Caculate the price of a 6 year bond with annual coupon payments, if the bond pay
ID: 2705712 • Letter: C
Question
Caculate the price of a 6 year bond with annual coupon payments, if the bond pays 4% in even years and pays 0% in odd year and has a YTM of 11% and face value of $100,000
Caculate the price of a 7 year bond with annual coupon payments, if the bond pays a coupon of 7% in even years and pays 0% in add years, if the YTM is 12% and the face value is $70,000.
You buy a 13 year bond with a 10% coupon rate, a YTM of 6% and a face value of $50,000. What will be your annualized holding period returm (HPR) on this investment be if you hold the bond for 13 years, the YTM on the bond when you sell it is 6% and you can reinvest coupons at 11%.
MUST SHOW WORK AND COMPLETE ALL 3 TO GET POINTS
Explanation / Answer
The value of a bond is the sum of its present value of payments (PMTs) plus the present value of its face value. All we need to solve this is the formula for the present value of a lump sum which is:
PV = FV / (1 + i) ^ n
For the first bond, i = 11%, the PMTs are 4% * $100,000 = $4,000, the first payment must be discounted 2 years, the second must be deducted 4 years, the third must be discounted 6 years, and the face value must be discounted 6 years, so the PV is $4,000 / (1 + 0.11) ^ 2 + $4,000 / (1 + 0.11) ^ 4 + $ 4,000 / (1 + 0.11) ^ 6 + $100,000 / (1 + 0.11) ^ 6 = $3,246.49 + $2,634.92 + $2,138.57 + $53,464.08 = $61,484.06.
For the second bond, i = 12%, the PMTs are 7% * $70,000 = $4,900, the first payment must be discounted 2 years, the second must be deducted 4 years, the third must be discounted 6 years, and the face value must be discounted 7 years, so the PV is $4,900 / (1 + 0.12) ^ 2 + $4,900 / (1 + 0.12) ^ 4 + $4,900 / (1 + 0.12) ^ 6 + $70,000 / (1 + 0.12) ^ 7 = $3,906.25 + $3,114.04 + $2,482.49 + $31,664.45 = $41,167.23.
The third bond does not pay every other year, so we will receive a payment every year. But, because we are reinvesting at a different rate than the YTM, we must get the FV of these cash flows. We can does this by calculating the FV of an annuity. The formula we must follow is
FV = PMT * ((1 + i) ^ n - 1) / i
When we plug into this, we should get $5,000 * ((1 + 0.12) ^ 13 - 1) / 0.12 = $140,145.55. Then we need to add the face value of the bond to this, which gives us a total future value of of $140,145.55 + $50,000 = $190,145.55. Now we must calculate the PV of the bond. To do so, we calculate that PV of the annuity, but we discount it at the coupon rate, not the reinvestment rate. The formula is
PV = PMT * (1 - (1 + i) ^ -n) / i
We plug in and get $5,000 * (1 - (1 + 0.06) ^ -13 / 0.06 = $44,623.41. Then we must add the PV of the face value of the bond which is found by using the lump sum formula I provided above. We plug in the values given in this problem and get $50,000 / (1 + 0.06) ^ 13 = $23,441.95. This gives us a total PV of $44,623.41 + $23,441.95 = $67,705.36. Finally, to calculate the annualized HPR for this we simply do: ($190,145.55 / $67,705.36) ^ (1 / 13) - 1 = 0.082673 = 8.27%. We expect this to be higher than the coupon rate because we are able to reinvest at a higher interest rate, so this answer seems reasonable.
Hope that helps! Please leave a comment if you need any further explanation.