# IR risk sensitivity to curve instruments

I need to understand if the 2 approaches are equivalent: assume I am constructing a yield curve with N instruments. I would like to compute IR delta for a product using this curve. One approach is to move every instrument's quote by 1bp, reconstruct the curve and revalue. Another one is to do one at a time, compute the impact of that particular instrument and sum the results across. Would those be equivalent? I don't get the same answers and I would like to confirm if this is due to FDM error or there is a math reason for the difference.

## 1 Answer

It sounds like you expect the sum of the single impacts to be equal to the impact you get when you move every instrument by 1bp. In theory this should be the case because (assuming two curve instruments for simplicity which happen to have same levels $$x$$) $$\frac{d}{dx}f(x,x)=\frac{\partial}{\partial x_1}f(x_1,x_2)\Big|_{x_1=x_2=x}+\frac{\partial}{\partial x_2}f(x_1,x_2)\Big|_{x_1=x_2=x}\,.$$ In practice however you are approxmating these derivatives by finite differences so that there will rarely be an equality. What you can expect is that the results are close. If they aren't you can try a move smaller than 1bp.