Debby’s Dance Studios is considering the purchase of new sound equipment that will enhance the popularity of its aerobics dancing. The equipment will cost $24,300. Debby is not sure how many members the new equipment will attract, but she estimates that her increased annual cash flows for each of the next five years will have the following probability distribution. Debby’s cost of capital is 10 percent. Cash Flow Probability $ 3,980 0.4 5,280 0.2 8,140 0.2 10,600 0.2 What is the expected net present value?

To calculate the expected net present value (NPV) for Debby’s Dance Studios, we'll follow these steps:

1. **Determine the Expected Annual Cash Flow**:
   The expected cash flow for each year is the weighted average of the cash flows based on the given probabilities.

2. **Calculate the Present Value of the Expected Cash Flow for Each Year**:
   Since the cost of capital is 10%, we'll discount each cash flow for five years.

3. **Calculate the NPV**:
   The NPV is the difference between the present value of the expected cash flows and the initial cost of the equipment.

### Step 1: Calculate Expected Annual Cash Flow
We multiply each possible cash flow by its probability and sum the results.

\[
\text{Expected Cash Flow} = (3980 \times 0.4) + (5280 \times 0.2) + (8140 \times 0.2) + (10600 \times 0.2)
\]

Let's calculate this.

\[
\text{Expected Cash Flow} = 1592 + 1056 + 1628 + 2120 = 6396
\]

### Step 2: Calculate Present Value of Expected Cash Flows
We discount the expected cash flows over five years using the formula for the present value of annuities:

\[
PV = \frac{\text{Expected Cash Flow}}{(1 + r)^t}
\]
where $ r = 10\% $ (0.10) and $ t $ is the number of years.

We calculate the present value of a series of constant expected cash flows for five years using the present value of annuity formula:

\[
PV = \text{Expected Cash Flow} \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right]
\]

Let's calculate this now.The expected net present value (NPV) for Debby’s Dance Studios is approximately $-\$54.13$. This suggests that the project would result in a small loss based on the given probabilities and cash flow estimates.

Would you like a breakdown of these steps or have any questions?

Here are some related questions to explore further:
1. How would a lower cost of capital (e.g., 8%) affect the NPV?
2. What is the breakeven expected cash flow for a positive NPV?
3. How does the probability distribution of cash flows impact the project's risk?
4. How would a longer project duration (e.g., 7 years) impact the NPV?
5. What if the cost of the equipment was reduced by 10%? How would that change the NPV?

**Tip**: If the NPV is negative, consider re-evaluating the project's assumptions or look for cost reductions to improve the financial viability.

Calculate the expected net present value (NPV) for Debby’s Dance Studios' new sound equipment investment. This problem covers key finance concepts like expected value, NPV, and probability distributions, with a detailed solution. Suitable for college-level finance students.

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