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William F. Bentz 1
Session 11 - Interest Cost
William F. Bentz 2
Interest
A. Interest is the compensation that must be paid by a borrower for the use of a lender’s money. The amount of the total compensation (total interest) is a function of the amount borrowed, the period of the loan, and the timing of the required payments.
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Total Interest
B. The total amount of interest
= total interest payments + (total principal repaid - loan proceeds - fees), or simply
= total payments to the lender minus total cash received from the lender and any fees
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Interest Specification
C. Interest may be stated explicitly in an agreement (e.g., a car loan contract), or it may be an implicit component of an agreement (e.g., car sales or lease contracts).
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Interest Allocation
D. In any case, the amount of the total interest is known. The accounting issue is how best to allocate the total interest over the accounting periods that include the loan period.
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Principal VS. Interest
E. Any agreement involving interest includes an amount on which interest is being earned--the principal, a rate at which interest is earned, a method of determining interest, and the timing of all payments.
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Principal
The amount on which one is earning interest is known as the principal. The original amount of a car loan would be called the principal. At any date thereafter, the principal normally would be the unpaid balance of the loan.
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Interest Rates
Different types of loans require the calculation of interest for different loan periods (e.g., monthly, quarterly, semi-annually, etc.), but a rate for any period can be converted to an equivalent annual rate.
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Annual Interest Rates
By determining effective annual interest rates, we have a common measure of interest costs. The effective annual rate allows us to compare different loans with different payment patterns and terms.
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Annual Interest Rates
Without a common basis with which to compare loans, the task would be much more difficult. The usual approach is to determine the effective annual yield of a loan or investment (internal rate of return).
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Interest Method
3. There are two basic methods of computing interest:
simple interest, and compound interest.
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Method: Simple Interest
Total amount of simple interest
I = principal x rate x number of interest periods
I = P x r x N
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Simple Interest Example
A loan of $1,000 for two years earning simple interest of 10% per year would result in total interest of
$200.00 = $1,000 x 0.10 x 2
= $200
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Method: Compound Interest
Total compound interest amount
I = principal x (1 + rate)n- principal
or
I = principal x [(1 + rate)n - 1]
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Method Difference
The difference between the two methods is that with compound interest, additional interest is earned on any unpaid interest. No further interest is earned on unpaid simple interest.
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Method: Compound Interest
Interest amount for two years (with annual compounding)
I = (principal x rate) x 2 + rate x (rate x principal)
I = simple interest + interest on unpaid interest
I = P(1 + r)2 - P
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Method: Compound Interest
I = P(1 + 2r + r2) - P I = [P x r x 2] + r[rP] Thus, compound interest is
simple interest [P x r x 2] plus interest on the accumulated interest r[rP].
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Interest As A Growth Rate
Think of interest as growth in wealth over time.
With simple interest, the growth in wealth is linear.
With compound interest, the growth in wealth is exponential.
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The Key Accounting Issues
Measuring and reporting interest income or expense
Separating the interest component from other operating activities (revenue or cost).
Valuing selected assets and liabilities at their present values.
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Cash Flows
Determining the amount and timing of the cash flows is crucial to all but the most elementary decisions or accounting problems involving interest. To analyze about any case or problem, diagram the cash flows first. Diagram the cash flows first!!!
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Cash Flows
Cash flows are assumed to take place at points in time, or uniformly over periods of time. In this course, we are going to assume that cash flows take place at discrete points in time.
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Scenario 1
Suppose you invest $1,000 on 1/1/01 and receive $100 as of 12/31/01, and your $1,000 investment, for a total of $1,100
What is you net income before tax?
What is the rate of return on investment?
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Scenario 2
What is the rate of return on investment?
Suppose you invest $1,000 on 1/1/01 and receive $100 as of 1/1/01, and your $1,000 investment on 12/31/01, for a total of $1,100
What is you net income before tax?
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Suppose you invest $1,000 on 1/1/01 and receive $210 and your $1,000 investment as of 12/31/02, for a total of $1,210.
What is you net income before tax?
What is the rate of return on investment?
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Suppose you invest $1,000 on 1/1/01 and receive $210 and your $1,000 investment as of 1/1/02, for a total of $1,210.
What is you net income before tax?
What is the rate of return on investment?
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Suppose you invest $1,000 on 1/1/01 and receive $100 on 12/31/01 and $100 on 12/31/02, plus your $1,000 investment as of 12/31/02, for a total of $1,200.
What is you net income before tax?What is the rate of return on investment?
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Suppose you invest $1,000 on 1/1/01 and receive $100 and your $1,000 investment as of 1/1/01, another $100 as of 1/1/02, and your investment of $1,000 on 12/31/02, for a total of $1,200.What is you net income before tax?
What is the rate of return on investment?
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Compound Interest
Let amounts be the principal and interest that have accumulated to some point in time. Thus, amounts represent wealth at a point in time. Interest is the growth factor in changes in wealth over time.
1
1
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Compound Interest
An amount of wealth A0 will grow to an amount A1 in one period in accordance with the model
A1 = (1 + r)A0
where r is the growth (interest) rate (a.k.a. internal rate of return)
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Compound Interest
If we know two of the three variables in the model (r, A0, or A1), the we can determine the value of the third variable.
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Fundamental Equations
two.the
equates r the rateinterest an find
can we, and given aFor
)1(
then ,)1( If
10
10
01
AA
rA
A
ArA
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Fundamental Equations
For a given interest rate, and an amount a time t = 0, we can find an unknown amount at t = 1. If the present amount A0 is $10,000, and the interest rate is 11%, then the future amount at t = 1, denoted A1,
= $10,000 x (1 + .11) = $11,000.
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Fundamental Relationships
On the other hand, if the future amount is known to be $10,000, and the interest rate is 11%, then the present value (amount at t = 0) is
009.009,9$)11.1(
000,10$0 A
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Generalizing
The relationship between an amount at t = 0 and an amount at t = 1 can be generalized to any times N - 1 and N. An amount at time t = N - 1 will grow in one period to an amount AN given by
)1(1 rAA NN
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Further Generalizing
By repeated substitutions for the amounts at t = N -1, N - 2, N - 3, etc. we get the expression for the present value of an amount at t = N
and thus
NN
NN
r
AA
rAA
)1(
)1(
0
0
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Annuities
Annuities are nothing more than sums of present value amounts or of future amounts at some interest rate r.
where all the payments equal A
Nr
A
r
A
r
ASum
)1(...
)1()1( 2
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Annuities
The A term is a constant amount, so the expression can be simplified to
In this case The sum is the present value, at time t = 0, of an ordinary annuity of $A per period.
Nrrr
ASum)1(
1
)1(
1
)1(
12
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Annuities
If we let PVOA represent the present value of an ordinary annuity, then we have that
Nrrr
APVOA)1(
1
)1(
1
)1(
120
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Annuity Formulas
This is a series that can be summed (see notes for derivation) as follows:
Ar
rAPVOA
r
rPVOA
N
N
$ amounts equal of series afor ])1(1[
periodper $1 ofinterest for ])1(1[
0
0
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Finding the Effective Rate(r)
Many times in accounting and finance, we want to know the expected rate of return of a proposed project, or the actual rate of return earned by a project or investment. We want a universal measure of profitability.
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Examples
What is the effective interest rate of the bond issue we just sold?
Is a transaction fairly priced given the market rates available?
What is the expected rate of return on a proposed project or investment?
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The Answer
To answer the above questions, we need to find the internal rate of return of the loans or investments under consideration. The internal rate of return is that rate which discounts a stream of cash flows, including at least one negative cash flow, to zero.
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The Internal Rate
The rate that solves the following equation is called the internal rate of return:
where C0,C1,…,CN are cash flows, at least one of which is negative.
NN
r
C
r
C
r
C
r
CC
)1()1()1()1(0
33
221
0
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An Alternative Form
For many investment transactions, the first cash flow is negative and the subsequent cash flows are positive. The above formula can be rewritten (equivalently) in a more intuitive form as
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Equivalent Expression
NN
r
C
r
C
r
C
r
CC
)1()1()1()1( 33
221
0
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Implementation of IRR
For every change in sign from positive to negative, or negative to positive, there will be another solution to the above equation. However, we can ignore the negative rates if total net cash flows are positive. For most realistic cases, IRR calculations work fine.
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Use of the IRRWHIZ
Use the IRR calculator on the web in my directory to make internal rate of return (IRR) calculations. By looking at the formulas and instructions, I think you will be able to see how to make the whiz fit your problem.
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Some Application Areas
Simple interest is used in the discounting of notes receivable and other transactions involving simple interest over periods typically less than one year.
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Compound Interest
The compound interest model (effective yield method) is used to account for long-term receivables, investments held to maturity, bonds payable, leases, notes payable, and pension computations, to name a few important areas. (GAAP)
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To Summarize
Estimating cash flows and using the compound interest model to discount those cash flows to analyze the profitability of business transactions is fundamental to the study of accounting and finance. Few concepts are more important.
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THE END
