Question

2.11 A company is considering investing in a new machine for its production line. Management has specified that $150,000 is available to invest in the machine. This money will be invested in a fund that earns 8% annually, compounded quarterly, to pay for the machine, its operation cost, and main- tenance costs. A machine has been identified that has a uniform annual operation cost of $1250. The maintenance cost of the machine is zero in the first year, $200 in the second year, then increasing by an additional $200 per year after that. The machine has a life of 7 years with a salvage value that is estimated to be 5% of its initial cost. Determine the maximum amount of money that can be spent to purchase this machine

Answer 1

Since $150000 is available total PV of the new machine expenses should be equal to $150000.

Interest rate = 8% compounded quarterly (hence n=4)

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

=(1+8%/4)^4 - 1

=(1+2%)^4 -1

=1.02^4-1

=1.0824-1

=0.0824 or 8.24%

PV of annual operating cost = P*(1-(1+r)^-n)/r

=1250*(1-(1+8.24%)^-7)/8.24%

=1250*(1-(1.0824)^-7)/0.0824

=1250*(1-0.5745)/0.0824

=1250*0.4255/0.0824

= $6454.88

PV of maintenance cost will be given by a gradient of $200 for 6 years as it starts from 2nd year

Hence PV at the end of year 1 = G*((1+i)^n-i*n-1)/(i^2*(1+i)^n)

=200*((1+8.24%)^6-8.24%*6-1)/(8.24%^2*(1+8.24%)^6)

=200*((1.0824)^6-0.0824*6-1)/(0.0824^2*(1.0824)^6)

=200*((1.6081-0.4944-1)/(0.0068*1.6081)

=200*(0.1137)/0.0109

=$2079.55

PV at the end of year 0=2079.55/(1+8.24%) = 2079.55/1.0824= $1921.24

If initial cost is C then Salvage value = 5%*C=0.05C

PV of salvage value = 0.05C/(1+8.24%)^7=0.05C/1.0824^7=0.05C/1.7407=0.0287C

Hence Total PV of cost =C+6454.88+1921.24-0.0287C=0.9713C+ 8376.12

Total PV of cost should be equal to 150000

Hence 0.9713C+ 8376.12 =150000

or, 0.9713C=150000 - 8376.12

or, 0.9713C=141623.88

or, C = 141623.88 /0.9713 = 145808.59

Hence maximum amount that can be spent on machine = $ 145808.59