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Engineering Economic Analysis 9th Edition Chapter 7 RATE OF RETURN ANALYSIS Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 1 Three Major Methods of Economic Analysis • PW - Present Worth • AW - Annual Worth • IRR - Internal Rate of Return If PW = A(P/A,i,n) Then (P/A,i,n) = PW/A Solve for (P/A,i,n) and look up interest in Compound Interest Tables Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 2 Internal Rate of Return Lender’s Viewpoint • The interest rate on the balance of a loan such that the unpaid loan balance equals zero when the final payment is made. Plan Year 1 2 3 4 5 IRR A 0 $ (5,000.00) $ 1,400.00 $ 1,320.00 $ 1,240.00 $ 1,160.00 $ 1,080.00 8.00% B $ (5,000.00) $ 400.00 $ 400.00 $ 400.00 $ 400.00 $ 5,400.00 C $ (5,000.00) $ 1,252.28 $ 1,252.28 $ 1,252.28 $ 1,252.28 $ 1,252.28 D $ (5,000.00) $ $ $ $ $ 7,346.64 8.00% 8.00% 8.00% Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 3 Internal Rate of Return Investor’s Viewpoint • The interest rate earned on the unrecovered investment such that the unrecovered investment equals zero at the end of the life of the investment. Year 0 1 2 3 4 5 IRR Cash flow ($5,000.00) $1,252.28 $1,252.28 $1,252.28 $1,252.28 $1,252.28 8.00% Unrecovered investment at beginning of year $5,000.00 $4,147.72 $3,227.25 $2,233.15 $1,159.52 Total Return on unrecovered investment Investment Unrecovered repayment at investment at end of year end of year $400.00 $331.81 $258.18 $178.65 $92.76 $852.28 $920.47 $994.10 $1,073.63 $1,159.52 $1,261.40 $5,000.00 $4,147.72 $3,227.25 $2,233.15 $1,159.52 ($0.00) Reset the cash flow to determine the IRR. Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 4 Calculating Rate of Return • The IRR is the interest rate at which the benefits equal the costs. IRR = i* • • • • • PW Benefit - PW Cost = 0 PW Benefit/PW Cost = 1 NPW = 0 EUAB - EUAC = 0 PW Benefit = PW Cost Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 5 Calculating IRR • PWB/PWC = 1 • 1252.28(P/A,i,5)/5000 = 1 • (P/A,i,5) = 5000/1252.28 = 3.9927 Example 7-1 From Compound Interest Tables Interest rate (P/A,i,5) 7% 4.100 8% 3.993 9% 3.890 Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 6 Calculating IRR EUAB - EUAC = 0 100 + 75(A/G,i,4) - 700(A/P, i, 4) = 0 Look up and iterate, book answer is 7% Example 7-2 Year 0 1 2 3 4 5 IRR Cash flow ($700.00) $100.00 $175.00 $250.00 $325.00 $0.00 6.91% Unrecovered investment at beginning of year $700.00 $648.37 $518.18 $303.99 $0.00 Total Return on unrecovered investment Investment Unrecovered repayment at investment at end of year end of year $48.37 $44.81 $35.81 $21.01 $0.00 $51.63 $130.19 $214.19 $303.99 $0.00 $150.00 $700.00 $648.37 $518.18 $303.99 $0.00 $0.00 Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 7 Calculating IRR Example 7-3 The iterations may be graphed and the true IRR will be indicated at the point where the NPW curve = 0. 0 1 2 3 4 5 IRR Trial interest rates 0 5 10 15 20 25 30 35 40 45 50 Cash flow ($100.00) $20.00 $30.00 $20.00 $40.00 $40.00 Return on unrecovered investment $100.00 $93.47 $76.07 $66.32 $35.25 13.47% Total Investment Unrecovered repayment at investment at end of year end of year $13.47 $12.59 $10.25 $8.93 $4.75 $6.53 $17.41 $9.75 $31.07 $35.25 $50.00 $100.00 $93.47 $76.07 $66.32 $35.25 ($0.00) $60.00 NPW $50.00 $50.00 $26.46 $9.24 ($3.49) ($12.97) ($20.06) ($25.37) ($29.36) ($32.34) ($34.54) ($36.16) $40.00 Net Present Value Year Unrecovered investment at beginning of year $30.00 $20.00 $10.00 $0.00 ($10.00) 0 5 10 15 20 25 30 35 40 45 50 ($20.00) ($30.00) ($40.00) ($50.00) Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 8 Calculating IRR - Bond Example 7-4a Period rate 1000 = 40(P/A,i,2) + 950(P/F,i,2) By lookup and interpolation i* 1.5% Nominal rate = 2 x 1.5% = 3% Effective rate = (1 + 1.5%)2 - 1 = 3.02% An easier way > 7-4a & 7b Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 9 Rate of Return (ROR) Analysis • Most frequently used measure of merit in industry • More accurately called Internal Rate of Return (IRR) Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 1 0 Calculating ROR • Where two mutually exclusive alternatives will provide the same benefit, ROR is performed using an incremental rate of return-DROR-on the difference between the alternatives. Two-alternative situation Decision DROR MARR Choose higher-cost alternative DROR <MARR Choose lower-cost alternative Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 1 1 ROR on Alternatives with Equivalent Benefits Example 7-5 MARR = 10% 0 1 2 3 4 5 IRR/period Trial interest rates 0 5 10 15 20 25 30 35 40 45 50 Cash flow - Cash flow alternative A alternative B ($1,000.00) $200.00 $200.00 $1,200.00 $1,200.00 $1,200.00 ($2,783.00) $1,200.00 $1,200.00 $1,200.00 $1,200.00 $1,200.00 48.72% 32.60% 8.01% NPW $217.00 $72.77 ($43.15) ($136.77) ($212.69) ($274.40) ($324.66) ($365.60) ($398.94) ($426.01) ($447.93) Total $217.00 $1,783.00 $300.00 $200.00 Net Present Value Year Cash flow on difference Unrecovered Return on Investment Unrecovered Cash flow investment at unrecovered repayment at investment at alternative B-A beginning of investment end of year end of year year ($1,783.00) $1,000.00 $1,783.00 $142.83 $857.17 $925.83 $1,000.00 $925.83 $74.17 $925.83 ($0.00) $0.00 ($0.00) ($0.00) $0.00 ($0.00) $0.00 ($0.00) ($0.00) $0.00 ($0.00) $0.00 ($0.00) ($0.00) $0.00 ($0.00) $100.00 $0.00 ($100.00) 0 5 10 15 20 25 30 35 40 45 50 ($200.00) ($300.00) ($400.00) ($500.00) Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 1 2 Criterion Is to Maximize Return Not ROR Examples 7- 6, 7 & 8 Year 0 1 2 3 4 5 IRR/period MARR = NPW = Trial interest rates 0 5 10 15 20 25 30 35 40 45 50 Cash flow - Cash flow Cash flow alternative A alternative B alternative B-A ($10.00) $15.00 $0.00 $0.00 $0.00 $0.00 ($20.00) $28.00 $0.00 $0.00 $0.00 $0.00 ($10.00) $13.00 $0.00 $0.00 $0.00 $0.00 50.00% 6.00 % $4.15 40.00% 30.00% $6.42 $2.26 NPW $3.00 $2.27 $1.65 $1.13 $0.69 $0.32 $0.00 ($0.27) ($0.51) ($0.71) ($0.89) Return on unrecovered investment $10.00 $0.00 $0.00 $0.00 $0.00 Total Investment repayment at end of year Unrecovered investment at end of year $3.00 $0.00 $0.00 $0.00 $0.00 $10.00 $0.00 $0.00 $0.00 $0.00 $0.00 $0.00 $0.00 $0.00 $0.00 $3.00 $10.00 $3.50 $3.00 Net Present Value What happens to NPW when MARR is varied? Try 6, 30 & 35%. Cash flow on difference Unrecovered investment at beginning of year $2.50 $2.00 $1.50 $1.00 $0.50 $0.00 ($0.50) 0 5 10 15 20 25 30 35 40 45 50 ($1.00) ($1.50) Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 1 3 Fixed Input and Differing Outputs Example 7-9 Cash flow on difference Year 0 1 2 3 4 5 IRR/period MARR = NPW = Trial interest rates 0 5 10 15 20 25 30 35 40 45 50 Cash flow - Cash flow alternative A alternative B ($1,000.00) $400.00 $350.00 $300.00 $250.00 $200.00 ($1,000.00) $300.00 $300.00 $300.00 $300.00 $300.00 17.47% 15.24% 7.00 % $257.75 $230.06 0.00% NPW $0.00 ($20.10) ($32.72) ($40.40) ($44.80) ($47.00) ($47.75) ($47.52) ($46.66) ($45.40) ($43.90) Return on unrecovered investment Investment Unrecovered repayment at investment at end of year end of year $0.00 ($0.00) ($0.00) ($0.00) ($0.00) Total ($100.00) ($50.00) $0.00 $50.00 $100.00 ($0.00) $100.00 $150.00 $150.00 $100.00 ($0.00) $0.00 ($27.69) $0.00 0 5 10 15 20 25 30 35 40 45 50 ($10.00) Net Present Value Choose the ranking so that the difference represents an increment of investment; goes from to + over time. Unrecovered Cash flow investment at alternative B-A beginning of year $0.00 ($100.00) $0.00 ($50.00) $100.00 $0.00 $150.00 $50.00 $150.00 $100.00 $100.00 ($20.00) ($30.00) ($40.00) ($50.00) ($60.00) Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 1 4 Analysis Period • Just as in PW and AW analysis, the analysis period must be considered: • Useful life of the alternative equals the analysis period • Alternatives have useful lives different from the analysis period • The analysis period is infinite, n = ¥ Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 1 5 Engineering Economic Analysis 9th Edition Chapter Appendix 7A Difficulties Solving for an Interest Rate Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 1 6 Multiple IRR Example 7A-1 Cash flow 0 1 2 3 4 5 $19.00 $10.00 ($50.00) ($50.00) $20.00 $60.00 IRR/period IRR/period 10.24% 47.30% Trial interest rates 0 5 10 15 20 25 30 35 40 45 50 Guess 10.00% 50.00% NPW $9.00 $3.28 $0.11 ($1.50) ($2.14) ($2.20) ($1.91) ($1.44) ($0.88) ($0.28) $0.32 Return on unrecovered investment ($19.00) ($30.95) $15.88 $67.51 $54.43 Total Investment repayment at end of year Unrecovered investment at end of year ($1.95) ($3.17) $1.63 $6.91 $5.57 $11.95 ($46.83) ($51.63) $13.09 $54.43 $9.00 ($19.00) ($30.95) $15.88 $67.51 $54.43 $0.00 $10.00 $8.00 Net Present Value Occurs when a cash flow produces more than one point at which NPW = 0 Year Unrecovered investment at beginning of year $6.00 $4.00 $2.00 $0.00 ($2.00) 0 5 10 15 20 25 30 35 40 45 50 ($4.00) Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 1 7 Cash Flow Rule of Signs • May be converted to a polynomial • Then, by Descartes’ rule Number of sign changes, m 0 1 Number of positive values of X 0 1 2 3 2 or 0 3 or 1 4 4, 2 or 0 Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 1 8 Cash Flow Rule of Signs Expands on This Notion • There may be as many positive values of “i” as there are sign changes in the cash flow. • Sign changes are counted when: • + to • - to + • A zero cash flow is ignored Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 1 9 Cash Flow Rule of Signs - Possibilities Number of sign changes, Number of positive values m of “i” 0 1 2 3 4 0 1 or 0 2, 1 or 0 3, 2, 1 or 0 4, 3, 2, 1 or 0 Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 2 0 Zero Sign Changes • Receiving a gift • Giving your friend a loan and not being paid back In either case, no “i” can be computed. Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 2 1 One Sign Change The Normal Situation Year 0 1 2 3 4 5 Sign change Guess 0 10 20 30 40 50 60 70 80 90 100 A ($50.00) $20.00 $10.00 B ($50.00) $20.00 $20.00 $11.00 C ($50.00) $20.00 $20.00 $20.00 $12.00 D $50.00 ($20.00) ($10.00) E $50.00 ($20.00) ($20.00) ($11.00) F $50.00 ($20.00) ($20.00) ($20.00) ($12.00) 1 1 1 1 1 1 IRR NPV NPV NPV NPV NPV NPV NPV NPV NPV NPV NPV IRR 1.09% 1.09% 1.09% 1.09% 1.09% 1.09% 1.09% 1.09% 1.09% 1.09% 1.09% IRR 17.64% 17.64% 17.64% 17.64% 17.64% 17.64% 17.64% 17.64% 17.64% 17.64% 17.64% IRR NPV NPV NPV NPV NPV NPV NPV NPV NPV NPV NPV IRR 1.09% 1.09% 1.09% 1.09% 1.09% 1.09% 1.09% 1.09% 1.09% 1.09% 1.09% IRR 17.64% 17.64% 17.64% 17.64% 17.64% 17.64% 17.64% 17.64% 17.64% 17.64% 17.64% Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 2 2 Two or More Sign Changes • More than one IRR may exist • Use an NPW plot to determine how many exist 0 1 2 3 4 5 $19.00 $10.00 ($50.00) ($50.00) $20.00 $60.00 IRR/period IRR/period 10.24% 47.30% Trial interest rates 0 5 10 15 20 25 30 35 40 45 50 Unrecovered investment at beginning of year Cash flow Guess 10.00% 50.00% NPW $9.00 $3.28 $0.11 ($1.50) ($2.14) ($2.20) ($1.91) ($1.44) ($0.88) ($0.28) $0.32 Return on unrecovered investment ($19.00) ($30.95) $15.88 $67.51 $54.43 Total Investment repayment at end of year Unrecovered investment at end of year ($1.95) ($3.17) $1.63 $6.91 $5.57 $11.95 ($46.83) ($51.63) $13.09 $54.43 $9.00 ($19.00) ($30.95) $15.88 $67.51 $54.43 $0.00 $10.00 $8.00 Net Present Value Year $6.00 $4.00 $2.00 $0.00 ($2.00) 0 5 10 15 20 25 30 35 40 45 50 ($4.00) Engineering Economic Analysis - Ninth Edition Newnan/Eschenbach/Lavelle Copyright 2004 by Oxford University Press, Inc. 2 3