Suppose a seven-year, $1000 bond with 7.5% coupon rate and semiannual coupons is trading with a yield to maturity of 8.21%.

a. (3 points) Is the bond currently trading at a discount, at par, or at a premium? Explain.

b. (5 points) If the yield to maturity of the bond rises to 8.7% (APR with semiannual compounding), what price will the bond trade for?

Let's address both parts of your question one by one.

### Part a: Is the bond trading at a discount, at par, or at a premium?

To determine whether the bond is trading at a discount, at par, or at a premium, we need to compare the bond's coupon rate with its yield to maturity (YTM):

- **Coupon rate**: 7.5% (annual rate, meaning 3.75% per semiannual period).
- **Yield to Maturity (YTM)**: 8.21% (annual yield, meaning 4.105% per semiannual period).

Since the YTM (8.21%) is higher than the coupon rate (7.5%), the bond must be trading at a **discount**. This happens because investors are demanding a higher return than the bond's coupon rate provides, which drives the bond price below its face value.

### Conclusion:  
The bond is trading at a **discount** because the YTM is higher than the coupon rate.

---

### Part b: Price of the bond if the YTM rises to 8.7%

The bond price is the present value of its future cash flows, which includes:

1. The semiannual coupon payments.
2. The face value at maturity.

We can use the formula for the price of a bond, which is the sum of the present value of the coupon payments and the present value of the face value (par value) at maturity:

\[
P = \sum_{t=1}^{2n} \frac{C}{(1 + y)^t} + \frac{F}{(1 + y)^{2n}}
\]

Where:
- $ P $ is the price of the bond.
- $ C $ is the semiannual coupon payment.
- $ F $ is the face value of the bond ($1000).
- $ y $ is the semiannual YTM (8.7% annual means 4.35% semiannual).
- $ n $ is the number of years to maturity (7 years).

Let's calculate the values step by step:

1. **Semiannual coupon payment**:  
   \[
   C = \frac{7.5\% \times 1000}{2} = 37.50
   \]

2. **Semiannual YTM**:  
   \[
   y = \frac{8.7\%}{2} = 4.35\%
   \]

3. **Number of periods (since it's semiannual for 7 years)**:  
   \[
   2n = 7 \times 2 = 14
   \]

We will now calculate the present value of the coupon payments and the face value:

- **Present Value of Coupons (PVC)**:  
  This is the present value of the annuity of the semiannual coupon payments:
  \[
  PVC = 37.50 \times \sum_{t=1}^{14} \frac{1}{(1 + 0.0435)^t}
  \]

- **Present Value of Face Value (PVF)**:  
  This is the present value of the $1000 to be received at maturity:
  \[
  PVF = \frac{1000}{(1 + 0.0435)^{14}}
  \]

Now, calculating each term:

\[
PVC = 37.50 \times \left( \frac{1 - (1 + 0.0435)^{-14}}{0.0435} \right) \approx 37.50 \times 10.034 = 376.28
\]
\[
PVF = \frac{1000}{(1 + 0.0435)^{14}} \approx \frac{1000}{1.84227} \approx 542.80
\]

### Bond Price Calculation:
\[
P = 376.28 + 542.80 = 919.08
\]

### Conclusion:
If the yield to maturity rises to 8.7%, the bond will trade at approximately **$919.08**.

---

### Summary:
- **Part a**: The bond is trading at a discount.
- **Part b**: If the YTM rises to 8.7%, the bond will trade at approximately **$919.08**.

Do you have any questions or need further details on these calculations?

### Related Questions:
1. What would the bond price be if the YTM fell to 7%?
2. How does the duration of the bond affect its price sensitivity to changes in YTM?
3. What would the bond price be if the coupons were paid annually instead of semiannually?
4. How would the bond price change if the time to maturity was only 3 years instead of 7 years?
5. How does inflation impact bond prices and yields?

### Tip:
As interest rates (YTM) increase, bond prices fall. This inverse relationship is crucial for understanding bond investments!

Suppose a seven-year, $1000 bond with 7.5% coupon rate and semiannual coupons is trading with a yield to maturity of 8.21%. a. Is the bond currently trading at a discount, at par, or at a premium? b. If the yield to maturity of the bond rises to 8.7%, what price will the bond trade for?

Explore how to determine whether a bond is trading at a discount, par, or premium based on its coupon rate and yield to maturity (YTM). This problem also demonstrates how to calculate the bond's price if the YTM rises to 8.7%, using present value formulas. Suitable for undergraduate finance students.

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