Question
The market price of a semi-annual pay bond is $994.96. It has 17.00 years to maturity and a coupon rate of 6.00%. Par value is $1,000. What is the effective annual yield?
Answers
GPT-4o mini
To calculate the effective annual yield (EAY) of the bond, we first need to determine several key components, including the semi-annual coupon payment, the number of periods until maturity, and the yield to maturity (YTM).
1. **Calculate the semi-annual coupon payment:**
- The bond has a par value of $1,000 and a coupon rate of 6.00%.
- Semi-annual coupon payment = \( \frac{6\% \times 1000}{2} = \frac{60}{2} = 30 \).
2. **Determine the number of periods until maturity:**
- The bond has 17.00 years to maturity and pays interest semi-annually.
- Total number of periods = \( 17 \times 2 = 34 \).
3. **Set up the cash flow equation for YTM:**
The bond price can be expressed with the following formula:
\[
P = \sum_{t=1}^{n} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^n}
\]
Where:
- \( P \) = market price of the bond = $994.96,
- \( C \) = semi-annual coupon payment = $30,
- \( F \) = face value of the bond = $1,000,
- \( n \) = total number of periods = 34,
- \( r \) = semi-annual YTM (yield per period).
This is an equation we need to solve for \( r \).
4. **Using trial and error or a financial calculator:**
You might typically use a financial calculator or Excel to find \( r \). For simplicity, let’s start with estimating YTM using bond formulas and approximation.
A good approximation for YTM, especially using \( P, C,\) and \( F \), can be:
\[
r \approx \frac{C + \frac{F - P}{n}}{\frac{F + P}{2}}
\]
Plugging in the values:
\[
r \approx \frac{30 + \frac{1000 - 994.96}{34}}{\frac{1000 + 994.96}{2}} = \frac{30 + \frac{5.04}{34}}{\frac{1994.96}{2}} = \frac{30 + 0.14824}{997.48} = \frac{30.14824}{997.48} \approx 0.03024 \text{ or } 3.024\%
\]
5. **Convert to effective annual yield (EAY):**
Since this is a semi-annual coupon bond, the effective annual yield can be calculated by:
\[
\text{EAY} = \left(1 + r\right)^2 - 1
\]
Substituting in our semi-annual yield:
\[
\text{EAY} = (1 + 0.03024)^2 - 1 \approx 1.03024^2 - 1 \approx 1.06168 - 1 \approx 0.06168
\]
Converting to percentage:
\[
\text{EAY} \approx 6.168\%
\]
**Conclusion:**
The effective annual yield (EAY) of the bond is approximately **6.17%**.
1. **Calculate the semi-annual coupon payment:**
- The bond has a par value of $1,000 and a coupon rate of 6.00%.
- Semi-annual coupon payment = \( \frac{6\% \times 1000}{2} = \frac{60}{2} = 30 \).
2. **Determine the number of periods until maturity:**
- The bond has 17.00 years to maturity and pays interest semi-annually.
- Total number of periods = \( 17 \times 2 = 34 \).
3. **Set up the cash flow equation for YTM:**
The bond price can be expressed with the following formula:
\[
P = \sum_{t=1}^{n} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^n}
\]
Where:
- \( P \) = market price of the bond = $994.96,
- \( C \) = semi-annual coupon payment = $30,
- \( F \) = face value of the bond = $1,000,
- \( n \) = total number of periods = 34,
- \( r \) = semi-annual YTM (yield per period).
This is an equation we need to solve for \( r \).
4. **Using trial and error or a financial calculator:**
You might typically use a financial calculator or Excel to find \( r \). For simplicity, let’s start with estimating YTM using bond formulas and approximation.
A good approximation for YTM, especially using \( P, C,\) and \( F \), can be:
\[
r \approx \frac{C + \frac{F - P}{n}}{\frac{F + P}{2}}
\]
Plugging in the values:
\[
r \approx \frac{30 + \frac{1000 - 994.96}{34}}{\frac{1000 + 994.96}{2}} = \frac{30 + \frac{5.04}{34}}{\frac{1994.96}{2}} = \frac{30 + 0.14824}{997.48} = \frac{30.14824}{997.48} \approx 0.03024 \text{ or } 3.024\%
\]
5. **Convert to effective annual yield (EAY):**
Since this is a semi-annual coupon bond, the effective annual yield can be calculated by:
\[
\text{EAY} = \left(1 + r\right)^2 - 1
\]
Substituting in our semi-annual yield:
\[
\text{EAY} = (1 + 0.03024)^2 - 1 \approx 1.03024^2 - 1 \approx 1.06168 - 1 \approx 0.06168
\]
Converting to percentage:
\[
\text{EAY} \approx 6.168\%
\]
**Conclusion:**
The effective annual yield (EAY) of the bond is approximately **6.17%**.