Minggu 2_Interest and Equivalence

8/13/2019 Minggu 2_Interest and Equivalence

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Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford Unversity Press, Inc. 1

Engineer ing Econo m ic Ana lys i s9th Edition

Minggu 2 INTEREST AND EQUIVALENCE

(Chapter 3)


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Economic Decision Components

Where economic decisions are immediate we need toconsider: amount of expenditure

taxes Where economic decisions occur over a considerable

period of time we also need to consider: 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 ofYear 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

The payment is called interest If you put $100 in a bank at 9% interest for one time periodyou will receive back your original $100 plus $9

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


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Simple Interest

Interest that is computed only on the originalsum or principal

Total interest earned = I = P x i x n Where

P present sum of money i interest rate n number of periods (years)

I = $100 x .09/period x 2 periods = $18


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Future Value of a Loan withSimple 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 = $100 (1 + .09 x 2) = $118

Would you accept payment with simple interest terms? Would a bank?


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Compound Interest

Interest that is computed on the originalunpaid debt and the unpaid interest

Total interest earned = I n = P (1+i) n - P Where

P present sum of money i interest rate n number of periods (years)

I2 = $100 x (1+.09) 2 - $100 = $18.81


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Future Value of a Loan withCompound Interest

Amount of money due at the end of a loan F = P(1+i) 1(1+i)2..(1+i) n or F = P (1 + i) n

Where F = future value

F = $100 (1 + .09) 2 = $118.81

Would you be more likely to accept payment withcompound interest terms? Would a bank?


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Comparison of Simple and CompoundInterest Over Time

If you loaned a friend moneyfor short period of time thedifference between simpleand compound interest isnegligible.If you loaned a friend moneyfor a long period of time thedifference between simpleand compound interest mayamount to a considerabledifference.

Short or long? When is the $ difference significant?You pick the t ime per iod.

Simple and compound interestSingle payment

Principal = 100.00Interest = 9.00%

PeriodSimple

amount factor Compound

amount factor

n

Find FsGiven P

Fs/P

Find F Given P

F/P0 100.000 100.0001 109.000 109.0002 118.000 118.8103 127.000 129.5034 136.000 141.1585 145.000 153.8626 154.000 167.7107 163.000 182.8048 172.000 199.2569 181.000 217.189

10 190.000 236.73611 199.000 258.04312 208.000 281.26613 217.000 306.58014 226.000 334.17315 235.000 364.24816 244.000 397.03117 253.000 432.76318 262.000 471.71219 271.000 514.16620 280.000 560.441

Check thetable tosee thedifferenceover time.


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Four Ways to Repay a Debt

Plan RepayPrincipal

Repay Interest Interest Earned

1 Equal annualinstallments

Interest onunpaid balance

Declines

2 End of loan Interest onunpaid balance

Constant

3 Equal annual installments Declines atincreasing rate

4 End of loan Compound andpay at end ofloan

Compounds atincreasing rateuntil end of loan


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Loan Repayment Four Options

This calculator is partiallycomplete. If you completethe calculator you can

earn 10 bonus points foryour team.

$5,000 Principal10.00% Interest rate (enter as .1 for 10%)

10 YearsPlan 1 Ent er 1 through 4

Principal payment Equal annual installmentsInterest payment EOY on unpaid principal

Years

Amountowed at thebeginningof the year

Interestowed forthat year

Total owedat the end

of year Principal payment

Total endof year

payment

1 5,000 500 5,500 500 1,0002 4,500 450 4,950 500 9503 4,000 400 4,400 500 9004 3,500 350 3,850 500 8505 3,000 300 3,300 500 800

6 2,500 250 2,750 500 7507 2,000 200 2,200 500 7008 1,500 150 1,650 500 6509 1,000 100 1,100 500 600

10 500 50 550 500 5502,750 5,000 7,750

1 5,000 500 5,500 0 5002 5,000 500 5,500 0 5003 5,000 500 5,500 0 5004 5,000 500 5,500 0 5005 5,000 500 5,500 5,000 5,500

2,500 5,000 7,500

1 5,000 500 5,500 314 8142 4,500 450 4,950 345 8143 4,000 400 4,400 380 8144 3,500 350 3,850 418 8145 3,000 300 3,300 459 814

2,000 1,915 4,069

1 5,000 500 5,500 -500 02 5,500 550 6,050 -550 03 6,050 605 6,655 -605 04 6,655 666 7,321 -666 05 7,321 732 8,053 7,321 8,053

3,053 5,000 8,053

Loan Repayment Option Calculator


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Equivalence (1)

When an organization is indifferent as to whether ithas a present sum of money now or the assurance ofsome other sum of money (or series of sums of

money) in the future, we say that the present sum ofmoney is equivalent to the future sum or series ofsums.

Each of the plans on the previous slide isequivalent because each repays $5000 atthe same 10% interest rate.


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Equivalence (2)

Example : Four plans for repayment of $5000in five years at 8% interest :- Plan 1 : At end of each year pay $1000

principle plus interest due.

(1) (2) (3) (4) (5) (6)

YearAmount owed

at beginning ofyear

Interest owedfor that year

Total owed atend of year

Principlepayment

Total end ofyear payment

[8% x (2)] [(2) + (3)}

1 5000 400 5400 1000 1400

2 4000 320 4320 1000 1320

3 3000 240 3240 1000 1240

4 2000 160 2160 1000 1160

5 1000 80 1080 1000 1080

Total 1200 5000 6200


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Equivalence (3)- Plan 2 : Pay Interest due at end of each year

and principal at end of five years(1) (2) (3) (4) (5) (6)

Year

Amountowed at

beginning ofyear


Total owedat end of

year

Principle

payment

Total end ofyear

payment

[8% x (2)] [(2) + (3)}

1 5000 400 5400 0 400

2 5000 400 5400 0 400

3 5000 400 5400 0 400

4 5000 400 5400 0 400

5 5000 400 5400 5000 5400

Total 2000 5000 7000


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Equivalence (4)- Plan 3 : Pay in five equal end-of-year

payments.(1) (2) (3) (4) (5) (6)

YearAmount owedat beginning

of year

Interestowed for

that year

Total owedat end of

year

Principle

payment

Total end ofyear

payment[8% x (2)] [(2) + (3)}

1 5000 400 5400 852 1252

2 4148 331 4479 921 1252

3 3227 258 3485 994 12524 2233 178 2411 1074 1252

5 1159 93 1252 1159 1252

Total 1260 5000 6260


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Equivalence (5)- Plan 4 : Pay interest and principal at the end

of periods.(1) (2) (3) (4) (5) (6)

YearAmount owedat beginning

of year

Interestowed for

that year

Total owedat end of

year

Principle

payment

Total end ofyear

payment[8% x (2)] [(2) + (3)}

1 5000 400 5400 0 0

2 5400 432 5832 0 0

3 5832 467 6299 0 04 6299 504 6803 0 0

5 6803 544 7347 5000 7347

Total 2347 5000 7347


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Equivalence (6) ratio = total interest paid /total amount owed

at the beginning of year.

Plan Total Interestpaid

Total amountowed at the

beginning of yearratio

1 $ 1200 $ 15000 0,08

2 2000 25000 0,08

3 1260 15767 0,08

4 2347 29334 0,08

From our calculations, we more easily see why the repaymentplans require the payment of different total sums of money,yet are actually equivalent to each other.


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Given the choice of these two plans which

would you choose?Year Plan 1 Plan 2

1 $1400 $4002 1320 4003 1240 4004 1160 4005 1080 5400

Total $6200 $7000

To make a choice the cash flows must bealtered so a comparison may be made.


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Technique of Equivalence

Determine a single equivalent value at a pointin time for plan 1.

Determine a single equivalent value at a point

in time for plan 2.

Both at the same interest rate.

Judge the relative attractiveness of thetwo alternatives from the comparableequivalent values.


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Repayment PlansEstablish the Interest Rate

1. Principal outstandingover time

2. Amount repaid overtime

As an example:

If F = P (1 + i)n

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

$5,0008.00%5

Plan 1Principal payment Equal annual

Interest payment EOY on unpai

Year s Amount owed at the beginning o f the year


Total owed atthe end of year

1 5,000 400 5,4002 4,000 320 4,3203 3,000 240 3,2404 2,000 160 2,1605 1,000 80 1,080

Totals 1,200

Interest paid over time 1,200Total owed over time 15,000

$4,876.639.00%

5Plan 1

Principal payment Equal annualInterest payment EOY on unpai

Year s Amount owed at the beginning o f the year


Total owed atthe end of year

1 4,877 439 5,3162 3,901 351 4,2523 2,926 263 3,1894 1,951 176 2,1265 975 88 1,063

Totals 1,317

Equivalence Calculator

= 8.00%


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Application of Equivalence Calculations

Interest rate 10.00%

Year A B C D0 $600 -$600 -$850 $8501 $115 -$115 -$80 $802 $115 -$115 -$80 $803 $115 -$115 -$80 $804 $115 -$115 -$80 $805 $115 -$115 -$80 $806 $115 -$115 -$80 $807 $115 -$115 -$80 $808 $115 -$115 -$80 $809 $115 -$115 -$80 $8010 $115 -$115 -$80 $80

P $1,306.63 ($1,306.63) ($1,341.57) $1,341.57 Present

worth

A $212.65 ($212.65) ($218.33) $218.33 Annualworth

F $3,389.05 ($3,389.05) ($3,479.68) $3,479.68 Futureworth

Alternative

Comparing alternatives

Pick analternative.Which wouldyou choose?

Change theinterest rate.Whathappens at8%,15%,3%?


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Interest Formulas

To understand equivalence, the underlyinginterest formulas must be analyzed.

Notation:I = Interest rate per interest periodn = Number of interest periodsP = Present sum of money (Present worth)F = Future sum of money (Future worth)


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Single Payment Compound Interest

Year BeginningbalanceInterest for

periodEndingbalance

1 P iP P(1+i)

2 P(1+i) iP(1+i) P(1+i) 2

3 P(1+i) 2 iP(1+i)2 P(1+i) 3

n P(1+i)n-1

iP(1+i)n-1

P(1+i)n

P at time 0 increases to P(1+i) n at the end of time n.Or a Future sum = present sum (1+i) n


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Notation forCalculating 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.


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Notation forCalculating a Present Value

P=F(1/1+i) n=F(1+i) -n is thesingle payment present worth factor .

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


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Single Payment Formulas (4)example :

An investor (owner) has an option to purchase a tract ofland that will be worth $ 10.000 in six years. If the valueof the land increases at 8% each year, how much souldthe investor be willing to pay now for this property ?

P = 10.000 (P/F,8%,6)= 10.000 (0,6302)= $6.302

P = ?

F = 10.000

n = 6i = 0,08

1 2 3 4 5 6


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MENCARI F DIKETAHUI P

Rumus F = P (1+i) n Dengan Excel F =FV(i%,n,,-P) Dengan tabel F = P(F/P,i,n )

Contoh:Bila sejumlah $500 didepositokan di bank, berapakah

jumlahnya 3 tahun yang akan datang kalau bunga bank 6%per tahun?

F = 500 (F/P,6%,3)= 500 (1,191)= $ 595,5

F = FV(6%,3,,-500) = ????P = 500

F = ?

n = 3i = 0,06

1 2 3

29


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MENCARI P DIKETAHUI F

Rumus Dengan Excel P =PV(i%,n,,-F) Dengan tabel P = F(P/F,i,n ) Contoh:Seorang investor dapat membeli tanah yang nilainya akan menjadi $

10.000 dalam 6 tahun mendatang. Bila nilai tanah naik sebesar 8%setiap tahun, berapakah harga yang harus dibayar investor ini saatsekarang?

P = 10.000 (P/F,8%,6) Dengan Excel:= 10.000 (0,6302) P=PV(8%,6,,-10000) = ????

= $6.302

1

n P F i

P = ?

F = 10.000

n = 6i = 0,08

1 2 3 4 5 6

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Examples

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

F=$5000 i=0.10 n=5 P=?

F=P(1+i) n=$5000(1+0.10) 5=$5000(1.611)=$8055

F=P(F/P,10,5)=$5000(1.611) =$8055

P=F(P/F,10,5)=$8055(.62092)=$5000

10.00%

n0 $1.00 $5,000.00 $1.00 $8,052.551 1.100 $5,500.00 0.90909 $7,320.502 1.210 $6,050.00 0.82645 $6,655.003 1.331 $6,655.00 0.75131 $6,050.004 1.464 $7,320.50 0.68301 $5,500.005 1.611 $8,052.55 0.62092 $5,000.006 1.772 $8,857.81 0.56447 $4,545.457 1.949 $9,743.59 0.51316 $4,132.238 2.144 $10,717.94 0.46651 $3,756.579 2.358 $11,789.74 0.42410 $3,415.07

10 2.594 $12,968.71 0.38554 $3,104.6111 2.853 $14,265.58 0.35049 $2,822.3712 3.138 $15,692.14 0.31863 $2,565.7913 3.452 $17,261.36 0.28966 $2,332.5414 3.797 $18,987.49 0.26333 $2,120.4915 4.177 $20,886.24 0.23939 $1,927.7216 4.595 $22,974.86 0.21763 $1,752.4717 5.054 $25,272.35 0.19784 $1,593.1518 5.560 $27,799.59 0.17986 $1,448.3219 6.116 $30,579.55 0.16351 $1,316.6620 6.727 $33,637.50 0.14864 $1,196.9625 10.835 $54,173.53 0.09230 $743.2230 17.449 $87,247.01 0.05731 $461.4840 45.259 $226,296.28 0.02209 $177.9250 117.391 $586,954.26 0.00852 $68.6060 304.482 $1,522,408.20 0.00328 $26.4572 955.594 $4,777,969.09 0.00105 $8.43100 13,780.612 $68,903,061.70 0.00007 $0.58

Compound Amount Factor

F/P

Single Amount Factor

Compound Interest Factors

Present Worth Factor

P/F


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18% Compounded Monthly

18% interest: Assume a yearly rate if not stated Compounded monthly: Indicates 12 periods/year [18%/year] / [12months/year] = 1.5% / month

Nominal Interest rate 9.00% @ 365 Periods/year Effective Interest rate 9.42% per year

Number of years 1.00

i n $1.00 $500.00 $1.00 $547.089.00% 1.00 1.090 $545.00 0.99975 $501.910.02% 365 1.094 $547.08 0.91394 $500.00

Effective vs Nominal Interest Comparator

Compound Amount Factor

F/P

Single Amount Factor Present Worth Factor

P/F

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Minggu 2_Interest and Equivalence