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How a Humped Yield Curve Shapes Forward and Future Spot Rates

Article Quant Q&A · Author: HumpedCurve
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Summary

This note considers how a humped USD interest-rate swap curve affects zero rates, forward rates, and modeled spot curves at future dates. The author describes bootstrapping a zero curve from swaps using a two-factor Hull–White model with constant parameters, and observes that the hump in swap rates appears in the zero and forward curves as well.

The question is whether future spot curves become increasingly inverted as time advances. The sole answer agrees that forward and spot rates will be decreasing, but offers no derivation, assumptions, or supporting calculations. The result should therefore be read as a brief response to the stated curve setup, rather than a general claim about all yield curves or interest-rate models. The note does not examine how parameter calibration, curve construction, or model dynamics might affect the shape of projected curves.

Key ideas

  • A hump in swap rates can also appear in bootstrapped zero and forward curves.
  • The author asks how that curve shape affects modeled spot rates at future dates.
  • The stated setup uses a two-factor Hull–White model with constant parameters.
  • The answer says future forwards and spot rates will be decreasing.
  • No derivation or evidence is provided for that conclusion.

Tags

Fixed incomeFuturesStatistics
Full text
# Implication of Humped Spot Curve on future spot curve(s)


# Implication of Humped Spot Curve on future spot curve(s)












I'm currently implementing a G++ model (Two Factor Hull & White model with constant parameters) on zero curve bootstrapped from USD IRS.

Currently, USD IRS is humped at 30 years; swap rate goes up until maturity of 30 years and starts to go down to maturity of 50 years. This causes the bootstrapped zero curve to be also inverted at 30 years. Furthermore, this causes the forward curve to be inverted as well.

My question is, if I model Spot Curve at some future points, i.e. P(t, T), would the future spot curve(s) become completely inverted as I increase t?

This is what I am observing, and it seems right as spot curve in the future depend on forward curve observed now (Please correct me if I am wrong). Thank you so much for any clarifications!

edit: changed "semi-inverted" to "humped"

## Answer by Edward Watson (score 2)

https://quant.stackexchange.com/a/63164

The forwards and the spot rates will be decreasing, that is correct.

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.