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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. Therefore, the following post contains three sections that will be analyzed as follows:
Anova Reports for the Mean and Variance: LP/Sim vs Bootlp/Bootsim.
Correlation Heatmaps: LP/Sim vs Bootlp/Bootsim.
Efficient Portfolios: All four methods compared.
Consequently, regarding Anova reports, the results are as follows² (click on image to access full report):
According to LP, none of the expected returns from the ETFs in Europe is significantly different from the others, it means that the difference among their returns is only coincidential. Again, as in the case of the USA Top10, Top95, and LATAM ETFs, to invest within one single region does not truly diversify risk, it means that, despite investing across sectors, within a same region the unsystematic risk is not overcome. Thus, may bootstrapping reveal something that solely LP can not? its Anova reports are as follows:
Results don't differ³ between them but with LP/Sim regarding the null hypothesis as it is rejected with a p-value near zero. However, let's analyze pairwise the HSD test to check if any dissimilarities arise between both runs:
Non-significantly different : 1.066 pairs (77%)
Significantly different : 312 pairs (23%)
Non-significantly different : 1.049 pairs (76%)
Significantly different : 329 pairs (24%)
As can be seen, proportions are around the 75/15 ratio, which could seem to differ from the 85/15 found in USA (it can't be confirmed they are different, why?). Consequently, if one would want to hold whole Europe an option could be EUEA.AS (EURO STOXX 50), its expected return counts with a significant difference against BX4.PA, BXX.PA, XSD2.L, 3ITS.MI, XACT-BEAR-2.ST, XACT-BEAR.ST, XBRMIB.MI, and 3BAL.MI all leveraged with the two latter being long while the former short.
Therefore, any of these pairs has a real diversification opportunity? at first glance, the answer would be not because the leveraged ETFs would either delete one's profits (short) or multiply his risk (long). Similarly, looking at the correlation matrix it is confirmed that none has an option with correlations at -0,4049, -0,4002, -0,4049, -0,5055, -0,0675, and 0,0814. A brief look at the heatmaps will help to identify how possible it is to diversify within the European ETFs :
From a graphical point of view both look pretty similar; however, there are slightly differences within their values e.g. the pair (BTP10.MI;BX4.PA) with LP records a correlation of -0,38 up from -0,13 recorded by Bootlp. Generally, LP computes higher correlations against bootstrapping; nonetheless, both ratify the non-possibility to diversify within a region: the former's only option would be BX4.PA:XACT-OBLIGATION.ST (-0,0008) but both are not significantly different thus such mixture raises risk.
In addition, this pairs do the same e.g. BTP10.MI is the ETF of Italy's Government bonds while BX4.PA is the leveraged inverse by two of the CAC 40 meaning that in a bear market investors would move funds towards these type of options thus move in same direction. About the latter promising option the case is exactly the same. The nagtive correlation -instead of being positive- may be the fact that these have been calculated as USD and not as EUR, the original currency.
Finally, let's have a look at the four scatter plots:
Sharpe Ratios (SR) go from -16,48% to 16,58%, up from -38,3% and 11,90%⁴ respectively.
Naive portfolio returns 0,14%, up from 0,04%, with a risk of 2,13% also up from 1,43%, at a SR of 5,42%, up from -2,46%.
Minimum variance portfolio returns -0,09%, down from -0,02%, with risk 0,88%, up from 0,73%, and SR -13,26%, unchanged from -13,37%.
Efficient portfolio returns 0,57%, unchanged from 0,51%, with risk 3,27%, down from 3,63%, and SR 16,58%, up from 11,92%, weights are:
SRs go from -0,56%, up from -15,42%, to 8,31%, up from 2,87%.
Minimum variance portfolio returns 0,05%, unchanged from -0,02%, with risk 1,34%, up from 0,85%, and SR 1,89%, up from -11,40%.
Efficient portfolio returns 0,24%, unchanged from 0,14%, with risk 2,62%, up from 2,29%, and SR 8,31%, up from 2,87%.
SRs go from -18,18%, up from -41,1%, to 12,59%, down from 15,8%.
Naive portfolio returns 0,10%, up from 0,06%, with a risk of 3,06%, up from 1,42%, at a SR of 2,43%, up from -1,10%.
Minimum variance portfolio returns -0,05%, unchanged from -0,02%, with risk 1,73%, up from 0,75%, and SR -4,46%, up from -12,28%.
Efficient portfolio returns 0,86%, up from 0,62%, with risk 6,61%, up from 3,43%, and SR 12,59%, down from 15,81%, weights are:
SRs go from -0,47%, up from -15,61%, to 6,02%, up from 2,45%.
Minimum variance portfolio returns 0,04%, unchanged from -0,03%, with risk 3,52%, up from 0,73%, and SR 0,30%, up from -14,87%.
Efficient portfolio returns 0,30%, up from 0,12%, with risk 4,64%, up from 1,82%, and SR 6,02%, up from 2,45%.
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; however, as this sample is big enough (checked by the degrees of freedom) the portfolio got with the Bootlp method could be suggested.
LP and Bootlp recommend XACT-BULL-2.ST in big proportion, this is a leveraged ETF by two of the OMXS30 Gross index from Sweden; therefore, according to the code "Pattern Search" it is expected a 1.04% weekly return for this index towards August 2022, it equals 37,81% up to 01/08/2022, looking at the real data between 20/12/2021 and that date this ETF returned 55,23% in USD⁵.
All SR ranges have seemingly shifted upwards, but is it enough to confirm that Europe is more profitable than USA? The answer is not, why?.
As in Top10 and Top95 USA, and Latam, to invest through sectors is not enough to diversify risk, it may be needed to invest through continents but such study has not been done yet.
¹ All ETF's whose historical prices are available on public domain.
² Sim and Bootsim are not shown here as simulation only makes a difference in the scatter plots.
³ Both bootstrapping methods can differ from each other as the option 'seed' to generate the random numbers is not used; therefore, each run is unique.
⁴ All comparisons are with USA Top95 ETFs.
⁵ Price USD 20/12/2021: 181.625
Price USD 01/08/2022: 117,0
(181.625-117)/117 = 0,5523
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.