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I am a bit confused with one problem,Though I have understood what bond duration and convexity but there is still a gap in understanding with respect to vanilla bonds and Zero coupon bonds(ZCB).I understand that bonds with a longer duration are more vulnerable to changes in interest rates but does this hold true when comparing a coupon bearing bond with a ZCB. I mean the question I have is
Which of the following two bonds is more price sensitive to changes in interest rates?
1. A par value bond. X, with a 5-year-to-maturity and a 10% coupon rate.
2. A zero coupon bond, Y, with a 5-year-to-maturity and a 10% yield-to-maturity.
a. Bond X because of the higher yield to maturity.
b. Bond X because of the longer time to maturity.
c. Bond Y because of the longer duration.
d. Both have the same sensitivity because both have the same yield to maturity.
Please throw some light on this.
Thanks
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If we use the definition of modified duration = -1/P (dP/dr), then we can derive
%change in price of bond = ModifiedDuration * % change in yield
Higher the "%change in price of bond" implies higher sensitivity to change in yield. Since the duration of zero coupon bond is higher it has a higher sensitivity to change in interest rate.
In this case, it is actually easy to compute as well. So, if you go by first principles,
Macaulay Duration of X = 1/100 [10/1.1 + 20/1.1^2 + 30/1.1^3 + 40/1.1^4 + 550/1.1^5] = 4.169857
Modified Duration of X = 4.169857/1.1 = 3.79077
Macaulay Duration of Y = 5
Modified Duration of Y = 5/1.1 = 4.5454
Therefore, it shows that the %change in Price of X = -3.79077 and %change Price of Y = -4.5454 (assume 1% change in yield).
(c) is the correct answer
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