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Multiple Rates of Return

This text discusses multiple rates of return in cash flow analysis, including the calculation of internal rate of return (IRR), rate of return (RoR), incremental analysis, and the use of MARR in decision making.

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Multiple Rates of Return

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  1. Multiple Rates of Return n Cash Flow Cumulative Adjusted 10% 20% 0 -$500 -$500 -$500 -$5001 1150 650 550 6002 - 660 - 10 0 0 (quadratic -500 1150 –660)  (1.2 1.1)(irr '(-500 1150 –660) 0.6)  10%; 20% rd

  2. Multiple RoR n 0 1 2 3 4 5 cf -50 20 -40 36.8 36.8 36.8 Compute RoR on internal investment with 10% external rate. New-cash flow -50 0 -18 36.8 36.8 36.820(1.1) = $22 added to –40 get –18 and 1 sign change. (IRR ‘(-50 0 –18 36.8 36.8 36.6))  14.91%. (IRR '(-50 20 -40 36.8 36.8 36.8))  15.38% rd

  3. Use rate of return (RoR) analysis for the following 3 mutually exclusive alternatives in reference to an unknown MARR. A B CFirst Cost $200 $300 $600Uniform annual benefits 59.7 77.1 165.2Useful life (years) 5 5 5End salvage 0 0 0Computed RoR 15% 9% 11.7% Incremental RoR B - A => 100 = 17.4(P/A, i%, 5) => i = -4.47% C - A => 400 = 105.5(P/A, i%,5) => i = 10% C - B => 300 = 88.1(P/A,i%,5) => i = 14.3% Conclude: if MARR  9% Choose C 9%  MARR  10% Choose C Reject B 10%  MARR  11.7% Choose A Reject B 11.7%  MARR  15% Choose A rd

  4. Multiple RoRs n 0 1 2 3 cf -1000 4100 -5580 2520 PW(20%) = -1000 +4100(1.2)-1 –5580(1.2)-2 +2520(1.2)-3 = -1000 + 3416.67 – 3875 + 1458.33 = 0 PW(40%) = -1000 +4100(1.4)-1 –5580(1.4)-2 +2520(1.4)-3 = 0 PW(50%) = -1000 +4100(1.5)-1 –5580(1.5)-2 +2520(1.5)-3 = 0. rd

  5. 7A-17 n 0 1 2 3 4 5 6 cf -1200 358 358 358 358 358 -394 External rate at 12%(IRR ‘(-1200 358 358 358 358 358 -394)) 7.22%(IRR ‘(-1200 358 358 358 358 358 -394))  43.96% (MIRR ‘(-1200 358 358 358 358 358 –394) 6 12)  9.5% At 12% the $358 in year 5 can be transformed to pay at n = 6. 358 * 1.12 = 400.96. (IRR ‘(-1200 358 358 358 358 6.21 0))  7.63% (list-pgf '(-1200 358 358 358 358 358 -394) 7.2175982)  -1.525879e-4 rd

  6. 7A-18 n 0 1 2 3 4 5 6 7 8 A -3570 1K 1K 1K -3170 1500 1500 1500 1500 (IRR ‘(-3570 1000 1000 1000 -3170 1500 1500 1500 1500)) 9.995%(Cum-add ‘(-3570 1000 1000 1000 –3170 1500 1500 1500 1500)) returns (-3570 -2570 -1570 -570 -3740 -2240 -740 760 2260) => unique RoR (list-pgf '(-3570 1000 1000 1000 -3170 1500 1500 1500 1500) 9.995)  0.058472 rd

  7. Incremental Analysis MARR = 8% A B A-BFirst Cost $100 $50 50UAB 19.93 11.93 8Life (years) 10 10 10RoR 15% 20% 9.6% => A 0 < MARR < 9.6% A is better If 9.6% < MARR < 20%, B is better. NPWA(9.6%) = $24.59 = NPWB(9.6%)A earned at B’s rate (20%) for the first $50 and at 9.6% for the next $50 (increment). rd

  8. Incremental Analysis MARR = 6% A B C D E 1st Cost 4000 2000 6000 1000 9000UAB 639 410 761 117 785Life (years) 20 20 20 20 20 RoR 15% 20% 11% 10% 6% Start with D, better than Do Nothing, Challenger is B.RoRB-D(UIRR 1000 293 20 0) 29.12% => B is better than DRoRA-B(UIRR 2000 229 20 0)  9.63% => A is better than BRoRC-A(UIRR 2000 122 20 0)  1.97% => A is better than CRoRE-A(UIRR 5000 146 20 0)  -4.65% => A is best rd

  9. Investment Decision Net cash flow: –1,000,000 2,300,000 –1,320,000 (2 years) MARR = 15%, quadratic roots => 10% and 20% RoRs NPW(15%) = -1000000 + 2300000(1.15)-1 + 1320000(1.15)-2 = $1890.36 > 0 => Invest cautiously. 2,300K -1320K(P/F, 15%, 1) = $1,152,173.91 New cash flow [–1,000,000 1,152,173.91 0] with RoR at 15.22% if MARR rate of 15% is used to transfer year-one amount to cover year-two amount. rd

  10. Higher IRR Not Sufficient Mutually Exclusive Alternatives n A B 0 -1000 -5000 1 2000 7000IRR 100% 40% PW(10%) $818.18 $1363.64, B is better rd

  11. View Point n 0 1 2 3 IRRA -3000 1350 1800 1500 25%B -12000 4200 6225 6330 17.4%B - A -9000 2850 4425 4830 15% (lending or investing)A – B 9000 -2850 -4425 -4830 15% (borrowing) Do you see why we strive to make the first difference negative? rd

  12. IRR on Incremental Investment • n A B A - B 0 -9000 -9000 01 480 5800 -53202 3700 3250 4503 6550 2000 4550 4 3780 1561 2219IRR 18% 20% 14.71%If MARR = 12%, then A is better rd

  13. Unequal Service Lives n A B B B - A0 -2000 -3000 -3000 -10001 1000 4000 1000 02 1000 1000 0 3 1000 4000 3000 MARR = 10% and can repeat service life. (IRR '(-1000 0 0 3000))  44.22% B is better. rd

  14. Infinite Cash Flow Find the rate of return for the following infinite cash flow: -18,976 3,225.92 3,225.92 3,225.92 … Ans. 17%. Perpetuity => RoR = 3225.92 / 18,976 (irr (cons -18976 (list-of 100 3225.92)))  16.992 rd

  15. Find X Given RoR Find minimum X to make at least a 10% return on investment. n 0 1 2 3 cf -2000 1000 X 1200 ans. $229.09 X = [2000 – 1000(1.1)-1 – 1200(1.1)-3] / (1.1)-2 rd

  16. Computing the MIRR • Compute the MIRR for the following cash flow using 6% for the borrowing rate and 12% for the investing rate. • n 0 1 2 3 cf -1000 500 900 -200 • (mirr '(-1000 500 900 -200) 6 12)  11.87% • (list-pgf '(1000 0 0 200) 6)  $1167.92 • (list-fgp '(0 500 900 0) 12)  $1635.20 • (igpfn 1167.92384 1635.20 3)  11.87% rd

  17. Compute the MIRR Find the MIRR for the following cash flow by using 5% for borrowing rate and 9% for the investment rate. n 0 1 2 3 4 5 cf -20 70 -15 30 -10 -20 P0 for the negatives at the borrowing rate Fn for the positives at the investing rate Then find i given P. F and n Ans. 18.52% 1/5/2020 rd rd

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