Question
The market price of a semi-annual pay bond is $973.90. It has 16.00 years to maturity and a yield to maturity of 7.20%. What is the coupon rate?
Answers
GPT-4o mini
To determine the coupon rate of the bond, we can use the present value formula for bonds. The price of a bond is the present value of its future cash flows, which consist of the annual coupon payments and the principal repayment at maturity.
Given:
- Market Price (P) = $973.90
- Time to Maturity (N) = 16 years
- Yield to Maturity (YTM) = 7.20% (this is an annual rate, but since it pays semi-annually, we will divide this by 2)
- Number of Payments (n) = 2 × N = 32 payments
To find the semi-annual yield:
\[
\text{Semi-Annual Yield} = \frac{YTM}{2} = \frac{7.20\%}{2} = 3.60\%
\]
\[
\text{Semi-Annual Yield in Decimal} = \frac{3.60}{100} = 0.036
\]
Let \(C\) be the semi-annual coupon payment and \(F\) be the face value of the bond (typically assumed to be $1000 unless specified otherwise).
The price of a bond can be expressed as follows:
\[
P = C \left( \frac{1 - (1 + r)^{-n}}{r} \right) + \frac{F}{(1 + r)^{n}}
\]
Where:
- \(P\) is the price of the bond
- \(C\) is the coupon payment
- \(r\) is the semi-annual yield (0.036)
- \(n\) is the total number of payments (32, in our case)
- \(F\) is the face value of the bond (assumed to be $1000)
Let’s assume \(F = 1000\) to simplify.
Rearranging the formula:
\[
973.90 = C \left( \frac{1 - (1 + 0.036)^{-32}}{0.036} \right) + \frac{1000}{(1 + 0.036)^{32}}
\]
Calculating \( (1 + 0.036)^{-32} \) and \( (1 + 0.036)^{32} \):
- \( (1 + 0.036)^{32} = (1.036)^{32} \approx 3.29064 \)
- \( (1 + 0.036)^{-32} \approx \frac{1}{3.29064} \approx 0.3047 \)
Now calculating:
\[
\frac{1000}{(1 + 0.036)^{32}} \approx \frac{1000}{3.29064} \approx 304.74
\]
Now we can set up the equation to calculate \(C\):
\[
973.90 = C \left( \frac{1 - 0.3047}{0.036} \right) + 304.74
\]
\[
\frac{1 - 0.3047}{0.036} = \frac{0.6953}{0.036} \approx 19.34
\]
Substituting this back into the equation gives:
\[
973.90 = C \times 19.34 + 304.74
\]
Solving for \(C\):
\[
973.90 - 304.74 = C \times 19.34
\]
\[
669.16 = C \times 19.34
\]
\[
C = \frac{669.16}{19.34} \approx 34.54
\]
Now that we have the semi-annual coupon payment \(C \approx 34.54\), we can calculate the coupon rate:
\[
\text{Coupon Rate} = \frac{C}{F} \times 2 = \frac{34.54}{1000} \times 2 \approx 0.06908 \text{ or } 6.91\%
\]
Thus, the coupon rate of the bond is approximately **6.91%**.
Given:
- Market Price (P) = $973.90
- Time to Maturity (N) = 16 years
- Yield to Maturity (YTM) = 7.20% (this is an annual rate, but since it pays semi-annually, we will divide this by 2)
- Number of Payments (n) = 2 × N = 32 payments
To find the semi-annual yield:
\[
\text{Semi-Annual Yield} = \frac{YTM}{2} = \frac{7.20\%}{2} = 3.60\%
\]
\[
\text{Semi-Annual Yield in Decimal} = \frac{3.60}{100} = 0.036
\]
Let \(C\) be the semi-annual coupon payment and \(F\) be the face value of the bond (typically assumed to be $1000 unless specified otherwise).
The price of a bond can be expressed as follows:
\[
P = C \left( \frac{1 - (1 + r)^{-n}}{r} \right) + \frac{F}{(1 + r)^{n}}
\]
Where:
- \(P\) is the price of the bond
- \(C\) is the coupon payment
- \(r\) is the semi-annual yield (0.036)
- \(n\) is the total number of payments (32, in our case)
- \(F\) is the face value of the bond (assumed to be $1000)
Let’s assume \(F = 1000\) to simplify.
Rearranging the formula:
\[
973.90 = C \left( \frac{1 - (1 + 0.036)^{-32}}{0.036} \right) + \frac{1000}{(1 + 0.036)^{32}}
\]
Calculating \( (1 + 0.036)^{-32} \) and \( (1 + 0.036)^{32} \):
- \( (1 + 0.036)^{32} = (1.036)^{32} \approx 3.29064 \)
- \( (1 + 0.036)^{-32} \approx \frac{1}{3.29064} \approx 0.3047 \)
Now calculating:
\[
\frac{1000}{(1 + 0.036)^{32}} \approx \frac{1000}{3.29064} \approx 304.74
\]
Now we can set up the equation to calculate \(C\):
\[
973.90 = C \left( \frac{1 - 0.3047}{0.036} \right) + 304.74
\]
\[
\frac{1 - 0.3047}{0.036} = \frac{0.6953}{0.036} \approx 19.34
\]
Substituting this back into the equation gives:
\[
973.90 = C \times 19.34 + 304.74
\]
Solving for \(C\):
\[
973.90 - 304.74 = C \times 19.34
\]
\[
669.16 = C \times 19.34
\]
\[
C = \frac{669.16}{19.34} \approx 34.54
\]
Now that we have the semi-annual coupon payment \(C \approx 34.54\), we can calculate the coupon rate:
\[
\text{Coupon Rate} = \frac{C}{F} \times 2 = \frac{34.54}{1000} \times 2 \approx 0.06908 \text{ or } 6.91\%
\]
Thus, the coupon rate of the bond is approximately **6.91%**.