Question

Consider the following. a. What is the duration of a four-year Treasury bond with a 4.5 percent semiannual coupon selling at par? b. What is the duration of a three-year Treasury bond with a 4.5 percent semiannual coupon selling at par? c. What is the duration of a two-year Treasury bond with a 4.5 percent semiannual coupon selling at par?

	a. Duration of the bond years b. Duration of the bond years c. Duration of the bond years

Answer 1

Let Par value = 1000
Price =1000
a.
Number of Periods =2*4=8
Semi annual YTM =4.5%/2 =2.25
Semi annual coupon =4.5%*1000/2 =22.25
Duration =Sum of (PV of Coupons*Time)/(Price*2)
Sum of (PV of Coupons*Time) =(22.5*1/(1+2.25%)^1)+(22.5*2/(1+2.25%)^2)+(22.5*3/(1+2.25%)^3+(22.5*4/(1+2.25%)^4)+(22.5*5/(1+2.25%)^5)+(22.5*6/(1+2.25%)^6)+(22.5*7/(1+2.25%)^7)+(1022.5*8/(1+2.25%)^8) =7410.2463

Duration =Sum of (PV of Coupons*Time)/(Price/2) =7410.2463/(1000*2)=3.71 years

b. Number of Periods =2*3=6
Semi annual YTM =4.5%/2 =2.25
Semi annual coupon =4.5%*1000/2 =22.25
Duration =Sum of (PV of Coupons*Time)/(Price*2)
Sum of (PV of Coupons*Time) =(22.5*1/(1+2.25%)^1)+(22.5*2/(1+2.25%)^2)+(22.5*3/(1+2.25%)^3+(22.5*4/(1+2.25%)^4)+(22.5*5/(1+2.25%)^5)+(1022.5*6/(1+2.25%)^6) =5679.4525

Duration =Sum of (PV of Coupons*Time)/(Price/2) =5679.4525/(1000*2)=2.84 years

c.  Number of Periods =2*2=4
Semi annual YTM =4.5%/2 =2.25
Semi annual coupon =4.5%*1000/2 =22.25
Duration =Sum of (PV of Coupons*Time)/(Price*2)
Sum of (PV of Coupons*Time) =(22.5*1/(1+2.25%)^1)+(22.5*2/(1+2.25%)^2)+(22.5*3/(1+2.25%)^3+(1022.5*4/(1+2.25%)^4)=3869.90

Duration =Sum of (PV of Coupons*Time)/(Price/2) =3869.90/(1000*2)=1.93 years