To determine how much [tex]$550 invested at the end of each quarter would be worth in 9 years at an annual interest rate of 4%, we can use the formula for the future value of an ordinary annuity. The formula is:
\[ FV = PMT \times \left( \frac{(1 + r)^n - 1}{r} \right) \]
where:
- \( FV \) is the future value of the annuity,
- \( PMT \) is the payment amount per period,
- \( r \) is the interest rate per period,
- \( n \) is the total number of periods.
Now, let's break down the problem step by step:
1. Identify the payment amount (\( PMT \)):
The payment amount each period is $[/tex]550.
2. Determine the total number of periods ([tex]\( n \)[/tex]):
Since payments are made quarterly (every 3 months) and the total investment duration is 9 years, we first find the number of quarters in 9 years:
[tex]\[
n = 9 \text{ years} \times 4 \text{ quarters/year} = 36 \text{ quarters}
\][/tex]
3. Compute the interest rate per period ([tex]\( r \)[/tex]):
The annual interest rate is 4%. Since the interest is compounded quarterly, we need to convert this annual rate into a quarterly rate:
[tex]\[
r = \frac{4\%}{4} = 1\%
\][/tex]
In decimal form, this is:
[tex]\[
r = \frac{4}{100 \times 4} = 0.01 = 0.04 / 4
\][/tex]
4. Substitute the known values into the annuity formula:
[tex]\[
FV = 550 \times \left( \frac{(1 + 0.01)^{36} - 1}{0.01} \right)
\][/tex]
5. Calculate the future value ([tex]\( FV \)[/tex]) (Show the critical intermediate steps without performing actual calculations):
[tex]\[
FV = 550 \times \left( \frac{(1 + 0.01)^{36} - 1}{0.01} \right)
\][/tex]
1. Calculate [tex]\( (1 + 0.01)^{36} \)[/tex]:
[tex]\[
1.01^{36}
\][/tex]
2. Subtract 1 from the result:
[tex]\[
\left( \frac{1.01^{36} - 1}{0.01} \right)
\][/tex]
3. Divide by the interest rate per period ([tex]\( 0.01 \)[/tex]):
[tex]\[
\frac{1.01^{36} - 1}{0.01}
\][/tex]
4. Multiply by the payment amount ([tex]\( 550 \)[/tex]):
[tex]\[
550 \times \left(\frac{1.01^{36} - 1}{0.01}\right)
\][/tex]
Performing these calculations, the future value of the annuity, rounded to the nearest cent, is:
[tex]\[
FV \approx \$23,692.28
\][/tex]
Therefore, investing [tex]$550 at the end of each quarter for 9 years at an annual interest rate of 4% will yield approximately $[/tex]23,692.28 after 9 years.
