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consider three mutually exclusive alternatives that have a uniform annual

-22 Consider three mutually exclusive alternatives that have a uniform annual benefit of $420. The analysis period is 8 years

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Answer #1
  • Basically first we must understand that as the projects have unequal lives, we will take the LCM of the lives to make them comparable
  • LCM of 2,4,8 is 8
  • So project A will be replaced 3 times in years 2,4 & 6 and project B will be replaced 1 time in year 4
  • Lets summarize the given information in tabular form for better understanding:
    • CF schedule
      Year A B C
      0 -770 -1406 -2563
      1 420 420 420
      2 420-770=-350 420 420
      3 420 420 420
      4 420-770=-350 420-1406=-986 420
      5 420 420 420
      6 420-770=-350 420 420
      7 420 420 420
      8 420 420 420
  • Keeping this in mind we will find the NPV of the 3 projects ate different discount rates by solving the following equation for each
    • 420 -350 420 -350 420 = -770 NPVA + + 2 + (1r)5 (1 r) 420 (1+r)2 (1r)3 (1r)4 -350 420 (1+r) (1r)6 (1r)8
    • -986 420 420 420 420 -1406 NPVB + (1+r)(1 r)2 420 + + + (1 r)3(1r)4 (1r)5 420 420 (1r)7 (1 +r)8 |(1r)6
    • 420 420 420 420 420 NPVc =2563 + + + (1 r)3 (1r)4 (1r)5 (1r)(1r)2 420 420 420 (1r)7 (1+r)8 |(1r)6
  • For values of r ranging from 0% to 100% at an interval of 5% we get the following NPV table for all the projects:
    • r A B C Decision
      0% $   280.00 $   548.00 $      797.00 Project C
      5% $     36.26 $   144.60 $      144.33 Project B
      10% $ -114.78 $ -114.23 $     -293.03 None
      15% $ -209.30 $ -282.79 $     -589.85 None
      20% $ -268.60 $ -393.70 $     -792.83 None
      25% $ -305.52 $ -467.00 $     -931.89 None
      30% $ -327.98 $ -515.31 $ -1,026.63 None
      35% $ -340.96 $ -546.72 $ -1,090.20 None
      40% $ -347.65 $ -566.53 $ -1,131.53 None
      45% $ -350.14 $ -578.27 $ -1,156.85 None
      50% $ -349.80 $ -584.34 $ -1,170.52 None
      55% $ -347.56 $ -586.37 $ -1,175.67 None
      60% $ -344.04 $ -585.52 $ -1,174.56 None
      65% $ -339.68 $ -582.61 $ -1,168.85 None
      70% $ -334.78 $ -578.20 $ -1,159.77 None
      75% $ -329.54 $ -572.73 $ -1,148.21 None
      80% $ -324.12 $ -566.50 $ -1,134.87 None
      85% $ -318.60 $ -559.74 $ -1,120.26 None
      90% $ -313.07 $ -552.62 $ -1,104.78 None
      95% $ -307.57 $ -545.26 $ -1,088.72 None
      100% $ -302.15 $ -537.76 $ -1,072.32 None
    • So now looking at the decisions, NPV is positive for all 3 projects only for rates between 0% and 5% and negative for all other discount rates
    • At 0% project C has the maximum NPV so it should be selected
    • At 5% NPV of B and C are almost the same and so either can be selected but just because it is slightly higher for B, B is the better project
    • For all other rates, non of the projects is worst investing as the cost of investment wont be recovered.
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