The Time Value of Money: Future Amounts and Present Values
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Transcript of 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
Year 1 Year 2 Year 3 Year 4 Year 5
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
The Future Amount of an The Future Amount of an AnnuityAnnuity
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 =
The Future Amount of an The Future Amount of an AnnuityAnnuity
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
Year 1 Year 2 Year 3 Year 4 Year 5
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
Year 1 Year 2 Year 3 Year 4 Year 5
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
Finance LeasesFinance Leases
$24,000 - $21,243 = $2,757 24 months = $114.88 per month
Appendix B-32
Finance LeasesFinance Leases
$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