The data is summarised using box plots. Box plots provide visually intuitive visualisations of characteristics of the return data’s distributions in the full sample ranging from 2006 - 2020.
Figure4.7plots the annualised mean of each asset’s returns. Clearly, the median S&P 500 had superior returns in the dataset compared to EuroSTOXX 50 or the broad indices.
Figure 4.7:Annualised mean return of all included data
Figure4.8plots the annualised volatility of each asset’s return series. Whilst offering seemingly higher returns, S&P500 firms also had higher volatility.
Figure 4.8:Annualised volatility of all included data
4 Data 27
Figure4.9plots the annualised semideviation of each asset’s return series. Compared to figure4.8, it is a striking difference that the disparity of the semideviation of S&P 500 firms and EuroSTOXX 50 firms on the one hand, and the volatility of S&P 500 firms and EuroSTOXX 50 firms on the other hand, is significantly smaller. This means, that the contribution from positive ’swings‘ to variance for firms in the S&P 500 was a more significant then elsewhere. This implies a strong, positive skewness, as well.
Figure 4.9:Annualised semideviation of all included data
Figure4.10plots the skewness of each asset’s return series. A Gaussian distribution would have zero skewness. In the dataset, S&P 500 firms mostly had strong positive skewness, while the broad indices all had negative skewness.
Figure 4.10:Return skewness of all included data
4 Data 28
Figure4.11plots the kurtosis of each asset’s return series. A Gaussian distribution has a kurtosis of three.
A probability distribution with kurtosis𝛾4 > 3 is called leptokurtic. Note that nearly all timeseries in the data are leptokurtic, making the approximation using Gaussian distributions problematic..
Figure 4.11:Return kurtosis of all included data. Note that nearly all distributions are leptokurtic
A further description of the dataset can be found in appendix section1. It shows that daily returns of 15%
are clearly not uncommon in the dataset, contrary what would be expected from a Gaussian distribution.
This extends to the lowest observed daily return for all assets in the dataset. The dataset includes the 2008 Great Financial Crisis, 2012 European Sovereign Debt Crisis and the 2020 crash resulting from the ongoing COVID-19 pandemic. Especially for EuroSTOXX 50 firms, there have been severe single-day losses in the dataset. Such events would be too rare to occur this common under a Gaussian distribution, further providing motivation to perform semivariance-based optimisation.
Results 5
T
he analyses will be discussed per dataset for each subanalysis. A number of statistics will be presented to evaluate performance. For instance, out-of-sample value at risk, and the lowest cumulative return, among others, will be used to proxy risk. For instance, in practice investors often trade using margin accounts and funds are pressured by their own investors. Deep cumulative losses can lead to premature abortion of the investment strategy. Detailed representations of the allocated portfolio weights can be found in appendix 2. It can be concluded that hierarchical risk parity and hierarchical downside risk parity manages to estimate portfolios in a more robust manner, i.e. the ultimately allocated weights vary less depending on the sample, when compared to other methods.In most cases, semideviation-based portfolios result in better out-of-sample performance. The downside risk-optimising allocations in a majority of the cases result in lower losses and value-at-risk estimates whilst achieving higher out-of-sample returns. Furthermore, the semideviation-based portfolios achieving higher out-of-sample Sharpe ratios is not uncommon. On the contrary, it seems that downside-risk based portfolios perform less strong in situations where there is a lot of diversification to achieve, for instance when applied on the broad indices.
5.1 Main Analysis
S&P 500
Table5.1shows the main results of the subanalysis 1 on the S&P 500 dataset. The probabilistic Sharpe Ratio is calculated with respect to the ‘variance-based counterpart’. In other words, for example, using training period 2 as a sample, there is a 90% probability that the minimum semivariance portfolio achieves a higher out-of-sample Sharpe Ratio than the minimum variance portfolio, as calculated over the same sample. The weights for each portfolio can be found in appendix2.
Table 5.1:S&P 500 results summary.
Allocation method Minimum Variance Minimum Semivariance Mean-Variance Mean-Semivariance HRP HDRP
Training period 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Annualised mean return 13.46% 13.98% 14.07% 13.71% 15.42% 16.2% 27.25% 39.97% 50.54% 24.66% 38.76% 46.46% 16.4% 17.13% 17.9% 16.56% 17.3% 17.8%
Annualised volatility 14.38% 13.42% 12.99% 14.55% 13.49% 13.14% 17.96% 18.39% 21.56% 17.26% 18.3% 20.1% 17.88% 18.16% 18.22% 18.17% 18.22% 18.47%
Annualised semideviation 9.7% 9.07% 8.9% 9.7% 8.97% 8.72% 12.34% 12.21% 14.54% 11.7% 11.98% 13.26% 12.75% 12.99% 13.05% 12.97% 13.02% 13.22%
Skewness 0.65 0.344 -0.087 0.933 0.69 0.513 -0.109 -0.217 -0.176 0.177 -0.03 0.117 -0.539 -0.62 -0.682 -0.552 -0.598 -0.635
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Table 5.1:S&P 500 results summary.
Allocation method Minimum Variance Minimum Semivariance Mean-Variance Mean-Semivariance HRP HDRP
Training period 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Kurtosis 18.413 16.698 15.831 21.449 18.773 17.887 9.125 13.462 12.504 11.334 13.554 13.075 21.835 21.846 21.372 22.187 21.732 21.529 Sharpe Ratio 0.95 1.043 1.079 0.956 1.13 1.209 1.432 1.922 2.006 1.364 1.882 2.0 0.939 0.962 0.996 0.935 0.968 0.98 Sortino Ratio 1.388 1.541 1.582 1.414 1.72 1.858 2.208 3.272 3.477 2.108 3.235 3.504 1.286 1.319 1.371 1.277 1.329 1.346 Diversification Ratio 1.661 1.829 2.01 1.652 1.85 2.003 1.597 1.708 1.63 1.607 1.695 1.69 1.626 1.615 1.62 1.611 1.608 1.605 Concentration Ratio 0.063 0.057 0.064 0.07 0.06 0.063 0.114 0.12 0.092 0.121 0.141 0.094 0.004 0.003 0.003 0.003 0.003 0.003 Probalistic Sharpe Ratio 0.539 0.9 0.964 0.151 0.344 0.474 0.474 0.529 0.412 VaR(5%) -1.16% -1.11% -1.09% -1.21% -1.14% -1.05% -1.65% -1.56% -1.93% -1.55% -1.5% -1.78% -1.54% -1.6% -1.55% -1.55% -1.57% -1.58%
VaR(1%) -2.67% -2.46% -2.24% -2.63% -2.35% -2.29% -3.25% -2.94% -3.79% -3.07% -2.99% -3.32% -2.98% -3.01% -3.0% -2.99% -3.01% -3.1%
Lowest Cumulative Return -2.8% -2.11% -3.38% -2.37% -1.92% -2.66% -8.35% -6.48% -7.46% -6.29% -4.81% -5.73% -6.15% -6.56% -7.21% -6.32% -6.56% -7.1%
From table5.1a number of conclusions can be drawn. In all but two cases∗, the downside-risk based port-folios realised higher mean returns. The mean-semivariance portfolio achieved lower out-of-sample volatility and all minimum-semivariance and mean-semivariance optimal portfolios achieved lower semideviation than their counterparts. The downside-risk based portfolios achieved similar levels of out-of-sample diversification as the difference between the diversification ratios are small. That is rather impressive, as the diversification ratio is based on the reduction in portfolio variance. The portfolios estimated based on downside risk also had similar levels of concentration as measured by the concentration ratio. The downside risk-based also portfolios managed to achieve slightly lower outer sample value-at-risk. Furthermore, downside-risk based portfolios consistently achieved less cumulative losses, except for one case†. Appendix3shows visualisations of the different portfolios’ performances on the S&P 500 firms. Although the semivariance-based portfolios did not outperform in terms of risk-adjusted returns, semivariance-based optimisation is successful in achieving less severe drawdowns.
EuroSTOXX 50
Table 5.2:EuroSTOXX 50 results summary.
Allocation method Minimum Variance Minimum Semivariance Mean-Variance Mean-Semivariance HRP HDRP
Training period 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Annualised mean return 3.84% 5.75% 7.35% 5.39% 6.35% 7.49% 19.68% 19.32% 19.58% 19.12% 18.66% 19.37% 5.45% 6.29% 5.88% 6.29% 6.12% 6.45%
Annualised volatility 14.02% 12.57% 12.33% 13.88% 12.58% 12.34% 18.56% 16.77% 16.28% 18.21% 16.27% 16.13% 16.18% 15.31% 15.42% 16.43% 15.3% 15.39%
Annualised semideviation10.28% 8.97% 8.86% 10.03% 8.88% 8.74% 12.79% 11.59% 11.53% 12.57% 11.26% 11.42% 11.84% 11.22% 11.29% 12.0% 11.2% 11.25%
Skewness -1.156 -0.793 -1.202 -0.941 -0.584 -0.811 -0.152 -0.369 -0.998 -0.161 -0.4 -0.985 -1.216 -1.294 -1.266 -1.209 -1.302 -1.232 Kurtosis 17.376 17.924 24.383 16.59 17.794 22.875 10.158 9.799 15.383 10.256 9.82 15.116 26.33 26.746 26.957 26.545 27.42 26.716
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∗Mean-variance estimated by period 2 HRP estimated by period 2
†Minimum variance portfolio estimated by period 3
5 Results 31
Table 5.2:EuroSTOXX 50 results summary.
Allocation method Minimum Variance Minimum Semivariance Mean-Variance Mean-Semivariance HRP HDRP
Training period 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Sharpe Ratio 0.339 0.508 0.637 0.448 0.552 0.647 1.061 1.138 1.18 1.052 1.134 1.179 0.41 0.476 0.449 0.454 0.465 0.484 Sortino Ratio 0.373 0.641 0.83 0.537 0.714 0.857 1.539 1.667 1.697 1.521 1.658 1.697 0.461 0.561 0.521 0.524 0.546 0.573 Diversification Ratio 1.417 1.541 1.517 1.406 1.52 1.491 1.23 1.326 1.277 1.239 1.349 1.286 1.416 1.454 1.45 1.406 1.45 1.448 Concentration Ratio 0.105 0.168 0.149 0.109 0.179 0.171 0.362 0.274 0.357 0.35 0.261 0.347 0.027 0.026 0.024 0.027 0.026 0.025
Probalistic Sharpe Ratio 0.999 0.878 0.589 0.42 0.466 0.495 0.87 0.398 0.803
VaR(5%) -1.3% -1.13% -1.08% -1.28% -1.14% -1.07% -1.72% -1.61% -1.45% -1.71% -1.58% -1.45% -1.41% -1.31% -1.29% -1.4% -1.28% -1.29%
VaR(1%) -2.83% -2.18% -2.11% -2.79% -2.17% -2.07% -3.42% -3.11% -2.81% -3.34% -3.02% -2.81% -3.23% -3.03% -3.02% -3.23% -2.92% -2.93%
Lowest Cumulative Return -8.5% -6.07% -5.74% -8.46% -5.82% -5.34% -7.62% -6.52% -5.74% -7.44% -6.25% -5.78% -11.99% -10.68% -11.35% -12.21% -11.16% -11.07%
Table5.2shows the main results of the subanalysis 1 on the EuroSTOXX 50 dataset. The weights for each portfolio can be found in appendix2. The minimum-semivariance portfolio managed to achieve remarkably better returns whilst not sacrificing in terms of volatility and semideviation compared to the minimum-variance portfolio, resulting in superior out-of-sample Sharpe ratios and relatively high (although not always statistically significant) probabilistic Sharpe ratios. However, the other portfolios performed similarly in terms of returns, volatility and semideviation, thus also resulting in similar Sharpe ratios and Sortino ratios.
This pattern extends to the out-of-sample value-at-risk estimates and lowest cumulative returns, although the downside-risk based portfolios were slightly superior on this aspect. The downside-risk based portfolios are remarkably similar to their counterparts in terms of concentration and diversification. Appendix 3 shows visualisations of the different portfolios’ performances. The minimum-semivariance outperform the minimum-variance portfolios in terms of returns, whilst experiencing less extreme drawdowns during the COVID-19 crash.
Broad Indices
Table 5.3:Broad Index results summary.
Allocation method Minimum Variance Minimum Semivariance Mean-Variance Mean-Semivariance HRP HDRP
Training period 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Annualised mean return 7.34% 7.4% 7.31% 7.92% 7.51% 7.66% 11.8% 11.71% 12.08% 4.2% 5.68% 4.83% 7.39% 7.4% 7.16% 7.23% 7.05% 7.2%
Annualised volatility 5.36% 5.24% 5.23% 5.42% 5.24% 5.25% 7.18% 7.31% 7.97% 2.79% 4.59% 5.28% 6.64% 6.51% 7.0% 7.35% 7.3% 7.59%
Annualised semideviation 4.07% 4.01% 4.01% 4.07% 3.99% 3.98% 5.35% 5.31% 5.68% 2.1% 3.36% 3.76% 5.07% 5.0% 5.38% 5.62% 5.56% 5.75%
Skewness -2.919 -3.129 -3.138 -2.685 -2.945 -2.929 -2.034 -1.476 -0.949 -2.755 -1.877 -0.766 -2.684 -2.89 -2.747 -2.563 -2.443 -2.236 Kurtosis 41.554 43.667 43.669 38.854 41.928 42.318 25.947 21.362 17.355 42.794 29.963 21.766 43.1 44.315 43.217 41.651 44.043 41.172 Sharpe Ratio 1.349 1.388 1.375 1.434 1.408 1.433 1.59 1.552 1.471 1.491 1.226 0.921 1.108 1.13 1.024 0.987 0.97 0.954 Sortino Ratio 1.804 1.844 1.823 1.946 1.885 1.922 2.204 2.205 2.129 2.004 1.69 1.284 1.458 1.48 1.332 1.287 1.268 1.251 Diversification Ratio 1.548 1.507 1.505 1.558 1.471 1.476 1.547 1.623 1.554 1.395 1.237 1.0 1.726 1.724 1.695 1.667 1.698 1.677 Concentration Ratio 0.422 0.401 0.392 0.443 0.416 0.422 0.405 0.394 0.527 0.366 0.419 0.57 0.213 0.218 0.174 0.173 0.181 0.153
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Table 5.3:Broad Index results summary.
Allocation method Minimum Variance Minimum Semivariance Mean-Variance Mean-Semivariance HRP HDRP
Training period 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
Probalistic Sharpe Ratio 0.736 0.557 0.66 0.251 0.001 0.0 0.115 0.057 0.235 VaR(5%) -0.39% -0.37% -0.37% -0.39% -0.38% -0.37% -0.58% -0.61% -0.72% -0.2% -0.35% -0.44% -0.45% -0.44% -0.48% -0.53% -0.51% -0.55%
VaR(1%) -0.75% -0.86% -0.87% -0.79% -0.83% -0.85% -1.18% -1.16% -1.35% -0.44% -0.63% -0.73% -1.01% -1.05% -1.06% -1.13% -1.07% -1.13%
Lowest Cumulative Return-2.15% -2.92% -3.08% -2.03% -2.77% -2.91% -4.36% -3.84% -2.82% -1.63% -0.48% -0.06% -4.62% -4.72% -5.04% -5.35% -4.88% -5.31%
Table5.3shows the main results of the subanalysis 1 on the selection of indices. All portfolio weights can be found in appendix2. The minimum-semivariance portfolios strongly outperformed the minimum-variance portfolios in terms of mean returns, whilst having similar levels of volatility and semideviation, resulting in superior Sharpe ratios and Sortino ratios. However, this statement cannot be extended to the other portfolios.
The mean-semivariance portfolios’ allocations resulted into a remarkably low-risk portfolios out-of-sample.
Appendix3shows plots of the different portfolios’ performances. The mean-semivariance optimal portfolios experienced less volatility and drawdowns due to their unexpectedly ‘risk-adverse’ allocation.