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Multiple Rates of ReturnMultiple Rates of Returnn 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%

Rate of Return vs. Present Worth

-12

-10

-8

-6

-4

-2

0

2

0 5 10 15 20 25 30 35

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Multiple RoRMultiple RoR

n 0 1 2 3 4 5cf -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%

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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

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Multiple RoRsMultiple RoRsn 0 1 2 3cf -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.

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7A-177A-17n 0 1 2 3 4 5 6cf -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

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7A-187A-18n 0 1 2 3 4 5 6 7 8A -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

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Incremental AnalysisIncremental 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).

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Incremental AnalysisIncremental Analysis

MARR = 6% A B C D E1st Cost4000 2000 6000 1000 9000UAB 639 410 761 117 785Life (years) 20 20 20 20 20RoR 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

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Investment DecisionInvestment 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.

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Higher IRR Not SufficientHigher IRR Not Sufficient

Mutually Exclusive Alternatives

n A B0 -1000 -50001 2000 7000IRR 100% 40%

PW(10%) $818.18 $1363.64, B is better

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View PointView 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?

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IRR on Incremental InvestmentIRR on Incremental Investment

n A B A - B0 -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

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Unequal Service LivesUnequal Service Lives

n A B B B - A0 -2000 -3000 -3000 -10001 1000 4000 1000 02 1000 1000 03 1000 4000 3000

MARR = 10% and can repeat service life.(IRR '(-1000 0 0 3000)) 44.22%

B is better.

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Infinite Cash FlowInfinite 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

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Find X Given RoRFind X Given RoR

Find minimum X to make at least a 10% return on investment.

n 0 1 2 3cf -2000 1000 X 1200 ans. $229.09

X = [2000 – 1000(1.1)-1 – 1200(1.1)-3] / (1.1)-2

Computing the MIRRComputing 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 3cf -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%

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Compute the MIRRCompute 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 5cf -20 70 -15 30 -10 -20P0 for the negatives at the borrowing rate

Fn for the positives at the investing rate

Then find i given P. F and nAns. 18.52%05/05/2305/05/23 rdrd 1717