To determine whether the company should replace its year-old machine, we need to calculate the Net Present Value (NPV) of replacing the machine. This involves comparing the cash flows associated with keeping the current machine versus replacing it with the new machine.

Step 1: Calculate the Cash Flows for the New Machine

1. Initial cost of the new machine: $165,000
2. Depreciation of the new machine:
   Since the new machine will be depreciated on a straight-line basis over 10 years with no salvage value: $\text{Annual Depreciation} = \frac{165,000}{10} = 16,500 \text{ per year}$
3. Gross margin produced by the new machine: $55,000 per year
4. Tax savings due to depreciation (tax shield):
   Tax rate = 40%, so the tax shield from depreciation is: $\text{Tax Shield} = 16,500 \times 0.40 = 6,600 \text{ per year}$
5. After-tax gross margin for the new machine: $\text{After-tax Gross Margin} = 55,000 \times (1 - 0.40) + 6,600 = 33,000 + 6,600 = 39,600 \text{ per year}$

Step 2: Calculate the Cash Flows for the Current Machine

1. Market value of the current machine today: $65,000
2. Depreciation of the current machine: $8,636 per year
3. Gross margin produced by the current machine: $22,000 per year
4. Tax shield due to depreciation: $\text{Tax Shield} = 8,636 \times 0.40 = 3,454.40 \text{ per year}$
5. After-tax gross margin for the current machine: $\text{After-tax Gross Margin} = 22,000 \times (1 - 0.40) + 3,454.40 = 13,200 + 3,454.40 = 16,654.40 \text{ per year}$

Step 3: Calculate the NPV of Replacing the Machine

1. Initial outlay: $\text{Initial Outlay} = \text{Cost of New Machine} - \text{Market Value of Old Machine} = 165,000 - 65,000 = 100,000$
2. Incremental after-tax cash flows: $\text{Incremental Cash Flows} = 39,600 - 16,654.40 = 22,945.60 \text{ per year for 10 years}$
3. NPV of incremental cash flows: $\text{NPV} = \sum_{t=1}^{10} \frac{22,945.60}{(1 + 0.12)^t} - 100,000$

Let me calculate this NPV.The NPV of replacing the year-old machine is $45,205 (rounded to the nearest dollar).

Since the NPV is positive, it indicates that replacing the current machine with the new one would add value to the company. Therefore, your company should replace the year-old machine.

Would you like further details or have any questions? Here are some related questions to consider:

1. How does the opportunity cost of capital affect the NPV calculation?
2. What would happen to the NPV if the gross margin of the new machine changes?
3. How does the tax rate influence the decision to replace the machine?
4. What impact does the salvage value of the machines have on the NPV analysis?
5. What alternative methods can be used to evaluate the replacement decision?

Tip: When evaluating investment decisions, always compare the NPV to zero. A positive NPV indicates the investment should be pursued, while a negative NPV suggests it should be avoided.