An investor wants to know the amount she should pay for an oil well expected to yield an annual return of $20,000 for the next 20 years, after which the well will be dry. Find the amount she should pay to yield a 12% annual return if a sinking fund earns 10% annually.

To find how much the investor should pay for the oil well, we need to break this problem into two components:

1. **The present value of the annual returns of $20,000** over 20 years at a 12% desired rate of return (also known as the "discount rate").
2. **The sinking fund** that earns 10% annually, which ensures that the capital is preserved by replacing the diminishing value of the well over time.

### Step 1: Calculate the Present Value (PV) of the Annual Returns

Since the investor wants to get a 12% return per year from the oil well, we can use the formula for the present value of an annuity to find out how much she should pay today to get a $20,000 annuity for 20 years. The formula for the present value of an annuity is:

\[
PV = P \times \left( 1 - (1 + r)^{-n} \right) \div r
\]

Where:
- $P = 20,000$ (annual return),
- $r = 0.12$ (annual return rate or discount rate),
- $n = 20$ (number of years).

Let’s calculate the present value of the $20,000 annual return:

\[
PV = 20000 \times \left( \frac{1 - (1 + 0.12)^{-20}}{0.12} \right)
\]

Let me calculate this value.The present value of the $20,000 annual return over 20 years at a 12% return rate is approximately **$149,388.87**.

### Step 2: Considering the Sinking Fund

A sinking fund is required to ensure that the capital is replaced over the lifetime of the well. The sinking fund earns 10% annually, and its purpose is to accumulate the total capital (i.e., the initial payment for the well) over the 20-year period.

The formula for the sinking fund payment per period (S) is:

\[
S = \frac{A}{\frac{(1 + i)^n - 1}{i}}
\]

Where:
- $A$ is the amount to be accumulated (the initial cost of the well, which is the unknown),
- $i = 0.10$ (the sinking fund rate),
- $n = 20$ (number of years).

Rearranging to solve for $A$:

\[
A = S \times \frac{(1 + i)^n - 1}{i}
\]

Let me compute this next.The total amount the investor should pay for the oil well, considering both the desired return of 12% and a sinking fund earning 10% annually, is approximately **$8,556,247.60**.

This amount accounts for both the present value of the well's future cash flows and the capital that will be replaced over the well's 20-year lifespan.

### Would you like further clarification or more details on this process?

Here are some related questions to consider:
1. How does a sinking fund work in different investment scenarios?
2. What are the differences between present value and future value in investment analysis?
3. How would a change in the oil well's expected yield affect the amount to be paid?
4. How is the annuity formula applied in real-world financial decisions?
5. What happens if the sinking fund rate is lower than 10%?

**Tip:** When evaluating investments, it's crucial to consider both the expected cash flow and the means of recovering or preserving capital over time.

Learn how to calculate the investment cost for an oil well that yields $20,000 annually over 20 years with a 12% desired return and a 10% sinking fund rate. This example involves the present value of annuities and sinking fund formulas, suitable for advanced high school or college-level students.

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