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For arithmetic returns, the arithmetic return of the portfolio is a weighted average of the individual arithmetic returns and the result is exact. For equally weighted portfolios all weights are equal. For logarithmic returns, you can use the same formula i.e. the logarithmic return of the portfolio is a weighted average of the individual logarithmic ...


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The arithmetic average of +100% in Year 1 and -100% in Year 2 is 0%, but I we all know the result is not a 0% return. So arithmetic returns are absurd to use in any real life context. Maybe in another universe they can serve some purpose.


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In practice, for heavily traded assets (above 60% quantile of average daily dollar volume), individual asset return is pretty scalable across different time frame by a factor of $\sqrt{T}$. However, for covariance among different assets, moving between different time frame is not linearly scalable (although it should be in math). This is known as "Epps ...


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If high frequency returns are iid and the mean and variance are finite and vthe variance is greater than zero then the Central Limit theorem holds Then, regardless of the distribution of the high returns, when aggregated over time the aggregated returns will tend in distribution to a Normal distribution. The Lindeberg-Lévy-Feller version of the Central Limit ...


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This paper states that heteroskedasticity is a stylized fact in daily as well as intra-day returns: https://statistik.econ.kit.edu/download/doc_secure1/HandbookITandFinan.pdf


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My main reference will be "Dan Xu, Christian Beck - Transition from lognormal to chi-square superstatistics for financial time series" Non-equilibrium statistical mechanics (more specifically, superstatistics) gives some ideas of explaining the relation between time frame and its distribution: "...to regard the time series as a superposition of local ...


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Surely, there is; search for aggregational gaussianity in Google Scholar or ScienceDirect. In fact, 5 minutes returns are leptokurtic and fat-tailed; then as you increase timeframe, returns become more and more normal. Yearly data is almost normal, if you have enough points.


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I would make a cumulative return chart, using a different color for each sector. Each line starts at 1, and each successive point is found by multiplying the previous point by (1+SectorReturn) for that sector. The horizontal axis shows dates. By looking at the lines on this chart you get a visual feel for what sectors performed best overall and also the ...


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There are several ways to do this: If you need the price for June 11 and the market is closed on that day, you can use the price for June 10th (which is known on June 11th). I would advise against using the price of June 12th because it is not known on the 11th, so you would be "looking into the future" which is a bad idea and can lead to subtle fallacies ...


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The CAPM model is based on the relationship existing between an asset and its benchmark market; assuming that the bitcoin could be thought as a currency, according to me, you should take the mean of returns over all the currencies traded and then regress the BTCUSD on the the average currency market returns. Indeed, although the EURUSD is one of the most ...


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You can apply the Kolmogorov-Smirnov test. I simply quote from the entry: "The two-sample K–S test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples." There is an R-implementation too.



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