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To solve the problem, we need to determine the expected Net Present Value (NPV) and the standard deviation about the expected value based on the provided cash flows and probabilities. Here's how we can approach it:

### Step 1: Calculate the Expected Cash Flow (ECF) for each period
For each period, the Expected Cash Flow (ECF) is calculated by multiplying each possible cash flow by its corresponding probability and then summing these products.

\[
\text{ECF}_t = \sum (\text{CFAT} \times \text{Prob})
\]

Where:
- \( t \) is the period (1, 2, or 3).
- \( \text{CFAT} \) is the cash flow in period \( t \).
- \( \text{Prob} \) is the probability of the corresponding cash flow.

Let's calculate it for each period.

**For Period 1:**

\[
\text{ECF}_1 = (1000 \times 0.1) + (1500 \times 0.3) + (2000 \times 0.4) + (2500 \times 0.2)
\]

**For Period 2:**

\[
\text{ECF}_2 = (0 \times 0.1) + (800 \times 0.3) + (1500 \times 0.4) + (2000 \times 0.2)
\]

**For Period 3:**

\[
\text{ECF}_3 = (0 \times 0.2) + (500 \times 0.5) + (1000 \times 0.3)
\]

### Step 2: Calculate the Present Value of Expected Cash Flows
Since the risk-free rate is 5%, we need to discount the expected cash flows to their present value:

\[
\text{PV}_t = \frac{\text{ECF}_t}{(1 + r)^t}
\]

Where:
- \( r = 0.05 \) (the risk-free rate)
- \( t \) is the period (1, 2, or 3).

### Step 3: Calculate the Expected NPV
The Expected NPV is simply the sum of the present values of the expected cash flows minus the initial outlay:

\[
\text{NPV} = \sum \text{PV}_t - \text{Initial Outlay}
\]

### Step 4: Calculate the Variance and Standard Deviation
To calculate the standard deviation, we first calculate the variance of the NPVs, using the expected cash flows and probabilities. Then, take the square root of the variance to get the standard deviation.

### Let's start with the calculations.

I'll now compute each of these steps.The calculations yield the following results:

1. **Expected NPV**: ₹361.73
2. **Standard Deviation of NPV**: ₹752.76

These results suggest that while the project has a positive expected NPV, indicating a potential for profitability, there is also a significant amount of risk, as reflected in the relatively high standard deviation.

Would you like to go into more detail on any of these calculations, or have any questions about the process?

Here are five additional questions that might be relevant:

1. How does changing the risk-free rate affect the NPV and standard deviation?
2. What would be the impact of a higher or lower initial outlay on the project's profitability?
3. How would the NPV change if the probability distributions of the cash flows were different?
4. What other financial metrics could be useful in evaluating this project besides NPV and standard deviation?
5. How does the correlation between different periods' cash flows impact the overall risk of the project?

**Tip:** When analyzing investment projects, always consider both the expected return (NPV) and the associated risk (standard deviation) to make well-informed decisions.

Learn how to calculate the expected Net Present Value (NPV) and standard deviation in finance. This detailed guide covers the steps to calculate Expected Cash Flow (ECF), Present Value (PV), NPV, variance, and standard deviation using provided cash flows and probabilities. Suitable for advanced finance or business students.

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