Comparison of Replicated Portfolio w/ Individual Strategies

We can thus assume that we do have some actual alpha in our 2/3 ensemble strategy. Let us instead try replicating the 2/3 ensemble by introducing the returns for our individual strategies to gain further insight into what contributes to the behaviour of the strategy:

Dep. Variable: 2/3 R-squared: 0.869

Model: OLS Adj. R-squared: 0.868

Method: Least Squares F-statistic: 749.9 Date: Wed, 13 May 2020 Prob (F-statistic): 0.00 Time: 09:10:43 Log-Likelihood: 6768.3

coef std err z P>|z| [0.025 0.975]

const 5.037e-05 8.29e-05 0.608 0.543 -0.000 0.000 FF Momentum 0.0242 0.019 1.304 0.192 -0.012 0.061 FF SMB / Size -0.0229 0.023 -0.982 0.326 -0.069 0.023 FF HML / Value -0.1038 0.029 -3.642 0.000 -0.160 -0.048 FF Market 0.1676 0.020 8.246 0.000 0.128 0.207 FF RMW 0.0234 0.035 0.670 0.503 -0.045 0.092 FF CMA 0.0576 0.043 1.340 0.180 -0.027 0.142

BAB 0.1183 0.025 4.725 0.000 0.069 0.167

XGBoost 0.4292 0.021 20.542 0.000 0.388 0.470

LSTM 0.2532 0.018 14.302 0.000 0.219 0.288

Momentum 0.1700 0.016 10.587 0.000 0.139 0.201 Omnibus: 107.142 Durbin-Watson: 1.954

Prob(Omnibus): 0.000 Jarque-Bera (JB): 502.301

Skew: 0.032 Prob(JB): 8.45e-110

Kurtosis: 5.776 Cond. No. 568.

The highest loadings are to our 3 individual strategies. This is expected, as those are the actual strategies that makes up the 2/3 ensemble. It’s also worth noting that XGBoost is by far the highest loading, which intuitively makes sense since this is the ”highest performing” strategy of the 3. It is also worth noticing that LSTM has a higher coefficient than momentum. So even though LSTM performed much worse, it is more important in the 2/3 ensemble. The 2/3 ensemble with HRP was able to outperform all our other strategies, so it is surprising that LSTM, with its relatively bad performance, can help in this regard.

After removing the insignificant coefficients in a stepwise fashion, we arrive at the following estimates:

Dep. Variable: 2/3 R-squared: 0.866

Model: OLS Adj. R-squared: 0.865

Method: Least Squares F-statistic: 1443.

Date: Wed, 13 May 2020 Prob (F-statistic): 0.00 Time: 09:10:43 Log-Likelihood: 6748.8

coef std err z P>|z| [0.025 0.975]

const 5.223e-05 8.45e-05 0.618 0.537 -0.000 0.000 FF Market 0.1894 0.020 9.399 0.000 0.150 0.229

BAB 0.1690 0.023 7.313 0.000 0.124 0.214

XGBoost 0.4314 0.021 20.228 0.000 0.390 0.473

LSTM 0.2331 0.017 13.981 0.000 0.200 0.266

Momentum 0.1786 0.015 11.559 0.000 0.148 0.209 Omnibus: 108.750 Durbin-Watson: 1.928 Prob(Omnibus): 0.000 Jarque-Bera (JB): 517.829

Skew: 0.031 Prob(JB): 3.59e-113

Kurtosis: 5.819 Cond. No. 271.

Once again, the FF Market and BAB portfolios are included, but with lower coefficients. This works out to the following composition:

Table 10: Composition of replicated portfolio with individual strategies

Constituents FF Market Betting Against Beta XGBoost LSTM Momentum Portfolio weights 0.1894x 0.1690x 0.4314x 0.2331x 0.1786x

The replicated portfolio in the table above is thus the closest thing we can construct to replicate the return movements of our 2/3 ensemble strategy when also using our individual strategies in isolation.

Notice that we still produce 0.013 annualized alpha in the ensemble strategy when compared to the replicated portfolio while maintaining a beta of 1. A comparison plot of the cumulative returns between the actual 2/3 ensemble and the replicated portfolio can be seen in Figure 58 below:

Figure 58: Cumulative returns between the 2/3 ensemble strategy and the replicated portfolio including the individual strategies

We observe that the 2/3 ensemble and the replicated portfolio exhibits similar behaviour though with

the 2/3 ensemble displaying a higher return on average. Additionally, in order to gauge whether the difference between the two portfolios are caused by a single event or gradually, a plot of the difference in cumulative returns along with the drift can be seen in Figure 59:

Figure 59: Difference in cumulative returns between the 2/3 ensemble strategy and the replicated portfolio including the individual strategies

Notice that the difference in cumulative returns are lower as compared to the replicated portfolio without the individual strategies, this is also evident from the higherR² value, which is also expected since these are the actual strategies that makes up the ensemble. We still observe a positive long-term drift and an annualized alpha of 0.013 that implies that the ensemble outperforms the combination of the individual strategies and portfolios in isolation.

9 Conclusion

In this final chapter, we will use our analysis to answer and discuss the questions from the thesis statement. Additionally, we will propose some future work that could potentially improve the results from this thesis. This thesis investigated the relative profitability of a long-only trading strategy based on combining an ensemble of models to confidently predict and identify stocks with high expected returns. Specifically, we have constructed three individual strategies that are trained on 94 different fundamental key-figures that describes the valuation of a company, along with several price-related variables for approximately 4200 stocks from December 1997. Every Monday, the strategies identifies top stocks based on their corresponding ranking method. From the top rankings lists, we examine portfolio formation with both a naive 1/N allocation and Hierarchical Risk Parity allocation.

9.0.1 The Individuals

The individual strategies were able to generate positive returns from start 2014 to end 2019. The XGBoost strategy with HRP allocation came out on top between the three strategies with a Sharpe ratio of 1.0, a Sortino ratio of 1.43, annual returns of 14.5% and an α of 0.07 against the S&P 500.

However, the momentum strategy with HRP allocation performed better with regards to volatility and risk, with the lowest annual volatility of 14.2%, lowest max. drawdown of -17.5% and the lowest β against the market at 0.59. The higher returns for XGBoost offsets the higher volatility, result-ing in a higher Sharpe. Furthermore, despite XGBoost havresult-ing a much higher loadresult-ing to the market than the momentum strategy, it still manages to produce the same amount of alpha, thus indicating that Extreme Gradient boost excels at producing excess returns. The LSTM model does not out-perform the two others on any of the metrics examined in this thesis. We can, however, conclude that the HRP method is significantly better on all metrics for all models, LSTM as well. Only the daily turnover for momentum with naive allocation beats the other models with 9.6%. Specifically, we wanted to show that the HRP allocation provides a better diversification than 1/N and that it is able to reduce volatility and risk. This was also the case for both momentum and XGBoost, but not LSTM.

In summation: In terms of excess returns, the XGBoost strategy was superior and greatly outper-formed the S&P500 index. The momentum strategy also provided great results, characterized by lower volatility with alphas similar to that of the XGBoost. The LSTM strategy did not prove to be as efficient as the other strategies, and barely produced the same returns as holding the S&P500 index.

The implementation of the HRP diversification was very successful, as it consistently produced better results in comparison to the Naive portfolios.