Flat Extrapolation at the Boundaries of a Yield Curve

Summary

The document clarifies what to do when a requested maturity falls outside the known terms used by a raw yield-curve interpolation formula. Such a case is extrapolation rather than interpolation, because the desired point lies beyond the range of observed maturities. The accepted answer describes the cited approach for maturities beyond the final known term: hold the rate constant at the rate for that last term.

For the short end, the answer says that if the first known term is set to zero, no extrapolation is needed for negative time because time to maturity cannot be negative. This is a narrowly scoped prescription tied to the cited curve-construction method; it does not compare flat extrapolation with alternative boundary assumptions or discuss how choices affect discount factors and valuation. The document offers no data-based test of the method, so users should treat it as a stated construction convention rather than evidence that a flat long end is suitable for every curve or application.

Key ideas

• A requested maturity outside the known maturity range requires extrapolation rather than interpolation.
• The described method holds the rate at the last observed rate for maturities beyond the final known term.
• Setting the first known term to zero avoids a negative-maturity extrapolation case.
• The answer states a construction convention but provides no comparison or empirical validation of boundary methods.

Tags

# Raw interpolation when the desired term is out of the know originals


# Raw interpolation when the desired term is out of the know originals












I was reading this paper regarding the yield curve construction and was programming the Raw Interpolation algorithm (page 7 equation 6) however I was wondering how to use the formula when the desired term is out of the originals. So the interpolation formula is: $r(t) = \frac{t-t_i}{t_{i+1}-t_i}*\frac{t_{i+1}}{t}*r(t_{i+1})+\frac{t_{i+1}-t}{t_{i+1}-t_i}*\frac{t_i}{t}r(t_i)$

where $t$ is the desired term, $t_i$ is the previous known term to the one desired and $t_{i+1}$ is the next one. To be clear, my question is how to use this interpolation when $t<t_i$ or $t>t_{i+1}$. At first I thought to set $t_i$ or $t_{i+1}$ (depending on $t$) to 0 but then it doesn't make sense if the original terms goes from 100 to 110 for example.

## Answer by JejeBelfort (score 1, accepted)

https://quant.stackexchange.com/a/44683

What you are interested in is called `extrapolation`.

In other words, you want to "extend" your function $r$ for $t < t_0$ and $t > t_n$.

What the author suggests on page 109, below equation (37), is to extrapolate "flat", that is:

$$r(t) = r(t_n), \space \forall t > t_n$$

Setting $t_0 = 0$ does not require extrapolation for $t < t_0$ as time cannot go negative.

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