What is $33,800 at 3.5% compounded semiannually for 5 years?
To calculate the future value of an amount compounded semiannually, we use the formula:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]
Where:
- A is the future value
- P is the principal amount (initial investment)
- r is the annual interest rate (in decimal form)
- n is the number of times interest is compounded per year
- t is the number of years
In this case, the principal amount (P) is $33,800, the annual interest rate (r) is 3.5% (or 0.035), the interest is compounded semiannually (n = 2), and the investment is for 5 years (t = 5).
Substituting the values into the formula:
\[A = 33800 \left(1 + \frac{0.035}{2}\right)^{(2 \cdot 5)}\]
Simplifying:
\[A = 33800 \left(1 + 0.0175\right)^{10}\]
Calculating the exponent:
\[A = 33800 \cdot 1.0175^{10}\]
Now we can solve this expression to find the future value:
\[A \approx 33800 \cdot 1.193058672\]
Calculating:
\[A \approx 40313.49\]
Therefore, $33,800 invested at 3.5% compounded semiannually for 5 years would grow to approximately $40,313.49.