Question

find the future value for the annuity due with the given rate.

payments of \\$220 for 6 years at 0.34\\% compounded quarterly

the future value of the annuity due is \\$ \box.
(do not round until the final answer. then round to the nearest cent as needed.)

Explanation:

🆕 New Concept Discovered: Future Value of an Annuity Due
Payments made at the start of each period.

Step 1: Identify the given values

An annuity due means payments are made at the beginning of each period. We extract the parameters from the problem:

• Periodic payment, \( PMT = 220 \)
• Annual interest rate, \( r = 0.34\% = 0.0034 \)
• Compounding frequency, \( m = 4 \) (compounded quarterly)
• Time in years, \( t = 6 \)

Step 2: Calculate period rate and total periods

The interest rate per compounding period \( i \) is:

$$ i = \frac{r}{m} = \frac{0.0034}{4} = 0.00085 $$

The total number of payment periods \( n \) is:

$$ n = t \times m = 6 \times 4 = 24 $$

Step 3: Apply the Future Value of an Annuity Due formula

The formula for the future value of an annuity due (\( FV_{\text{due}} \)) is:

$$ FV_{\text{due}} = PMT \times \frac{(1 + i)^n - 1}{i} \times (1 + i) $$

Substitute the values into the formula:

$$ FV_{\text{due}} = 220 \times \frac{(1 + 0.00085)^{24} - 1}{0.00085} \times (1 + 0.00085) $$

Step 4: Perform the calculation

First, calculate the growth factor:

$$ 1 + i = 1.00085 $$

$$ (1.00085)^{24} \approx 1.0205938 $$

Next, calculate the numerator:

$$ (1.00085)^{24} - 1 \approx 0.0205938 $$

Divide by \( i \):

$$ \frac{0.0205938}{0.00085} \approx 24.228014 $$

Multiply by the payment amount and the extra period of interest:

$$ FV_{\text{due}} \approx 220 \times 24.228014 \times 1.00085 $$

$$ FV_{\text{due}} \approx 5330.163 \times 1.00085 \approx 5334.694 $$

Rounding to the nearest cent gives \( 5334.69 \).

Answer:

5334.69