Solution

Given Data:

1. Initial investment: $10 million
2. Annual revenue: $3 million for 5 years
3. Discount rate: $5\%$

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a) Break-Even Period

Break-even occurs when the cumulative cash flows equal the initial investment. Let’s calculate the cash flow for each year:

• Cumulative Cash Flows (Non-Discounted):

$\text{Year 1: } 3 \text{ million}$ $\text{Year 2: } 3 + 3 = 6 \text{ million}$ $\text{Year 3: } 6 + 3 = 9 \text{ million}$ $\text{Year 4: } 9 + 3 = 12 \text{ million}$

The break-even point occurs between Year 3 and Year 4 since $9 million < $10 million and $12 million > $10 million.

Interpolation for exact break-even period: $\text{Break-Even Period} = 3 + \frac{(10 - 9)}{3} = 3.33 \text{ years}$

Answer: Break-even occurs in 3.33 years.

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b) Discounted Cash Flow and NPV

Discounted Cash Flow (DCF):

The formula for discounted cash flow is: $\text{DCF}_t = \frac{\text{Cash Flow}_t}{(1 + r)^t}$ where $r = 0.05$.

$\text{Year 1: } \frac{3}{(1 + 0.05)^1} = \frac{3}{1.05} \approx 2.857 \text{ million}$ $\text{Year 2: } \frac{3}{(1 + 0.05)^2} = \frac{3}{1.1025} \approx 2.723 \text{ million}$ $\text{Year 3: } \frac{3}{(1 + 0.05)^3} = \frac{3}{1.157625} \approx 2.593 \text{ million}$ $\text{Year 4: } \frac{3}{(1 + 0.05)^4} = \frac{3}{1.21550625} \approx 2.470 \text{ million}$ $\text{Year 5: } \frac{3}{(1 + 0.05)^5} = \frac{3}{1.27628156} \approx 2.352 \text{ million}$

Total DCF (Sum of discounted cash flows):

$\text{Total DCF} = 2.857 + 2.723 + 2.593 + 2.470 + 2.352 = 13.00 \text{ million (approximately)}$

Net Present Value (NPV):

$\text{NPV} = \text{Total DCF} - \text{Initial Investment}$ $\text{NPV} = 13.00 - 10 = 3.00 \text{ million}$

Answer: DCF = $13 million, NPV = $3 million

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c) Internal Rate of Return (IRR)

The IRR is the discount rate ($r$) at which the NPV becomes zero. This requires solving the equation:

$0 = -10 + \frac{3}{(1 + r)^1} + \frac{3}{(1 + r)^2} + \frac{3}{(1 + r)^3} + \frac{3}{(1 + r)^4} + \frac{3}{(1 + r)^5}$

Using numerical methods (or a financial calculator), we find: $\text{IRR} \approx 14.87\%$

Answer: IRR = 14.87%

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Summary of Answers:

a) Break-even period: 3.33 years
b) Discounted Cash Flow: $13 million, NPV: $3 million
c) IRR: 14.87%

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Would you like to see a graph or additional explanations?

Follow-up Questions:

1. How is the break-even period influenced by the discount rate?
2. What does a positive NPV imply about a project’s feasibility?
3. How would changing the annual revenue affect the IRR and NPV?
4. Can the project sustain profitability if the discount rate rises to 10%?
5. What are the implications if the IRR is lower than the discount rate?

Tip:

To quickly estimate IRR or NPV for larger datasets, consider using financial software like Excel or specialized tools for cash flow analysis.