Question

You purchased machine A one year ago at a cost of $2,000. You planned to use the machine for 5 years (its useful life) and will need a machine for this function, for the indefinitely long term. However, it has not performed properly and costs $200 per month for repairs, adjustments, and shutdowns. Machine B performs the same functions, but you are confident that it will cost $50 per month for repairs, adjustments, and shutdowns. It costs $3,500.

a. Using an 8% annual interest rate, compute the present worth of initial cost and repairs, adjustments, and shutdowns for both machine A and machine B.

b. Using an 8% annual interest rate, compute the incremental annual net equivalent value of machine B compared with machine A.

c. At this same interest rate, what is the most that machine B repairs could cost per month before it is not an economically justified replacement for machine A.

Answer 1

Machine A was purchased year ago for $2000, and since then incurring $200/month on repairs, adjustments and shutdowns.

So, we need to calculate first the present value of this investments and monthly expenses incurred over the last year.

PV of Machine A: Initial Purchase Amount * (1 + Annual Interest Rate) ^N

Annual Interest Rate = 8%

N = Number of years

Hence, PV of Machine A: $2000 * (1 + 0.08) ^1

                PV of Machine A: $2160 -----------------------(1)

We can treat the monthly expenses as annuities and calculate the present value by using Future Value Factor of Annuity formula i.e.:

FVFA = PMT * [{(1+ R) ^N – 1}/R]

PMT = Payment Per Period i.e. cost per month $200

R = Rate of interest i.e. 8% per annum.
Since these are monthly expenses hence we’ll consider interest rate for our calculations as (8%/12) 0.666667%

N = Time periods which in this case is 12

FVFA      = $200 * [{(1+0.00666667) ^12 – 1}/0.00666667]

                = $200 * 12.44998435

                = $2489.997-------------------------------------(2)

Since the useful life of the machine is 5 years and 1 year has already passed by, hence we need to calculate the present value of the month repair, adjustments and shutdown costs to be incurred for the next 4 years.

Since these expenses are uniformly distributed on a monthly basis, we can use the Present Value Factor of the annuity formula i.e.:

PVFA = PMT * [{1 - (1+R) ^-N} / R]

PMT = Payment Per Period i.e. cost per month $200

R = Rate of interest i.e. 8% per annum.
Since these are monthly expenses hence we’ll consider interest rate for our calculations as (8%/12) 0.666667%

N = Time periods which in this case is 48 (12 months * 4 years)

PVFA     = $200 * [{1 - (1+0.0066667) ^-48} / 0.0066667]

                = $200 * 40.96191

PVFA     = $8192.382---------------------------------------(3)

So, the present worth of Machine A         = (1) + (2) + (3)

                                                                                =$2160 + $2489.997 + $8192.382

                                                                                = $12842.38

Machine B, which involves initial purchase cost of $3500 will incur $50/month on repairs, adjustments and shutdowns.

Present Worth of Machine B = Initial Purchase cost + Present Value of monthly cost on repairs, adjustments and shutdowns

We can apply the Present Value Factor of Annuity formula to calculate the present value of monthly costs for the next 5 years.

PVFA = PMT * [{1 - (1+R) ^-N} / R]

PMT = Payment Per Period i.e. cost per month $50

R = Rate of interest i.e. 8% per annum.
Since these are monthly expenses hence we’ll consider interest rate for our calculations as (8%/12) 0.666667%

N = Time periods which in this case is 60 (12 months * 4 years)

PVFA     = $50 * [{1 - (1+0.0066667) ^-60} / 0.0066667]

                = $50 * 49.3184

                = $2465.92

Present Worth of Machine B       = $3500 + $2465.92

                                                                = $5965.92

Part A)

Present worth of Machine A = $12842.38

Present worth of Machine B = $5965.92

Part B)

Incremental annual net equivalent value of machine B compared with machine A:

                                                = Present worth of Machine A – Present worth of Machine B

                                                = $12842.38 - $5965.92

                                                =$6876.46

Part C)

Here we need to calculate the monthly cost of repairs, adjustments and shutdown which will equate the present worth of Machine A & Machine B

We can approach the question like this:

Step 1 – Deduct the initial cost of Machine B with the Present Worth of Machine A

                = $12842.38 - $3500

                = $9342.38

Step 2 – Apply the Present Value Factor of Annuity Formula, and derive the value of PMT using the equation

PVFA = PMT * [{1 - (1+R) ^-N} / R]

$9342.38 = PMT * [{1 - (1+0.0066667) ^-60} / 0.0066667]

$9342.38 = PMT * 49.3184

PMT = $9342.38 / 49.3184

PMT = $189.4299085

This means if monthly costs on repairs, adjustments and shutdowns, exceeds $189.4299085 it becomes not an economically justified replacement for machine A