Question
A tax-exempt municipal bond with a coupon rate of 5.00% has a market price of 99.08% of par. The bond matures in 7.00 years and pays semi-annually. Assume an investor has a 38.00% marginal tax rate. The investor would prefer otherwise identical taxable bond if it's yield to maturity was more than _____%
Answers
GPT-4o mini
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%**.
### 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%**.
123
what value do i enter for N, i/y, PV, PMT and FV to solve on a finance calculator?
GPT-4o mini
To use a financial calculator to solve for the yield to maturity (YTM) of a municipal bond, you need to correctly define the values for each parameter. Here's how you would set up the parameters based on the provided information:
1. **N (Number of Periods)**: This is the total number of coupon payments until maturity.
- Given the bond matures in 7 years and pays semi-annually, you would enter \( N = 7 \times 2 = 14 \).
2. **I/Y (Interest Rate per Period)**: This is what you are solving for, so you will leave this blank or set it as \( 0 \) or use a guess.
3. **PV (Present Value)**: This is the current market price of the bond (entered as a negative value since it's a cash outflow).
- Given the bond price is 99.08% of par (assuming par value is $1,000), you will enter \( PV = -990.80 \).
4. **PMT (Payment)**: This is the coupon payment received each period.
- With a coupon rate of 5.00% and a par value of $1,000, the semi-annual coupon payment is \( PMT = \frac{5\% \cdot 1000}{2} = 25 \).
5. **FV (Future Value)**: This is the face value of the bond, which is received at maturity.
- You will enter \( FV = 1000 \).
### Summary of values entered into the financial calculator:
- **N**: \( 14 \) (semi-annual periods over 7 years)
- **I/Y**: Leave this to solve for YTM (or can set to 0 for some calculators)
- **PV**: \( -990.80 \) (current market price)
- **PMT**: \( 25 \) (coupon payment)
- **FV**: \( 1000 \) (face value)
After entering these values, you would compute the I/Y (YTM). The output will be the semi-annual yield, which you would then double to find the annual yield YTM.
1. **N (Number of Periods)**: This is the total number of coupon payments until maturity.
- Given the bond matures in 7 years and pays semi-annually, you would enter \( N = 7 \times 2 = 14 \).
2. **I/Y (Interest Rate per Period)**: This is what you are solving for, so you will leave this blank or set it as \( 0 \) or use a guess.
3. **PV (Present Value)**: This is the current market price of the bond (entered as a negative value since it's a cash outflow).
- Given the bond price is 99.08% of par (assuming par value is $1,000), you will enter \( PV = -990.80 \).
4. **PMT (Payment)**: This is the coupon payment received each period.
- With a coupon rate of 5.00% and a par value of $1,000, the semi-annual coupon payment is \( PMT = \frac{5\% \cdot 1000}{2} = 25 \).
5. **FV (Future Value)**: This is the face value of the bond, which is received at maturity.
- You will enter \( FV = 1000 \).
### Summary of values entered into the financial calculator:
- **N**: \( 14 \) (semi-annual periods over 7 years)
- **I/Y**: Leave this to solve for YTM (or can set to 0 for some calculators)
- **PV**: \( -990.80 \) (current market price)
- **PMT**: \( 25 \) (coupon payment)
- **FV**: \( 1000 \) (face value)
After entering these values, you would compute the I/Y (YTM). The output will be the semi-annual yield, which you would then double to find the annual yield YTM.