You will be spending $50,000 now. So, to realize a 12% investment yield, you want to make sure that the present value of the residual value (at a 12% rate) plus the present value of whatever the annual savings will be (at a 12% rate) will equal the initial $50,000 cost.

Therefore $50,000 = Present Value of Residual Value + Present Value of Annual Savings

Present Value of Residual Value = Residual Value * PV Factor

Present Value of Residual Value = $10,000 * 0.57

Present Value of Residual Value = $5,700

$50,000 = $5,700 + Present Value of Annual Savings

Present Value of Annual Savings = 50,000 – 5,700 = $44,300

Now that we know what the present value of the annual savings is, we need to determine what the annual savings actually is:

Present Value of Annual Savings = Annual Savings * PV Annuity Factor

$44,300 = Annual Savings * 3.6

Annual Savings = $44,300 / 3.6 = $12,305.56