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Roll Yield and Basis

If Basis = S - F(T) (per page 315 of Schweser Book1) and Roll Yield is your change in Basis how is the formula for Roll Yield = [F(T)0 - F(T)1] - [S0 - S1] per pg. 338?

And if in an efficient market your basis = carrying cost (pg. 335); how dow F(T) = S + carrying cost (pg. 306) if Basis = S - F(T)?

I feel like Basis should be calced as F(T) - S. Am I thinking about this wrong? Should basis just be a magnitude representing the gain on changes in future and spot prices assuming constant carrying cost in a backwardated market?

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Basis is equal to S - F(T), always keep that in mind 

Now on page 338 - you made a mistake in your formula it is F(T)-1  & S-1 . So think about it this way:

Roll Yield = Expiring Contract (which you will close) - New Contract ( which you will open today at time 0) 

                = Basis-1  -  Basis

                = S-1 - F(T)-1   - [S0 - F(T)0]

                = S-1 - F(T)-1  - S0 + F(T)0

Re-arrange this and you will get the formula on page 338

Roll Yield = F0 - F-1 - (S0 - S-1

The Absolute value of the basis & carrying cost are equal. F(T) will converge to S and your profit/Loss in an efficient market will be the carrying cost. In an inefficient market (especially for physical commodities as opposed to financial assets) the carrying costs are unobservable(i.e. we don’t know how much each participant stored and how much each is paying in storage). So participants that have a competitive advantage in storage costs, and a high convenience yield will earn Alpha.