Question

1. The maintenance cost for a certain machine is 1500 TL per year for the first 5 years (from n=1 to n=5), 2200 TL in year 6, and from year 7 until year 35 the maintenance cost per year increases 6% with respect to the amount of the previous year. At an interest rate of 8% per year, what is the equivalent equal payment series?

Answer 1

The equivalent equal payment series can be an annuity which gives the same present value at the same rate of interest as is given by compunding the given series of investments on the machine. This can be calculated by first finidng out the present value of this payment series, and then finding out the amount of annual installments for 35 years that will give the same present value.

The sum can be solved in the method given below (the picture below shows all the three formulae for calculating present value that we willbe using in this sum. Here, PV means present value, PVA means present value of annuity, PVAg means present value of growing annuity, C is the annual installment paid, r is the interest rate given that is 8%, g is the growth rate given that is 6%, and n is the number of years).

The payment for the first five years can be considered as a simple annuity. Its present value can be calculated using the formula in the picture. The payment for the 6th year is a standalone figure and its present value has to be calculated separately, and the payment for the years 7 through 35 is a growing annuity. The net present value of this payment series can be calculated as follows -

Once the net present value is calculated, we will use the same formula of present value annuity to calculate the equivalent series of equal annual payment that will yeild the same present value as this one.