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When we talk about the single-period CAPM, the return in a particular period t can be defined as $(P_t - P_{t-1})/P_{t-1}$. Investors plan at t-1 and get a payoff at t.

After this period, the same mechanics take place. Does the “new” Pt-1 have to necessarily be equal to Pt? In other words, is the return in each period calculated separately, or is the “opening” price in a period necessarily the “closing” price of the previous period? Or does the “payoff” price potentially differ from the new “equilibrium” price? Can this change given a fixed return distribution? If the return distribution changes, it obviously changes. Does this depend on whether the model is “static” or “dynamic”?

If it is the case that these prices can differ, how does the model incorporate the potential of an investor purchasing the model at time t, and then the price falling (at the new t-1)?

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As you write yourself, the CAPM is a one-period model. It states the equilibrium in a single-period economy. It does not in itself say anything about the next period. A suggestion could be to have a look at Pedersen (2013) "Betting against beta" which extends to multi periods (by an over-lapping generations model) and constrained agents. They reach the intuitive conclusion in proposition 1, that "everyone" wants high-beta stocks, so you have to "bet against beta" to increase alpha.

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