# Understanding the formulation for the SDF chosen in a paper

In a Stanford paper they claim the SDF is an affine transformation of the tangency portfolio by citing a textbook and then say a valid formulation of the SDF can be given by $$M_{t+1} = 1 - \sum_{i=1}^N \omega_{t,i}R^e_{t+1,i}$$.

I'm trying to understand a bit more about how they came up with their formulation. For example, why "$$1 -$$" and not "$$1 -$$", also why not wirte "$$+\sum_{i=1}^N \omega_{t,i}R^e_{t+1,i}$$". I have no reason for these alternative formulations but I'm assuming there's a bit of intuition behind there formulation that I'm not following.

So far the reasons I've come up with is "$$-\sum_{i=1}^N \omega_{t,i}R^e_{t+1,i}$$" so that the tangency portfolio has larger positive weights on high risk assets. If that's a satisfactory reason, then I'm just trying to figure out why "$$1-$$". That's going to scale the optimal weights but I don't the reason why for a particular scaling.