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Let's break down each of the problems step by step.

### (a) Calculation for the semiannual settlement with 6% interest:

The company will receive $6,000 every six months for 13 years. To calculate how much the insurance company must set aside now, we need to use the **Present Value of an Annuity formula**:

\[
PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]

Where:
- $P = 6,000$ (the payment every period),
- $r = 6\% \text{ annual interest compounded semiannually} = \frac{6\%}{2} = 0.03$ (interest per period),
- $n = 13 \text{ years} \times 2 = 26$ periods (since there are 2 periods per year).

Substitute the values into the formula:

\[
PV = 6000 \times \left( \frac{1 - (1 + 0.03)^{-26}}{0.03} \right)
\]

Calculating this will give us the present value the insurance company must set aside now.

---

### (b) Changing the settlement to $3,000 and 8% interest compounded quarterly:

Here, the settlement is now $3,000 every three months for 13 years. We'll use the same formula, but with the following changes:
- $P = 3,000$,
- $r = 8\% \text{ annual interest compounded quarterly} = \frac{8\%}{4} = 0.02$ (interest per quarter),
- $n = 13 \text{ years} \times 4 = 52$ periods (since there are 4 periods per year).

Substitute the values:

\[
PV = 3000 \times \left( \frac{1 - (1 + 0.02)^{-52}}{0.02} \right)
\]

---

### (c) Settlement at the beginning of each 3-month period:

This is an annuity due instead of an ordinary annuity. The formula for the present value of an annuity due is:

\[
PV_{\text{due}} = PV_{\text{ordinary}} \times (1 + r)
\]

We already calculated the present value for the ordinary annuity in part (b). So, multiply that value by $ (1 + r) $, where $r = 0.02$ (quarterly interest rate), to account for payments made at the beginning of each period.

---

Let me calculate each of these values for you.Here are the results for each part of the problem:

### (a) The present value the insurance company must set aside for the $6,000 payments every six months at 6% interest compounded semiannually is:
\[
PV \approx 107,261.05
\]

### (b) The present value the insurance company must set aside for the $3,000 payments every three months at 8% interest compounded quarterly is:
\[
PV \approx 96,434.85
\]

### (c) If the $3,000 payments were made at the beginning of each 3-month period rather than at the end, the present value the insurance company must set aside is:
\[
PV_{\text{due}} \approx 98,363.55
\]

If you have any questions or want further details, feel free to ask!

---

Here are 5 related questions you might find helpful:
1. How would the present value change if the interest rate was compounded monthly instead of semiannually or quarterly?
2. What would be the impact of increasing the interest rate on the present value of the annuity?
3. How can you calculate future value instead of present value in these scenarios?
4. What happens if the number of years is shortened to 10 years for both scenarios?
5. How would a one-time lump sum compare to the structured settlement over time?

**Tip:** The more frequently interest is compounded, the higher the present value of the annuity due compared to an ordinary annuity.

Apollo Enterprises has been awarded an insurance settlement of $6,000 at the end of each 6 month period for the next 13 years. (a) As the accountant, calculate how much (in $) the insurance company must set aside now at 6% interest compounded semiannually to pay this obligation to Apollo. (b) How much (in $) would the insurance company have to invest now if the Apollo settlement was changed to $3,000 at the end of each 3 month period for 13 years and the insurance company earned 8% interest compounded quarterly? (c) How much (in $) would the insurance company have to invest now if the Apollo settlement was paid at the beginning of each 3 month period rather than at the end?

Solve Apollo Enterprises' insurance settlement problem by calculating the present value of $6,000 semiannual payments and $3,000 quarterly payments using compound interest at 6% and 8%, respectively. Learn how to compute annuities due with detailed solutions. Suitable for college-level finance students.

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