Chapter 6 Chapter 6

29
Duration = ∑XW / ∑W = 34,064 / 6,799 = 5.01 (it is as good as 5)
The investor’s time horizon matches with the duration of the bond. Hence, the
change in the market rates could not change the return; in other words, the bond
investment remains immunized against the interest rate risk (also known as
systematic risk) as the duration of the bond investment matched with investor’s
time horizon.
Q. No. 35 : Find the duration of an irredeemable bond.
Answer: Let the face value of irredeemable bond is Rs.10 and it carries 10%
coupon.
X W XW
1 1.(1/1+YTM) 1 x 1.(1/1+YTM)
2 1.(1/1+YTM) 2 2 x 1.(1/1+YTM) 2
3 1.(1/1+YTM) 3 3 x 1.(1/1+YTM) 3
4 1.(1/1+YTM) 4 4 x 1.(1/1+YTM) 4
5 ………………… …………………
6 ………………… …………………
7 ………………… …………………
8 ………… …… ………… ……
And so on ………… …… ………… ……
∑W = B ∑XW = A
∑XW
A
Duration = ---------- = ----------
∑W
B
Where B =(1/1+YTM)+(1/1+YTM) 2 +(1/1+YTM) 3 +(1/1+YTM) 4 +…= 1/YTM
Where A =
1.(1/1+YTM) + 2.(1/1+YTM) 2 +3.(1/1+YTM) 3 +4.(1/1+YTM) 4 + ………
= (1/1+YTM) + (1/1+YTM) 2 + (1/1+YTM) 2 + (1/1+YTM) 3
+(1/1+YTM) 3 +(1/1+YTM) 3 + (1/1+YTM) 4 +(1/1+YTM) 4
+(1/1+YTM) 4 +(1/1+YTM) 4 + ……….
= (1/1+YTM) + (1/1+YTM) 2 + (1/1+YTM) 3 +(1/1+YTM) 4 + ………
+ (1/1+YTM) 2 + (1/1+YTM) 3 + (1/1+YTM) 4 …………
+ (1/1+YTM) 3 + (1/1+YTM) 4 + …………
+(1/1+YTM) 4 + ………
=[1/YTM]+[1/YTM(1+YTM)]+ 1/YTM(1+YTM) 2 + (1/YTM(1+YTM) 3
+ ……………
= [(1/YTM)x{(1)+(1/1+YTM) + (1/1+YTM) 2 + (1/1+YTM) 3 + ………}]
= (1/YTM) x {(1) + (1/YTM) }
= (1/YTM) x (YTM +1)/YTM