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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.

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