To calculate the future value of an ordinary annuity, we can use the following formula:

$FV = P \times \frac{(1 + \frac{r}{n})^{n \cdot t} - 1}{\frac{r}{n}}$

where:

• $FV$ = Future Value
• $P$ = Periodic Payment
• $r$ = Annual Interest Rate (as a decimal)
• $n$ = Number of compounding periods per year
• $t$ = Term (in years)

Let’s calculate the future value for each case based on the provided data.

Case 1: Monthly Payments of $850 for 13 years, 7 months at 4.00% interest, compounded monthly

• $P = 850$
• $r = 0.04$
• $n = 12$
• $t = 13 + \frac{7}{12} = 13.5833$ years

$FV = 850 \times \frac{(1 + \frac{0.04}{12})^{12 \times 13.5833} - 1}{\frac{0.04}{12}}$

Case 2: Semi-Annual Payments of $550 for 3 years, 6 months at 5.25% interest, compounded semi-annually

• $P = 550$
• $r = 0.0525$
• $n = 2$
• $t = 3.5$ years

$FV = 550 \times \frac{(1 + \frac{0.0525}{2})^{2 \times 3.5} - 1}{\frac{0.0525}{2}}$

Case 3: Quarterly Payments of $650 for 3 years, 6 months at 3.00% interest, compounded quarterly

• $P = 650$
• $r = 0.03$
• $n = 4$
• $t = 3.5$ years

$FV = 650 \times \frac{(1 + \frac{0.03}{4})^{4 \times 3.5} - 1}{\frac{0.03}{4}}$

Let's calculate these values.The future values for each of the annuities are as follows:

1. Monthly Payments of $850 for 13 years, 7 months at 4.00% interest, compounded monthly: $183,646.27
2. Semi-Annual Payments of $550 for 3 years, 6 months at 5.25% interest, compounded semi-annually: $4,166.81
3. Quarterly Payments of $650 for 3 years, 6 months at 3.00% interest, compounded quarterly: $9,557.21

Would you like a detailed breakdown of each calculation or further assistance?

Here are five related questions for exploration:

1. How does the compounding frequency impact the future value of an annuity?
2. What would be the future value if interest rates were to change slightly in each case?
3. How would the results differ if these were annuities due instead of ordinary annuities?
4. Can the formula be adjusted for varying payment amounts over time?
5. What if the term included additional months rather than rounding to the nearest half-year?

Tip: The future value of an annuity grows significantly with higher compounding frequencies due to more frequent application of interest.