Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford Unversity Press, Inc.1 Engineering Economic Analysis 9th

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

Engineering Economic Analysis9th Edition

Chapter 3

INTEREST AND EQUIVALENCE


Economic Decision Components

• Where economic decisions are immediate we need to consider:• amount of expenditure • taxes

• Where economic decisions occur over a considerable period of time we also need to consider:• interest• inflation


Computing Cash Flows

• Cash flows have:• Costs (disbursements) > a negative number• Benefits (receipts) > a positive number

Example 3-1End of Year Cash flow

0 (1,000.00)$ 1 580.00$ 2 580.00$


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

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


Simple Interest

• Interest that is computed only on the original sum 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


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 = $100 (1 + .09 x 2) = $118

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


Compound Interest

• Interest that is computed on the original unpaid debt and the unpaid interest

• Total interest earned = In = 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


Future Value of a Loan with Compound 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 with compound interest terms?• Would a bank?


Comparison of Simple and Compound Interest Over Time

•If you loaned a friend money 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 and compound interest may amount to a considerable difference.

Short or long? When is the $ difference significant?You pick the time period.

Simple and compound interest Single payment

Principal = 100.00Interest = 9.00%

PeriodSimple

amount factorCompound

amount factor

n

Find Fs Given P

Fs/PFind 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 the table to see the difference over time.


Four Ways to Repay a Debt

Plan RepayPrincipal

Repay Interest Interest Earned

1 Equal annual installments

Interest on unpaid balance

Declines

2 End of loan Interest on unpaid balance

Constant

3 Equal annual installments Declines at increasing rate

4 End of loan Compound and pay at end of loan

Compounds at increasing rate until end of loan


Loan Repayment – Four Options

This calculator is partially complete. If you complete the calculator you can earn 10 bonus points for your team.

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

10 YearsPlan 1 Enter 1 through 4

Principal payment Equal annual installmentsInterest payment EOY on unpaid principal

Years

Amount owed at the beginning of the year

Interest owed for that year

Total owed at the end

of yearPrincipal payment

Total end of 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 8006 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


Equivalence

• When an organization is indifferent as to whether it has a present sum of money now or the assurance of some other sum of money (or series of sums of money) in the future, we say that the present sum of money is equivalent to the future sum or series of sums.

Each of the plans on the previous slide is equivalent because each repays $5000 at the same 10% interest rate.


Given the choice of these two plans which would you choose?

Year Plan 1 Plan 2

1 $1400 $400

2 1320 400

3 1240 400

4 1160 400

5 1080 5400

Total $6200 $7000

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


Technique of Equivalence

• Determine a single equivalent value at a point in 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 the two alternatives from the comparable equivalent values.


Repayment Plans Establish the Interest Rate

1. Principal outstanding over time

2. Amount repaid over time

As an example:

If F = P (1 + i)n

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

$5,0008.00%

5Plan 1


Years Amount owed at the beginning of the year


Total owed at the 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 1,200

Total owed over time 15,000

$4,876.639.00%

5Plan 1


Years Amount owed at the beginning of the year


Total owed at the 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%


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) $1,341.57Present worth

A $212.65 ($212.65) ($218.33) $218.33Annual worth

F $3,389.05 ($3,389.05) ($3,479.68) $3,479.68Future worth

Alternative

Comparing alternativesPick an alternative. Which would you choose?

Change the interest rate. What happens at 8%,15%,3%?


Interest Formulas

• To understand equivalence, the underlying interest formulas must be analyzed.

• Notation:I = Interest rate per interest period

n = Number of interest periods

P = Present sum of money (Present worth)

F = Future sum of money (Future worth)


Single Payment Compound Interest

YearBeginning balance

Interest for period

Ending balance

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


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.


Notation forCalculating 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)


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


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/yearEffective 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

Documents

Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford Unversity Press, Inc.1 Engineering Economic Analysis 9th