# Stress testing fixed income Yield curve with Nelson Siegel

I am attempting to stress test the Zero coupon Yield curve using The Nelson Siegel model as described in the following papers :

Since i am studying a low rate environment,i noticed that the methods used to determine the shocks does not take into consideration the current level of the rates (because i think the shocks applied to low rate environment should differ from high rate environment )

How can i include the rates level when determining the Yield curve shocks ?

More generally (not just in NS context), suppose that some rate is $$R_{now}$$ and that you want to apply a shock comparable to the rate changing from $$R_{old}$$ to $$R_{new}$$.

The most naive approach is to treat interest rates as you would equity or commodity prices. Suppose $$R_{old}=1\%$$, and $$R_{new}=R_{now}=2\%$$ - the rate has gone up $$1\%$$ or "doubled". Applying the same shock, the rate becomes $$R_{now}*R_{new}/R_{old}=4\%$$. It needs to up up $$2\%$$ to "double" again. No, this is like doubling the temperature in Fahrenheit - not good.

A less naive approach is $$R_{now}+R_{new}-R_{old}=2\%$$. It went up another $$1\%$$. This works for this example, but suppose $$R_{old}=13\%$$, $$R_{new}=10\%$$, $$R_{now}=1\%$$. You want to replicate the shock when it went down $$3\%$$. Would it go from $$1\%$$ to $$-2\%$$? Negative rates are OK, but you don't want rates $$\le-1$$.

To make the shocks comparable across different levels, you'd bump to something like $$\exp(\ln(1+R_{now})*\ln(1+R_{new})/\ln(1+R_{old}))-1$$...

Have a look at Basel document. The section 98.56 and on describe derivation of the interest rate shocks. 16 years may be too long depending on your portfolio, but I think you can shorten the period and start from there.

Caveat: I did not try it myself yet, but will revisit this topic soon and might be able to share my findings. I asked a question related to the document a while ago, but did not receive any answers.

UPDATE

Alright, I did finish some quick and dirty calculations on the comment mentioned above.

I took the US Treasury Yield Curve as of August 30, 2019 and proceeded with Basel proposed method. There are some interesting outputs:

1. Since the current yield curve is flat(-ish), for Basel's Flattener I actually got inverted yield curve;
2. Given the current yield environment, Basel's Steepener also looks reasonable;
3. I now wonder if the "no arbitrage" case is applicable for these scenarios or not. Any input/comment would be appreciated.