To calculate the firm's component cost of debt (before and after taxes) for purposes of calculating the WACC, we use the Yield to Maturity (YTM) formula, as the bond's current market price is different from its face value. The YTM represents the bond's pre-tax cost of debt. Since the company makes semi-annual payments, we'll adjust the coupon rate and time accordingly.

Given Information:

• Coupon Rate (Annual) = 9%
• Coupon Payment (Semi-annual) = $\frac{9\%}{2} \times 1000 = 45$
• Face Value (Par Value) = $1,000
• Bond Price (Current Market Price) = $689.86
• Years to Maturity = 20 years
• Semi-annual Periods = $20 \times 2 = 40$
• Tax Rate = 30%

We will use the following formula to estimate YTM iteratively or by approximation (since there is no closed-form solution for YTM):

$P = \sum_{t=1}^{N} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^N}$

Where:

• $P$ = Current bond price = $689.86
• $C$ = Semi-annual coupon payment = $45
• $F$ = Face value = $1,000
• $N$ = Total number of semi-annual periods = 40
• $r$ = Semi-annual yield (to be solved for)

Since the YTM is difficult to solve algebraically, we'll approximate it using a financial calculator or trial and error methods. For now, we'll estimate that the semi-annual YTM is around 7% (this is approximate based on standard formulas).

1. Pre-Tax Cost of Debt:

The YTM calculated above is on a semi-annual basis, so to convert it to an annual nominal YTM: $\text{Annual YTM} = 2 \times \text{Semi-annual YTM} = 2 \times 7\% = 14\%$

Thus, the pre-tax cost of debt is approximately 14%.

2. After-Tax Cost of Debt:

The after-tax cost of debt is calculated using the following formula: $\text{After-tax cost of debt} = \text{Pre-tax cost of debt} \times (1 - \text{Tax rate})$ $\text{After-tax cost of debt} = 14\% \times (1 - 0.30) = 14\% \times 0.70 = 9.8\%$

Thus, the firm's after-tax cost of debt is approximately 9.8%.

Summary:

• Pre-tax cost of debt (YTM): 14%
• After-tax cost of debt: 9.8%

Would you like me to dive deeper into how YTM is approximated or provide more details on the impact on WACC?

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Follow-up Questions:

1. How does the bond's price affect the firm's WACC?
2. How does the coupon rate influence the YTM calculation?
3. What is the relationship between tax rate and after-tax cost of debt?
4. Why is the cost of debt adjusted for taxes when calculating WACC?
5. How would a change in interest rates impact the bond's market price?

Tip: A bond’s price below its par value typically indicates that the YTM is higher than the coupon rate.