the Stridewell Wholesale Shoe Company hired Sammy Sossa. Sammy is expected to work for 25 years before retirement on December 31, 2040. Annual retirement payments will be paid at the end of each year during his retirement period, expected to be 20 years. The first payment will be on December 31, 2041. During 2016 Sammy earned an annual retirement benefit estimated to be $2,000 per year. The company plans to contribute cash to a pension fund that will accumulate to an amount sufficient to pay Sammy this benefit. Assuming that Stridewell anticipates earning 6% on all funds invested in the pension plan, how much would the company have to contribute at the end of 2016 to pay for pension benefits earned in 2016?

PVA = $2,000 (annuity amount) X 11.46992 $22,940=PV = $22,940 (future amount) X .24698 $5,666=

Present value of an ordinary annuity of $1: n = 20, i = 6%

Present value of $1: n = 24, i = 6%

This is a deferred annuity.

Summary of Time Value of Money Concepts

End of Chapter 6