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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.

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    $\begingroup$ 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. $\endgroup$
    – AKdemy
    Commented Oct 7, 2022 at 23:22
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    $\begingroup$ @Akdemy I think the formulation is slightly different than the standard finite difference equation or the slope because the denominator is not $0.01EQ_k$ $\endgroup$
    – Alper
    Commented Oct 8, 2022 at 10:28
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    $\begingroup$ @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. $\endgroup$
    – AKdemy
    Commented Oct 8, 2022 at 19:24

1 Answer 1

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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%.

It is precisely the goal of rescaling the sensitivities of cash equity or FX positions to equity price and to exchange rates so the sensitivities would equal to the mark to market. It shows the "rise" in mark to market if the underlying price or rate "runs" 100%. It shows how much you'd lose/gain if a long/short position becomes worthless. For a linear cash position, you could as well just bump by 100%, whereas for non-linear ones, you get more meaningful sensitivities bumping only a little, both up and down, and then rescaling.

For sensitivities to interest rate, first, we measure the sensitivity to the interest rates going up 1 basis point, rather than down; and secondly rescale the sensitivity to an unrealistically large interest rate hike of 100%, ignoring any non-linearity. However, when perturbing very high interest rates, such as Argentine peso, that's been around 60-70% per year lately, the 1 bp bump gets lost in numerical noise, so you may instead prefer to bump by more than 1 bp. Conversely, in some rare cases you may prefer to bump by less than 1bp. It's more art than science:), but remember to divide the mtm change by the same bump size that you actually used.

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