1111111 Intermediate Accounting, Ninth Edition Kieso and Weygandt Prepared by Catherine Katagiri,...

1111111

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.

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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?

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• 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?

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• 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.

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• 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!

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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.

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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.

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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.

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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.

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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)

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– 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.

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• 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.

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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).

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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)

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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.

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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

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• 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

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• 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

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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.


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• 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 )

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• 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”

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• 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

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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!

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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.

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• 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%.

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• 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.

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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

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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)

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• 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)

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• 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!

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• 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

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• 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 ) )

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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)

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• 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)

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• 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!

1111111 Intermediate Accounting, Ninth Edition Kieso and Weygandt Prepared by Catherine Katagiri,...

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