1. 2/22/2016rd Multiple Rates of Return nCash FlowCumulativeAdjusted 10% 20% 0-$500-$500 -$500 -$500 1 1150 650 550 600 2- 660 - 10 0 0 (quadratic -500 1150 –660)  (1.2 1.1) (irr '(-500 1150 –660) 0.6)  10%; 20%

2. 2/22/2016rd Multiple RoR n 01 2 3 4 5 cf-5020-4036.836.836.8 Compute RoR on internal investment with 10% external rate. New-cash flow -50 0 -18 36.8 36.836.8 20(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%

3. 2/22/2016rd Use rate of return (RoR) analysis for the following 3 mutually exclusive alternatives in reference to an unknown MARR. A B C First Cost$200$300$600 Uniform annual benefits 59.7 77.1165.2 Useful life (years) 5 5 5 End salvage 0 0 0 Computed 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

4. 2/22/2016rd Multiple RoRs n 0 1 2 3 cf-10004100-55802520 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.

5. 2/22/2016rd 7A-17 n 0 1 2 3 4 5 6 cf-1200358358358358358-394 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

6. 2/22/2016rd 7A-18 n 0 1 2 3 4 5 6 7 8 A -3570 1K1K 1K -3170 1500 150015001500 (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

7. 2/22/2016rd Incremental Analysis MARR = 8%ABA-B First Cost$100$50 50 UAB19.9311.93 8 Life (years)1010 10 RoR15%20% 9.6% => A 0 < MARR < 9.6% A is better If 9.6% < MARR < 20%, B is better. NPW A (9.6%) = $24.59 = NPW B (9.6%) A earned at B’s rate (20%) for the first $50 and at 9.6% for the next $50 (increment).

8. 2/22/2016rd Incremental Analysis MARR = 6% ABCDE 1 st Cost40002000600010009000 UAB639410761117785 Life (years)2020202020 RoR15%20%11%10%6% Start with D, better than Do Nothing, Challenger is B. RoR B-D (UIRR 1000 293 20 0)  29.12% => B is better than D RoR A-B (UIRR 2000 229 20 0)  9.63% => A is better than B RoR C-A (UIRR 2000 122 20 0)  1.97% => A is better than C RoR E-A (UIRR 5000 146 20 0)  -4.65% => A is best

9. 2/22/2016rd 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.

10. 2/22/2016rd Higher IRR Not Sufficient Mutually Exclusive Alternatives n A B 0-1000-5000 1 2000 7000 IRR 100% 40% PW(10%) $818.18$1363.64, B is better

11. 2/22/2016rd View Point n0123 IRR A-3000135018001500 25% B-12000420062256330 17.4% B - A-9000285044254830 15% (lending or investing) A – B 9000 -2850-4425-483015%(borrowing) Do you see why we strive to make the first difference negative?

12. 2/22/2016rd IRR on Incremental Investment n A B A - B 0-9000 -9000 0 1 480 5800 -5320 2 3700 3250 450 3 6550 2000 4550 4 3780 1561 2219 IRR 18% 20% 14.71% If MARR = 12%, then A is better

13. 2/22/2016rd Unequal Service Lives nA B BB - A 0-2000-3000-3000-1000 1 1000 4000 1000 0 2 1000 1000 0 3 1000 4000 3000 MARR = 10% and can repeat service life. (IRR '(-1000 0 0 3000))  44.22% B is better.

14. 2/22/2016rd 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

15. 2/22/2016rd Find X Given RoR Find minimum X to make at least a 10% return on investment. n 0 1 2 3 cf-20001000 X1200ans. $229.09 X = [2000 – 1000(1.1) -1 – 1200(1.1) -3 ] / (1.1) -2

16. Computing the MIRR Compute the MIRR for the following cash flow using 6% for the borrowing rate and 12% for the investing rate. n0123 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% 2/22/2016rd

17. 2/22/2016rd Compute the MIRR Find the MIRR for the following cash flow by using 5% for borrowing rate and 9% for the investment rate. n0123 4 5 cf-20 70 -15 30 -10-20 P 0 for the negatives at the borrowing rate F n for the positives at the investing rate Then find i given P. F and n Ans. 18.52% 2/22/2016rd17