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The Top10 traded ETFs in USA are compounded by SQQQ, SPY, XLF, QQQ, UVXY, TQQQ, VXX, EEM, IWM, and XLE. Pairs as (TQQQ;SQQQ) are expected to obviously have a near perfect negative correlation of minus one, while so does the pair (SPY;XLF) but in the positive direction. Even though the Top10 ETFs are not expected to generate great diversification alternatives, it is of importance to find out how significantly different are one from each other as they are commonly used by speculators.

The picture at the beginning of this post shows that, apparently, all of them count with same expected return and risk (variance) then there are two hypotheses to test and the first one would be: "all means are equal". The Anova report in the sheet "Summary" from the following Excel report:

Gives the answer: "At a 95% confidence level, the hypothesis of all means being equal can be rejected as there is a probability of 0,034% that the difference in the observed means is coincidential". As the Anova report indicates that at least one of the means is significantly different, a Honest Significant Difference (HSD) test is undertook to spot this or these means. All 45 compared pairs but (TQQQ;SQQQ) count with a confident interval that goes over zero; therefore, the expected ETF SQQQ is the only with a significant difference.

What does it mean? If a speculator or investor holds in his portfolio two or more of this Top10 ETFs, he would be holding alternatives that yield significant equality in expected returns. Thus, to complement the assessment a Minitab MC Test for the Variance is also computed. As the 45 compared pairs yield a test p-value of 0.2335, it can be affirmed that all ETFs in this study count with the same level of risk at a 95% confidence level. Consequently, it is not adviced to use them as a diversification tool.

As can be seen in the above correlation heatmap, all values are extremely high compared with a desired negative value around zero (for more discussion on Markowitz definitions visit this link). Therefore, as no diversification can be suggested for the Top10 ETFs in USA and all of them represent significant same return at a certain level of risk, it should be analyzed which ones offer the best Sharpe ratio to pick them up for speculative purposes (the risk free rate taken is the Treasury Yield for a 10-y bond).

The Sharpe ratio values go from -0,2973, for the minimum risk portfolio at an expected return of -1,5192%, to 0,2212 for QQQ. Even though the right process would be to run a MC Test for Proportions, the highest ratio can be selected as such tests have been already evaluated for the mean and risk (elaborated explanation of why a visual selection is enough in this link). The following scatter trace shows the location in terms of risk and return for the Top10 ETFs in USA, five portfolios equally separated, Naive, and Market Portfolio:

Regarding the market portfolio, the program "MGM_v3_2_decimals" (developed by ML Perspective) yields to accurate results between 33% and 50% of the times the algorithm is run. This percentage could have not been improved since the first version "MGM_v1_0_fractions", therefore, further actions have been taken to solve it and get success rate improved on later versions. However, it may also be due to a limitation of the algorithm itself coming from the provider as for small samples work well.

## Markov Chains Projections

Each ETF sheet counts with six tables (for elaboration on each table visit this link), the category column is the one that will determine, over time, changes in trends that can be easily identified with this approach rather than on the classic one based on the quotes themselves. Main outcomes are as follows:

SQQQ, most likely weekly return/loss: between -3,00 USD and 13,96 USD (65,19%). If in one week the ETF loses up to 36,92 USD, it is highly probable that the following week will record a loss of at least 3,00 USD. On the long run, it is expected weekly returns from -19,96 USD to 13,96 USD (92,08%). Odds for the positive side 1,08.

SPY, if in a week records returns of 15,55 USD or higher, one can expect to get a certain return on the following week between -11,85 USD and 15,55 USD with odds for the positive side of 1,5.

XLF, if a return between 2,09 USD or higher is recorded, a correction of up to -2,48 USD can be expected to be certain. Unlike SPY, odds for the positive side on the long run are 0,39.

All these calculations are based on probabilities, which can fail sometimes; however, the developed algorithm to reach those numbers has been thought to reduce such failures to their lowest level.

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