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The **payback period** is the time it takes for the cumulative cash flows to equal the initial investment. In this case, the initial investment is the cash outflow at Year 0, which is \(-65,219.00\). We will determine how long it takes for the sum of the cash inflows (years 1, 2, and 3) to cover this amount.

### Cash Flow Summary:

- Year 0: \(-65,219.00\)
- Year 1: \(31,724.00\)
- Year 2: \(32,196.69\)
- Year 3: \(33,307.47\)

### Step-by-step Calculation:

1. **Year 0:** Initial outflow = \(-65,219.00\)
   
2. **Year 1:** Cumulative cash flow after Year 1 = \(-65,219.00 + 31,724.00 = -33,495.00\)
   - The project still hasn't recovered the initial investment.

3. **Year 2:** Cumulative cash flow after Year 2 = \(-33,495.00 + 32,196.69 = -1,298.31\)
   - The project still hasn't fully recovered, but the gap is much smaller.

4. **Year 3:** Cumulative cash flow after Year 3 = \(-1,298.31 + 33,307.47 = 32,009.16\)
   - The project recovers the investment sometime during Year 3.

### Payback Period Calculation:

We know the payback happens between Year 2 and Year 3. To find the exact time during Year 3, we calculate how much of Year 3 is needed to cover the remaining amount.

- Amount left to recover after Year 2: \(1,298.31\)
- Cash flow in Year 3: \(33,307.47\)

So, the fraction of Year 3 required is:

\[
\text{Fraction of Year 3} = \frac{1,298.31}{33,307.47} \approx 0.039\ (\text{or about 3.9\% of Year 3})
\]

Therefore, the **payback period** is approximately:

\[
2 + 0.039 = 2.039 \ \text{years} \ \text{(or roughly 2 years and 14 days)}
\]

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Would you like more details or have any questions? Here are some related questions:

1. How is the discounted payback period calculated?
2. How does the payback period compare to other investment criteria like NPV?
3. What are the limitations of using the payback period as a decision-making tool?
4. Can you explain the relationship between cash flows and the project's risk?
5. How does the payback period affect liquidity analysis?

**Tip:** The payback period does not account for the time value of money. To incorporate that, you would use the **discounted payback period**.

Learn how to calculate the payback period for an investment with a detailed step-by-step solution. This financial problem involves analyzing cash flows over multiple years, determining the recovery of initial investment, and understanding key finance concepts such as cash flow and investment risk. Suitable for undergraduate finance students.

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