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Chapter 3 - Interest

and EquivalenceClick here for Streaming Audio To

Accompany Presentation (optional)

EGR 403 Capital Allocation Theory

Dr. Phillip R. Rosenkrantz

Industrial & Manufacturing Engineering Department

Cal Poly Pomona
http://video.csupomona.edu/PRRosenkrant/EGR403Lecture-05.asxhttp://video.csupomona.edu/PRRosenkrant/EGR403Lecture-05.asxhttp://video.csupomona.edu/PRRosenkrant/EGR403Lecture-05.asxhttp://video.csupomona.edu/PRRosenkrant/EGR403Lecture-05.asx

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EGR 403 - Cal Poly Pomona - SA5 2

EGR 403 - The Big Picture Framework:Accounting& Breakeven Analysis

Time-value of money concepts - Ch. 3, 4

Analysis methods

Ch. 5 - Present Worth

Ch. 6 - Annual Worth

Ch. 7, 8 - Rate of Return (incremental analysis)

Ch. 9 - Benefit Cost Ratio & other techniques

Refining the analysis

Ch. 10, 11 - Depreciation & Taxes

Ch. 12 - Replacement Analysis

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Economic Decision Components

Where economic decisions are immediate we needto consider: amount of expenditure

taxes

Where economic decisions occur over aconsiderable period of time we need to also

consider the consequences of: interest inflation

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Computing Cash Flows

Cash flows have:

Costs (disbursements) a negative number

Benefits (receipts) a positive number

Example 3-1

End of

Year Cash flow

0 (1,000.00)$

1 580.00$

2 580.00$

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Time Value of Money

Money has value Money can be leased or rented

Thepayment is called interest If you put $100 in a bank at 9% interest for one time

period you will receive back your original $100 plus $9

Original amount to be returned = $100

Interest to be returned = $100 x .09 = $9

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Simple Interest

Interest that is computed only on the original

sum or principal

Total interest earned = I = P i n , where:

P = present sum of money, or principal (example:

$1000)

i = interest rate (10% interest is a .10 interest rate)

n = number of periods (years) (example: n = 2 years)

I = $1000 x .10/period x 2 periods = $200

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Future Value of a Loan With

Simple Interest Amount of money due at the end of a loan

F = P + P i n or F = P (1 + i n )

Where,

F = future value

F = $1000 (1 + .10 x 2) = $1200

Simple interest is not used today

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Compound Interest Over Time

If you loaned a friendmoney for short period

of time the difference

between simple and

compound interest is

negligible.

If you loaned a friend

money for a long period

of time the difference

between simple andcompound interest may

amount to a considerable

difference.

P n i% F

1000 1 10% $1,100.00

1000 2 10% $1,210.001000 3 10% $1,331.00

1000 10 10% $2,593.74

1000 20 10% $6,727.50

1000 30 10% $17,449.40

1000 40 10% $45,259.26

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Nominal and Effective Interest

Interest rates are normally given on an annual basis withagreement on how often compounding will occur (e.g., monthly,

quarterly, annually, continuous).

Nominal interest rate /year ( r )the annual interest rate w/o

considering the effect of any compounding (e.g., r = 12%).

Interest rate /period ( i )the nominal interest rate /year

divided by the number of interest compounding periods (e.g.,

monthly compounding: i = 12% / 12 months/year = 1%).

Effective interest rate /year ( ieffor APR )the annual interest

rate taking into account the effect of the multiple compounding

periods in the year. (e.g., as shown later, r = 12% compounded

monthly is equivalent to 12.68% year compounded yearly.

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Interest Rates (contd)

We use i for the periodic interest rate Nominal interest rate = r (an annual rate)

Number of compounding periods/year = m

r = i * m, and i = r / m

Let r = .12 (or 12%)

Interest Period m = interest periods /

year

i = interest rate / period

Annual 1 .12Quarter 4 .03

Month 12 .01

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Plan 1 Equal annual

principal paymentsYear Balance P i Payment

1 5000 1000 500 1500

2 4000 1000 400 1400

3 3000 1000 300 1300

4 2000 1000 200 1200

5 1000 1000 100 1100

6500

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Plan 2Annual interest + balloon

payment of principalYear Balance P i Payment

1 5000 500 500

2 5000 500 500

3 5000 500 500

4 5000 500 500

5 5000 5000 500 500

7500

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Plan 3Equal annual payments

(installments)Year Balance P i Payment

1 5000.00 819.00 500.00 1319

2 4181.00 900.90 418.10 1319

3 3280.10 990.99 328.01 1319

4 2289.11 1090.09 228.91 1319

5 1199.02 1199.10 119.90 1319

6595

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Plan 4Principal & interest at

end of the loanYear Balance P i Payment

1 5000 0 500 0

2 5500 0 550 0

3 6050 0 605 0

4 6655 0 665.50 0

5 7320.50 0 732.05 8052.55

8052.55

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Which plan would you choose?

Total Principal + Interest Paid

Plan 1 = $6500

Plan 2 = $7500

Plan 3 = $6595

Plan 4 = $8052.55

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Equivalence

When an organization is indifferent as to whether

it has a present sum of money now or, with

interest the assurance of some other sum of money

in the future, or a series of future sums of money,we say that the present sum of money is

equivalentto the future sum or series of future

sums.

Each of the four repayment plans are equivalent

because each repays $5000 at the same 10% interest rate.

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To further illustrate this concept, given the choice

of these two plans which would you choose?

$7000$6200Total

540010805

40011604

40012403

40013202

$400$14001

Plan 2Plan 1Year

To make a choice the cash flows must be

altered so a comparison may be made.

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Technique of Equivalence

Determine a single equivalent value at apoint in time for plan 1.

Determine a single equivalent value at apoint in time for plan 2.

Both at the same interest rate

Judge the relative attractiveness of the two

alternatives from the comparable equivalentvalues. You will learn a number of methods

for finding comparable equivalent values.

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Analysis Methods that Compare

Equivalent Values Present Worth Analysis (Ch. 5) - Find the equivalent value of

cash flows at time 0.

Annual Worth Analysis (Ch. 6) - Find the equivalent annual

worth of all cash flows.

Rate of Return Analysis (Ch. 7, 8) - Compare the interest rate

(ROR) of each alternatives cash flows to a minimum value you

will accept.

Future Worth Analysis (Ch. 9) - Find the equivalent value of

cash flows at time in the future.

Benefit/Cost Ratio (Ch. 9) - Use equivalent values of cash flows

to form ratios that can be easily analyzed.

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Interest Formulas To understand equivalence the underlying

interest formulas must be analyzed. We willstart with Single Payment interest formulas.

Notation:

i = Interest rate per interest period.

n = Number of interest periods.

P = Present sum of money (Present worth, PV).

F = Future sum of money (Future worth, FV).

If you know any three of the above variables

you can find the fourth one.

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For example, given F, P, and n, find

the interest rate (i) or ROR

Principal outstanding over time (P)

Amount repaid (F) at n future periods from now

We know F, P, and n and want to find the interest rate

that makes them equivalent:

If F = P (1 + i)n

Then i = (F/P)1/n - 1

This value of i is theRate Of Return or ROR for

investing the amount P to earn the future sum F

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Cash Flow Diagrams We use cash flow

diagrams to helporganize the data for

each alternative.

Down arrow -

disbursement cash flow Up arrow - Income cash

flow

n = number of

compounding periodsin the problem

i = interest rate/period

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Notation for

Calculating a Future Value

Formula:

F=P(1+i)n is the

single payment compound amount factor.

Functional notation:

F=P(F/P, i, n) F = 5000(F/P, 6%, 10)

F =P(F/P) which is dimensionally correct.

Find the factor values in the tables in the

back of the text.

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Using the Functional Notation and

Tables to Find the Factor Values

F = 5000(F/P, 6%, 10)

To use the tables:

Step 1: Find the page with the 6% tableStep 2: Find the F/P column

Step 3: Go down the F/P column to n = 10

The Factor shown is 1.791, therefore:F = 5000 (1.791) = $8955

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Notation for

Calculating a Present Value P=F(1/1+i)n=F(1+i)-n is the

single payment present worth factor

Functional notation:

P=F(P/F, i, n) P=5000(P/F, 6%, 10)

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Example: P=F(P/F, i, n)

F = $1000, i = 0.10, n = 5, P = ?

Using notation: P = F(P/F, 10%, 5)= $1000(.6209) = $620.90

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egr403_sv5_chapter3 (1)