For this post, I will be creating a script to download pricing data for the S&P 500 stocks, calculate their historic returns and volatility and then proceed to use the K-Means clustering algorithm to divide the stocks into distinct groups based upon said returns and volatilities.

So why would we want to do this you ask? Well dividing stocks into groups with “similar characteristics” can help in portfolio construction to ensure we choose a universe of stocks with sufficient diversification between them.

The concept behind K-Means clustering is explained here far more succinctly than I ever could, so please visit that link for more details on the concept and algorithm

I’ll deal instead with the actual Python code needed to carry out the necessary data collection, manipulation and analysis.

First things first, we need to collect the data – lets run our imports and create a simple data download script that scrapes the web to collect the tickers for all the individual stocks within the S&P 500.

from pylab import plot,show from numpy import vstack,array from numpy.random import rand import numpy as np from scipy.cluster.vq import kmeans,vq import pandas as pd import pandas_datareader as dr from math import sqrt from sklearn.cluster import KMeans from matplotlib import pyplot as plt sp500_url = 'https://en.wikipedia.org/wiki/List_of_S%26P_500_companies' #read in the url and scrape ticker data data_table = pd.read_html(sp500_url) tickers = data_table[0][1:][0].tolist() prices_list = [] for ticker in tickers: try: prices = dr.DataReader(ticker,'yahoo','01/01/2017')['Adj Close'] prices = pd.DataFrame(prices) prices.columns = [ticker] prices_list.append(prices) except: pass prices_df = pd.concat(prices_list,axis=1) prices_df.sort_index(inplace=True) prices_df.head()

This gets up something resembling the following:

We can now start to analyse the data and begin our K-Means investigation…

Our first decision is to choose how many clusters do we actually want to separate the data into. Rather than make some arbitrary decision we can use an “Elbow Curve” to highlight the relationship between how many clusters we choose, and the Sum of Squared Errors (SSE) resulting from using that number of clusters.

We then plot this relationship to help us identify the optimal number of clusters to use – we would prefer a lower number of clusters, but also would prefer the SSE to be lower – so this trade off needs to be taken into account.

Lets run the code for our Elbow Curve plot.

#Calculate average annual percentage return and volatilities over a theoretical one year period returns = prices_df.pct_change().mean() * 252 returns = pd.DataFrame(returns) returns.columns = ['Returns'] returns['Volatility'] = prices_df.pct_change().std() * sqrt(252) #format the data as a numpy array to feed into the K-Means algorithm data = np.asarray([np.asarray(returns['Returns']),np.asarray(returns['Volatility'])]).T X = data distorsions = [] for k in range(2, 20): k_means = KMeans(n_clusters=k) k_means.fit(X) distorsions.append(k_means.inertia_) fig = plt.figure(figsize=(15, 5)) plt.plot(range(2, 20), distorsions) plt.grid(True) plt.title('Elbow curve')

The resulting plot with the above data is as follows:

So we can sort of see that once the number of clusters reaches 5 (on the bottom axis), the reduction in the SSE begins to slow down for each increase in cluster number. This would lead me to believe that the optimal number of clusters for this exercise lies around the 5 mark – so let’s use 5.

# computing K-Means with K = 5 (5 clusters) centroids,_ = kmeans(data,5) # assign each sample to a cluster idx,_ = vq(data,centroids) # some plotting using numpy's logical indexing plot(data[idx==0,0],data[idx==0,1],'ob', data[idx==1,0],data[idx==1,1],'oy', data[idx==2,0],data[idx==2,1],'or', data[idx==3,0],data[idx==3,1],'og', data[idx==4,0],data[idx==4,1],'om') plot(centroids[:,0],centroids[:,1],'sg',markersize=8) show()

This gives us the output:

Ok, so it looks like we have an outlier in the data which is skewing the results and making it difficult to actually see what is going on for all the other stocks. Let’s take the easy route and just delete the outlier from our data set and run this again.

#identify the outlier print(returns.idxmax())

Returns BHF

Volatility BHF

dtype: object

Ok so let’s drop the stock ‘BHF and recreate the necessary data arrays.

#drop the relevant stock from our data returns.drop('BHF',inplace=True) #recreate data to feed into the algorithm data = np.asarray([np.asarray(returns['Returns']),np.asarray(returns['Volatility'])]).T

So now running the following piece of code:

# computing K-Means with K = 5 (5 clusters) centroids,_ = kmeans(data,5) # assign each sample to a cluster idx,_ = vq(data,centroids) # some plotting using numpy's logical indexing plot(data[idx==0,0],data[idx==0,1],'ob', data[idx==1,0],data[idx==1,1],'oy', data[idx==2,0],data[idx==2,1],'or', data[idx==3,0],data[idx==3,1],'og', data[idx==4,0],data[idx==4,1],'om') plot(centroids[:,0],centroids[:,1],'sg',markersize=8) show()

gets us a much clearer visual representation of the clusters as follows:

Finally to get the details of which stock is actually in which cluster we can run the following line of code to carry out a list comprehension to create a list of tuples in the (Stock Name, Cluster Number) format:

details = [(name,cluster) for name, cluster in zip(returns.index,idx)] for detail in details: print(detail)

This will print out something resembling the below (I havn’t included all the results for brevity)

SO there you have it, we now have a list of each of the stocks in the S&P 500, along with which one of 5 clusters they belong to with the clusters being defined by their return and volatility characteristics. We also have a visual representation of the clusters in chart format.

If anyone has any questions or comments, as always feel free to leave them below.

Cheers!

## 7 comments

Sir, could you please tell what atr the x and y labels in the lsat graph? Thank you

Do you mean the cluster scatter chart? If so the x-axis is returns and y-axis is volatility. If you mean the Elbow Curve chart, then the x-axis is the number of clusters and the y-axis is the SSE (Sum of Squared Errors)

hi, do you have any research regarding to this article? id be really appreciated if you could share it with me. Thank you very much.

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Hello,

Thank you for this! I have tried it for my specific set of securities, and I found it weird that with the same dataset, it gives me different clusters by security every time I run the notebook.

Any insight as to why ? Is it the same for your dataset ?

Thanks!

Hi John,

This is actually expected behaviour to a degree as KMEANS by default selects the initial cluster centroid positions at random and works from there to optimise. The results can indeed rest on the location of those initial random centroid placements.

You can read more and how to overcome it by setting your “random_state” here:

https://stackoverflow.com/questions/25921762/changes-of-clustering-results-after-each-time-run-in-python-scikit-learn

The Stock BHF was removed from SP500 in 2019. Therefore, your example shows the following for outliers:

Returns NKTR

Volatility CTVA

In this case, the k-means step would have to be executed again removing each outlier? This would be repeated until no outliers exist? Given this result, would adding another (unbiased) feature help to improve the clustering?