# FRTB Delta risk sensitivity definitions

Items 21.19 to 21.24 from FRTB's documentation define delta risk sensitivities for each risk class.

For instance, delta equity risk $$s_{k}$$ is defined as:

$$s_{k} = \frac{V_{i}(1.01EQ_{k})-V_{i}(EQ_{k})}{0.01}$$

where:

$$k$$ is the asset

$$EQ_{k}$$ is the market value of equity $$k$$

$$V_{i}$$ is the market value of the instrument i as a function of the spot price of equity $$k$$

I think I'm missing the reason for that 0.01 in the denominator. For a position on a stock it would cancel out with the numerator. For non-linear instruments that won't happen, but the meaning of such a measure isn't clear nevertheless.

• It is a standard finite difference equation, See for example this question. It follows the definition of a slope, which is defined as change in y over change in x. Commented Oct 7, 2022 at 23:22
• @Akdemy I think the formulation is slightly different than the standard finite difference equation or the slope because the denominator is not $0.01EQ_k$ Commented Oct 8, 2022 at 10:28
• @Alper you are right. In most cases, it's actually quite different in terms of numbers. Rather unprofessional comment - thanks for pointing out my error. Commented Oct 8, 2022 at 19:24

The goal of dividing by the bump $$\delta$$ is to rescale the sensitivity (slope) $$s$$ to a 100% bump. If $$i$$ is just a linear cash position with notional $$n$$, $$V(nx)=nx$$, and $$s=\frac{V((1+\delta)nx)-V(nx)}{\delta}=nx.$$It's more convenient to scale to 100% than to some other arbitrary bump size like 1% or .1%.