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.