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

Question

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 1

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.