Consider two bonds: bond XY and bond ZW . Bond XY has a face value of $1,000 and 10 years to maturity and has just been issued at par. It bears the current market interest rate of 7% (i.e. this is the yield to maturity for this bond). Bond ZW was issued 5 years ago when interest rates were much higher. Bond ZW has face value of $1,000 and pays a 13% coupon rate. When issued, this bond had a 15-year, so today its remaining maturity is 10 years. Both bonds make annual coupon payments.
a) (5 points) What is the price of Bond ZW , given that market interest rates are 7%? b) (15 points) Compute the duration for bond bonds (use Excel).
Price of the bond can be calculated by discounting the future cash flows at the current rate of interest:
Note: when the coupon rate > than the discount rate, the price > par and vice versa
Duration of the two bonds are calculated below:
Bond ZW | |||||
Year | CF | PV | PV x t | ||
1 | 130 | 130*1/1.09^(1)= | 119.266055 | 119.266055045872*1= | 119.266055 |
2 | 130 | 130*1/1.09^(2)= | 109.4183991 | 109.418399124653*2= | 218.8367982 |
3 | 130 | 130*1/1.09^(3)= | 100.3838524 | 100.383852407938*3= | 301.1515572 |
4 | 130 | 130*1/1.09^(4)= | 92.09527744 | 92.0952774384755*4= | 368.3811098 |
5 | 130 | 130*1/1.09^(5)= | 84.49108022 | 84.4910802187849*5= | 422.4554011 |
6 | 130 | 130*1/1.09^(6)= | 77.51475249 | 77.5147524942981*6= | 465.088515 |
7 | 130 | 130*1/1.09^(7)= | 71.11445183 | 71.1144518296312*7= | 497.8011628 |
8 | 130 | 130*1/1.09^(8)= | 65.24261636 | 65.2426163574599*8= | 521.9409309 |
9 | 130 | 130*1/1.09^(9)= | 59.85561134 | 59.8556113371191*9= | 538.700502 |
10 | 130 | 130*1/1.09^(10)= | 54.9134049 | 54.9134048964396*10= | 549.134049 |
10 | 1000 | 1000*1/1.09^(10)= | 422.4108069 | 422.410806895689*10= | 4224.108069 |
SUM | 1256.706308 | SUM | 8226.86415 | ||
Macaulay's duration | 8226.86415/1256.706308 = 6.546369742 | ||||
Modified duration | Macaulay's duration/(1.07)= 6.118102563 |
Bond XY | |||||
Year | CF | PV | PV x t | ||
1 | 70 | 70*1/1.09^(1)= | 64.22018349 | 64.2201834862385*1= | 64.22018349 |
2 | 70 | 70*1/1.09^(2)= | 58.91759953 | 58.9175995286592*2= | 117.8351991 |
3 | 70 | 70*1/1.09^(3)= | 54.0528436 | 54.0528436042745*3= | 162.1585308 |
4 | 70 | 70*1/1.09^(4)= | 49.58976477 | 49.5897647745638*4= | 198.3590591 |
5 | 70 | 70*1/1.09^(5)= | 45.49519704 | 45.4951970408842*5= | 227.4759852 |
6 | 70 | 70*1/1.09^(6)= | 41.73871288 | 41.7387128815451*6= | 250.4322773 |
7 | 70 | 70*1/1.09^(7)= | 38.29239714 | 38.2923971390322*7= | 268.04678 |
8 | 70 | 70*1/1.09^(8)= | 35.13063958 | 35.1306395770938*8= | 281.0451166 |
9 | 70 | 70*1/1.09^(9)= | 32.22994457 | 32.2299445661411*9= | 290.0695011 |
10 | 70 | 70*1/1.09^(10)= | 29.56875648 | 29.5687564826982*10= | 295.6875648 |
10 | 1000 | 1000*1/1.09^(10)= | 422.4108069 | 422.410806895689*10= | 4224.108069 |
SUM | 871.646846 | SUM | 6379.438266 | ||
Macaulay's duration | 6379.438266/871.646846=7.318833649 | ||||
Modified duration | Macaulay's duration/(1.07)= 6.840031448 |
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