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Intermediate Accounting, Ninth EditionIntermediate Accounting, Ninth Edition
Kieso and WeygandtKieso and Weygandt
Prepared byPrepared by
Catherine Katagiri, CPACatherine Katagiri, CPA
The College of Saint RoseThe College of Saint Rose
Albany, New YorkAlbany, New York
John Wiley & Sons, IncJohn Wiley & Sons, Inc..
Chapter 6: Accounting and the Chapter 6: Accounting and the Time Value of MoneyTime Value of Money
After studying this chapter you should be able to:After studying this chapter you should be able to:
• Identify accounting topics where time value of money is
relevant.
• Distinguish between simple and compound interest.
• Learn how to use appropriate compound interest tables.
• Identify variables fundamental to solving interest
problems.
• Solve future and present value of 1 problems.
• Solve future value of ordinary and annuity due problems.
• Solve present value of ordinary and annuity due problems.
33333
IntroductionIntroduction
• Concept of the time value of money is very important especially
when interest rates are high and/or time periods are
long.
• Simply put--You are not indifferent as to when you
receive or pay an identical sum of money. That is, a
dollar received or paid today (the present) is not worth
a dollar received or paid tomorrow (the future).
• Someone owes you, a rational person, $100. They say
to you--Which would you prefer: I pay you $100 today
or I pay you $100 one year from now?
44444
IntroductionIntroduction
• Your response should be--Pay me $100 today
(present). WHY? Assuming there is no inflation, would
your answer change?
• Again, your response should be to be paid today. You
could take the funds today, invest them, earn a return
and have more than $100 a year from now. How much
more than $100 would depend on the return you could
earn (interest).
• Suppose, instead, someone offers you $100 one year
from today. You wish the funds now (today), however.
Will you accept less than $100 in settlement of the
debt?
55555
IntroductionIntroduction
• Considering the interest rates, you will accept less than
$100 today to settle the debt. WHY?
• If you had the settlement today you could invest it, earn a
return and have $100 one year from now. If the interest rate
is high you would accept less to settle the debt than if the
interest rate was low.
Simply put, at any interest rate:
• A dollar received today is worth more than a
dollar in the future.
• A dollar in the future is worth less than a dollar
today.
66666
IntroductionIntroduction
• The concept of the time value of money is pervasive. We
see this concept in many topics including (to name a few):
– Leases
– Pensions
– Non-interest bearing notes
– Installment contracts
– Valuation of bonds
– Sinking fund provisions
• The FASB will simplify calculations by stating, at times, the
time value of money is to be ignored. You must realize the
import of that assumption as well!
77777
InterestInterest
• Definitions of interest:
– A fee for the use of money
– Principle x Interest Rate x Time = InterestPrinciple x Interest Rate x Time = Interest
• Principle: The amount borrowed or invested.
• Interest rate: A percentage of the outstanding
principle. Always expressed as an annual
rate.
• Time: The number of years or fraction thereof
that the principle is outstanding.
88
InterestInterest
– Selection of appropriate interest rate is
critical and, at times, very difficult.
Would you be a wealthy person if you
could accurately predict the interest
rate?
– The interest for us will be a given figure.
In practice it can be the hardest figure to
derive accurately.
99
InterestInterest
– Elements of the interest rate: The interest rate is the sum of
• Pure rate of interest (system risk) Initial rate charged
assuming no possibility of default and no inflation.
• Credit risk rate of interest (individual risk) Additional rate
charged as a result of an individual entity’s risk of default.
• Expected inflation rate of interest (inflation premium)
Additional rate charged to compensate for the decrease in
the purchasing power of the dollar over time.
10
InterestInterest
– Simple interestSimple interest is computed on the principle
only. That is, interest is earned and removed.
The interest does not earn interest.
– Compound interestCompound interest is computed on the
principle and any accumulated interest.
Both the principal and interest then earn
interest.
11
Types of ProblemsTypes of Problems
• For our calculations we will assume compound
interest. The greater the number of periods and
the higher the interest rate, the greater will be the
effect of compounding interest.
• Types of problems we will be working with:
– Single SumSingle Sum. One sum ($1) will be received or
paid either in the
• Present (Present Value of a Single Sum or
PVPV)
• Future (Future Value of a Single Sum or FVFV)
12
Types of ProblemsTypes of Problems
– Types of annuity problemsTypes of annuity problems:
• Ordinary annuity (OAOA)
– A series of equal payments (or rents) received or
paid at the end of a period, assuming a constant
rate of interest.
– Value today of a series of equal, end-of-period
payments received in the future is the Present Value
of an Ordinary Annuity or PV-OAPV-OA.
– Value in the future of a series of equal, end-of-
period payments received in the future is the Future
Value of an Ordinary Annuity or FV-OA.FV-OA.
1313
Types of ProblemsTypes of Problems
• Annuity Due (ADAD)
– A series of equal payments (or rents) received or
paid at the beginning of a period, assuming a
constant rate of interest.
– Value today of a series of equal, beginning-of-
period payments received in the future is the
Present Value of an annuity due or PV-ADPV-AD.
– Value in the future of a series of equal, beginning-
of-period payments received in the future is the
Future Value of an annuity due or FV-AD.FV-AD.
1414
Types of ProblemsTypes of Problems
Note:Note: The difference between an ordinary annuity and an annuity due is that:
Each rent or payment compounds (interest added) one more period in a annuity due, future value situation.
– Each rent or payment is discounted (interest
removed) one less period under the annuity due
situation.
– Given the same i, n and periodic rent, the annuity
due will always yield a greater present value (less
interest removed) and a greater future value (more
interest added).
1515
Calculation VariablesCalculation Variables
• There will always be at least four variables in any
present or future value problem. Three of the
four will be known and you will solve for the
fourth.
– Single sum problemsSingle sum problems:
• n = number of compounding periods
• i = interest rate
• PV = Value today of a single sum ($1)
• FV = Value in the future of a single sum ($1)
1616
Calculation VariablesCalculation Variables
– Annuity problemsAnnuity problems:
• n = number of payments or rents
• i = interest rate
• R = Periodic rent received or paid
And either:
• FV-OA or AD = Value in the future of a series of
future payments (either ordinary or due). OR
• PV-OA or AD= Value today of a series of payments in
the future (either ordinary or due).
• Note: The “n” and the “i” must match. That is, if the time period is semi-annual then so must the interest rate. Interest rates are assumed to be annual unless otherwise stated so you may have to adjust the rate to match the time period.
1717
Single Sum ProblemsSingle Sum Problems
Let’s review the derivation of the single sum formula:Let’s review the derivation of the single sum formula:
• Suppose you have $100 today (present) and wish to deposit
it at 10% for three periods, in this case years. What is the
value of this single sum in the future?
– At the end of the first year (n = 1): (Compound interest)
$100 + $100(.1) = $110
– At the end of the second year ( n = 2)
$110 + $110(.1) = $121
– At the end of third year (n = 3)
$121 + $121(.1) = $133 (rounded)
– So the future value of $100 three years hence at 10% =
$133
1818
Single Sum ProblemsSingle Sum Problems
• I realize there is a simpler way to approach this:
$100 (1 +.1)(1+.1)(1+.1) = $133
$100 (1+.1)3 = $133
Generally for any “n” and “i” the single sum
formula would be:
PV (1+ i)n = FV
$100 (1+.1)3 = $133
1919
Single Sum ProblemsSingle Sum Problems
• Not wishing to have to constantly raise the term to
the required power I name the term (1+ i)n, the
Future Value Factor for a single term (FVFFVFn,i n,i )). I
then employ the table on page 320-321. The table
is the result of the required multiplications at
various “n” and “i” and is to be read vertically for
the “n” and horizontally for the “i”.
• To solve my problem using the table:
PV(FVFPV(FVFn,in,i ) = FV ) = FV where n = 3 and i = 10%
$100 (1.331) = $133
2020
Note: Note:
For single sum problems the “n” refers to periods For single sum problems the “n” refers to periods not necessarily defined as years! The period may be not necessarily defined as years! The period may be annual, semi-annual, quarterly or another time annual, semi-annual, quarterly or another time frame.frame.
In manual calculations the use of the table is In manual calculations the use of the table is strongly recommended. It enhances both speed and strongly recommended. It enhances both speed and accuracy.accuracy.
Single Sum ProblemsSingle Sum Problems
2121
Single Sum ProblemsSingle Sum Problems
• Now suppose instead of $100 today I am to receive $133
three years from now. Again the interest rate is 10%. I
don’t want to wait three years for my money. How much
will I accept today in lieu of the future payment?
Going back to the general formula for a single sum:
PV(FVFPV(FVFn,in,i ) = FV ) = FV
• I realize I can isolate the term I wish to solve for on one
side of the equation:
PV = FV divided by (FVFn,i )
and rearranging terms:
PV = FV ( 1/ FVFn,i )
2222
Single Sum ProblemsSingle Sum Problems
• Not wishing to have to constantly raise the term to the
required power and then divide it into 1, I name the term 1/
FVFn,i the Present Value Factor for a single sum (PVF (PVF n,in,i ) ). I
restate the formula
PV = FV (PVF PV = FV (PVF n,in,i ) )
• I then employ the table on page 322-323. The table is the
result of the required multiplications and division at various
“n” and “i” and is to be read vertically for the “n”read vertically for the “n”
and horizontally for the “i”horizontally for the “i”
23
Single Sum ProblemsSingle Sum Problems
• To solve my problem using the table:
Please look up the values on the tables as we go along!Please look up the values on the tables as we go along!
n = 3, i = 10%, FV = $133PV = FV (PVF PV = FV (PVF n,in,i ) )
PV = $133 (.75132)
PV = $100
24
Single Sum ProblemsSingle Sum Problems
Note: Review the tables and note their characteristics. They are very logical.
All sums in the future are worth less than themselves in the present. All factors on the present value of a single sum table are less than one.
All factors on the future value of a single sum table are greater than one. All present sums are worth more than themselves in the future.
Notice how the factors change dramatically as the “i” increases and the “n” lengthens!
25
Single Sum ProblemsSingle Sum Problems
Single Sum problems, other unknowns.
– Suppose you have $6,000 today (PV ) and you
need $9,000 five years hence. What rate of
annual interest must you earn to achieve your
goal?
Note: This can be solved as either a future or present value of a single sum problem. The formulas are reciprocals of each other.
26
Single Sum ProblemsSingle Sum Problems
• To solve as a future value of a single sum problem:
PV(FVFPV(FVFn,in,i ) = FV ) = FV where n = 5 and i = ?
$6,000 (FVFn,i ) = $9,000
FVFn,i = $9,000/$6,000 = 1.500
Looking on the future of a single sum table (page
321) for n = 3 and a FVF of 1.500, I find the
corresponding “i” to be between 8-9%.
27
Single Sum ProblemsSingle Sum Problems
• Alternatively, suppose I have $750 today. How long will I
have to wait to have $1,200 when the interest rate is 10%? I
will solve this as a PV problem.
PV = FV (PVF PV = FV (PVF n,in,i ) )
$750 = $1,200 (PVF n,i ) where n = ? and i = 10%
PVF n,i = $750/$1,200 = .625
Reading on the present value of a single sum table (page 323)
for 10% the “n” for the factor .625 is between 4-5 periods.
A more precise answer may be derived through the use of the mathematical technique of interpolation.
28
Annuity ProblemsAnnuity Problems
• Suppose I am to receive three equal $100 payments (R) each
at the end of the period (in this case a period is a year) when
the interest rate is a constant 15%. This is an ordinary
annuity since payments are at the end of the period. What is
the value to me at the end of the third year from receiving
this annuity? This is a future value of an ordinary annuity
problem. How do I go about solving it?
– I realize an annuity is really a series of single sums.
– If I take the FV for each single sum and add them I will
have the value of the entire stream of payments.
– I will use the FV formula which is:
PV(FVFPV(FVFn,in,i ) = FV ) = FV
29
Annuity ProblemsAnnuity Problems
Rent #Rent # Cmpd periodsCmpd periods SumSum FVFFVF FVFV All at i = 15% All at i = 15%
1 2 $100 1.322 $132
2 1 $100 1.150 $115
3 0 $100 1.000 $100
Totals 3.472 $347
• This is tedious! I notice I am multiplying a constant rent ($100)
by a changing interest factor. What if I added the three factors
and multiplied the total by the rent? That would be less work!
I’ll call the sum of the appropriate factors the FVF-AO FVF-AO n,in,i. I’ll
derive the general formula where R equals the constant rent:
FV-OAFV-OA = R (FVF-AO = R (FVF-AO n,in,i ) ) when I = 15% and n = 3 payments
FV-OA = $100 (3.472) = $347 (rounded)
30
Annuity ProblemsAnnuity Problems
• Where do I get these summed factors? From the future value of
an ordinary annuity table on pages 324-325. The addition for the
appropriate “n” and “i” has already been done. The annuity
tables are derived directly from the single sum tables.
• What if the same annuity had been set up but the rents were
received at the beginning rather than the end of the period?
Then you would have the future value of an annuity due case.
– I realize the only difference between the ordinary and due
situations is the timing of the rents. Each rent has one more
period to compound under the due situation than under the
ordinary annuity. I adjust the formula for one more interest
period:
FV-ADFV-AD = R (FVF-AO = R (FVF-AO n,in,i ) (1 + i) ) (1 + i)
31
Annuity ProblemsAnnuity Problems
• Is adjusting this factor this way logical? If I increase the factor
for one more period’s interest the FV will go up for the annuity.
That is what I want as the FV of an annuity due should be
greater than the ordinary one. To find the FV-AD in this case:
FV-ADFV-AD = R (FVF-AO = R (FVF-AO n,in,i ) (1 + i) ) (1 + i) where n = 3 payments
and i = 15%
FV-AD = $100 (3.472)(1 + .15) = $399 (rounded)
Note: The text does not have a table for the future value of an annuity due. If doing the calculations manually, approach the adjustment needed intuitively. Always apply a test of reasonableness in all calculations!
32
Annuity ProblemsAnnuity Problems
• Let’s take the case of the present value of an
ordinary annuity. That is, what is a stream of
future payments worth today? What sum do you
need today to draw out a series of equal
payments and have nothing left over? This
situation is very common in retirement cases or
the payment of debt.
Bonds Payable
ABC Company
33
Annuity ProblemsAnnuity Problems
• Suppose you are to pay three rents of $500, each at the
end of the next three years. The interest rate is 8%. How
much should I set aside today to have the required rents?
– Again I realize an annuity is simply a series of single
sums. I take the same approach as before in that I
discount (remove) interest from each of the rents.
– I take the present value of each, add, and I will have the
required sum (total present value) that I will need.
I use the formula: PV = FV (PVF PV = FV (PVF n,in,i ) )
34
Annuity ProblemsAnnuity Problems
Rent #Rent # Disc periodsDisc periods SumSum PVFPVF PV PV All at i = 8%All at i = 8%
1 1 $500 .92593 $463
2 2 $500 .85734 $429
3 3 $500 .79383 $397
Totals 2.5771 $1,289
• This is tedious! I notice I am multiplying a constant rent ($500)
by a changing interest factor. What if I added the three factors
and multiplied the total by the rent? That would be less work!
I’ll call the sum of the appropriate factors the PVF-AOPVF-AO n,in,i. I’ll
derive the general formula where R is the constant rent:
PV-OAPV-OA = R (PVF-AO = R (PVF-AO n,in,i) ) for n = 3 payments and i = 8%
PV-OA = $500 (2.5771) = $1,289 (rounded)
35
Annuity ProblemsAnnuity Problems
• Where do I get these summed factors? From the present
value of an ordinary annuity table on pages 326-327. The
addition for the appropriate “n” and “i” has already been
done. The annuity tables are derived directly from the
single sum tables.
• What if the same annuity had been set up but the rents were
received at the beginning rather than the end of the period?
Then you have the present value of an annuity duepresent value of an annuity due case.
– I realize the only difference between the ordinary and
due situations is the timing of the rents. Each rent has
one less period to discount under the due situation than
under the ordinary annuity. I adjust the formula for one
more interest period:
PV-ADPV-AD = R (PVF-AO = R (PVF-AO n,in,i)(1+i) )(1+i)
36
Annuity ProblemsAnnuity Problems
• Is adjusting this factor this way logical? If I increase the
factor the PV will go up for the annuity. That is what I want
as the PV of an annuity due should be greater than the
ordinary one. Each rent has one less period of interest
removed from it! To find the PV-AD in this case:
PV-ADPV-AD = R (PVF-AO = R (PVF-AO n,in,i)(1+i) )(1+i) where n = 3 payments and i = 8%
PV-AD = $500 (2.5771)(1 + .08) = $1,391 (rounded)
Note: The text does have a table for the present value of an annuity due. But again, if you are without the table, approach the adjustment needed intuitively. Always apply a test of reasonableness in all calculations!