Download Chapter 6 (5E)

Chapter 6
TIME VALUE OF MONEY
CONCEPTS
McGraw-Hill /Irwin
© 2009 The McGraw-Hill Companies, Inc.
Slide 2
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)
6-2
Slide 3
Compound Interest
Original balance
$ 1,000.00
Assume
we
deposit
$1,000
in
a
bank
that
earns
First year interest
60.00
6% interest compounded annually.
Balance, end of year 1
$ 1,060.00
Balance, beginning of year 2
Second year interest
Balance, end of year 2
$ 1,060.00
63.60
$ 1,123.60
What is the balance
Balance, beginning
of yearin3
our account at the
Third year interest
end of three years?
Balance, end of year 3
$ 1,123.60
67.42
$ 1,191.02
6-3
Slide 4
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 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
6-4
Slide 5
Future Value of a Single Amount
Using
Value way,
of $1we
Table,
we. find
Writingthe
in aFuture
more efficient
can say
...
the factor for 6% and 3 periods is 1.19102.
3
So, we
can solve=our
problem
like this.
..
$1,191.02
$1,000
× [1.06]
FV = $1,000 × 1.19102 Number
of
n
FV = FVPV= $1,191.02
(1 + i) Compounding
Periods
Future
Value
Amount
Invested at
the
Beginning of
the Period
Interest
Rate
6-5
Slide 6
Present 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.
6-6
Slide 7
Present Value of a Single Amount
Remember our equation?
FV = PV (1 + i)
n
We can solve for PV and get . . . .
PV =
FV
n
(1 + i)
6-7
Slide 8
Present 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 today in order to
accumulate $20,000 in 5 years, if you can
earn 8% interest compounded annually?
6-8
Slide 9
Present 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.
6-9
Slide 10
Solving for Other Values
FV = PV (1 +
Future
Value
Present
Value
n
i)
Interest
Rate
Number
of 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.
6-10
Slide 11
Determining 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%
Present Value of $1 Table
b.
4.0%
$1,000 = $1,092 × ?
c.
4.5%
$1,000 ÷ $1,092 = .91575
d.
5.0%
Search the PV of $1 table
in row 2 (n=2) for this value.
6-11
Slide 12
Accounting Applications of Present Value
Techniques—Single Cash Amount
Monetary assets and monetary
liabilities are valued at the present
value of future cash flows.
Monetary
Assets
Monetary
Liabilities
Money and claims to
receive money, the
amount which is fixed
or determinable
Obligations to pay
amounts of cash, the
amount of which is
fixed or determinable
6-12
Slide 13
No Explicit Interest
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.
6-13
Slide 14
Expected Cash Flow Approach
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 Interest
Present Value
6-14
Slide 15
Basic Annuities
An annuity is a series of
equal periodic payments.
6-15
Slide 16
Ordinary Annuity
An annuity with payments at the end of the
period is known as an ordinary annuity.
Today
1
2
3
4
$10,000
$10,000
$10,000
$10,000
End of year 1
End of year 2
End of year 3
End of year 4
6-16
Slide 17
Annuity Due
An annuity with payments at the beginning of
the period is known as an annuity due.
Today
1
2
3
$10,000
$10,000
$10,000
$10,000
Beginning
of year 1
4
Beginning
of year 2
Beginning
of year 3
Beginning
of year 4
6-17
Slide 18
Future 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.
6-18
Slide 19
Future 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?
Am ount of annuity
$
2,500.00
Future value of ordinary annuity of $1
(i = 8%, n = 10)
Future value
×
14.4866
$
36,216.50
6-19
Slide 20
Future 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.
6-20
Slide 21
Future 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.
Amount of annuity
$ 10,000
FV of annuity due of $1
(i=6%, n=4)
Future value
× 4.63710
$ 46,371
6-21
Slide 22
Present 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?
6-22
Slide 23
Present Value of an Ordinary Annuity
Today
1
2
3
4
$10,000
$10,000
$10,000
$10,000
PV1
PV2
PV3
PV4
6-23
Slide 24
Present Value of an Ordinary Annuity
PV1
PV2
PV3
PV4
Total
Annuity
$ 10,000
10,000
10,000
10,000
PV of $1
Factor
0.90909
0.82645
0.75131
0.68301
3.16986
Present
Value
$ 9,090.90
8,264.50
7,513.10
6,830.10
$ 31,698.60
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.
6-24
Slide 25
Present Value of an Ordinary Annuity
PV1
PV2
PV3
PV4
Total
Annuity
$ 10,000
10,000
10,000
10,000
PV of $1
Factor
0.90909
0.82645
0.75131
0.68301
3.16986
Present
Value
$ 9,090.90
8,264.50
7,513.10
6,830.10
$ 31,698.60
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
6-25
Slide 26
Present 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.
b.
c.
d.
$153,981
$171,190
$167,324
$174,680
PV of Ordinary Annuity $1
Payment
$ 20,000.00
PV Factor
× 8.55948
Amount
$171,189.60
6-26
Slide 27
Present 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.
Amount of annuity
$ 10,000
PV of annuity due of $1
(i=6%, n=4)
Present value of annuity
× 3.67301
$ 36,730
6-27
Slide 28
Present 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.
6-28
Slide 29
Present Value of a Deferred Annuity
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?
Present
Value?
1/1/09
1
2
12/31/09
1
Payment
$
12,500
12,500
12/31/10
2
$12,500
$12,500
12/31/11
3
12/31/12
4
PV of $1
i = 12%
0.71178
0.63552
$
$
12/31/13
PV
8,897
7,944
16,841
n
3
4
6-29
Slide 30
Present Value of a Deferred Annuity
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?
Present
Value?
1/1/09
12/31/09
1
12/31/10
2
$12,500
$12,500
12/31/11
3
12/31/12
4
12/31/13
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.
6-30
Slide 31
Present Value of a Deferred Annuity
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?
Present
Value?
1/1/09
Payment
$
12,500
12/31/09
1
PV of
Ordinary
Annuity of $1
n=2, i = 12%
1.69005
12/31/10
2
$
PV
21,126
$12,500
$12,500
12/31/11
3
12/31/12
4
Future Value
$
21,126
12/31/13
PV of $1
n=2, i = 12%
0.79719
$
PV
16,841
6-31
Slide 32
Solving for Unknown Values in Present Value
Situations
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.
6-32
Slide 33
Solving 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?
Present
Value
$700
Today
End of
Year 1
End of
Year 2
End of
Year 3
End of
Year 4
6-33
Slide 34
Solving 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?
Present value
$ 700.00
PV of ordinary annuity of $1
(i=8%, n=4)
Annuity amount
÷ 3.31213
$ 211.34
6-34
Slide 35
Accounting Applications of Present Value
Techniques—Annuities
Because financial instruments
typically specify equal periodic
payments, these applications quite
often involve annuity situations.
Long-term
Bonds
Long-term
Leases
Pension
Obligations
6-35
Slide 36
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
Face value of the bond
Interest (annuity)
Price of bonds
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?
Table
PV of $1
n=10; i=6%
PV of
Ordinary
Annuity of $1
n=10; i=6%
Table
Value
Amount
0.5584 $ 1,000,000
7.3601
Present
Value
$
558,400
$
368,005
926,405
50,000
6-36
Slide 37
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.
6-37
Slide 38
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.
6-38
End of Chapter 6
McGraw-Hill /Irwin
© 2009 The McGraw-Hill Companies, Inc.