Doisneau 18​-year bonds have an annual coupon interest of 11 ​percent, make interest payments on a semiannual​ basis, and have a ​$1 comma 000 par value.  If the bonds are trading with a​ market's required yield to maturity of 13 ​percent, are these premium or discount​ bonds?  Explain your answer.  What is the price of the​ bonds?a.  If the bonds are trading with a yield to maturity of 13​%, then ​ (Select the best choice​ below.)
A.
the bonds should be selling at a discount because the​ bond's coupon rate is less than the yield to maturity of similar bonds.
B.
there is not enough information to judge the value of the bonds.
C.
the bonds should be selling at par because the​ bond's coupon rate is equal to the yield to maturity of similar bonds.
D.
the bonds should be selling at a premium because the​ bond's coupon rate is greater than the yield to maturity of similar bonds.

### Step 1: Understanding the Problem

We are analyzing 18-year bonds with an **annual coupon interest rate of 11%**, making **semiannual payments** and having a **$1,000 par value**. The **market's required yield to maturity (YTM) is 13%**, also compounded semiannually. We need to determine whether these bonds are selling at a **premium** or a **discount** and find their **price**.

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### Step 2: Concept of Premium vs. Discount Bonds

A bond can either sell at a **premium** or a **discount**, depending on how its coupon rate compares to the market's required YTM:
- If the **coupon rate** is **higher than the YTM**, the bond sells at a **premium** because it offers better returns than the market.
- If the **coupon rate** is **lower than the YTM**, the bond sells at a **discount** because it offers lower returns compared to what the market demands.
- If the coupon rate is equal to the YTM, the bond will sell **at par value**.

---

### Step 3: Comparing Coupon Rate and YTM

- The bond has a **coupon rate of 11%**.
- The **market YTM is 13%**.
  
Since the **coupon rate (11%) is lower** than the **YTM (13%)**, the bond offers a lower return than the market requires. Therefore, the bond should be selling at a **discount**.

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### Step 4: Calculating the Bond Price

To calculate the price of the bond, we use the **present value formula** for bond pricing. The price of a bond is the sum of the present value of its coupon payments and the present value of its par value, discounted at the market's required yield to maturity (YTM).

#### Bond Pricing Formula:
\[
\text{Price} = \left( \sum_{i=1}^{n} \frac{C}{(1 + r)^i} \right) + \frac{F}{(1 + r)^n}
\]
Where:
- $C$ = Coupon payment per period = $\frac{\text{Coupon Rate} \times \text{Par Value}}{\text{Periods per Year}}$
- $F$ = Face value (par value of the bond) = $1,000
- $r$ = Market yield per period = $\frac{\text{YTM}}{\text{Periods per Year}}$
- $n$ = Total number of periods = $\text{Years to Maturity} \times \text{Periods per Year}$

Let’s calculate the price.

- **Coupon payment per period**:
  - Annual coupon rate: 11%
  - Semiannual payment = $ \frac{11\% \times 1000}{2} = 55 $

- **Market yield per period**:
  - Annual YTM = 13%
  - Semiannual YTM = $ \frac{13\%}{2} = 6.5\% = 0.065 $

- **Number of periods**: 
  - 18 years × 2 periods per year = 36 periods

Now, let’s calculate the price using these values.

The price of the bond is approximately **$862.09**.

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### Step 5: Final Answer to the Questions

- The bonds are trading at a **discount** because the **coupon rate (11%)** is **lower** than the **market's required yield to maturity (13%)**.
- The price of the bond is approximately **$862.09**.

### Best Choice:  
**A.** The bonds should be selling at a discount because the bond's coupon rate is less than the yield to maturity of similar bonds.

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Do you want more details or have any questions?

### Follow-up Questions:
1. How does increasing the coupon rate affect the bond's price?
2. What happens to the bond's price if the YTM decreases to 10%?
3. How do changes in interest rates in the market influence bond prices?
4. What is the relationship between bond price and duration?
5. How would the bond price change if payments were made annually instead of semiannually?

### Tip:  
The further a bond is from its maturity date, the more sensitive its price is to changes in interest rates.

Doisneau 18​-year bonds have an annual coupon interest of 11 ​percent, make interest payments on a semiannual​ basis, and have a ​$1,000 par value. If the bonds are trading with a​ market's required yield to maturity of 13 ​percent, are these premium or discount​ bonds? Explain your answer. What is the price of the​ bonds?

Learn how to determine whether an 18-year bond with an 11% coupon and 13% yield to maturity is selling at a discount or premium. Explore key concepts like bond pricing, yield to maturity, and the present value of future payments. This problem is suitable for undergraduate finance students.

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