# Ch5 - Chapter 5: Comparison Methods Part 2 Internal Rate of...

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Chapter 5: Comparison Methods Part 2 Example 5-1. Consider two mutually exclusive investments. The first costs \$10 today and returns \$13 in one year. The second costs \$1000 and returns \$1250 in one year. Which is the preferred investment? Your MARR is 20%. (You cannot take “multiples” of the investments.)

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Example 5-2. Revisit the 1-storey versus 2-storey example: N = 20 years MARR = 4%/yr (a)Calculate the Internal Rates of Return for each building. (b)Choose the better plan, based on an IRR analysis. 1-Storey 2-Storey First Cost \$250,000 \$400,000 Annual Rent Revenue \$26,000 per year \$40,000 per year Annual Cost \$4,000 per year \$6,000 per year
Generalizing IRR Analysis: One Independent Proposal • given estimates of receipts R t , disbursements D t ( R 0 = 0 ) • find i = IRR as the discount rate such that: 0 ) 1 ( ... ) 1 ( ) 1 ( ) ( 2 2 2 1 1 1 0 0 = + + + + + + + = T T T i D R i D R i D R D R i PW • if i MARR, then the proposal is acceptable • if i < MARR, then the proposal is rejected Important Question: Does IRR procedure agree with Present Worth Procedure ?

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Does IRR Method Agree with PW Method ? (One Proposal) Assume: 1. Proposal is profitable without discounting: 2. First cost for t = 0, followed by positive net cash flows: = > = = T t t t i PW D R 0 0 ) 0 ( ) ( T t D R D D R t t , ... , 2 , 1 for 0 , 0 0 0 0 = > < = i i=IRR , where PW ( i )=0 -D 0 PW(0) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 ) ( ... 1 ) ( 2 1 ) ( 1 ) ( ... 1 ) ( 1 ) ( 1 3 2 2 2 1 1 2 2 2 1 1 1 0 < + + + = + + + + + + + = + T T T T T T i D R T i D R i D R i D R i D R i D R D di d di i PW d Therefore, ( ) ( ) 0 1 1 1 0 1 ) ( ... 1 ) ( lim D i D R i D R D T T T i = + + + + +
If IRR MARR , then geometry of the graph implies PW ( MARR ) 0: If IRR < MARR , then geometry of the graph implies PW ( MARR ) < 0: i =MARR i =IRR i =MARR i =IRR

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PW < 0 IRR < MARR PW = 0 IRR = MARR PW > 0 IRR > MARR Conclusion: Under the assumptions , PW and IRR methods agree. = > = = T t t t i PW D R 0 0 ) 0 ( ) ( T t D R D D R t t , ... , 2 , 1 for 0 , 0 0 0 0 = > < = The assumptions, again:
Alternative Definitions of IRR It is the interest rate such that: PW = 0, or PW(receipts) = PW(disbursements) FW = 0, or FW(receipts) = FW(disbursements) AW = 0, or AW(receipts) = AW(disbursements) T T T T T T T i D i D D i R i R R i D R i D R i D R D R i PW ) 1 ( ... ) 1

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## This note was uploaded on 05/17/2011 for the course MSCI 261 taught by Professor Bonkoo during the Spring '09 term at Waterloo.

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Ch5 - Chapter 5: Comparison Methods Part 2 Internal Rate of...

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