You find the following Treasury bond quotes. To calculate the number of years until maturity, assume that it is currently May 2022. All of the bonds have a par value of $1,000 and pay semiannual coupons.

Rate	Maturity Month/Year	Bid	Asked	Change	Ask Yield
??question mark	May 36	103.5462	103.6340	+.3015	2.329
6.252	May 41	104.4952	104.6409	+.4293	??question mark
6.163	May 51	??question mark	??question mark	+.5405	4.031
In the above table, find the Treasury bond that matures in May 2036. What is the coupon rate for this bond?

Note: Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.

To find the coupon rate for the Treasury bond that matures in May 2036, we can use the information provided in the table.

Here are the key steps to calculate the coupon rate:

### Step 1: Interpret the Information

- **Maturity**: May 2036
- **Bid**: 103.5462
- **Asked**: 103.6340
- **Ask Yield**: 2.329%
- **Par value**: $1,000
- **Price (based on Ask)**: 103.6340% of $1,000 = $1,036.34

Since the bond is trading above par (its price is greater than $1,000), we know the bond has a **coupon rate higher than the yield to maturity** (which is 2.329%).

### Step 2: Use the Yield-to-Maturity (YTM) Formula

The yield to maturity (YTM) can be approximated using this formula for bonds with semiannual payments:

\[
YTM \approx \frac{C + \frac{F - P}{T}}{\frac{F + P}{2}}
\]

Where:
- $ C $ = Annual coupon payment (which we want to find)
- $ F $ = Face value of the bond ($1,000)
- $ P $ = Price of the bond (ask price = $1,036.34)
- $ T $ = Time to maturity in years (2036 - 2022 = 14 years)
- YTM = Ask yield (2.329% as a decimal, 0.02329)

Since the bond pays semiannual coupons, we need to adjust for this, and the formula becomes:

\[
0.02329 = \frac{\frac{C}{2} + \frac{1,000 - 1,036.34}{14 \times 2}}{\frac{1,000 + 1,036.34}{2}}
\]

### Step 3: Solve for the Coupon Payment

Rearranging the formula to solve for $ C $ (the annual coupon payment):

1. Calculate the fraction part for the price and par value:

\[
\frac{1,000 - 1,036.34}{28} = \frac{-36.34}{28} = -1.2979
\]

2. Calculate the denominator (average of face value and price):

\[
\frac{1,000 + 1,036.34}{2} = 1,018.17
\]

3. Now plug these into the YTM formula:

\[
0.02329 = \frac{\frac{C}{2} - 1.2979}{1,018.17}
\]

Multiply both sides by 1,018.17:

\[
0.02329 \times 1,018.17 = \frac{C}{2} - 1.2979
\]

\[
23.692 = \frac{C}{2} - 1.2979
\]

Add 1.2979 to both sides:

\[
23.692 + 1.2979 = \frac{C}{2}
\]

\[
24.9899 = \frac{C}{2}
\]

Multiply both sides by 2 to solve for $ C $:

\[
C = 49.9798
\]

Thus, the **annual coupon payment** is approximately **$49.98**.

### Step 4: Calculate the Coupon Rate

Now, to find the coupon rate, divide the annual coupon payment by the par value of the bond:

\[
\text{Coupon Rate} = \frac{49.98}{1,000} = 0.04998
\]

Convert this to a percentage:

\[
\text{Coupon Rate} = 4.998\% \approx 5.00\%
\]

### Final Answer:
The **coupon rate** for the Treasury bond maturing in May 2036 is **5.00%**.

---

Would you like further details or have any questions?

Here are some related questions to expand on this topic:

1. How does the ask yield differ from the coupon rate of a bond?
2. What happens to bond prices when interest rates rise or fall?
3. Why is the bond trading above par value in this case?
4. How does the time to maturity affect the bond's sensitivity to interest rates?
5. What is the relationship between the bond’s YTM and the market price?

**Tip:** Remember, when a bond's price is higher than its face value, it typically means the coupon rate is higher than the market interest

Learn how to calculate the coupon rate for a Treasury bond maturing in May 2036. This problem covers key concepts in finance, such as bond pricing, yield to maturity (YTM), and coupon rate calculation. Detailed step-by-step instructions included, perfect for college-level finance students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation