A $1,000 bond with semiannual coupons is redeemable for $1,100 in ten years. It has a nominal coupon rate of 6%. The bond is purchased to yield 8% per annumcompounded semiannually. Find the amount for accumulation of discount in the fifth coupon.

To solve the problem of finding the accumulation of discount in the fifth coupon of the bond, we need to first break down the bond pricing, coupon payments, and how the yield affects the discount.

### Problem Breakdown:
- **Face Value of Bond:** $1,000
- **Redemption Value:** $1,100
- **Coupon Rate:** 6% nominal per annum (with semiannual payments)
- **Yield Rate:** 8% per annum compounded semiannually
- **Term:** 10 years

### Steps to Solve:

1. **Calculate the semiannual coupon payment:**
   \[
   C = \frac{6\% \times 1000}{2} = \frac{60}{2} = 30
   \]
   So, the semiannual coupon payment is $30.

2. **Determine the number of periods:**
   Since the bond pays semiannually for 10 years, the total number of periods is:
   \[
   n = 10 \times 2 = 20
   \]

3. **Calculate the semiannual yield rate:**
   The yield is 8% per annum compounded semiannually, so the semiannual yield rate is:
   \[
   i = \frac{8\%}{2} = 4\% = 0.04
   \]

4. **Calculate the price of the bond:**
   The price $ P $ of the bond can be calculated using the formula:
   \[
   P = C \times \left( \frac{1 - (1+i)^{-n}}{i} \right) + \frac{R}{(1+i)^n}
   \]
   where $ R $ is the redemption value.

   Plugging in the values:
   \[
   P = 30 \times \left( \frac{1 - (1+0.04)^{-20}}{0.04} \right) + \frac{1100}{(1+0.04)^{20}}
   \]

   Calculating these:
   \[
   P = 30 \times 13.5903 + \frac{1100}{2.1911} \approx 407.709 + 502.08 = 909.789
   \]

5. **Calculate the amount of discount in the fifth coupon:**
   The discount in each coupon is the difference between the present value of the remaining coupons (plus redemption value) before and after the coupon payment.

   For the fifth coupon, we compute the present value of the remaining cash flows just before and after the fifth coupon.

   **Just before the fifth coupon:**
   Present value of 16 remaining coupons plus redemption:
   \[
   PV_{before} = 30 \times \left( \frac{1 - (1+0.04)^{-16}}{0.04} \right) + \frac{1100}{(1+0.04)^{16}} = 30 \times 12.8493 + 502.08 = 885.479
   \]

   **Just after the fifth coupon:**
   \[
   PV_{after} = PV_{before} - 30 = 885.479 - 30 = 855.479
   \]

   The **accumulation of discount** in the fifth coupon is:
   \[
   \text{Accumulation of discount} = 885.479 - 855.479 = 30
   \]

Therefore, the accumulation of discount in the fifth coupon is approximately **$30**.

Would you like further details or have any questions?

Here are 5 related questions:
1. How would the bond price change if the yield increased to 10%?
2. How do changes in the redemption value impact the bond's price?
3. What is the accumulated value of all coupons by the end of the bond's term?
4. How do semiannual compounding and annual compounding affect the yield calculation?
5. Can you derive a formula for the accumulation of discount in any coupon number?

**Tip:** Understanding how yield and coupon rates interact helps in assessing bond value under varying market conditions.

Learn how to calculate the accumulation of discount in the fifth coupon of a $1,000 bond with a 6% nominal coupon rate and an 8% per annum yield compounded semiannually over a 10-year term. Understand the impact of coupon payments and yield rates on bond valuation.

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