Matakuliah: D0762 – Ekonomi Teknik Tahun: 2009 RATE OF RETURN ANALYSIS Course Outline 7.

Matakuliah : D0762 – Ekonomi TeknikTahun : 2009

RATE OF RETURN ANALYSISCourse Outline 7

Outline

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• Definition• ROR Facts• ROR for Single project• ROR for Multiple project• Spreadsheet

Refererences- Engineering Economy – Leland T. Blank, Anthoy J.

Tarquin p.200-246- Engineering Economic Analysis, Donald G. Newman, p.

163-196- Engineering Economy, William G. Sulivan, p.137-194,

p. 212-284

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Definition

• Synonym: IRR (Internal Rate of Return)

• Popular measurement on investment worth

• Which one represent the correct interpretation of ROR?

• Rate of Return on the un-recovered balance

• Rate of Return on the initial balance

• ROR (i*) is the interest rate earned on the un-recovered balance or the interest rate paid on the unpaid balance of a loan in which the final payment or receipt brings the terminal value to exactly equal “0”

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ROR Facts

• If i…

• > MARR, investment is justified

• = MARR, investment is justified (indifferent decision)

• < MARR, investment is not justified

• i is ranges …. –100% < i ≤ +• –100%: means total lost of capital

• >0%: means positive return on investment

• Some CF might have multiple ROR

• If there is a reinvestment option, use the composite rate 4KGA-Spr09®Reserved

Single Project

• Equation for computing ROR – Present Worth of benefits – PW of costs = 0– PW of benefits/PW of costs = 1– Net Present Worth = 0– EUAB – EUAC = 0 – PW of cost = PW of benefits

Note :EUAB : equivalent uniform annual benefitEUAC : equivalent uniform annual cost

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Example. 1An investment $8200 investment returned $2000 per year over a five – year useful life. What was the rate of return on the investment ?Solution

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1.4

2000

8200)5,,/(

18200

)5,,/(2000

1cost ofPW

benefits ofPW

iAP

iAP

From interest table (P/A,i,5):

i (P/A,i,5)

6% 4.212

7% 4.100

8% 3.993

The rate of return for the investment is 7%:

ROR Calculation

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• Trial & Error…

• Draw the CF Diagram

• Set up the equivalence equation and set equal to 0

• Select values of i and solve the equation

• Repeat until you find the i which give a balanced equation

• Sometimes might need to interpolate to find the approximate value of i*

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Example 7.2

The table shows an investment cash flow

Find rate of return for the investment above

Solution :• EUAB –EUAC = 0• 100 + 75(A/G,i, 4) -7000(A/P,i,4) = 0

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Year Cashflow

0 -$7000

1 +100

2 +175

3 +250

4 +325

There are two different interest factor. Solve the equation by

trial and error

Solution for Example 7.2.Try i = 5%

EUAB –EUAC = 0100 + 75(G/A,5%, 4) -7000(A/P,5%,4) = 0100 + 75(1.439) -7000(0.282) = 0208 – 197 = +6The EUAB –EUAC > 0, too low. If interest rate is increased, EUAC will increase.

Try i = 8%EUAB –EUAC = 0100 + 75(A/G,8%, 4) -7000(A/P,8%,4) = 0100 + 75(1.404) -7000(0.3019) = 0205 – 211 = -6The EUAB –EUAC < 0, too high

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Try i = 7%EUAB –EUAC = 0100 + 75(A/G,7%, 4) -7000(A/P,7%,4) = 0100 + 75(1.416) -7000(0.2952) = 0206 – 206 = 0The Rate Of Return = 7%

• So far we have learnt IRR not ERR (External Rate of Return)

• The difference between IRR and ERR is…

Un-recovered balance versus positive CF generated becomes released/external funds

• Solve it by…

Basic guesses (must performed both!):

• Descartes’ rule: sign change in the series of net CF

• Norstrom’s rule: sign change in the series of cumulative CF

Graphically

Better way: Composite Rate of Return (CRR)10

Multiple Values of ROR

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Example 101

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• For the CF below, how many ROR at most we could have? Use Descartes’ Rule!

1 2 3 4 5 6 Max i* values

- + + + - - 2

+ - + - + + 4

- + + + + + 1

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Composite Rate of Return (CRR or ERR)

• CRR/ERR/RIC: the unique ROR for a project which assumes that net positive CF, which represent money not immediately needed by the project are reinvested at the reinvestment rate “c”

• To summarize… any positive CF available in year X

• Let’s consider the funds released from a project in calculating the overall ROR of a project

• Reinvestment rate, “c”

• Composite rate of return =i’

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Equation for CRR

• Ft = Ft-1 (1+i) + Ct

Where t = 1, 2, …, n

n = total years in project

Ct = net CF in year t

i = c, if Ft-1 > 0

i’, if Ft-1 < 0

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Example 1

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• Find the ROR!

ROR = 16.82% on the un-recovered investment balances over 5 years

-$10,000

0 1 2 3 4 5

+$8,000

+$9,000

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Example 2

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• Reinvestment rate, c = 15%. What is the CRR?

• Answer…• F0 = 50,

• F1 = -142.50

• F2 = -142.50 (1+i’) + 50

• F3 = F2 (1+i’) + 100

• Set F3 = 0 to find the i’

Year Cash Flow, $

0 50

1 -200

2 50

3 100

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Example 3 - CF Diagram• Purchase price: P - $800/bond. Bond interest at 4% paid semiannually

for $1000 face value. Life = 20 years. If you pay the $800 per bond, what is the ROR (yield) on this investment?

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…. …. ….0 1 2 3 4 39 40

$800

F40 = $1000

A= $1000(0.04/2) = $20.00 every 6 months for 20 years

A = +$20/6 monthsFrom the bond purchaser’s perspective

Pay $800 per bond to receive the $20each 6-months in interest cash flow plus $1,000 at the end of 40 time periods. What is the ROR of this cash flow?

Example 3 (Cont’d)

• Equation:

0 = -$800 + $20(P/A, i*, 40) + $1000 (P/F, i*, 40)

• Solve for i*, we get 2.87% per semiannual

• Not done yet, thus find the …

Nominal ROR/year = (2.87%)(2) = 5.74%/year

Effective ROR/year = 5.82%/year

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Multiple Projects

• Incremental Analysis, Introduction• ROR on Extra Investment• ROR Analysis• Multiple Alternatives

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Incremental Analysis

• MARR Definition

• Example:

A company uses a MARR of 16% per year. The company has $90,000 available for investment and that two alternatives (A and B) are being evaluated. Alternative A requires an investment of $50,000 and will yield an IRR of 35% per year.

Alternative B requires $85,000 and will yield an IRR of 29% per year. Which alternative will be the best?

• Overall ROR(A) = [50k (.35) + 40k (.16)]/90k = 26.6%

• Overall ROR(B) = [85k (.29) + 5k (.16)]/90k = 28.3%

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Tabulation for Incremental CF For 2 Alternatives

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• Equal lives versus Unequal lives

• ROR Analysis on incremental CF:

• Need to use LCM (no matter what!)

• Larger initial investment alternative B!

• Incremental CF = CFB – CFA

• Check the sign changes like in Descartes’ and Norstorm’s rules

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ROR on Extra Investment

• Decision:

• Do-Nothing alternative

• Equivalent worth of the savings > equivalent worth of the extra investment using company’s MARR

DO the extra investment

• If the extra investment is not justified by the savings

Choose LOWER first-cost proposal

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ROR Analysis Procedure

• Sort the alternative by initial investment in an ascending order

• Develop CF and incremental CF series

• Draw if necessary

• Count the # of sign changes

• Set up PW equation for the incremental CF & find i*B-A

• If i*B-A < MARR: choose A,

• Otherwise: choose B

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Example 8.3

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• A leather clothes manufacturer is considering the purchase of one new industrial sewing machine, which is either semiautomatic or fully automatic. Which machine should be selected if the MARR is 15% per year? The estimates are listed in the table below.

• SORT! Incremental CF # sign Δ (max #ROR) PW Incremental CF trial & error

• 12.65% < MARR choose the lower-cost (Semiautomatic)

Fully Automatic Semiautomatic

First cost, $ 13,000 8,000

Annual disbursement, $ 1,600 3,500

Salvage value, $ 2,000 0

Life, years 5 10

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• What if…

• MARR is 12.65%, which alternative is better?

• MARR = 10%, which one will you choose?

• MARR = 20%, semiautomatic or full automatic?

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Example 8.3

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Multiple Alternatives• Criteria:

Select one alternative that requires the largest investment AND indicated that the extra investment over another acceptable alternative is justified

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Multiple Alternatives Procedure1. Sort them in an ascending order2. Determine the nature of the CF series

• Some positive CF: do nothing (defender) vs. lowest-initial investment alternative. Go to step 3

• All negative CF: lowest-initial investment (defender) vs. next-higher investment3. Find the i* of the defender

• If i* < MARR, remove the lowest-investment alternative• Compute the next one. Repeat until i* ≥ MARR, this alternative defender and

compares it with the next one4. Find the annual incremental CF between the challenger and defender5. Find i* using PW-based or AW6. If i* ≥ MARR, challenger new defender, o/w next challenger vs. defender7. Repeat until only 1 alternative remains, OPTIMAL

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Example 8.7 (T. Blank, p. 247)

Four different prefab-building locations have been suggested, of which only one will be selected. Cost and annual net cash-flow information are detailed in table below. The annual net cash-flow series vary due to differences in maintenance, labor costs, transportation charges, etc. If the MARR is 10%, use ROR analysis to select the one economically-best location

• Answer:• Sort C, A, B, D• Start comparing one by one, i = 9.63%; 10.49%, 17.28%, 8.55%• Choose B

Location A B C D Building cost, $ -200,000 -275,000 -190,000 -350,000 Annual Cash flow, $ +22,000 +35,000 +19,500 +42,000 Life, years 30 30 30 30

Spreadsheet Example

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