To determine the yield to maturity (YTM) at which an investor would prefer a taxable bond over the tax-exempt municipal bond, we first need to calculate the YTM of the municipal bond.
Step 1: Calculate the Yield to Maturity (YTM) of the Municipal Bond
Given:
- Coupon Rate: 5.00% (annual)
- Coupon Payment: \( \frac{5% \cdot 1000}{2} = 25 \) (semi-annual payment)
- Market Price: 99.08% of par = \( 0.9908 \times 1000 = 990.80 \)
- Maturity: 7 years (or 14 semi-annual periods)
Now, we need to solve for the YTM using the following equation for a bond:
\[ P = \sum \left( \frac{C}{(1 + YTM/2)^t} \right) + \frac{F}{(1 + YTM/2)^N} \]
Where:
- \( P \) = price of the bond = 990.80
- \( C \) = coupon payment = 25
- \( F \) = face value of the bond = 1000
- \( N \) = total number of payments = 14
- \( t \) = each period from 1 to 14
This equation cannot be solved algebraically and requires numerical methods or a financial calculator.
Using a financial calculator or Excel, finding the YTM typically gives:
- YTM (semi-annually) ≈ 5.08%
To find the annualized YTM:
\[ YTM = 2 \cdot \text{semi-annual YTM} \approx 2 \cdot 0.0508 = 0.1016 \text{ or } 10.16% \]
Step 2: Compare with Taxable Bond Yield
Next, to compare against a taxable bond, we need to adjust the municipal bond yield considering the investor's tax rate of 38.00%:
We use the formula for converting the municipal bond's YTM to an equivalent taxable yield:
\[ \text{Taxable Yield} = \frac{\text{Tax-Exempt Yield}}{1 - \text{Tax Rate}} \]
Substituting the values:
\[ \text{Taxable Yield} = \frac{0.0508}{1 - 0.38} \approx \frac{0.0508}{0.62} \approx 0.08194 \text{ or } 8.19% \]
Answer
Therefore, the investor would prefer an otherwise identical taxable bond if its yield to maturity was more than 8.19%.