To find the value of an investment compounded quarterly, we use the formula for compound interest:

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

Where:

• $A$ is the final amount
• $P = 10,000$ is the initial principal (investment)
• $r = 0.0405$ is the annual interest rate (4.05%)
• $n = 4$ is the number of times the interest is compounded per year (quarterly)
• $t = 1$ is the time in years

Step 1: Plug the values into the formula

$A = 10,000 \left(1 + \frac{0.0405}{4}\right)^{4(1)}$ $A = 10,000 \left(1 + 0.010125\right)^4$ $A = 10,000 \left(1.010125\right)^4$

Step 2: Calculate $\left(1.010125\right)^4$

$A = 10,000 \times 1.041051$

Step 3: Calculate the final amount

$A = 10,410.51$

Thus, the value of the investment compounded quarterly is $10,410.51.

Would you like further details, or do you have any questions?

Here are some related questions:

1. What would the value be if compounded weekly instead of quarterly?
2. How does the frequency of compounding affect the final amount?
3. What is the formula for continuous compounding?
4. What is the effective annual rate (EAR) for each of these compounding frequencies?
5. How would this change if the interest rate was 5% instead of 4.05%?

Tip: The more frequently interest is compounded, the higher the final amount will be!