Mercury Corp is considering a project that will provide an annual cash flow of $43,000 starting at the end of year 4 and continuing until the end of year 17. The project requires 3 investments of $57000.0 each in the present and the following 2 years. The discount rate is 12.6% p.a..

Part A: Compute the present value of positive cash flows in years 4 through 17: (2 marks)

$

(Please answer to 2 decimal places)

Part B: Compute the present value of the investment costs between now and year 2: (2 marks)

$

(Please answer to 2 decimal places)

Part C: Compute the net present value (NPV): (2 marks)

$

Please answer to 2 decimal places)

To solve this problem, we need to compute the present value (PV) of both the cash inflows (Part A) and the investment costs (Part B), and then find the Net Present Value (NPV) by subtracting the present value of costs from the present value of cash inflows.

### Part A: Present Value of Positive Cash Flows (Years 4 through 17)

The cash flows start at the end of year 4 and continue until the end of year 17, so there are 14 years of cash flows. The formula for the present value of an annuity is:

\[
PV_{\text{cash flows}} = C \times \left[\frac{1 - (1 + r)^{-n}}{r}\right] \times (1 + r)^{-t}
\]

Where:
- $ C = 43,000 $ (annual cash flow)
- $ r = 12.6\% $ or 0.126 (discount rate)
- $ n = 14 $ (number of periods)
- $ t = 3 $ (since the cash flows start at the end of year 4, discount them back to year 0)

Let's compute:

\[
PV_{\text{cash flows}} = 43,000 \times \left[\frac{1 - (1 + 0.126)^{-14}}{0.126}\right] \times (1 + 0.126)^{-3}
\]

First, compute the annuity factor:

\[
\text{Annuity factor} = \frac{1 - (1 + 0.126)^{-14}}{0.126} = \frac{1 - (1.126)^{-14}}{0.126} \approx 7.3045
\]

Now, compute the present value of the annuity:

\[
PV_{\text{cash flows}} = 43,000 \times 7.3045 \times (1.126)^{-3}
\]

\[
PV_{\text{cash flows}} = 313,093.5 \times 0.7077 \approx 221,600.70
\]

So, the present value of the positive cash flows is **$221,600.70**.

### Part B: Present Value of the Investment Costs (Years 0 through 2)

The investment costs are $57,000 each at year 0, year 1, and year 2. The present value of these costs is simply the sum of the present values of each of these individual cash outflows:

\[
PV_{\text{costs}} = 57,000 + \frac{57,000}{(1 + 0.126)^1} + \frac{57,000}{(1 + 0.126)^2}
\]

Let's calculate each term:

1. Present value of year 0 cost: 
   \[
   PV_{\text{cost 0}} = 57,000
   \]

2. Present value of year 1 cost: 
   \[
   PV_{\text{cost 1}} = \frac{57,000}{1.126} \approx 50,622.91
   \]

3. Present value of year 2 cost: 
   \[
   PV_{\text{cost 2}} = \frac{57,000}{(1.126)^2} \approx 44,956.06
   \]

Now, sum these up:

\[
PV_{\text{costs}} = 57,000 + 50,622.91 + 44,956.06 \approx 152,578.97
\]

So, the present value of the investment costs is **$152,578.97**.

### Part C: Compute the Net Present Value (NPV)

The NPV is calculated by subtracting the present value of costs from the present value of cash inflows:

\[
\text{NPV} = PV_{\text{cash flows}} - PV_{\text{costs}} = 221,600.70 - 152,578.97 \approx 69,021.73
\]

So, the Net Present Value (NPV) is **$69,021.73**.

### Final Answers:

- **Part A: Present Value of Positive Cash Flows:** \$221,600.70
- **Part B: Present Value of Investment Costs:** \$152,578.97
- **Part C: Net Present Value (NPV):** \$69,021.73

---

Would you like more details or have any questions? Here are some follow-up questions you might consider:

1. How would changing the discount rate affect the NPV?
2. What if the project started generating cash flows earlier? How would that impact the NPV?
3. What is the impact on NPV if the cash flows increase by 5% annually?
4. How sensitive is the NPV to the timing of the initial investments?
5. What would happen to NPV if the project life extends beyond 17 years?
6. How would an upfront investment of $171,000 instead of spreading it over three years impact the NPV?
7. What other factors should be considered when making investment decisions based on NPV?
8. How does NPV compare to other investment appraisal techniques like IRR or Payback Period?

**Tip:** Always double-check cash flow timing and discount rates when calculating NPV to ensure accuracy.

Learn how to compute the net present value (NPV) and present value (PV) for cash flows and investments using a detailed financial analysis. This problem involves calculating the PV of positive cash flows from year 4 to year 17, the PV of investment costs from year 0 to year 2, and then deriving NPV. Suitable for advanced finance students and professionals.

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