You are buying right now a zero-coupon bond. It has exactly 8 years to maturity, and you expect the YTM to be 6% over the bond’s life. Even though you intended to hold it to maturity, you end up selling it after 3 years. By that time, the bond’s YTM has risen to 8% due to changes in market conditions.
Your tax rate is 40% on regular income, and 15% on capital gains.
a) What is your annual average after-tax return for this bond over the three-year holding period? (8 Pts) (Note: Please do the calculation exactly, not as an approximation)
b) What would have been the bond’s price today under the initial YTM assumptions if the time to maturity were to be 7 years and 200 days rather than 8 years as stated above. What would be the accrued interest to be added to the purchase invoice? (5 Pts) (Note: Assume a 360-day year)
c) The result of the calculation in b) is either higher or lower than the result in a). Tell me whether it is higher or lower, and explain to me why it is. (3 Pts)
An EXCEL spreadsheet is very useful for these types of calculations.
Assume you have a bond that pays $100 after 8 years. Calculate how much one would pay for such a bond at 1) t=0 when YTM=6%, 2) t=5 and YTM=6%, and 3)t=5 when YTM=8%. As I understand, the difference between 1) and 2) is treated as ordinary income, and between 2) and 3) as a capital gain. (I presume you have the formula for converting YTM into market prices). You should have all the parts needed to answer a)
Part b seems to repeat part a cept for a shorter maturity.
I'd like to know it as well...:)
To calculate the annual average after-tax return for the bond over the three-year holding period, we need to consider the tax implications of both the coupon payments and the capital gain/loss from selling the bond.
a) Here are the steps to calculate the annual average after-tax return:
Step 1: Determine the purchase price of the bond:
The purchase price of the bond can be calculated using the present value formula:
Purchase Price = Face Value / (1 + Yield to Maturity) ^ Years to Maturity
Face Value = $100 (given)
Yield to Maturity = 6% = 0.06 (given)
Years to Maturity = 8 (given)
Purchase Price = $100 / (1 + 0.06) ^ 8
Purchase Price ≈ $63.61
Step 2: Determine the selling price of the bond:
The selling price of the bond can be calculated using the present value formula, considering the change in yield to maturity:
Selling Price = Face Value / (1 + Yield to Maturity at Selling) ^ Remaining Years to Maturity
Yield to Maturity at Selling = 8% = 0.08 (given)
Remaining Years to Maturity = 8 - 3 = 5 (calculated)
Selling Price = $100 / (1 + 0.08) ^ 5
Selling Price ≈ $68.06
Step 3: Calculate the total coupon payments over the three-year holding period:
Since the bond is a zero-coupon bond, there are no periodic coupon payments. However, we need to consider the accrued interest.
Accrued Interest = Purchase Price * Yield to Maturity * Holding Period
Holding Period = 3 years (given)
Accrued Interest = $63.61 * 0.06 * 3
Accrued Interest ≈ $11.47
Step 4: Calculate the tax on coupon payments:
Tax on coupon payments = Accrued Interest * Tax Rate (Regular Income Tax Rate)
Tax Rate (Regular Income Tax Rate) = 40% (given)
Tax on coupon payments = $11.47 * 0.4
Tax on coupon payments ≈ $4.59
Step 5: Calculate the tax on capital gain/loss:
Capital Gain/Loss = Selling Price - Purchase Price - Accrued Interest
Capital Gain/Loss = $68.06 - $63.61 - $11.47
Capital Gain/Loss ≈ $-6.02 (negative as there is a loss)
Tax on capital gain/loss = Capital Gain/Loss * Tax Rate (Capital Gains Tax Rate)
Tax Rate (Capital Gains Tax Rate) = 15% (given)
Tax on capital gain/loss = $-6.02 * 0.15
Tax on capital gain/loss ≈ $-0.90 (negative as there is a loss)
Step 6: Calculate the after-tax proceeds from selling the bond:
After-tax Proceeds = Selling Price - Tax on capital gain/loss
After-tax Proceeds = $68.06 - $-0.90
After-tax Proceeds ≈ $68.96
Step 7: Calculate the annual average after-tax return:
Annual Average After-tax Return = (After-tax Proceeds - Purchase Price) / Holding Period
Annual Average After-tax Return = ($68.96 - $63.61) / 3
Annual Average After-tax Return ≈ $1.78
Therefore, the annual average after-tax return for this bond over the three-year holding period is approximately $1.78.
b) To calculate the bond's price today under the initial YTM assumptions with a time to maturity of 7 years and 200 days, we can use the same present value formula:
Purchase Price = Face Value / (1 + Yield to Maturity) ^ Years to Maturity
Years to Maturity = 7 years + 200 days = 7 + (200/360) years
Purchase Price = $100 / (1 + 0.06) ^ (7 + (200/360))
Purchase Price ≈ $64.52
To calculate the accrued interest to be added to the purchase invoice, we use the formula:
Accrued Interest = Purchase Price * Yield to Maturity * (Days to Maturity / Days in a Year)
Days to Maturity = 200 (given)
Days in a Year = 360 (given)
Accrued Interest = $64.52 * 0.06 * (200/360)
Accrued Interest ≈ $6.03
Therefore, the bond's price today would be approximately $64.52, and the accrued interest to be added to the purchase invoice would be approximately $6.03.
c) The result of the calculation in part b (with a shorter maturity) is lower than the result in part a (original holding period) because a shorter time to maturity decreases the total value of the bond. As a result, the bond's price would be lower, resulting in a lower after-tax return.