Question

Equipment maintenance costs for manufacturing explosion-proof pressure switches are projected to be $125,000 in year one and increase by 2.5% each year through year five. What is the equivalent annual worth of the maintenance costs at an interest rate of 10% per year, compounded MONTHLY?

Please do not use excel and show formulas.

Answer 1

Equivalent annual worth of maintenance cost = PV of maintenance cost/ PV factor opf annuity @ 10% for 5 Years

The interest rate in this case is 10% compounded monthly, we need to convert this to effective interest rate.

The effective annual interest rate is the interest rate that is actually earned or paid on an investment, loan or other financial product due to the result of compounding over a given time period. It is also called the effective interest rate, the effective rate or the annual equivalent rate.

So if nominal interest rate (i), number of compounding in a year is (m), effective interest will be

Effective interest rate = (1 + i/m) ^m -1

Where,

Nominal interest rate (i) = 0.1 per year

Number of compounding in a year (m) = 12

Lets put all the values in the formula

Effective interest rate = ( 1 + 0.1/12) ^12 - 1

                                              = ( 1 + 0.0083333333) ^12 - 1

                                              = ( 1.0083333333) ^12 - 1

                                              = 1.10471 - 1

                                              = 0.10471

So annual effective interest rate is 10.47% per year

Now we will calculate PV of maintenance cost

PV of growing annuity = P/(r – g) [1 – {(1+g)/(1+r)}ⁿ]

                                      = 125000/(.1047 – 0.025) [1 – {(1 + 0.025)/(1 + 0.1047)}ⁿ]

                                      = 125000/(0.0797) [1 – {(1.025)/(1.1047)}⁵]

                                      = (1568381.43) [1 – {0.92785}⁵]

                                      = (1568381.43) [1 – {0.68768]

                                      = (1568381.43) [0.312316]

                                      = 489830.61

Now we need to find PV factor of annuity @ 10.47% for 5 years

PV of annuity = P[1- (1+ r)^-n]/ r

= [1- (1+ 0.1047)^-5]/ 0.1047

= [1- (1.1047)^-5]/ 0.1047

= [1- 0.60782453982412]/ 0.1047

= [0.39217546017588/ 0.1047]

= [3.74570640091576]

= 3.7457

Let’s put the values in the formula to find the Equivalent annual worth of maintenance cost

Equivalent annual worth of maintenance cost = 489830.61/3.7457

                                                                              = 130771.44

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Hope this answer your query.

Feel free to comment if you need further assistance. J