Question 1196651: Occam Industrial Machines issued 225,000 zero coupon bonds five years ago. The bonds originally had 30 years to maturity with a yield to maturity of 5.2 percent. Interest rates have recently decreased, and the bonds now have a yield to maturity of 4.6 percent. The bonds have a par value of $1,000 and semiannual compounding. If the company has a market value of equity of $120 million, what weight should it use for debt when calculating the cost of capital?
Answer by proyaop(69)   (Show Source):
You can put this solution on YOUR website! To determine the weight of debt when calculating the cost of capital, we first need to calculate the market value of the bonds. Here's a step-by-step approach:
---
### **Step 1: Bond Valuation Formula**
The market value of a zero-coupon bond is calculated as:
\[
PV = \frac{FV}{(1 + r)^t}
\]
Where:
- \( PV \): Present value (market price of the bond)
- \( FV \): Face value of the bond (\( \$1,000 \))
- \( r \): Yield to maturity (per compounding period, semiannual)
- \( t \): Total number of compounding periods remaining
---
### **Step 2: Identify the Given Information**
- Number of bonds: \( 225,000 \)
- Face value per bond: \( \$1,000 \)
- Total face value: \( 225,000 \times 1,000 = \$225,000,000 \)
- Yield to maturity (current): \( 4.6\% \) annually, or \( 4.6\% / 2 = 2.3\% = 0.023 \) per semiannual period
- Years to maturity remaining: \( 30 - 5 = 25 \) years
- Total compounding periods remaining: \( 25 \times 2 = 50 \) periods
---
### **Step 3: Calculate the Market Price of One Bond**
\[
PV = \frac{1,000}{(1 + 0.023)^{50}}
\]
1. Calculate \( (1 + 0.023)^{50} \):
\[
(1 + 0.023)^{50} \approx 3.2094
\]
2. Calculate \( PV \) for one bond:
\[
PV = \frac{1,000}{3.2094} \approx 311.63
\]
---
### **Step 4: Calculate the Total Market Value of Debt**
The total market value of the bonds is:
\[
\text{Total Market Value} = 225,000 \times 311.63 \approx 69,116,750
\]
---
### **Step 5: Calculate the Weight of Debt**
The weight of debt is calculated as:
\[
\text{Weight of Debt} = \frac{\text{Market Value of Debt}}{\text{Market Value of Debt + Market Value of Equity}}
\]
Given:
- Market value of equity = \( \$120,000,000 \)
- Market value of debt = \( 69,116,750 \)
\[
\text{Weight of Debt} = \frac{69,116,750}{69,116,750 + 120,000,000} \approx \frac{69,116,750}{189,116,750} \approx 0.3654
\]
---
### **Final Answer**
The weight of debt to use when calculating the cost of capital is approximately:
\[
\boxed{36.54\%}
\]
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