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Answer :
Let's solve this step-by-step using the information provided.
1. Identify the Cash Flows:
- Initial cost in Year 0: [tex]\(-\$150,000\)[/tex]
- Annual savings from Year 1 to Year 6: [tex]\(\$42,500\)[/tex] each year
- Overhaul cost at the end of Year 4: [tex]\(-\$25,000\)[/tex]
- Salvage value at the end of Year 6: [tex]\(\$7,500\)[/tex]
2. Construct the Cash Flow Timeline:
- Year 0: [tex]\(-\$150,000\)[/tex] (initial investment)
- Year 1: [tex]\(\$42,500\)[/tex]
- Year 2: [tex]\(\$42,500\)[/tex]
- Year 3: [tex]\(\$42,500\)[/tex]
- Year 4: [tex]\(\$42,500 - \$25,000 = \$17,500\)[/tex] (savings minus overhaul cost)
- Year 5: [tex]\(\$42,500\)[/tex]
- Year 6: [tex]\(\$42,500 + \$7,500 = \$50,000\)[/tex] (savings plus salvage value)
3. Net Present Value (NPV) Calculation:
The NPV formula is:
[tex]\[
NPV = \sum \left( \frac{\text{Cash Flow}_{t}}{(1+\text{rate})^t} \right)
\][/tex]
where [tex]\( \text{Cash Flow}_{t} \)[/tex] is the cash flow at time [tex]\( t \)[/tex] and [tex]\(\text{rate}\)[/tex] is the cost of capital (12% = 0.12).
- Calculate NPV:
[tex]\[
NPV = -150,000 + \frac{42,500}{1.12^1} + \frac{42,500}{1.12^2} + \frac{42,500}{1.12^3} + \frac{17,500}{1.12^4} + \frac{42,500}{1.12^5} + \frac{50,000}{1.12^6}
\][/tex]
4. Internal Rate of Return (IRR):
IRR is the discount rate that makes the NPV of all cash flows from the investment equal to zero—or the breakeven point for the investment. It's generally calculated using a financial calculator or software.
5. Present Value Index (Profitability Index):
The Present Value Index (PVI) or Profitability Index is calculated as:
[tex]\[
\text{PVI} = \frac{\text{NPV} + \text{Initial Investment}}{\text{Initial Investment}}
\][/tex]
Now, to compute the exact numbers, you could use a financial calculator or an Excel spreadsheet to plug in these values to compute the NPV, IRR, and PVI.
If you have more specific questions on any step or need further explanations, feel free to ask!
1. Identify the Cash Flows:
- Initial cost in Year 0: [tex]\(-\$150,000\)[/tex]
- Annual savings from Year 1 to Year 6: [tex]\(\$42,500\)[/tex] each year
- Overhaul cost at the end of Year 4: [tex]\(-\$25,000\)[/tex]
- Salvage value at the end of Year 6: [tex]\(\$7,500\)[/tex]
2. Construct the Cash Flow Timeline:
- Year 0: [tex]\(-\$150,000\)[/tex] (initial investment)
- Year 1: [tex]\(\$42,500\)[/tex]
- Year 2: [tex]\(\$42,500\)[/tex]
- Year 3: [tex]\(\$42,500\)[/tex]
- Year 4: [tex]\(\$42,500 - \$25,000 = \$17,500\)[/tex] (savings minus overhaul cost)
- Year 5: [tex]\(\$42,500\)[/tex]
- Year 6: [tex]\(\$42,500 + \$7,500 = \$50,000\)[/tex] (savings plus salvage value)
3. Net Present Value (NPV) Calculation:
The NPV formula is:
[tex]\[
NPV = \sum \left( \frac{\text{Cash Flow}_{t}}{(1+\text{rate})^t} \right)
\][/tex]
where [tex]\( \text{Cash Flow}_{t} \)[/tex] is the cash flow at time [tex]\( t \)[/tex] and [tex]\(\text{rate}\)[/tex] is the cost of capital (12% = 0.12).
- Calculate NPV:
[tex]\[
NPV = -150,000 + \frac{42,500}{1.12^1} + \frac{42,500}{1.12^2} + \frac{42,500}{1.12^3} + \frac{17,500}{1.12^4} + \frac{42,500}{1.12^5} + \frac{50,000}{1.12^6}
\][/tex]
4. Internal Rate of Return (IRR):
IRR is the discount rate that makes the NPV of all cash flows from the investment equal to zero—or the breakeven point for the investment. It's generally calculated using a financial calculator or software.
5. Present Value Index (Profitability Index):
The Present Value Index (PVI) or Profitability Index is calculated as:
[tex]\[
\text{PVI} = \frac{\text{NPV} + \text{Initial Investment}}{\text{Initial Investment}}
\][/tex]
Now, to compute the exact numbers, you could use a financial calculator or an Excel spreadsheet to plug in these values to compute the NPV, IRR, and PVI.
If you have more specific questions on any step or need further explanations, feel free to ask!
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