Paxton has two investments; investment A required a $9,430 buy-in 3 years ago and is expected to payoff $21,777 13 years from today. Investment B had a $468 buy-in required 4 years ago and is expected to payoff $7,182 12 years from today. What is the return of the higher-risk investment? Assume interest is compounded quarterly.

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To determine the return (rate of return) of the higher-risk investment, we will use the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $ A $ is the future value (payoff),
- $ P $ is the initial investment (buy-in),
- $ r $ is the annual interest rate (what we are solving for),
- $ n $ is the number of compounding periods per year (quarterly means $ n = 4 $),
- $ t $ is the time in years.

### Step 1: Identify the higher-risk investment
Based on the scenario, the risk level isn't explicitly stated. However, typically, the smaller the initial investment, the higher the risk, so we assume **Investment B** is the higher-risk one.

### Step 2: Apply the compound interest formula to Investment B
For Investment B:
- $ A = 7,182 $ (future value),
- $ P = 468 $ (initial investment),
- $ n = 4 $ (quarterly compounding),
- $ t = 16 $ (total time from buy-in to payoff: 4 years ago + 12 years into the future).

The formula becomes:
\[
7,182 = 468 \left(1 + \frac{r}{4}\right)^{4 \times 16}
\]
Simplifying:
\[
7,182 = 468 \left(1 + \frac{r}{4}\right)^{64}
\]

### Step 3: Solve for $ r $
We will now solve for $ r $, the annual rate of return. Let's compute this.The return of the higher-risk investment (Investment B) is approximately **17.44%**.

Do you want more details or have any questions?

Here are 5 related questions to explore:
1. How would the return change if the buy-in for Investment B was higher?
2. What is the return of Investment A under similar calculations?
3. How does the compounding frequency affect the rate of return?
4. Can we use the same formula for monthly or yearly compounding periods?
5. What factors generally make an investment high-risk?

**Tip:** The more frequent the compounding periods (e.g., quarterly vs. annually), the higher the effective interest rate for the same nominal rate.

Paxton has two investments; investment A required a $9,430 buy-in 3 years ago and is expected to payoff $21,777 13 years from today. Investment B had a $468 buy-in required 4 years ago and is expected to payoff $7,182 12 years from today. What is the return of the higher-risk investment? Assume interest is compounded quarterly.

Learn how to calculate the return on the higher-risk investment using the compound interest formula. This problem involves exponential functions and quarterly compounding over 16 years. Suitable for students in Grades 10-12.

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