Exam success document with mathematical equations and text.

Problem Analysis and Solution

The problem involves a scenario where a company, S.A., is considering two investment projects: Project A and Project B. The goal is to evaluate these projects using financial metrics such as Net Present Value (NPV), Internal Rate of Return (IRR), and Payback Period. Additionally, the company must decide which project to choose based on these evaluations.

#### Step 1: Understand the Cash Flows
- Project A: Initial investment = €50,000; Annual cash inflow = €20,000 for 4 years.
- Project B: Initial investment = €80,000; Annual cash inflow = €30,000 for 4 years.
- Discount rate = 10%.

#### Step 2: Calculate NPV for Both Projects

The formula for NPV is:
\[
NPV = -\text{Initial Investment} + \sum_{t=1}^{n} \frac{\text{Cash Flow}_t}{(1 + r)^t}
\]
where \( r \) is the discount rate and \( n \) is the number of periods.

##### Project A
- Initial investment = €50,000
- Annual cash flow = €20,000
- Discount rate = 10%
- Number of periods = 4

\[
NPV_A = -50,000 + \frac{20,000}{1.1^1} + \frac{20,000}{1.1^2} + \frac{20,000}{1.1^3} + \frac{20,000}{1.1^4}
\]

Calculate each term:
\[
\frac{20,000}{1.1^1} = 18,181.82
\]
\[
\frac{20,000}{1.1^2} = 16,528.93
\]
\[
\frac{20,000}{1.1^3} = 14,935.39
\]
\[
\frac{20,000}{1.1^4} = 13,577.63
\]

Sum the discounted cash flows:
\[
18,181.82 + 16,528.93 + 14,935.39 + 13,577.63 = 63,223.77
\]

Calculate NPV:
\[
NPV_A = -50,000 + 63,223.77 = 13,223.77
\]

##### Project B
- Initial investment = €80,000
- Annual cash flow = €30,000
- Discount rate = 10%
- Number of periods = 4

\[
NPV_B = -80,000 + \frac{30,000}{1.1^1} + \frac{30,000}{1.1^2} + \frac{30,000}{1.1^3} + \frac{30,000}{1.1^4}
\]

Calculate each term:
\[
\frac{30,000}{1.1^1} = 27,272.73
\]
\[
\frac{30,000}{1.1^2} = 24,793.39
\]
\[
\frac{30,000}{1.1^3} = 22,539.44
\]
\[
\frac{30,000}{1.1^4} = 20,490.40
\]

Sum the discounted cash flows:
\[
27,272.73 + 24,793.39 + 22,539.44 + 20,490.40 = 95,095.96
\]

Calculate NPV:
\[
NPV_B = -80,000 + 95,095.96 = 15,095.96
\]

#### Step 3: Calculate IRR for Both Projects

The IRR is the discount rate that makes the NPV equal to zero. We solve for \( r \) in the equation:
\[
NPV = 0
\]

##### Project A
\[
0 = -50,000 + \frac{20,000}{1 + r} + \frac{20,000}{(1 + r)^2} + \frac{20,000}{(1 + r)^3} + \frac{20,000}{(1 + r)^4}
\]

Using a financial calculator or software, we find:
\[
IRR_A \approx 18.92\%
\]

##### Project B
\[
0 = -80,000 + \frac{30,000}{1 + r} + \frac{30,000}{(1 + r)^2} + \frac{30,000}{(1 + r)^3} + \frac{30,000}{(1 + r)^4}
\]

Using a financial calculator or software, we find:
\[
IRR_B \approx 16.35\%
\]

#### Step 4: Calculate Payback Period for Both Projects

The payback period is the time it takes for the cumulative cash inflows to equal the initial investment.

##### Project A
- Initial investment = €50,000
- Annual cash flow = €20,000

Cumulative cash flows:
\[
\text{Year 1: } 20,000
\]
\[
\text{Year 2: } 20,000 + 20,000 = 40,000
\]
\[
\text{Year 3: } 40,000 + 20,000 = 60,000
\]

Payback period:
\[
\text{Payback Period}_A = 2 + \frac{50,000 - 40,000}{20,000} = 2 + 0.5 = 2.5 \text{ years}
\]

##### Project B
- Initial investment = €80,000
- Annual cash flow = €30,000

Cumulative cash flows:
\[
\text{Year 1: } 30,000
\]
\[
\text{Year 2: } 30,000 + 30,000 = 60,000
\]
\[
\text{Year 3: } 60,000 + 30,000 = 90,000
\]

Payback period:
\[
\text{Payback Period}_B = 2 + \frac{80,000 - 60,000}{30,000} = 2 + 0.67 = 2.67 \text{ years}
\]

#### Step 5: Decision Based on NPV, IRR, and Payback Period

- NPV: Project B has a higher NPV (€15,095.96) compared to Project A (€13,223.77).
- IRR: Project A has a higher IRR (18.92%) compared to Project B (16.35%).
- Payback Period: Project A has a shorter payback period (2.5 years) compared to Project B (2.67 years).

Given that the company's primary objective is to maximize shareholder value, the NPV is the most reliable metric for comparing mutually exclusive projects. Therefore, Project B should be chosen because it has a higher NPV.

#### Final Answer
\[
\boxed{B}
\]