A taxable bond with a coupon rate of 5.00% has a market price of 98.45% of par. The bond matures in 20.00 years ans pays semi-annually. Assume an investor has a 23.00% marginal tax rate. The investor would prefer otherwise identical tax-exempt bond if it's yield to maturity was more than _____%

Answer 1

To find out the yield to maturity (YTM) of the taxable bond and then compare it to the yield of a tax-exempt bond, we can use the following steps:

1. Calculate the Yield to Maturity (YTM) of the Taxable Bond:

   • The bond's coupon payment (C) can be calculated as: \[ C = \frac{5% \times 1000}{2} = 25 \]
   • The bond is currently priced at 98.45% of par value, which means its price (P) is: \[ P = 0.9845 \times 1000 = 984.50 \]
   • The bond matures in 20 years, which means it has 40 periods of payments (since it pays semi-annually).

   We can use a financial calculator or a spreadsheet to solve for YTM, but for illustrative purposes, we can use an approximation formula for YTM: \[ \text{YTM} \approx \left( \frac{C + \frac{F - P}{n}}{(F + P) / 2} \right) \] where:

   • \( C = 25 \) is the coupon payment,
   • \( F = 1000 \) is the face value,
   • \( P = 984.50 \) is the current market price,
   • \( n = 40 \) is the total number of periods.

   Plugging in values: \[ \text{YTM} \approx \left( \frac{25 + \frac{1000 - 984.50}{40}}{(1000 + 984.50) / 2} \right) \]

   First calculate the terms:

   • \( \frac{1000 - 984.50}{40} = \frac{15.50}{40} = 0.3875 \).
   • So now the YTM formula becomes: \[ \text{YTM} \approx \left( \frac{25 + 0.3875}{(1000 + 984.50) / 2} \right) \]
   • Calculate the average price:
   • \( (1000 + 984.50) / 2 = 992.25 \).

   Now calculate the YTM: \[ \text{YTM} \approx \frac{25 + 0.3875}{992.25} \approx \frac{25.3875}{992.25} \approx 0.0256 \quad \text{or} \quad 2.56% \] Since this might come out as annual yield, to find the semi-annual yield, we consider it in its semi-annual format and then multiply by 2.

2. Adjust for Taxes: The after-tax yield on the taxable bond can be calculated by: \[ \text{After-tax YTM} = \text{YTM} \times (1 - \text{Tax Rate}) \]

   • Given the tax rate of 23%, we have: \[ \text{After-tax YTM} = 0.0256 \times (1 - 0.23) = 0.0256 \times 0.77 \approx 0.0197 \quad \text{or} \quad 1.97% \]

3. Determine Required Yield for Tax-Exempt Bond: Set the after-tax yield of the taxable bond equal to the yield of the tax-exempt bond: \[ \text{Tax-exempt yield} > 1.97% \]

Therefore, the investor would prefer otherwise identical tax-exempt bond if its yield to maturity was more than 1.97%.

Answer 2

What value do I enter for N, i/y, PV, PMT and FV to solve on a finance calculator?

Answer 3

To solve for the yield to maturity (YTM) of the taxable bond using a financial calculator, you'll need to input the values for \( N \), \( i/y \), \( PV \), \( PMT \), and \( FV \) correctly. Here’s how to enter the values:

• N (Number of periods): Since the bond matures in 20 years and pays semi-annually, the total number of periods (\( N \)) is: \[ N = 20 \times 2 = 40 \]

• i/y (Interest rate per period): This is what we are solving for (YTM), so you leave it blank or enter a temporary guess.

• PV (Present Value, or current price of the bond): The present value is the market price of the bond, which you found to be 984.50. Note that this value should be entered as a negative number since it represents an outflow (you’re paying this amount to buy the bond): \[ PV = -984.50 \]

• PMT (Periodic payment or coupon payment): The bond has a coupon rate of 5.00%, which means it pays \( 25 \) every six months: \[ PMT = 25 \]

• FV (Future Value, or face value of the bond): The face value of the bond is: \[ FV = 1000 \]

With these values, here is a summary of what you would enter into a financial calculator:

• \( N = 40 \)
• \( PV = -984.50 \)
• \( PMT = 25 \)
• \( FV = 1000 \)
• \( i/y = ?\) (solve for this)

After entering these values, press the button to compute \( i/y \) (which represents the yield to maturity). The calculated \( i/y \) will give you the semi-annual yield, which can be further multiplied by 2 if you'd like to express it as an annual yield.

Remember that this yield will be before tax, which you can then adjust with the marginal tax rate if needed, as discussed earlier.