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.


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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