Extrapolating Long-Term Treasury Yields with Swap Spreads

Summary

The document considers how to estimate Treasury yields beyond the longest available bond maturities, using a theoretical 40- or 50-year yield as an example. The questioner reports that Nelson–Siegel and Svensson curve fits gave unsatisfactory, unstable long-end estimates, with substantial variation over short periods. The post does not provide a systematic comparison of extrapolation models or validation results.

One response cautions that mechanically extending a fitted curve can ignore market behavior, including the tendency for long-end forward rates to rise. For a Treasury estimate, it suggests using a quoted long-dated swap rate and an assumed Treasury–swap spread, either modeling the spread’s behavior or holding it constant. The implied Treasury point can then be added to the observed data before fitting a curve. This is a practical proposal, not a demonstrated universal method: its usefulness depends on the intended application and on the spread assumption, and the document supplies no empirical test of that assumption.

Key ideas

• Long-maturity yield estimates can be unstable when extrapolated from a shorter observed curve.
• A response warns that simple extrapolation may fail to reflect long-end forward-rate behavior.
• A long-dated swap quote combined with an assumed Treasury–swap spread can imply a Treasury yield.
• The resulting estimate can be added as a curve point, but the spread assumption is not validated in the document.

Tags

# Bond curve extrapolation


# Bond curve extrapolation












What are the best methods to extrapolate bond yields from an existing curve that doesn't extend quite this far?

For example, how would one come about finding a theoretical bond yield for a 40 or 50 year US Treasury Bond, when no bond exists of a maturity of much more than 30 years? I guess you could remove credit risk from corporate ultra-long bonds, but this might prove difficult.

I explored using curve fitting models such as Nelson-Siegel and Svensson, but the results are a little unsatisfactory and highly volatile, with theoretical yields at the 50 year mark varying by as more than 50bps depending on the date at which the curve is calculated (over a short time frame).

## Answer by imachabeli (score 2)

https://quant.stackexchange.com/a/7889

It really depends on how/where do you plan to use final values. I would not use extrapolation since it will ignore market realities. Forward rates across long end tend to be increasing while dumb extrapolation might give you the opposite result.

In case of treasuries one can use treasury and swap spread and while you do not have 50 Y treasuyy one can find quotes for 50Y swap. You can imply some dynamics of spread or keep it constant and effectively 50Y swap minus spread will be your 50Y treasury yield and than fit curve with that data point. If you have bloomberg check USBE30.

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