Download - 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
Slide 2
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)
Slide 3
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?
Slide 4
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
Slide 5
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
Slide 6
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.
Slide 7
6-7
Present Value of a Single AmountPresent Value of a Single Amount
Remember our equation?
FV = PV (1 + i) n
We can solve for PV and get . . . .
FV
(1 + i)nPV =
Slide 8
6-8
Present Value of a Single AmountPresent 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?
Slide 9
6-9
Present Value of a Single AmountPresent Value of a Single Amount
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.
Slide 10
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
Slide 11
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.
Slide 12
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
Slide 13
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
Slide 14
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
Slide 15
6-15
An annuity is a series of equal periodic payments.
Basic AnnuitiesBasic Annuities
Slide 16
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
Slide 17
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
Slide 18
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.
Slide 19
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?
Slide 20
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.
Slide 21
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.
Slide 22
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?
Slide 23
6-23
PV1PV2PV3PV4
$10,000 $10,000 $10,000 $10,000
1 2 3 4Today
Present Value of an Ordinary AnnuityPresent Value of an Ordinary Annuity
Slide 24
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$
Present Value of an Ordinary AnnuityPresent Value of an Ordinary Annuity
Slide 25
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$
Present Value of an Ordinary AnnuityPresent 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
Slide 26
6-26
Present Value of an Ordinary AnnuityPresent Value of an Ordinary Annuity
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
Slide 27
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.
Slide 28
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.
Slide 29
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
Slide 30
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
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
Slide 31
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
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
Slide 32
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.
Slide 33
6-33
Solving for Unknown Values in Present Value Solving for Unknown Values in Present Value SituationsSituations
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
Slide 34
6-34
Solving for Unknown Values in Present Value Solving for Unknown Values in Present Value SituationsSituations
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?
Slide 35
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
Slide 36
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?
Slide 37
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.
Slide 38
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.
End of Chapter 6End of Chapter 6