The Time Value of Money: Future Amounts and Present Values

McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.

The Time Value of Money:The Time Value of Money:Future Amounts and Present Future Amounts and Present ValuesValuesAppendix B

Appendix B-2

The ConceptThe Concept

An amount of money available

today can be safely invested to accumulate to a larger amount in

the future.

Appendix B-3

The ConceptThe ConceptDifferent Time Values of the "Same Money"

500540

583630

680

400

450

500550

600

650

700

1 2 3 4 5

Time (in years)

Bal

ance

($)

0 1 2 3 4

Assume you invest $500 in a savings account that earns interest at the rate of 8% per year. This graph

illustrates the growth in your savings account balance at the end of each of the next four years.

$500 × 1.08$540 × 1.08

$583 × 1.08$630 × 1.08

Appendix B-4

Relationships between Relationships between Present Values and Future Present Values and Future AmountsAmounts

In this example, your initial investment of $500 is the present value. It is invested for four years at 8% interest. Over the four years, the value of your

investment increases to $680, the future amount.

Different Time Values of the "Same Money"

500540

583630

680

400

450

500550

600

650

700

1 2 3 4 5

Time (in years)

Bal

ance

($)

0 1 2 3 4

$500 × 1.08$540 × 1.08

$583 × 1.08$630 × 1.08

Appendix B-5

Applications of the Time Applications of the Time Value of Money ConceptValue of Money Concept

Determine the amount to which an investment will

accumulate over time

Determine the amount that must be invested

every period to accumulate a required

future amount

Determine the present value of cash flows

expected to occur in the future

Investors, accountants, and other decision makers apply the time value of money in three basic ways.

Appendix B-6

Future AmountsFuture Amounts

Year 1 Year 2 Year 3 Year 4 Year 5

Present Value

Future Amount

A future amount is simply the dollar amount to which a

present value will accumulate over time.

Appendix B-7

Future AmountsFuture Amounts

1% 1.50% 5% 6% 8% 10%1 1.010 1.015 1.050 1.060 1.080 1.1002 1.020 1.030 1.103 1.124 1.166 1.2103 1.030 1.046 1.158 1.191 1.260 1.3314 1.041 1.061 1.216 1.262 1.360 1.4645 1.051 1.077 1.276 1.338 1.469 1.611

Interest Rates

Table FA-1Future Amount of $1 after n Periods

Number of

Periods (n )

Assume you invest $500 in a savings account that earns interest at the rate of 8% per year. What will be

the future amount at the end of 4 years?

$500 Present Value × 1.360 Factor = $680 Future Amount

Appendix B-8

1% 1.50% 5% 6% 8% 10%1 1.010 1.015 1.050 1.060 1.080 1.1002 1.020 1.030 1.103 1.124 1.166 1.2103 1.030 1.046 1.158 1.191 1.260 1.3314 1.041 1.061 1.216 1.262 1.360 1.4645 1.051 1.077 1.276 1.338 1.469 1.611

Interest Rates

Table FA-1Future Amount of $1 after n Periods

Number of

Periods (n )

Computing the Required Computing the Required InvestmentInvestment

Assume you need $680 at the end of 4 years. If you can invest at 8% per year, what is the present value?

$680 Future Amount1.360 Factor$500 Present Value =

Appendix B-9

The Future Amount of an The Future Amount of an AnnuityAnnuity


Future Amount

An annuity is a series of equal

periodic payments.

Appendix B-10

1% 1.50% 5% 6% 8% 10%1 1.000 1.000 1.000 1.000 1.000 1.0002 2.010 2.015 2.050 2.060 2.080 2.1003 3.030 3.045 3.153 3.184 3.246 3.3104 4.060 4.091 4.310 4.375 4.506 4.6415 5.101 5.152 5.526 5.637 5.867 6.105

Interest Rates

Table FA-2Future Amount of $1 Paid Periodically for n Periods

Number of

Periods (n )

Assume you invest $500 in a savings account at the end of each of the next 4 years. The account earns interest at the rate of 8% per year. What will be the balance in your account at the end of 4

years?$500 Periodic Payment × 4.506 Factor =$2,253 Future Amount of an

Annuity


Appendix B-11

1% 1.50% 5% 6% 8% 10%1 1.000 1.000 1.000 1.000 1.000 1.0002 2.010 2.015 2.050 2.060 2.080 2.1003 3.030 3.045 3.153 3.184 3.246 3.3104 4.060 4.091 4.310 4.375 4.506 4.6415 5.101 5.152 5.526 5.637 5.867 6.105

Interest Rates

Table FA-2Future Amount of $1 Paid Periodically for n Periods

Number of

Periods (n )

Assume you need $2,253 at the end of 4 years. If you can invest at 8% per year, what is the amount of

required periodic payment?$2,253 Future Amount of an Annuity

4.506 Factor$500 Periodic Payment =


Appendix B-12

Interest Periods of Less Interest Periods of Less than One Yearthan One Year

In our computations, we have assumed that interest

is paid (compounded) or payments are made annually. Investment payments or interest

payments may be made on a more frequent basis,

such as monthly, quarterly, or semiannually.

Appendix B-13

Present ValuePresent Value


Present Value

Future Amount

The present value is today’s value of funds to be

received in the future.

Appendix B-14

Present ValuesPresent Values

1% 1.50% 5% 6% 8% 10%1 0.990 0.985 0.952 0.943 0.926 0.9092 0.980 0.971 0.907 0.890 0.857 0.8263 0.971 0.956 0.864 0.840 0.794 0.7514 0.961 0.942 0.823 0.792 0.735 0.6835 0.951 0.928 0.784 0.747 0.681 0.621

Interest Rates

Table PV-1Present Values of $1 Due in n Periods

Number of

Periods (n )

What would you pay today for the opportunity to receive $680 in 4 years, assuming an 8% interest rate?

$680 Future Amount × .735 Factor = $500 Present Value (rounded)

Appendix B-15

What is the Appropriate What is the Appropriate Discount Rate?Discount Rate?

All investments involve some degree of risk

that actual future cash flows may turn out to be

less than expected. Investors will require a

rate of return that justifies taking this risk.

Appendix B-16

The Present Value of an The Present Value of an AnnuityAnnuity


Present Value

Appendix B-17

The Present Value of an The Present Value of an AnnuityAnnuity

1% 1.50% 5% 6% 8% 10%1 0.990 0.985 0.952 0.943 0.926 0.9092 1.970 1.956 1.859 1.833 1.783 1.7363 2.941 2.912 2.723 2.673 2.577 2.4874 3.902 3.854 3.546 3.465 3.312 3.1705 4.853 4.783 4.329 4.212 3.993 3.791

Interest Rates

Table PV-2Present Value of $1 to Be Received Periodically for n Periods

Number of

Periods (n )

Assume you need cash flows of $500 at the end of each of the next 4 years. If your investment earns interest at the rate of 8% per year, what amount do you need to invest today to achieve

your cash flow needs?

$500 Periodic Payment × 3.312 Factor =$1,656 Present Value of an Annuity

Appendix B-18

Discount Periods of Less Discount Periods of Less than One Yearthan One Year

The present value tables can be used

with discount periods of any length, but the

discount rate must be for that length of

time.

Appendix B-19

Valuation of Financial Valuation of Financial InstrumentsInstruments

Cash Equity Contracts

Accountants use the phrase financial instruments to describe cash, equity investment in another business, and

any contracts that call for receipts or payments of cash.

Whenever the present value of a financial instrument differs significantly from the sum of the expected future cash

flows, the instrument is recorded in the accounting records at its present value—not at the expected amount of the

future cash receipts or payments.

Appendix B-20

Valuation of Financial Valuation of Financial InstrumentsInstrumentsInvestments in

Securities

Accounts Receivable

Accounts Payable

Appendix B-21

Interest-Bearing Interest-Bearing Receivables and PayablesReceivables and Payables

Interest-bearing receivables and payables initially are recorded in accounting

records at the present value of the future cash flows—also called the “principal

amount” of the obligation. This present value is often substantially less than the sum of the expected future

amounts.

Appendix B-22

““Non-Interest-Bearing” Non-Interest-Bearing” NotesNotes

If the difference between the

present value of a note and its face

amount is material, the note initially is

recorded at its present value.

Appendix B-23

““Non-Interest-Bearing” Non-Interest-Bearing” NotesNotesAssume that on 1 January 2009, Elron Corporation purchases land from

U.S. Development Company. As full payment for this land, Elron issues a $300,000 installment note payable, due in 3 annual installments of $100,000, beginning 31 December 2009. There is no mention of an

interest rate. Elron should use the present value of this note—not the face amount—in determining the cost of the land and reporting its

liability. Assume that a realistic interest rate for financing land over a 3 year period currently is 10% per year.

Interest Period

Payment Date

Annual Payment

Interest Expense

(10%)*

Reduction in Unpaid Balance

Unpaid Balance

Issue date 1 Jan. 2009 248,700$ 1 Dec. 31, 2009 100,000$ 24,870$ 75,130$ 173,570 2 Dec. 31, 2010 100,000 17,357 82,643 90,927 3 Dec. 31, 2011 100,000 9,073 90,927 -

* Interest expense is determined by multiplying 10% times the last unpaid balance. In the last period, interest expense is equal to the amount of the final payment minus the remaining unpaid balance. This compensates for using factors carried to only three decimal places.

Appendix B-24

““Non-Interest-Bearing” Non-Interest-Bearing” NotesNotes

Date Account Titles and ExplanationPR Debit Credit

20091 Jan Land 248,700

Notes Payable 248,700

31 Dec Interest Expense 24,870Notes Payable 75,130

Cash 100,000

GENERAL JOURNAL

Appendix B-25

Market Prices of BondsMarket Prices of BondsCalculate the Present Value of the Lump-sum Maturity

Payment (Face Value)

Calculate the Present Value of the Annuity Payments

(Interest)

On 1 January 2009, Driscole Corporation issues $1,000,000 of

10-year, 10% bonds when the going market rate of interest is 12%. Interest is paid semiannually beginning on 30 June 2009.

Because bond interest is paid semiannually, we must use 20 semiannual periods as the life of the bond issue and a 6% semiannual market rate of

interest in our present value calculations.

Appendix B-26

Market Prices of BondsMarket Prices of Bonds

Cash Flow Table Table Value Amount

Present Value

Face value of the bondPV-1

n=20; i=6% 0.312 1,000,000$ 312,000$

Interest (annuity) PV-2 n=20; i=6% 11.47 50,000 573,500

Market price of bonds 885,500$

Calculate the Present Value of the Lump-sum Maturity

Payment (Face Value)

Calculate the Present Value of the Annuity Payments

(Interest)

On 1 January 2009, Driscole Corporation issues $1,000,000 of

10-year, 10% bonds when the going market rate of interest is 12%. Interest is paid semiannually beginning on 30 June 2009.

Appendix B-27

Market Prices of BondsMarket Prices of Bonds

Date Account Titles and ExplanationPR Debit Credit

20091 Jan Cash 885,500

Discount on Bonds Payable 114,500Bonds Payable 1,000,000

31 Dec Bond Interest Expense 55,725Cash 50,000Discount on Bonds Payable 5,725

GENERAL JOURNAL

Appendix B-28

Finance LeasesFinance LeasesA finance lease is regarded as a sale of the

leased asset by the lessor to the lessee.

At the date of this sale, the lessor recognizes sales revenue equal to the present value of the future lease payments receivable, discounted at a realistic rate of

interest. The lessee also uses the present value of the future payments to determine the cost of the leased

asset and the valuation of the related liability.

Appendix B-29

Finance LeasesFinance LeasesAssume that on 1 December, Pace Tractor uses a finance

lease to finance the sale of a tractor to Kelly Grading Company. The tractor was carried in Pace Tractor’s

perpetual inventory records at a cost of $15,000. Terms of the lease call for Kelly Grading Company to make 24 monthly payments of $1,000 each, beginning on 31

December. These lease payments include an interest charge of 1% per month. At the end of the 24-month

lease, title to the tractor will pass to Kelly Grading Company at no additional cost.

Let’s look at the entries for Pace Tractor.

Appendix B-30

Finance LeasesFinance Leases

$1,000 Periodic Payment × 21.243 Factor= $21,243 Present Value of an Annuity

Appendix B-31


$24,000 - $21,243 = $2,757 24 months = $114.88 per month

Appendix B-32


$1,000 Periodic Payment × 21.243 Factor = $21,243 Present Value of an Annuity

Appendix B-33

Obligations for Obligations for Postretirement BenefitsPostretirement BenefitsAny unfunded obligation for

postretirement benefits appears in

the balance sheet at the present value of the expected future

cash outlays to retired employees.

Appendix B-34

End of Appendix BEnd of Appendix B

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The Time Value of Money: Future Amounts and Present Values