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If prior posts have been checked by the reader, it will be already known that the four methodologies used are: Linear programming (LP), Simulation (Sim), Bootstrapping plus LP (Bootlp), and Bootstrapping plus sim (Bootsim). As stock quotes are stochastic processes, hidden behaviors will only be revealed as approaches are compared to each other, at least two needed to use one as a benchmark. A simple example, the pictureÂ¹ below that has been selected as this post's header:

According to LP, none of the expected returns from the LATAM ETFs quoting in USA is significantly different from the others meaning that if by investing in one of them one gets a higher/lower return than others, the difference would be only coincidential. However, here a more interesting case has been revealed: while bootlp gave similar results bootsim didn't, the latter recorded that at least one of the means is not different from the others by chance but statistically significant:

It is not task of this post neither website to be a teacher on the field; then, main takeaways are as follows:

The pair ('MEXX', 'BZQ') are significantly different.

The CI merely goes over zero, why?

The algorithm doesn't use feed to generate the bootstrap, therefore, every run differs.

Additionally, to check out why three out of four tests didn't find the significantly different pair, the boxplots give the clue on explaining this resultÂ²:

First outcome, not major differences around the mean or median. Secondly, LP shows a slightly less scattered expected returns against Bootlp, the reason lies on the fact that while LP has total degrees of freedom of 3401, Bootlp records 18719. Lastly, there are three ETFs expected to count with a significant higher risk form the others BRZU, MEXX, and UBR, all with pronounced outliers. A strongly based statement can be said after the anova report for the variance is inspected:

While the p-value with the LP and Sim method gives 0.19 indicating no significant difference in risks among the LATAM ETFs traded in USA, Bootlp and Bootsim find it with p-values at 0,003 and 0,001. According to the latter, The risk of EWZ is significantly different from that one of MEXX and BZQ; indeed, 37 pairs count with signficant difference. Moreover, as in the prior case analyzed, LATAM ETFs don't give major diversification opportunities, the following heatmaps show no distinct results within methodologies:

No sense to show the other heatmaps as all of them have only slight differences. For the other heatmaps:

At the same time, the missing comprehensive report for Bootlp can be accessed here.

Finally, let's have a look at the four scatter plots:

Sharpe Ratios (SR) go from -16,84% to 5,02%.

Naive portfolio returns 0,098% with a risk of 4,60% at a SR of 1,06%.

Minimum variance portfolio returns -0,01% with risk 1,23% and SR -4,83%.

Efficient portfolio returns 0,3% with risk 4,99% and SR 5,02%, composition:

EWW: 3,98%

ARGT: 86,6%

FLMX: 4,31%

MEXX: 5,10%

SRs go from -0,78% to 2,67%.

Minimum variance portfolio returns -0,04% with risk 2,81% and SR -0,42%.

Efficient portfolio returns 0,19% with risk 5,26% and SR 2,67%.

SRs go from -16,83% to 4,32%.

Naive portfolio returns 0,01% with a risk of 4,85% at a SR of -0,68%.

Minimum variance portfolio returns -0,12% with risk 1,01% and SR -16,83%.

Efficient portfolio returns 0,27% with risk 5,25% and SR 4,31%, there is an implied higher risk asociated as this is not a portfolio but a security: ARGT.

SRs go from 1,11% to 4,49%.

Minimum variance portfolio returns 0,11% with risk 2,80% and SR 2,07%.

Efficient portfolio returns 0,27% with risk 5,01% and SR 4,49%.

Otucomes:

There is not best methodology, it depends on the decision maker's profile e.g. a +40 years old married with children investor may choose to follow the simulation method as it is more concervative while a single one on the 20s would take a decision based on the Bootlp method.

Despite prior statement, are the efficient portfolio's return, risk and SR signficant different from each other? Here it would be correct to say that results are not conclusive as the sample is not big enough (only 18 ETFs).

Best option could be the portfolio with expected return 0,11%, 2,20% risk and SR 2,75% that invests mostly in Argentinian and Brazilian securities.

Â¹ This post includes comprehensive Excel reports by clicking on the images.

Â² Sim and Bootsim are not shown here as simulation only makes a difference in the scatter plots.

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.