The market price of a semi-annual pay bond is $970.83. It has 22.00 years to maturity and a yield to maturity of 7.47%. What is the coupon rate?



To find the coupon rate of a bond, we can use the following bond pricing formula:

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

Where:
- $ P $ = Price of the bond = $970.83
- $ C $ = Semi-annual coupon payment
- $ r $ = Yield per period (semi-annual yield to maturity)
- $ n $ = Total number of periods (number of semi-annual periods to maturity)
- $ F $ = Face value of the bond (assumed to be $1,000)

We need to solve for $ C $, which will allow us to find the coupon rate.

### Step-by-step breakdown:

1. **Yield per period**: Since this is a semi-annual bond, we divide the yield to maturity (YTM) by 2.
\[
r = \frac{7.47\%}{2} = 3.735\% = 0.03735
\]

2. **Number of periods**: Since the bond pays semi-annually and has 22 years to maturity, the number of periods is:
\[
n = 22 \times 2 = 44 \text{ periods}
\]

3. **Bond pricing formula**:
We use the bond price formula, rearranged to isolate $ C $. Given the bond's current price and the yield to maturity, we can solve for $ C $, then calculate the annual coupon rate.

We will solve this equation numerically since it involves multiple terms. Let me compute the coupon payment, then convert it to the coupon rate.The annual coupon rate of the bond is approximately **7.20%**.

Would you like further details on how this was calculated, or do you have any additional questions?

Here are 5 related questions for deeper understanding:
1. How does the yield to maturity affect bond pricing?
2. What is the relationship between bond price and coupon rate?
3. How would the bond's price change if the yield to maturity increased?
4. What is the difference between nominal yield and yield to maturity?
5. How are semi-annual coupon payments different from annual payments?

**Tip:** A bond’s price and its yield to maturity (YTM) are inversely related. When YTM goes up, the bond price goes down, and vice versa.

The market price of a semi-annual pay bond is $970.83. It has 22.00 years to maturity and a yield to maturity of 7.47%. What is the coupon rate?

Learn how to calculate the coupon rate of a semi-annual pay bond priced at $970.83 with a yield to maturity of 7.47% and 22 years to maturity. This step-by-step solution covers bond pricing formulas, yield per period, and more. Suitable for finance students and advanced high school learners.

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