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As I see it, a reasonable possibility is that it is a directional bet; money will only be made if the direction of the jump is correct, else the bet is lost. So if the random outcome is $X \in \{-J, 0, J\}$ with $\mathbb{P}(+J) = \mathbb{P}(-J) = \varphi /2 \;$ such that $\mathbb{P}(\pm J) = \varphi$, then it follows that $\mathbb{E}[X] = 0$, and $\mathbb{...


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Generally speaking, you pose a system of linear equations that is undetermined. If you provided the initial 3 quarterly returns together with the cumulative returns then yes, all other quarterly returns can be backed out. Else, you could try to approximate the other returns using the pseudo-inverse. This answer here has a very nice example on that topic. ...


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