Effective duration of a zero

CFA Level II Fixed Income
#1

Hi,

Why is the effective duration of a zero-coupon bond equal to its maturity under continuous compounding? I bet there’s an simple answer here that I just can’t see!

Thanks in advance.

#2

If it’s zero coupon, the only payment you get is at maturity - the repayment of the principal at par.

#3

Thanks, I understand why the Macaulay duration of a zero-coupon is equal to its maturity (because the weighted average of its payments is at maturity), but what I don’t understand is why the effective duration is equal to its maturity.

#4

In general,

Modified duration = Macaulay duration / (1 + r)

where r is the YTM for one period. In the case of continuous compounding, r = 0%.

And, of course, when a bond has no embedded options,

Effective duration = Modified duration

QED

#5

Thanks s2000, can you help explain why the 1-period YTM is 0% under continuous compounding?

#6

Because the period has length zero.

It’s a calculus thing.

#7

Ah I get it now, thanks so much!

#8

My pleasure.