Chapter 6 Lecture
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Transcript of Chapter 6 Lecture
PowerPoint Authors:Susan Coomer Galbreath, Ph.D., CPACharles W. Caldwell, D.B.A., CMAJon A. Booker, Ph.D., CPA, CIACynthia J. Rooney, Ph.D., CPA
Time Value of Money Concepts
6
Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin
6 - 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 - 3
Compound 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?
6 - 5
Future Value of a Single AmountThe 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 - 6
Future 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
6 - 7
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
Future Value of a Single Amount
6 - 8
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 - 9
Present Value of a Single Amount
Remember our equation?
FV = PV (1 + i) n
We can solve for PV and get . . . .
FV
(1 + i)nPV =
6 - 10
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 todaytoday in order to
accumulate $20,000 in 5 years, if you can earn 8% interest compounded annually?
6 - 11
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, at8% annual interest, you will have
$20,000 at the end of 5 years.
Present Value of $1 Table
6 - 12
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 - 13
Determining the UnknownInterest 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.
6 - 14
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 noninterest-bearing
notes.
Accounting Applications of Present Value Techniques—Single Cash Amount
6 - 15
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 Approach
6 - 16
Basic Annuities
An annuity is a seriesof equal periodic
payments.
Period 1 Period 2 Period 3 Period 4
$10,000 $10,000 $10,000 $10,000
6 - 17
An annuity with payments at the end of the period is known as an ordinary annuity.
Ordinary 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
6 - 18
Annuity Due
An annuity with payments at the beginning of the period is known as an annuity 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
6 - 19
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 - 20
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?
6 - 21
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 - 22
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.
6 - 23
Present Value of an Ordinary Annuity
You wish to withdraw $10,000 at the endof 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 - 24
PV1PV2PV3PV4
$10,000 $10,000 $10,000 $10,000
1 2 3 4Today
Present Value of an Ordinary Annuity
6 - 25
Present Value of an Ordinary Annuity
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$
6 - 26
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
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 - 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.
6 - 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 - 29
Present Value of a Deferred AnnuityOn January 1, 2011, you are considering an investment
that will pay $12,500 a year for 2 years beginning on December 31, 2013. If you require a 12% return on
your investments, how much are you willing to pay for this investment?
1/1/11 12/31/11 12/31/12 12/31/13 12/31/14 12/31/15
Present Value? $12,500 $12,500
1 2 3 4
6 - 30
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.
Present Value of a Deferred Annuity
1/1/11 12/31/11 12/31/12 12/31/13 12/31/14 12/31/15
Present Value? $12,500 $12,500
1 2 3 4
On January 1, 2011, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2013. If you require a 12% return on
your investments, how much are you willing to pay for this investment?
6 - 31
Present Value of a Deferred Annuity
1/1/11 12/31/11 12/31/12 12/31/13 12/31/14 12/31/15
Present Value? $12,500 $12,500
1 2 3 4
On January 1, 2011, you are considering an investment that will pay $12,500 a year for 2 years beginning on December 31, 2013. If you require a 12% return on
your investments, how much are you willing to pay for this investment?
6 - 32
Solving for Unknown Valuesin Present Value SituationsIn present value problems involving
annuities, there are four variables:
Present value of an Present value of an ordinary annuity or ordinary annuity or Present value of an Present value of an
annuity dueannuity due
The amount of the The amount of the annuity paymentannuity payment
The number of The number of periodsperiods The interest rateThe interest rate
If you know any three of these,the fourth can be determined.
6 - 33
Solving for Unknown Valuesin 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 - 34
Solving for Unknown Valuesin 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 - 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 - 36
Valuation of Long-term BondsCalculate 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 $1n=10; i=6% 7.3601 50,000 368,005
Price of bond issue 926,405$
On June 30, 2011, Ebsen Electric issued 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 December 31, 2011. What was the price of the bond
issue?
6 - 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 - 38
Valuation of Long-term Leases
On January 1, 2011, Todd Furniture Company signed a 20-year non-cancelable lease for a new retail showroom. The lease
agreement calls for annual payments of $25,000 for 20 years beginning on January 1, 2011. The appropriate rate of interest for
this long-term lease is 8%. Calculate the value of the asset acquired and the liability assumed by Todd (the present value of an
annuity due at 8% for 20 years).
6 - 39
Valuation 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.
6 - 40
Valuation of Pension Obligations
On January 1, 2011, Todd Furniture Company hired a new sales manger for the new showroom. The sales manager is expected to work 30 years before retirement on December 31, 2040. Annual
retirement benefits will be paid at the end of each year of retirement, a period that is expected to be 25 years. The sales
manager will earn $2,500 in annual retirement benefits for the first year worked, 2011. How much must Todd contribute to the
company pension fund in 2011 to provide for $2,500 in annual pension benefits for 25 years that are expected to begin in 30
years. Todd’s pension fund is expected to earn 5%.
6 - 41
Valuation of Pension Obligations
This is a two part calculation. The first part requires the computation of the present value of a 25-year ordinary annuity of $2,500 as of December 31, 2040. Next we calculate the present
value of the December 31, 2040 amount. This second present value is the amount Todd will contribute in 2011 to fund the
retirement benefit earned by the sales manager in 2011.