The problem asks to determine how much needs to be deposited now to yield an annuity payment of $500, compounded semiannually, over 14 years, given a 6% annual interest rate.

Step-by-step breakdown:

1. Interest Rate (Compounded Semiannually):
   The annual interest rate is 6%, but since it's compounded semiannually, the interest rate per period is: $r = \frac{6\%}{2} = 3\% = 0.03$

2. Number of Periods:
   Since the payments are made every six months for 14 years, the total number of periods is: $n = 14 \times 2 = 28 \, \text{periods}$

3. Present Value of an Annuity Formula:
   To calculate how much should be deposited, we use the present value of an annuity formula: $PV = P \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right]$ Where:

   • $PV$ is the present value (amount to be deposited).
   • $P$ is the periodic payment, which is $500.
   • $r$ is the interest rate per period, 0.03.
   • $n$ is the number of periods, 28.

4. Substituting the Values: $PV = 500 \times \left[ \frac{1 - (1 + 0.03)^{-28}}{0.03} \right]$

Let me calculate this value for you.The amount that needs to be deposited now to yield an annuity payment of $500 every six months for 14 years, with a 6% interest rate compounded semiannually, is approximately $9,382.05.

Would you like more details on any step, or do you have any other questions?

Here are 5 related questions you might want to explore:

1. How would the result change if the interest rate were compounded quarterly instead of semiannually?
2. How does the present value change if the interest rate is 5% instead of 6%?
3. What if the payments were made monthly instead of semiannually?
4. How can we adjust the formula for annuity payments that begin at the end of each period instead of the beginning?
5. How does inflation affect the future value of the deposited amount?

Tip: When dealing with annuities, it's important to pay attention to whether payments are made at the beginning or end of periods, as it affects the formula used.