To solve this, let's work through each part step by step.

Part a: NPV with a 12% discount rate

The Net Present Value (NPV) formula is:

$NPV = \sum \frac{C_t}{(1 + r)^t} - C_0$

Where:

• $C_t$ = Cash inflow at time $t$
• $r$ = Discount rate (12% = 0.12)
• $t$ = Time period
• $C_0$ = Initial investment (100,000)

We know the following:

• Initial outlay: $C_0 = 100,000$
• Cash inflows per year: $C_t = 20,000$
• Project duration: 8 years
• Discount rate: 12%

The formula for NPV becomes:

$NPV = -100,000 + \sum_{t=1}^{8} \frac{20,000}{(1 + 0.12)^t}$

Let's calculate this sum:

$NPV = -100,000 + 20,000 \left( \frac{1}{1.12^1} + \frac{1}{1.12^2} + \dots + \frac{1}{1.12^8} \right)$

We can use the present value of an annuity formula since the cash inflows are constant:

$PV_{\text{annuity}} = C \times \frac{1 - (1 + r)^{-n}}{r}$

Where:

• $C = 20,000$ (cash inflow)
• $r = 0.12$ (discount rate)
• $n = 8$ (years)

Substituting these values into the formula:

$PV_{\text{annuity}} = 20,000 \times \frac{1 - (1.12)^{-8}}{0.12}$

Now let's calculate:

$PV_{\text{annuity}} = 20,000 \times \frac{1 - 0.4039}{0.12} = 20,000 \times \frac{0.5961}{0.12} = 20,000 \times 4.9675 = 99,350$

Now, subtract the initial investment of $100,000$:

$NPV = 99,350 - 100,000 = -650$

Part a Answer:

The NPV is -650.

Since the NPV is negative, the project should not be accepted because it does not add value to the firm.

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Part b: NPV with a 13% discount rate

Now, we repeat the same process using a discount rate of 13%:

$PV_{\text{annuity}} = 20,000 \times \frac{1 - (1.13)^{-8}}{0.13}$

Let's calculate the present value factor:

$PV_{\text{annuity}} = 20,000 \times \frac{1 - 0.3970}{0.13} = 20,000 \times \frac{0.6030}{0.13} = 20,000 \times 4.6385 = 92,770$

Subtract the initial investment of $100,000$:

$NPV = 92,770 - 100,000 = -7,230$

Part b Answer:

The NPV is -7,230.

Since the NPV is negative, the project should not be accepted because it does not add value to the firm.

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Part c: IRR

To find the IRR, we set NPV to zero and solve for the discount rate:

$0 = -100,000 + 20,000 \times \frac{1 - (1 + IRR)^{-8}}{IRR}$

We need to solve this equation for IRR. This typically requires trial and error or a financial calculator, but we can estimate the IRR by trying different rates.

Using an iterative process (or financial calculator), the IRR is approximately 12.8%.

Part c Answer:

The IRR is 12.8%.

Since the IRR (12.8%) is greater than the required discount rate of 12%, the project should be accepted because the IRR adds value to the firm.

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Final Answers:

1. a: NPV at 12% = -650 → The project should not be accepted because it does not add value.
2. b: NPV at 13% = -7,230 → The project should not be accepted because it does not add value.
3. c: IRR = 12.8% → The project should be accepted because the IRR is higher than the required discount rate.

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Would you like more details on any of these calculations or concepts? Here are some related questions you might find useful:

1. How does changing the discount rate impact the NPV of a project?
2. What are the advantages and disadvantages of using IRR as a decision-making tool?
3. How do you calculate NPV for a project with varying cash inflows over time?
4. What are the assumptions underlying the NPV method?
5. How can you interpret a project with an IRR lower than the required rate?

Tip: Always ensure the discount rate used for NPV and IRR reflects the project's risk and opportunity cost of capital.