ml-portfolio-optimization

ML based Portfolio Management with Macro-Financial Indicators

CS7641 Project - Group 11

I. Introduction

Literature Review

In rapidly evolving financial markets, traditional investment strategies are increasingly inadequate for risk management and return forecasting [1]. Research has highlighted the importance of macroeconomic influence on stock prices [2]. The lack of data and computing resources constrained the integration of machine learning into investment portfolio management. [3]. However, technological advancements now enable the inclusion of non-financial information such as macroeconomic trends and social media sentiment into predictive models [4][5].

Dataset

II. Problem Definition

Problem

The efficacy of traditional portfolio management techniques solely depend on historical stock prices and basic financial indicators. These models fail to account for the broader economic context, resulting in inadequate portfolio allocations and higher risk exposure. To achieve near-perfect forecasts, it is imperative to build more sophisticated frameworks using wider ranges of data sources.

Motivation:

The motivation behind this project stems from the potential for ML models to address the shortcomings of existing portfolio optimization approaches. By integrating macroeconomic indicators and possibly sentiment analysis into portfolio optimization, we aim to enhance the accuracy and robustness of investment strategies, allowing for better risk management and return maximization and providing investors with valuable insights into market dynamics.

III. Methods

Collected datasets have different formats and intervals of records. The examples of preprocessing methodologies are listed below:

We utilized supervised/unsupervised models useful for the time series datasets. Our current models include K-means and DBSCAN clustering for unsupervised and Autoregressive Integrated Moving Average (ARIMA) and Long Short Term Memory Model (LSTM) for supervised. The stock data has been divided into two sets: training data covering the period from 2018 to 2022, and test data for the year 2023.

We chose K-means particularly because it is simple and fast, and we did not feel the need to obtain probabilities from a soft clustering model like GMM. We plan to use K-means primarily to analyze stock similarity, facilitating the selection of stocks across various clusters to enhance portfolio diversification. Some challenges that using a clustering algorithm like K-means poses for this, however, is determining a proper distance function that accurately identifies similarities between stocks.

We chose to implement DBSCAN for our other unsupervised learning method. We chose DBSCAN particularly because it is density-based and we don’t need to specify the number of clusters beforehand. We plan to use DBSCAN to also analyze stock similarity and compare it to the performance of K-means. Like K-means, we also need to determine a proper distance function, but we also have to choose an optimal epsilon radius and minimum points per cluster.

We primarily utilized sklearn to train our K-means, PCA and DBSCAN models and for performance metrics for clustering, and we also utilized pandas and numpy for general data manipulation.

To initially simplify data preprocessing, we limited our model to 2 features and experimented with what we chose for the features, such as mean returns, volatility, GDP growth, unemployment rate, etc. We mainly compared stocks via mean returns and volatility, which is calculated via the mean and variance of each stock’s percent change in value over the last 5 years. We chose mean returns and volatility primarily because they were simple stock-specific indicators. We also attempted to compare stocks via macro-economic indicators such as GDP growth, unemployment rate, and CPI.

We also compared the performance of these chosen features to feature extraction via PCA. We wanted to compare the manually selected features we chose to machine learning methods of selecting features. We chose PCA specifically because PCA is a popular unsupervised learning method of performing feature extraction.

Through the elbow method, we confirmed that the most optimal number of clusters without PCA was 3, and the optimal number of clusters with PCA was 4.

For K-means, we realized that our choice of macro-economic indicators such as GDP, CPI, unemployment rate may not be the best candidates for features for unsupervised clustering since these economic indicators don’t differentiate between stocks. In an effort to utilize more stock specific economic indicators to increase the performance of our clustering algorithm, we assessed the balance sheet per stock as our feature data set. We manually scrapped the balance sheet data from the following source [7]. Revenue Growth (YoY), shares change, gross margin, operating margin, profit margin, free cash flow margin, EBITDA margin, cash growth, and depth growth are utilized for clustering. None of the balance sheet data outperformed the volatility in terms of silhouette score. The comparison of the silhouette score is shown below:

One main reason volatility shows the best clustering is that the balance sheet is less responsive to market changes than the volatility. In addition, the balance sheet data could be influenced by how a company chooses to report its finances, which causes noise in the data. Therefore, we primarily focused on optimizing volatility for Kmeans.

We explored using K-means to cluster 10 of most popular stock indices. Although K-means doesn’t select stocks directly, we altered the algorithm to choose the stocks with the minimum volatility for each cluster. We estimated our K-values by seeing which data points on the graphs were closest to each other. Our K-means algorithm for mean returns vs volatility for k=3 selected JPM, NVDA, and VOO,and our K-means algorithm with PCA for k=4 selected JPM, NVDA, TSLA, and VOO. Since PCA turned out to output quite similar results compared to mean returns vs volatility, this confirmed our assumptions that mean returns and volatility were good measurements to categorize stocks.

We utilized DBSCAN to improve silhouette score using a similar approach to K-means. We chose the minimum number of points per cluster to be 3 since when we plotted it out, we saw a group of 3 as the smallest cluster. We also estimated the epsilon radius through the elbow method.

Through the elbow method, we found that the most optimal epsilon radius without PCA would be somewhere between 1.25 and 1.5.

We assessed the silhouette scores for DBSCAN, using volatility and balance sheet data, similar to our evaluation with K-means. Volatility consistently outperformed balance sheet data, with scores at least 2.7 times higher. This finding aligns with the results from our K-means analysis, further confirming that volatility is highly sensitive to changes in stock prices.

Because volatility was the most sensitive to stock prices, we decided to also cluster stocks for DBSCAN in terms of means returns vs volatility. It turns out that an epsilon of 1.25 and 1.5 produces the exact same result, with two selected clusters. Choosing stocks per cluster with the highest volatility, both selected NVDA and VOO.

Unfortunately, PCA wasn’t as useful for DBSCAN. No matter what epsilon we chose, applying PCA always resulted in a large number of clusters as outliers (denoted as –1 in the plot below).

For the supervised learning aspect of our project, we opted for the ARIMA (Autoregressive Integrated Moving Average) model, renowned as one of the most widely used models for forecasting linear time series data. Its extensive adoption stems from its reliability, efficiency, and capability in forecasting short-term fluctuations in stock market movements. ARIMA models are a popular tool in time series analysis, particularly in forecasting future values based on historical data. By incorporating lagged moving averages, ARIMA models provide a method to smooth out fluctuations and identify patterns in time series data. However, their reliance on past observations inherently assumes that future behavior will resemble the past, making them susceptible to inaccuracies in volatile market conditions like financial crises or during periods of rapid technological advancements. While ARIMA models offer valuable insights, it’s essential to supplement their predictions with other analytical techniques and consider the broader economic and technological landscape for more robust forecasting in dynamic markets. We conducted exploratory data analysis (EDA) on a selection of 10 ticker stocks, focusing on their adjusted closing prices from 2018 to 2024. Our EDA encompassed various aspects, including the visualization of stock volatility, daily returns, quarterly returns, annual returns, adjusted closing prices over the years, and a covariance matrix heatmap of portfolio returns.

While preparing our data for modeling, we rigorously checked for seasonality and stationarity, both crucial conditions for employing the ARIMA model. Seasonality, characterized by regular and predictable patterns that repeat over a calendar year, can adversely impact regression models. To ensure stationarity, a prerequisite for ARIMA, we employed differencing techniques to remove trends and seasonal structures from the data. The Augmented Dickey-Fuller (ADF) test emerged as a pivotal tool in our analysis, widely employed in time series analysis for assessing stationarity [6]. This statistical test aims to determine the presence of a unit root within a series, indicating non-stationarity. The test formulates null and alternative hypotheses, with the null hypothesis positing the existence of a unit root, and the alternative hypothesis suggesting its absence. Failure to reject the null hypothesis indicates non-stationarity, implying the series may exhibit either linear or difference stationary characteristics.

In addition to checking for seasonality and stationarity, we performed seasonal decomposition to extract the underlying time series components, namely Trend and Seasonality, from our data. To mitigate the magnitude of values and address any growing trends within the series, we initially applied a logarithmic transformation. Subsequently, we calculated the rolling average of the log-transformed series, aggregating data from the preceding 12 months to compute mean consumption values at each subsequent point in the series. Following data preprocessing, we partitioned the dataset into training and testing subsets, preserving the chronological order of observations to maintain the time-series nature of the data.

For tuning the hyperparameters of the ARIMA model, denoted as a tuple (p, d, q), we employed a systematic approach. Here, ‘p’ represents the number of autoregressive terms, ‘d’ denotes the number of nonseasonal differences, and ‘q’ signifies the number of lagged forecast errors in the prediction equation. To determine suitable values for ‘p’ and ‘q’, we analyzed autocorrelation and partial autocorrelation plots. Specifically, we utilized the partial autocorrelation function (PACF) plot to identify the value of ‘p’, as the cutoff point in the PACF plot indicates the autoregressive term. Conversely, the autocorrelation function (ACF) plot aided in determining the value of ‘q’, with the cutoff point in the ACF plot corresponding to the moving average term [6].

For each of the stock, we analyzed whether linear differencing would work, or the price data needs to be transformed using logarithm for stable modelling using ARIMA. Then, each stock was investigated for stationarity and seasonal trends. All timeseries showed stationarity after single differencing, passing the ADF test.

Additionally, we leveraged grid search methods and the auto_arima function to systematically explore various combinations of hyperparameters and select the optimal configuration for our ARIMA model. This comprehensive approach ensured robust parameter tuning, facilitating the development of an effective forecasting model for our time series data.

The table below shows the values of the parameters (p,d,q) for each of the stock’s time series [6]

Stock (p,d,q) Log Data
AAPL (1,1,0) True
AMD (2,1,2) True
AMZN (0,1,0) False
F (0,1,0) True
GOOG (1,1,1) False
INTC (0,1,2) False
JPM (3,1,2) False
MSFT (0,1,1) False
MS (2,1,0) True
NVDA (2,1,0) True
TSLA (0,1,0) True
VOO (2,1,2) False

Another supervised learning model that we utilized was LSTM. While LSTMs is a Recurrent Neural Network, their ability to retain information over lengthy sequences enables them to capture long-term dependencies, perhaps making them more successful for modelling complicated temporal patterns such as stock price swings than typical RNNs. An investment portfolio comprised of stocks is constructed by leveraging LSTM-based predictions of stock prices and fine-tuned weight optimization. For implementation we use sklearn and PyTorch. To ensure compatibility with LSTM, we utilize sklearn’s MinMaxScaler to normalize the data, essential for scaling it within the 0 to 1 range, a prerequisite for LSTM model training. Initially, we prepare the data, then employ NumPy to convert it into an array. If we designate today’s adjusted close price as the response variable, the features will consist of the adjusted close prices from the preceding 100 days. To generate predictions, we first apply the MinMaxScaler again to normalize the data and adjust the shape as necessary. Then, we proceed to perform predictions on each ticker individually to obtain their respective loss and predicted values.

The LSTM model parameters are configured as follows:

The following explains the design decisions for the LSTM model:

input_size: 1

hidden_size: 1

num_layers: 1

batch_first: True

num_classes: 1

learning_rate: 0.001

optimizer: Adam

loss_function: MSELoss()

num_epochs: 10000

IV. (Potential) Results and Discussion

Project Goals

Maximize accuracy and efficiency of ML techniques while minimizing computational resources, optimize model complexity to ensure robust predictions without overfitting, and enhance portfolio optimization through effective integration of non-financial information.

Ultimately, we found that feature extraction via PCA performed slightly better than mean returns vs volatility in Silhouette, Davies Bouldin, and Calinski Harabasz scores for K values from 3 to 9. We chose these metrics because all three of these didn’t require ground truth labels. Silhouette most matches our observation that k=3 is the best choice for K. However, Davies Bouldin and Calinski Harabasz metrics suggest that k=9 is better especially for PCA, which contradicts what we have seen through the elbow method. Therefore, we suspect that PCA may be overfitted since the top principal components are not as well defined, so they may capture irrelevant details in the dataset.

Overall, we conclude the current K-means algorithm with and without PCA is an average performer since it is able to reach a value slightly above 0.5 for Silhouette score for k=3 and k=4, and a Davies Bouldin score below 0.4. Additionally, the low Calinski Harabasz score for k=3 may show that k=3 is not as overfitted as k=9.

For DBSCAN, we found that mean returns vs volatility performed better than feature extraction via PCA. When choosing mean returns and volatility as features, we found the highest silhouette score yet of 0.737. Additionally, the Davies Bouldin score was 0.315, while the Calinski Harabasz score was 48.6. This could mean that DBSCAN method might not be as overfitted as the K-means method with PCA. DBSCAN with PCA did not perform as well, with a silhouette score of 0.496, a Davies Bouldin score of 1.079, and a Calinski Harabasz score of 5.827.

We conclude that DBSCAN without PCA is likely to be a better performer than K-means because of the higher silhouette score, similar Davies Bouldin score, and lower Calinski Harabasz score. DBSCAN with PCA also shouldn’t be considered since it produced many outliers instead of forming them into a cluster, which is also reflected in its performance scores.

We utilized functions from the statsmodel library for implementing the ARIMA model. Based on the partial autocorrelation (PACF) and autocorrelation (ACF) plots, along with grid search methodology, we determined the optimal parameters for the ARIMA model to be (p, d, q) = (1,1,0).

Furthermore, examination of the plot for rolling mean and standard deviation revealed an increasing trend in both metrics. Additionally, with a p-value exceeding 0.05 and the test statistic surpassing the critical values, indicating non-stationarity, the time series data is deemed to be non-linear. Consequently, to address this non-linearity, the natural logarithm of the time series was applied, as shown below.

As observed, the application of log-transformation to the time series has induced a slight linearity, rendering it amenable to modelling. This transformation has effectively mitigated the non-linear trends present in the original data.

For LSTM based portfolio optimizer, here was the methods involved.

  1. Each time series was standardized using a Min Max Scaler so that there is no explosion of gradients in the Neural Network.
  2. Then, each timeseries was pre-processed to generate sequences of of length 100, so that the 100 values would be the input features to the LSTM, and the output is the 101st value.
  3. Each of these training rows are fed into the LSTM framework, and the model is trained to predict the next value in the time series.

The trained models are then used to predict the time series for the next sequence of unseen data.For this, the last sequence of training data of length 100 (last 100 days of 2022) is fed into the LSTM. And the output time series (after reverse scaling) is the forecasted time series for the stock.Then, this data is used as an input for the MPT optimizer, maximizing the predicted sharpe ratio of the portfolio. One of the predictions for the stocks can be seen below.

For each of the methods, here are optimal portfolios selected:

Model AAPL AMD AMZN F GOOG INTC JPM MSFT MS NVDA TSLA VOO
Benchmark (MPT) 0 0.575 0 0 0 0 0 0 0 0 0.425 0
K-Means (non PCA) 0 0 0 0 0 0 0.33 0 0 0.33 0 0.33
K Means (PCA) 0 0 0 0 0 0 0.25 0 0 0.25 0.25 0.25
ARIMA + MPT 0 0 0 0 0.946 0 0 0 0.0013 0.041 0 0
          0.041              
DBSCAN 0 0 0 0 0 0 0 0 0 0.5 0 0.5
NN + LSTM 0 0 0.015 0.451 0 0.0009 0.0036 0.089 0.043 0.225 0 0.137

Lastly, we employed Modern Portfolio Theory (MPT) as the benchmark model for comparison, which provides a framework for constructing investment portfolios that aim to optimize the trade-off between risk and return. The goal of this algorithm is to find the best asset allocation within a portfolio in order to maximize the Sharpe ratio, a risk-adjusted return metric. The benchmark approach begins by allocating equal amounts to all assets in the portfolio. It then uses optimization techniques, such as Sequential Least Squares Quadratic Programming (SLSQP), to iteratively adjust the asset allocations while adhering to limitations, such as the requirement that the overall allocation add to 1.0. The objective function aims to maximize the negative Sharpe ratio, hence maximizing the Sharpe ratio itself. By using this benchmark model to historical price data, we can determine the best asset allocations that would have resulted in the maximum risk-adjusted returns for the specified time period. Comparing the results of this benchmark model to those of our supervised and unsupervised models allows us to assess the performance and efficacy of our suggested approaches for forecasting and portfolio optimization.

For the different supervised and unsupervised models utilized in this report, here is a table comparing the metrics vs the benchmark model, implementing the maximization of Sharpe ratio using modern portfolio theory.

Model Mean Daily Returns Std Dev of Daily Returns Sharpe Ratio Treynor Ratio Beta Alpha Cumulative Return
Benchmark 0.0037 0.0267 2.205 0.00167 2.103 0.0016 1.301
K Means (w/o PCA) 0.00243 0.0148 2.609 0.00165 1.351 0.0011 0.782
K Means (w PCA) 0.00262 0.0161 2.588 0.00160 1.514 0.0011 0.859
ARIMA + MPT 0.00223 0.0187 1.937 0.00141 1.478 0.0008 0.688
DBSCAN 0.00274 0.0166 2.621 0.00176 1.444 0.0013 0.9117
LSTM + MPT 0.00280 0.0168 2.650 0.00177 1.469 0.0013 0.9396

Next Steps

While this paper has explored the application of Kmeans, DBSCAN, ARIMA, and LSTM models for portfolio optimization, there remains ample room for further evaluation and comparison. Future research could delve deeper into the performance of these models across different market conditions, asset classes, and time horizons. Conducting rigorous backtesting and sensitivity analysis can provide more insights into the robustness and stability of these models in real-world investment scenarios.

While we attempted to include data beyond historical stock prices with the macroeconmic indicators, we were not able to successfully integrate them within the overall model. Incorporating such additional data sources could enhance the predictive power of the models. By integrating a broader range of information, researchers can potentially uncover new patterns and correlations that contribute to more accurate portfolio optimization strategies.

References

[1] J. C. Van Horne and G. G. C. Parker, “The Random-Walk Theory: An Empirical Test,” Financial Analysts Journal, vol. 23, no. 6, pp. 87–92, 1967.
[2] L. Lania, R. Collage and M. Vereycken, “The Impact of Uncertainty in Macroeconomic Variables on Stock Returns in the USA,” Journal of Risk Financial Manag. *, vol. 16, no. 3, pp. 189, 2023. DOI: 10.3390/jrfm16030189
[3] M. Lim, “History of AI Winters,” *Actuaries Digital. Accessed: Feb. 13, 2024. [Online]. Available: https://www.actuaries.digital/2018/09/05/history-of-ai-winters/
[4] V. S. Rajput and S. M. Dubey, “Stock market sentiment analysis based on machine learning,” in 2016 2nd International Conference on Next Generation Computing Technologies (NGCT), Oct. 2016, pp. 506–510. DOI: 10.1109/NGCT.2016.7877468
[5] J. Bollen, H. Mao, and X. Zeng, “Twitter mood predicts the stock market,” Journal of Computational Science, vol. 2, no. 1, pp. 1–8, Mar. 2011. DOI: 10.1016/j.jocs.2010.12.007
[6] Hayes, A. (2024, February 23). Autoregressive integrated moving average (ARIMA) prediction model. Investopedia. https://www.investopedia.com/terms/a/autoregressive-integrated-moving-average-arima.asp [7] Macrotrends LLC. (n.d.). The long term perspective on markets. Macrotrends. https://www.macrotrends.net/

Gantt Chart

Proposal Contribution Table

Name Proposal Contributions
Sai - Implemented LSTM
  - Report and Video recording
Jungyoun Kwak - Collected balance data and Implemented DBSCAN
  - Report and Video recording
Prabhanjan Nayak - Finalized models, repository structure and description, and report.
  - Team management for all final deliverables.
Kaushik Arcot - Compiled code for modelling LSTM for all stocks
  - Completed code and compiled metrics of LSTM and analyzing portfolio allocations for results and discussion section.
Daniel Wu - PCA feature reduction for DBSCAN
  - Wrote the methods for DBSCAN