2. You are comparing two pieces of equipment with different life expectancies. Their cashflow streams are as below. In order to compare them on an equal basis, we assume that we will run both pieces of equipment forever by replacing them every 4 and 5 years, respectively.
A. (10) What is the net PV of equipment A’s cashflow, if run just once?
B. (10) What is the net PV of equipment B’s cashflow, if run just once?
C. (20) What is the net PV of equipment A’s cashflow, if run forever?
D. (20) What is the net PV of equipment B’s cashflow, if run forever?
Homework Answers
A) Project A NPV is calculated below for just one run:
| Year | CF | Discount Factor | Discounted CF | ||
| 0 | $-35.00 | 1/(1+0.1)^0= | 1 | 1*-35= | $ -35.00 |
| 1 | $ 20.00 | 1/(1+0.1)^1= | 0.909090909 | 0.909090909090909*20= | $ 18.18 |
| 2 | $ 12.00 | 1/(1+0.1)^2= | 0.826446281 | 0.826446280991735*12= | $ 9.92 |
| 3 | $ 14.00 | 1/(1+0.1)^3= | 0.751314801 | 0.751314800901578*14= | $ 10.52 |
| 4 | $ 8.00 | 1/(1+0.1)^4= | 0.683013455 | 0.683013455365071*8= | $ 5.46 |
| NPV = Sum of all Discounted CF | $ 9.08 | ||||
B) Project B NPV is calculated below for just one run:
| Year | CF | Discount Factor | Discounted CF | ||
| 0 | $-45.00 | 1/(1+0.1)^0= | 1 | 1*-45= | $ -45.00 |
| 1 | $ 21.00 | 1/(1+0.1)^1= | 0.909090909 | 0.909090909090909*21= | $ 19.09 |
| 2 | $ 12.00 | 1/(1+0.1)^2= | 0.826446281 | 0.826446280991735*12= | $ 9.92 |
| 3 | $ 15.00 | 1/(1+0.1)^3= | 0.751314801 | 0.751314800901578*15= | $ 11.27 |
| 4 | $ 10.00 | 1/(1+0.1)^4= | 0.683013455 | 0.683013455365071*10= | $ 6.83 |
| 5 | $ 16.00 | 1/(1+0.1)^5= | 0.620921323 | 0.620921323059155*16= | $ 9.93 |
| NPV = Sum of all Discounted CF | $ 12.04 | ||||
According to one run NPV, project B has higher NPV and if the projects are mutually exclusive then B should be selected over A
If the projects have a life of 4 and 5 years so their LCM is 20 years, we can calculate the replacement method NPV for 20 years for each project and compare, the result will be the same if they are replaced forever.
C) Project A Replacement NPV:
| Year | CF | Discount Factor | Discounted CF | ||
| 0 | $-35.00 | 1/(1+0.1)^0= | 1 | 1*-35= | -35.00 |
| 1 | $ 20.00 | 1/(1+0.1)^1= | 0.909090909 | 0.909090909090909*20= | 18.18 |
| 2 | $ 12.00 | 1/(1+0.1)^2= | 0.826446281 | 0.826446280991735*12= | 9.92 |
| 3 | $ 14.00 | 1/(1+0.1)^3= | 0.751314801 | 0.751314800901578*14= | 10.52 |
| 4 | 8-35 = $-27.00 | 1/(1+0.1)^4= | 0.683013455 | 0.683013455365071*-27= | -18.44 |
| 5 | $ 20.00 | 1/(1+0.1)^5= | 0.620921323 | 0.620921323059155*20= | 12.42 |
| 6 | $ 12.00 | 1/(1+0.1)^6= | 0.56447393 | 0.564473930053777*12= | 6.77 |
| 7 | $ 14.00 | 1/(1+0.1)^7= | 0.513158118 | 0.513158118230706*14= | 7.18 |
| 8 | 8-35 =$-27.00 | 1/(1+0.1)^8= | 0.46650738 | 0.466507380209733*-27= | -12.60 |
| 9 | $ 20.00 | 1/(1+0.1)^9= | 0.424097618 | 0.424097618372485*20= | 8.48 |
| 10 | $ 12.00 | 1/(1+0.1)^10= | 0.385543289 | 0.385543289429531*12= | 4.63 |
| 11 | $ 14.00 | 1/(1+0.1)^11= | 0.350493899 | 0.350493899481392*14= | 4.91 |
| 12 | 8-35 = $-27.00 | 1/(1+0.1)^12= | 0.318630818 | 0.318630817710357*-27= | -8.60 |
| 13 | $ 20.00 | 1/(1+0.1)^13= | 0.28966438 | 0.289664379736688*20= | 5.79 |
| 14 | $ 12.00 | 1/(1+0.1)^14= | 0.263331254 | 0.26333125430608*12= | 3.16 |
| 15 | $ 14.00 | 1/(1+0.1)^15= | 0.239392049 | 0.239392049369163*14= | 3.35 |
| 16 | 8-35 = $-27.00 | 1/(1+0.1)^16= | 0.217629136 | 0.217629135790149*-27= | -5.88 |
| 17 | $ 20.00 | 1/(1+0.1)^17= | 0.197844669 | 0.197844668900135*20= | 3.96 |
| 18 | $ 12.00 | 1/(1+0.1)^18= | 0.17985879 | 0.179858789909214*12= | 2.16 |
| 19 | $ 14.00 | 1/(1+0.1)^19= | 0.163507991 | 0.163507990826558*14= | 2.29 |
| 20 | $ 8.00 | 1/(1+0.1)^20= | 0.148643628 | 0.148643628024143*8= | 1.19 |
| NPV = Sum of all Discounted CF | 24.39 | ||||
In the years of replacement the CF will be 8 inflow and 35 outflow so 8-35 = -27 or total 27 outflow
d) Project B Replacement NPV:
| Year | CF | Discount Factor | Discounted CF | ||
| 0 | $-45.00 | 1/(1+0.1)^0= | 1 | 1*-45= | -45.00 |
| 1 | $ 21.00 | 1/(1+0.1)^1= | 0.909090909 | 0.909090909090909*21= | 19.09 |
| 2 | $ 12.00 | 1/(1+0.1)^2= | 0.826446281 | 0.826446280991735*12= | 9.92 |
| 3 | $ 15.00 | 1/(1+0.1)^3= | 0.751314801 | 0.751314800901578*15= | 11.27 |
| 4 | $ 10.00 | 1/(1+0.1)^4= | 0.683013455 | 0.683013455365071*10= | 6.83 |
| 5 | $-29.00 | 1/(1+0.1)^5= | 0.620921323 | 0.620921323059155*-29= | -18.01 |
| 6 | $ 21.00 | 1/(1+0.1)^6= | 0.56447393 | 0.564473930053777*21= | 11.85 |
| 7 | $ 12.00 | 1/(1+0.1)^7= | 0.513158118 | 0.513158118230706*12= | 6.16 |
| 8 | $ 15.00 | 1/(1+0.1)^8= | 0.46650738 | 0.466507380209733*15= | 7.00 |
| 9 | $ 10.00 | 1/(1+0.1)^9= | 0.424097618 | 0.424097618372485*10= | 4.24 |
| 10 | $-29.00 | 1/(1+0.1)^10= | 0.385543289 | 0.385543289429531*-29= | -11.18 |
| 11 | $ 21.00 | 1/(1+0.1)^11= | 0.350493899 | 0.350493899481392*21= | 7.36 |
| 12 | $ 12.00 | 1/(1+0.1)^12= | 0.318630818 | 0.318630817710357*12= | 3.82 |
| 13 | $ 15.00 | 1/(1+0.1)^13= | 0.28966438 | 0.289664379736688*15= | 4.34 |
| 14 | $ 10.00 | 1/(1+0.1)^14= | 0.263331254 | 0.26333125430608*10= | 2.63 |
| 15 | $-29.00 | 1/(1+0.1)^15= | 0.239392049 | 0.239392049369163*-29= | -6.94 |
| 16 | $ 21.00 | 1/(1+0.1)^16= | 0.217629136 | 0.217629135790149*21= | 4.57 |
| 17 | $ 12.00 | 1/(1+0.1)^17= | 0.197844669 | 0.197844668900135*12= | 2.37 |
| 18 | $ 15.00 | 1/(1+0.1)^18= | 0.17985879 | 0.179858789909214*15= | 2.70 |
| 19 | $ 10.00 | 1/(1+0.1)^19= | 0.163507991 | 0.163507990826558*10= | 1.64 |
| 20 | $ 16.00 | 1/(1+0.1)^20= | 0.148643628 | 0.148643628024143*16= | 2.38 |
| NPV = Sum of all Discounted CF | 27.05 | ||||
In the years of replacement the CF will be 16 inflow and 45 outflow so 16-45= -29 or total 29 outflow
So even with replacement, project B has higher NPV
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