Question

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. Assume identical replacements and construct a choice table for interest rates from 0% to 100%. OfficeSt (a) De fro (b) If pla (a) Assume doing nothing is allowed. (b) Assume A, B, or C must be chosen. 8-26 Consi A Initia B Unif $770 Useful life (years) $2563 $1406 Initial cost b 4 2 Use 6.4% 7.5% Rate of return 6.0% Al

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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.