I am really struggling to prove to myself that when we can estimate the one-year holding period return for a three-year zero by using the following estimation:

S3 - Duration2*(f1,3- S3)

Where Sn is spot rate.

What I don't get is why duration is subscript to 2 instead of 3? I thought we can measure capital gain/loss approximately by taking the duration multiplied by yield change. How does (f1,3 -S3) measure the change in yield? Well let's say it's becoming a 2-year zero. Then why does it match to a duration of 2?

To me what is difficult is the fact that we are getting a different bond altogether (changing yield and changing duration), while the typical interpretation of yield change is constant duration and a parallel shift in interest rates.

  • $\begingroup$ Any link to where you found such estimate? $\endgroup$
    – Rustam
    Oct 16, 2013 at 5:45
  • $\begingroup$ Solomon Brothers' Yield Curve series part 3. I realized that this relationship is an identity (when ignoring convexity), but I am still uncomfortable with the interpretation of the subscript. If anything, when duration is 2, then f1,3 must be known..., which means we already know the price of the bond and hence holding period return.... which further the point that this is an identity... $\endgroup$ Oct 17, 2013 at 16:00
  • $\begingroup$ wilmott.com/messageview.cfm?catid=11&threadid=7633 $\endgroup$ Oct 17, 2013 at 16:00

1 Answer 1


I know the answer now... I don't know how I missed it :(

Sn - (n-1)(f1n - Sn) = S1 Sn = S1 + (n-1)(f1n-Sn)

where n-1 is Dur_n-1

S1 is known, therefore it's the intercept. And hence:

Sn is a linear function of Dur_n1


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