Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Time Value of Money Concepts 6
Mar 31, 2015
Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
Time Value of Money
Concepts
6
6-2
Time Value of Money
Interest is therent paid for the useof money over time.
That’s right! A dollartoday is more valuable
than a dollar to bereceived in one year.
6-3
Learning Objectives
Explain the difference between simple and compound interest.
6-4
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)
6-5
Compound Interest
Compound interest includes interest not only on the initial investment but also on the
accumulated interest in previous periods.
Principal Interest
6-6
Assume we will save $1,000 for three years and earn 6% interest compounded annually.
What is the balance inour account at the
end of three years?
Compound Interest
6-7
Compound Interest
6-8
Learning Objectives
Compute the future value of a single amount.
6-9
Future 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 will save $1,000 for three years and earn 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
6-10
Writing in a more efficient way, we can say . . . .
$1,000 × 1.06 × 1.06 × 1.06 = $1,191.02
or
$1,000 × [1.06]3 = $1,191.02
Future Value of a Single Amount
6-11
$1,000 × [1.06]3 = $1,191.02
We can generalize this as . . .
FV = PV (1 + i)n
FutureValue
FutureValue
Present Value
Present Value
InterestRate
InterestRate
Numberof
Compounding Periods
Numberof
Compounding Periods
Future Value of a Single Amount
6-12
Find the Future Value of $1 table in
your textbook.
Future Value of a Single Amount
Find the factor for 6% and 3 periods.
6-13
Find the factor for 6% and 3 periods.
Solve our problem like this. . .
FV = $1,000 × 1.19102
FV = $1,191.02
FV $1
Future Value of a Single Amount
6-14
Learning Objectives
Compute the present value of a single amount.
6-15
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.
Present Value of a Single Amount
6-16
Remember our equation?
FV = PV (1 + i) n
We can solve for PV and get . . . .
FV
(1 + i)nPV =
Present Value of a Single Amount
6-17
Find the Present Value of $1 table in
your textbook.
Hey, it looks familiar!
Present Value of a Single Amount
6-18
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?
Present Value of a Single Amount
6-19
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.
Present Value of a Single Amount
6-20
Learning Objectives
Solving for either the interest rate or the number of compounding periods when present value and future value of a single amount are
known.
6-21
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
6-22
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%
Determining the Unknown Interest Rate
6-23
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%
Determining the Unknown Interest Rate
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.
6-24
Monetary assets and monetary liabilities are valued at the
present value of future cash flows.
Accounting Applications of Present Value Techniques—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
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 a loan of this type to use
as the interest rate.
No Explicit Interest
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 Cash Flow
× Risk-Free Rate of InterestPresent Value
Expected Cash Flow Approach
6-27
Learning Objectives
Explain the difference between an ordinary annuity and an annuity due.
6-28
An annuity is a series of equal periodic payments.
Basic Annuities
6-29
An annuity with payments at the end of the period is known as an ordinary annuity.
EndEnd EndEnd
Ordinary Annuity
6-30
An annuity with payments at the beginning of the period is known as an annuity due.
Beginning Beginning Beginning
Annuity Due
6-31
Learning Objectives
Compute the future value of both an ordinary annuity and an annuity due.
6-32
Future Value of an Ordinary Annuity
To find the future value of an
ordinary annuity, multiply the
amount of a single payment or receipt by the future value
of an ordinary annuity factor.
6-33
We plan to invest $2,500 at the end of each of the next 10 years. We can earn 8%, compounded
annually, on all invested funds.
What will be the fund balance at the end of 10 years?
Future Value of an Ordinary Annuity
6-34
Future Value of an Annuity Due
To find the future value of an annuity
due, multiply the amount of a single payment or receipt by the future value
of an ordinary annuity factor.
6-35
Compute the future value of $10,000 invested at the beginning of each of the
next four years with interest at 6% compounded annually.
Future Value of an Annuity Due
6-36
Learning Objectives
Compute the present value of an ordinary annuity, an annuity due, and a deferred
annuity.
6-37
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?
Present Value of an Ordinary Annuity
6-38
PV1PV2PV3PV4
$10,000 $10,000 $10,000 $10,000
1 2 3 4Today
Present Value of an Ordinary Annuity
6-39
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 Annuity
6-40
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$
Can you find this value in the Present Value of Ordinary Annuity of $1 table?
Present Value of an Ordinary Annuity
More Efficient Computation $10,000 × 3.16986 = $31,698.60
6-41
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
Present Value of an Ordinary Annuity
6-42
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
Present Value of an Ordinary Annuity
6-43
Compute the present value of $10,000 received at the beginning of each of the
next four years with interest at 6% compounded annually.
Present Value of an Annuity Due
6-44
In a deferred annuity, the first cash flow is expected to occur more than one
period after the date of the agreement.
Present Value of a Deferred Annuity
6-45
On January 1, 2006, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2008. If you require a 12% return on
your investments, how much are you willing to pay for this investment?
1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10
Present Value? $12,500 $12,500
1 2 3 4
Present Value of a Deferred Annuity
6-46
On January 1, 2006, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2008. If you require a 12% return on
your investments, how much are you willing to pay for this investment?
1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10
Present Value? $12,500 $12,500
1 2 3 4
Present 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.
6-47
On January 1, 2006, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2008. If you require a 12% return on
your investments, how much are you willing to pay for this investment?
1/1/06 12/31/06 12/31/07 12/31/08 12/31/09 12/31/10
Present Value? $12,500 $12,500
1 2 3 4
Present Value of a Deferred Annuity
6-48
Learning Objectives
Solve for unknown values in annuity situations involving present value.
6-49
In present value problems involving annuities, there are four variables:
Solving for Unknown Values in Present Value Situations
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.
6-50Solving 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?
Today End ofYear 1
Present Value $700
End ofYear 2
End ofYear 3
End ofYear 4
6-51Solving 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?
6-52
Learning Objectives
Briefly describe how the concept of the time value of money is incorporated into the
valuation of bonds, long-term leases, and pension obligations.
6-53
Because financial instruments typically specify equal periodic
payments, these applications quite often involve annuity situations.
Accounting Applications of Present Value Techniques—Annuities
Long-term Bonds
Long-term Leases
Pension Obligations
6-54
Valuation 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, 2006, 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, 2006. What is the price of
the bonds?
6-55
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.
6-56
Valuation of Pension Obligations
Some 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.
6-57
End of Chapter 6