What is the coupon rate of an eight-year, $10,000 bond with semiannual coupons and a price of $9006, if it has a yield to maturity of 5.6%?

Question 11Answer

a.
4.04%

b.
3.91%

c.
5.87%

d.
4.89%

To find the coupon rate of the bond, we can follow these steps:

### Step 1: Identify the given variables
- **Bond price (P):** $9,006
- **Par value (F):** $10,000
- **Time to maturity (T):** 8 years
- **Yield to maturity (YTM):** 5.6% per year, or 0.056 (with semiannual payments, YTM per period = 0.056/2 = 0.028)
- **Number of periods (n):** Since the bond has semiannual coupons and matures in 8 years, there will be $ 8 \times 2 = 16 $ periods.
- **Coupon payment (C):** This is what we are trying to find. The coupon rate is $ \frac{C}{F} \times 100 $.

### Step 2: Use the bond price formula
The bond price formula for a bond with semiannual coupon payments is:

\[
P = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) + F \times (1 + r)^{-n}
\]

Where:
- $ P $ is the bond price,
- $ C $ is the coupon payment,
- $ F $ is the face value,
- $ r $ is the yield per period (YTM / 2),
- $ n $ is the number of periods.

### Step 3: Substitute the known values
We can substitute the known values into the formula:

\[
9,006 = C \times \left( \frac{1 - (1 + 0.028)^{-16}}{0.028} \right) + 10,000 \times (1 + 0.028)^{-16}
\]

### Step 4: Solve for $ C $
I'll now calculate the value of $ C $.# Given variables
P = 9006  # bond price
F = 10000  # face value
r = 0.056 / 2  # yield per period (semiannual YTM)
n = 8 * 2  # number of periods (semiannual)

# Solve for coupon payment C using the bond price formula
# P = C * [(1 - (1 + r)^(-n)) / r] + F * (1 + r)^(-n)

import sympy as sp

# Define the coupon payment C as a symbol
C = sp.Symbol('C')

# Bond price formula equation
bond_price_eq = sp.Eq(C * ((1 - (1 + r)**(-n)) / r) + F * (1 + r)**(-n), P)

# Solve for C
coupon_payment = sp.solve(bond_price_eq, C)[0]
coupon_paymentThe semiannual coupon payment $ C $ is approximately \$202.07.

### Step 5: Calculate the coupon rate
To find the annual coupon rate, we use the formula:

\[
\text{Coupon rate} = \frac{C \times 2}{F} \times 100
\]

Substituting the values:

\[
\text{Coupon rate} = \frac{202.07 \times 2}{10,000} \times 100 \approx 4.04\%
\]

Thus, the coupon rate is approximately **4.04%**.

The correct answer is **(a) 4.04%**.

Would you like more details on this? Here are some related questions:

1. How does the yield to maturity affect bond pricing?
2. What is the difference between coupon rate and yield to maturity?
3. How do semiannual coupons change bond calculations?
4. What happens to a bond's price when interest rates change?
5. Can a bond have a negative yield?

**Tip:** The bond price decreases when the yield to maturity is higher than the coupon rate.

Learn how to calculate the coupon rate of an eight-year $10,000 bond with semiannual coupons, priced at $9,006 and a yield to maturity of 5.6%. Understand the bond price formula, semiannual coupon payments, and the relationship between bond price, yield, and coupon rate.

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