To find the quoted annual interest rate required for an investment to grow from $134,000 to $459,000 over 15 years with monthly compounding, we can use the compound interest formula:

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

Where:

• $A$ is the final amount ($459,000)
• $P$ is the initial principal ($134,000)
• $r$ is the annual interest rate (quoted rate we're solving for)
• $n$ is the number of times the interest is compounded per year (12 for monthly)
• $t$ is the number of years (15)

Step 1: Set up the equation

$459,000 = 134,000 \left( 1 + \frac{r}{12} \right)^{12 \times 15}$

Step 2: Solve for $r$

First, divide both sides by 134,000: $\frac{459,000}{134,000} = \left( 1 + \frac{r}{12} \right)^{180}$

Simplify the left side: $3.4269 \approx \left( 1 + \frac{r}{12} \right)^{180}$

Now, take the natural logarithm (ln) of both sides to eliminate the exponent: $\ln(3.4269) = 180 \cdot \ln\left(1 + \frac{r}{12}\right)$

Next, solve for $r$. I'll compute this for you.The quoted annual interest rate required for the investment to grow from $134,000 to $459,000 in 15 years with monthly compounding is approximately 8.24%.

Would you like more details on any step or have further questions? Here are 5 related questions you might explore:

1. How would the interest rate change if the investment period were 10 years instead of 15?
2. What is the impact of different compounding frequencies on the final amount?
3. How much would you need to invest initially to reach $459,000 in 15 years with the same interest rate?
4. How does continuous compounding affect the required interest rate for the same growth?
5. What if the interest were compounded quarterly instead of monthly?

Tip: The more frequently interest is compounded, the higher the effective annual interest rate (EAR) becomes.