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The first method gives you the return of a a price-weighted average, like the Dow Jones average. So I suppose it is OK to use. The second method gives you a rebalanced EW (equal weighted) average: you initially invest the same amount (say 1000 dollars) in each stock and then you rebalance to equal weights at each point where you do the calculation. ...

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Variance is additive also for any other distribution. The reason that variance is additive is because of the assumption of independent increments - ie, the change in underlying value between 2 moments is assumed to be independent of the change in value between 2 later moments

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If you use log-returns, then it is true that the return over n periods is the sum of the returns over each subperiod (e.g. the 10 day return is the sum of 10 1-day returns) $$R = \sum_{i=1}^n r_i.$$ If we now look at the variance of $R$ then we get $$VAR(R) = VAR( \sum_{i=1}^n r_i ),$$ if we assume that the returns are uncorrelated then we get $$VAR(R) ... 1 You should clarify a bit your question. Your first computation with$$Var((1+R_t)(1+R_{t+1})-1) is ambiguous. What do you mean by $Var$ here? You have time subscripts $t$ and $t + 1$ so unless you specify which filtration you compute your variance, it is unclear what you're computing. If you are computing at $t = 0$, then this is not a two period return ...

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A couple quick thoughts. Do the PCA on changes or log-changes in your series. That is often how PCA is conducted in fixed-income settings. You're large move in wights corresponds to outlier moves in the blue series. Given the assumptions of a PCA, I would consider whether your dataset has suffered from any breakpoints, regime changes or other rare events ...

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Instead of wild guesses about R's/python's future in the community, here some facts: The following query on StackExchange Data Explorer counts the number of questions that have <r> or <python> tags. If you scroll down on one of the three webpages provided below, you can see a graph with data on a monthly basis. You can easily run this query on ...

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The major advantage of Python (w/ pandas) over R is that Python supports OOP (object-oriented programming). It makes sense to organize a large code base using a hierarchy of classes. Python also supports the notion of polymorphism so that we can use well-known design patterns (e.g., Strategy, Observer, etc.) in our code.

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