To solve this, we will use the following formulas for future value of an annuity:

Future Value of an Ordinary Annuity

For an ordinary annuity (payments made at the end of each period), the future value is calculated using the formula:

$FV_{\text{ordinary}} = P \times \left[ \frac{(1 + r)^n - 1}{r} \right]$

Where:

• $FV_{\text{ordinary}}$ = future value of the annuity
• $P$ = annuity payment ($3,500)
• $r$ = interest rate per period (8% or 0.08 annually)
• $n$ = number of periods (7 years)

Future Value of an Annuity Due

For an annuity due (payments made at the beginning of each period), the future value is:

$FV_{\text{due}} = P \times \left[ \frac{(1 + r)^n - 1}{r} \right] \times (1 + r)$

This is similar to the ordinary annuity, but we multiply the entire term by $(1 + r)$ because the payments are made at the beginning of each period, allowing them to earn one more period of interest.

Let's go ahead and calculate these values.### a. Future Value Calculations:

1. Future Value of an Ordinary Annuity:
   $FV_{\text{ordinary}} = \$31,229.81$

2. Future Value of an Annuity Due:
   $FV_{\text{due}} = \$33,728.20$

b. Comparison:

The future value of an annuity due is higher than that of an ordinary annuity ($33,728.20 vs. $31,229.81). This is because with an annuity due, payments are made at the beginning of each period, allowing each payment to earn interest for one additional period compared to an ordinary annuity.

Preferable Investment:

An annuity due is preferable as an investment because the payments have more time to grow due to the earlier deposit schedule. Over time, this results in a higher future value, all else being equal.

Would you like more details on these calculations, or have any other questions?

Related Questions:

1. What happens if the interest rate changes from 8% to 10%?
2. How does increasing the deposit period to 10 years affect the future value?
3. What is the future value if the annuity amount changes to $4,000?
4. How would the future value change with monthly payments instead of annual payments?
5. How can the present value of the same annuity be calculated?

Tip:

Always prefer an annuity due if you want to maximize growth, as payments earn interest for a longer period compared to ordinary annuities.