dstefan
06-06-2008, 12:19 AM
Chapter 2, Problem 8
The continuously compounded 6-month, 12-month, 18-month, and 24-month zero rates are 5%, 5.25%, 5.35%, and 5.5%, respectively. What is the par yield for a 2-year semiannual coupon bond?
Solution:
Par yield is the coupon rate C that makes the value of the bond equal to its face value, and can be found by solving
100 = \frac{C}{2} ~100 ~e^{-r(0,0.5) 0.5} ~+~ \frac{C}{2} ~100 ~e^{-r(0,1 ) } ~+~ \frac{C}{2} ~100 ~e^{-r(0,1.5) 1.5} ~+~ \left( 100 ~+~ \frac{C}{2} ~100 \right)~e^{-r(0,2) 2}
Thus,
C ~=~ \frac{2 (1 - e^{-r(0,2) 2})}{e^{-r(0,0.5) 0.5} + e^{-r(0,1)} + e^{-r(0,1.5) 1.5} + e^{-r(0,2) 2}}
The value of the par yield is C = 0.05566075, i.e., 5.566075%.
The continuously compounded 6-month, 12-month, 18-month, and 24-month zero rates are 5%, 5.25%, 5.35%, and 5.5%, respectively. What is the par yield for a 2-year semiannual coupon bond?
Solution:
Par yield is the coupon rate C that makes the value of the bond equal to its face value, and can be found by solving
100 = \frac{C}{2} ~100 ~e^{-r(0,0.5) 0.5} ~+~ \frac{C}{2} ~100 ~e^{-r(0,1 ) } ~+~ \frac{C}{2} ~100 ~e^{-r(0,1.5) 1.5} ~+~ \left( 100 ~+~ \frac{C}{2} ~100 \right)~e^{-r(0,2) 2}
Thus,
C ~=~ \frac{2 (1 - e^{-r(0,2) 2})}{e^{-r(0,0.5) 0.5} + e^{-r(0,1)} + e^{-r(0,1.5) 1.5} + e^{-r(0,2) 2}}
The value of the par yield is C = 0.05566075, i.e., 5.566075%.