Using Duration to Approximate a Bond’s One-Year Holding Return

Summary

The document examines an estimate for the one-year holding-period return on a three-year zero-coupon bond. It asks why the duration used in the estimate corresponds to the bond’s remaining two years at the end of the holding period, and how the forward rate compares with the current three-year spot rate as a yield change. The question highlights a limitation of applying the usual duration rule: that rule approximates a price response to a yield change for a bond whose duration is held fixed.

The accepted response rearranges the spot and forward rate relationship to express the initial spot rate as an intercept and the forward-minus-spot spread as a slope scaled by the remaining duration. This offers an algebraic interpretation of the estimate rather than a general duration derivation. The discussion is brief and does not address coupon bonds, changing curve shapes, or the accuracy of the approximation under market conditions.

Key ideas

• A one-year holding period shortens a three-year zero-coupon bond’s remaining maturity to two years.
• The estimate relates the forward-minus-spot rate spread to the bond’s remaining duration.
• The answer rewrites the relationship as a linear expression with the initial spot rate as its intercept.
• The document does not establish how well this reasoning extends beyond the stated zero-coupon example.

Tags

# Bond duration as estimation to holding return


# Bond duration as estimation to holding return












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.

## Answer by PrinnySquad (score 0, accepted)

https://quant.stackexchange.com/a/9202

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.