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Bond Sensitivity to Yield-Curve Steepening

Article Quant Q&A · Author: Bazman
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Summary

The document examines a proposed rule that relates a bond’s sensitivity to yield-curve slope changes to the logarithm of its maturity, and asks whether a duration-based expression is reliable. The answer tests sensitivity under a simple linear continuously compounded rate curve, where the rate is an intercept plus a slope times maturity. The discount factor is therefore exponential in both maturity and the slope parameter.

Differentiating that discount factor with respect to the slope gives a sensitivity proportional to the negative square of the cash-flow time, multiplied by the discount factor. The response says the same structure carries over to a bond as a collection of discounted cash flows, and connects the effect to negative convexity sensitivity. This provides a mathematical result for the stated curve specification, rather than evidence from market data or a practical performance comparison. The answer does not define the symbol in the questioned alternative formula or establish how either proposed maturity or duration approximation performs across real bonds and curve models.

Key ideas

  • Under a linear continuously compounded rate curve, the discount factor depends quadratically on cash-flow time and slope.
  • Its slope sensitivity scales with the negative square of maturity, weighted by the discount factor.
  • Bond sensitivity follows by aggregating the discounted cash-flow sensitivities.
  • The response links the result to negative convexity sensitivity but gives no empirical test of alternative rules.

Tags

Fixed incomeDerivatives pricingRisk management
Full text
# Formula to calculate the sensitivity of a bond to curve steepening


# Formula to calculate the sensitivity of a bond to curve steepening












https://github.com/jerryxyx/TreasuryFutureTrading/blob/master/README.pdf

On page 2 of the pdf above, the sensitivity of a bond to increases in the slope of the curve, is given as $\ln(T)$, where $T$ is the maturity of the bond.

- Is this relationship exact, i.e. derived from the bond pricing equation, or simply a rule of thumb?

- If it is simply a rule of thumb, does anyone know how well it works in practice?

On page 4 the sensitivity of the bond to the slope is defined as $S\ln(T)$ (unfortunately $S$ is not defined), they say then that this expression can be approximated by $D \ln(D)$, where D is the bond's duration.

- Does anyone know what S could stand for?

- How well do one of both of there expressions on p4 work in capturing the sensitivity of the bond to changes in curve steepness?

Thanks Baz

## Answer by Kermittfrog (score 2)

https://quant.stackexchange.com/a/70391

I cannot quite follow that ansatz, maybe we do not agree on the definition of slope:

Let $r(\tau)\equiv a+b\tau$ be the discount rate curve for a continuously compounded rate as a function of the time to maturity $\tau$. The discount factor for a cash flow with time to maturity $\tau$ is $D(\tau)=e^{-r(\tau)\tau}=e^{-a\tau-b\tau^2}$. The sensitivity of the discount factor w.r.t. the slope parameter $b$ is

$$ \frac{\partial D(\tau)}{\partial b}=-\tau^2D(\tau) $$

which scales with $\tau^2$. The same holds true for a bond as a series of discounted cash flows; furthermore, the sensitivity is proportional to the negative of the bond's convexity, $\frac{\partial D(\tau)}{\partial b}\propto -\frac{\partial^2 D(\tau)}{\partial r^2}$

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.