Question

M Digem Mines must choose between two alternative machines A and B which perform the same function, but which have lives of 2 and 4 years, respectively. The initial cost of machine A is $30,000 and its annual operating costs are expected to be $6,000. The initial cost of machine B is $42,000 and its annual operating costs are expected to be $9,000. Assume that the projects are repeatable and there are no constraints on the availability of funds. Digem will use a discount rate of 10% p.a. to evaluate the two machines.

a) Calculate the net present values of the two machines using the constant chain of replacement assumption and the lowest common multiple method. Which machine should Digem choose and why?

b) Redo your analysis in part (a) using the constant chain of replacement assumption and the perpetuity method. Which machine should Digem choose now? Comment on any differences between the method used here and that used in part (a).

Answer 1

Ans a)

The life span of Machine A is 02 Years and Machine B is 04 Years. So Minimum Common Multiple is 04.

Here Project A will be repeated 04 / 02 = 02 Times Where Project B will be repeated 02 / 02 = 01 Time only.

present values of the two machines using the constant chain of replacement assumption and the lowest common multiple methods.

For Machine A :

This Project Will repeated 02 Times ,

Here Initial Cost= 30,000

annual operating costs are expected to be = $6,000.

Here in Year 02, roject L will Be repeated again.

NPV(A) = NPV1 + NPV2

NPV2 = NPV1 / ( 1+ r)^2

NPV1 = PV of All Cash Flows

= 30,000 + 6,000 / (1.1) + 6,000 / (1.1)^2

= 40,413.22

NPV 2 = 40,413.22 / (1.1)^2 = 33,399.36

NPV(A) = NPV1 + NPV2 = 40,413.22 + 33,399.36 = 73812.58 ~ 73813

For Machine B :

Project B will be repeated01 Time only.

Here Initial Cost=$42,000

annual operating costs are expected to be =  $9,000.

NPV(B) =  PV of All Cash Flows

= 42,000 + 9,000 / (1.1) + 9,000 / (1.1)^2 + 9,000 / (1.1)^3   + 9,000 / (1.1)^4

= 70528.78 ~ 70529

Ans: net present values of of the two machines using the constant chain of replacement assumption and the lowest common multiple method

Machine A = 73813

Machine B = 70529

As both are related to cost. So we will choose machine with the Lowest NPV

Digem choose Machine B (Ans)

Ans b)

perpetuity method

For Machine A :

Here Initial Cost= 30,000

annual operating costs are expected to be =  $6,000.

Here PV of Cash flows for Machine A will be its NPV

= 30,000 + 6,000 / 1.1 +  6,000 / (1.1)^2 +  { 30,000/  (1.1)^2 +  6,000 / (1.1)^3 + 6,000 / (1.1)^4 } +  { 30,000/ (1.1)^4 + 6,000 / (1.1)^5 + 6,000 / (1.1)^6 } + ................... to perpetuity

= 30,000 / 1 - { 1 / (1.1)^2} +   6,000 / {1 - 1 / 1.1 } * 1/1.1 [ We are using formula of Sum in Perpetuity = A / 1 - r ]

= 172857.14 + 60000 ~ 232856

For MachineB :

Here Initial Cost=$42,000

annual operating costs are expected to be =  $9,000.

Life Span = 04 Years

Here PV of Cash flows for Machine A will be its NPV

= 42000 + 9,000 / 1.1 + 9,000 / (1.1)^2  + 9,000 / (1.1)^3  + 9,000 / (1.1)^4

+ [  42000/ (1.1)^4 + 9,000 / (1.1)^5  + 9,000 / (1.1)^6  + 9,000 / (1.1)^7 + 9,000 / (1.1)^8 ] .... to Perpetuity

= 42000 / 1 - { 1 / (1.1)^4} +   9,000 / {1 - 1 / 1.1 } * 1/1.1 [ We are using formula of Sum in Perpetuity = A / 1 - r ]

= 132,497.73 + 90,000 = 222,497.73 ~ 222,498

Ans :

The NPVs are $232,856 (project A) and $222,498 (project B).

As both are related to cost. So we will choose machine with the Lowest NPV

Digem choose Machine B  (Ans)

Here in Part A calculated only common Multiple up to 04 Years but in Part B till Perpetuity