The IRR or internal rate of return is the factor by which cash flows associated with the new machine have to be discounted so that the net cash flow over the entire life of the machine is equal to 0. The payback period is the time required for the net cash flow to become 0 given the opportunity cost.
It is given that the cost of the new machine is $100000, it generates inflows of $365000 over a 4 year period and the opportunity cost is 12%.
Let the IRR be represented by r.
`100000 = 365000*(1/(1+r) + 1/(1+r)^2 + 1/(1+r)^3 + 1/(1+r)^4)`
=> `100000/365000 = (1/(1+r))*((1/(1+r))^4 - 1)/(1/(1+r)-1)`
=> `100000/365000 = (1-(1/(1+r))^4)/r`
Solving the equation derived for r, gives r = 364.21%
The initial outflow for the machine is $100000. The inflow at the end of the first year is $365000. Discounted at the rate of 12%, it is equal to `365000/1.12~~ 325892.85` which is greater than the initial outflow. The break-even period is one year.
The IRR of the machine is 364.21% and the break-even period is 1 year.