Discounted payback: RPA Energy Company is installing new equipment at a cost of $10 million. Expected cash flows from this project over the next five years will be $1,550,000, $2,550,000, $4,125,000, $6,350,000, and $7,000,000. The company's discount rate for such projects is 11 percent. What is the project's discounted payback period?

To find the **discounted payback period** for the RPA Energy Company project, we need to determine how many years it takes for the project's discounted cash flows to repay the initial investment of $10 million.

The discounted cash flows are computed using the formula:

\[
\text{Discounted Cash Flow} = \frac{\text{Cash Flow}}{(1 + r)^t}
\]

where:
- $ r $ is the discount rate (11%, or 0.11),
- $ t $ is the year (1, 2, 3, etc.).

We will discount each year's cash flow, then accumulate them until the sum reaches or exceeds the initial investment of $10 million. 

Let's calculate this step by step.

### Step 1: Discount the Cash Flows

#### Year 1:
\[
\text{Discounted Cash Flow}_1 = \frac{1,550,000}{(1 + 0.11)^1} = \frac{1,550,000}{1.11} = 1,396,396.40
\]

#### Year 2:
\[
\text{Discounted Cash Flow}_2 = \frac{2,550,000}{(1 + 0.11)^2} = \frac{2,550,000}{1.2321} = 2,069,658.88
\]

#### Year 3:
\[
\text{Discounted Cash Flow}_3 = \frac{4,125,000}{(1 + 0.11)^3} = \frac{4,125,000}{1.367631} = 3,016,544.08
\]

#### Year 4:
\[
\text{Discounted Cash Flow}_4 = \frac{6,350,000}{(1 + 0.11)^4} = \frac{6,350,000}{1.518059} = 4,182,136.18
\]

#### Year 5:
\[
\text{Discounted Cash Flow}_5 = \frac{7,000,000}{(1 + 0.11)^5} = \frac{7,000,000}{1.684169} = 4,156,551.41
\]

### Step 2: Accumulate the Discounted Cash Flows

Now we sum the discounted cash flows year by year to see when they exceed $10 million:

- **After Year 1**: $1,396,396.40
- **After Year 2**: $1,396,396.40 + $2,069,658.88 = $3,466,055.28
- **After Year 3**: $3,466,055.28 + $3,016,544.08 = $6,482,599.36
- **After Year 4**: $6,482,599.36 + $4,182,136.18 = $10,664,735.54

At the end of **Year 4**, the cumulative discounted cash flows exceed $10 million. Therefore, the project pays back in **4 years**.

Thus, the **discounted payback period** is **4 years**.

Would you like more details or have any other questions?

Here are 5 follow-up questions you might consider:
1. How does the discounted payback period differ from the regular payback period?
2. How would a change in the discount rate affect the discounted payback period?
3. What is the project's internal rate of return (IRR)?
4. How do we interpret the significance of the discounted payback period in investment decisions?
5. Can you compare the discounted payback period to the net present value (NPV)?

**Tip**: The discounted payback period considers the time value of money, unlike the regular payback period, which only looks at cash flows without discounting.

Learn how to calculate the discounted payback period for RPA Energy Company's new project, involving a $10 million investment and a discount rate of 11%. This detailed step-by-step guide covers the discounted cash flows for five years and explains how to find the payback period using the time value of money principle.

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