You purchased a newly issued 7% 40-year $1,000 bond 8 years ago.  The bond pays interest each six months.  You incur financial difficulties and are forced to sell the bond.  You  have just received the 16th semiannual interest payment.  The prevailing rate on similar bonds is 8.75%.  What is your $1,000 bond worth today?

To determine the current value of your bond, we need to calculate the present value of the remaining cash flows from the bond. These cash flows consist of the remaining interest payments and the principal repayment at the end of the bond's term. Given that the prevailing market rate is now 8.75%, we will use this rate to discount the future cash flows.

Here are the steps:

1. **Determine the remaining interest payments and principal repayment:**
   - The bond pays 7% annually, which means it pays 3.5% semiannually (7% / 2).
   - The bond's face value is $1,000, so each semiannual interest payment is $ \$1,000 \times 3.5\% = \$35 $.
   - The bond has 40 years to maturity, but you've held it for 8 years, so there are 32 years remaining.
   - Since interest is paid semiannually, there are $ 32 \text{ years} \times 2 = 64 $ remaining semiannual payments.

2. **Calculate the present value of the remaining interest payments:**
   - The prevailing semiannual market rate is $ 8.75\% / 2 = 4.375\% $.
   - The present value of an annuity (the interest payments) can be calculated using the formula:
     \[
     PV_{\text{interest}} = PMT \times \left( 1 - (1 + r)^{-n} \right) / r
     \]
     where:
     - $ PMT = \$35 $ (semiannual interest payment)
     - $ r = 0.04375 $ (semiannual market rate)
     - $ n = 64 $ (number of remaining semiannual periods)

3. **Calculate the present value of the principal repayment:**
   - The present value of a single future amount (the principal) can be calculated using the formula:
     \[
     PV_{\text{principal}} = F / (1 + r)^n
     \]
     where:
     - $ F = \$1,000 $ (face value of the bond)
     - $ r = 0.04375 $ (semiannual market rate)
     - $ n = 64 $ (number of remaining semiannual periods)

Let's perform the calculations:

### Step 2: Present Value of Interest Payments
\[
PV_{\text{interest}} = 35 \times \left( 1 - (1 + 0.04375)^{-64} \right) / 0.04375
\]

### Step 3: Present Value of Principal Repayment
\[
PV_{\text{principal}} = 1000 / (1 + 0.04375)^{64}
\]

### Calculations:
First, calculate $(1 + 0.04375)^{-64}$:
\[
(1 + 0.04375)^{-64} = (1.04375)^{-64} \approx 0.046946
\]

Now, calculate the present value of the interest payments:
\[
PV_{\text{interest}} = 35 \times \left( 1 - 0.046946 \right) / 0.04375 \approx 35 \times 21.8353 \approx 764.24
\]

Next, calculate the present value of the principal repayment:
\[
PV_{\text{principal}} = 1000 \times 0.046946 \approx 46.95
\]

Finally, add the two present values to get the bond's current value:
\[
PV_{\text{bond}} = PV_{\text{interest}} + PV_{\text{principal}} \approx 764.24 + 46.95 \approx 811.19
\]

Therefore, the current value of your $1,000 bond, given the prevailing market rate of 8.75%, is approximately **$811.19**.

Would you like more details or have any questions? 

Here are 5 numbered questions you might consider next:
1. How do changes in market interest rates affect bond prices?
2. What are the differences between bonds with semiannual and annual interest payments?
3. How do you calculate the yield to maturity for a bond?
4. What are the risks associated with investing in long-term bonds?
5. How does inflation impact bond prices and yields?

**Tip:** When investing in bonds, always consider the current market interest rates and how they compare to the bond's coupon rate to understand the bond's potential price volatility.

Learn how to calculate the current value of a 7% 40-year bond with semiannual interest payments. This problem involves determining the present value of remaining cash flows using prevailing market interest rates. Suitable for advanced learners or financial professionals.

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