# Optimal Portfolio Construction Using Machine Learning

By Lamarcus Coleman

In this post, we will learn about the Stereoscopic Portfolio Optimization (SPO) framework and how it can be used to improve a quantitative trading strategy.

We'll also review concepts such as
• Gaussian Mixture Models,
• K-Means Clustering, and
• Random Forests.
Our objective is to determine whether we can reject the null hypothesis that the Stereoscopic Portfolio Optimization (SPO) model is not a viable option for creating optimal short-term portfolios.

### What is the Stereoscopic Portfolio Optimization (SPO) Framework?

The Stereoscopic Portfolio Optimization (SPO) Framework is an idea I introduced in a paper entitled "Applying Machine Learning Ensembles to Market Microstructure to Achieve Portfolio Optimization".

You can download the paper and files at the end of the article.

In the paper, I refer to traditional techniques such as the Efficient Frontier, as being top-down approaches to achieve an optimal portfolio.

Traditionally, a large part of portfolio optimization has centred on finding the proper balance for allocations to different asset classes based on the mean-variance tradeoff.

What the SPO Framework does:
The Stereoscopic Portfolio Optimization Framework introduces the idea of bottom-up optimization via the use of machine learning ensembles applied to some market microstructure component.

In the text volatility was the microstructure component used but other components such as order arrival rates, liquidity, can be substituted into the framework.

Creating an SPO Framework:
These bottom-up techniques are combined with top-down approaches to creating the Stereoscopic Portfolio Optimization (SPO) Framework.

Premise of the SPO Framework:
The premise of the framework is that a portfolio is the sum of n market microstructures.

The equation below was given as a novel way of explaining this logic.

This equation is a novel way of stating that the static state of a portfolio is a product of the component market microstructures and their weights.

Where,

'an' represents the nth asset's microstructure;
'wn' represents the weight of the nth asset;
'et' is an error term and representative of randomness.

The absolute value signs were used as an indication of non-linearity.

The equation below was offered as a novel representation of the dynamic state of a portfolio.

The equation above is stating that the dynamic state of the portfolio is not only dependent upon the respective components' microstructures but their relative correlations as well.

The correlations in this context don't refer to prices but the relationships between the actual microstructure components. For instance, how does one asset respond when shocks occur in another asset, or when liquidity changes, etc.

The idea of the correlations of microstructure components can be written as a conditional probability. where,

an is the nth asset's microstructure, and
q represents the qth asset's microstructure.

The correlation terms can thus be expressed in matrix form:

where,

R is the correlation matrix, and
m is a n×1 vector of our microstructure components

We can thus update our novel equation to the following:

Ensembles were used consistent with the idea that a portfolio is the composition of n market microstructures.

The strength of ensembles is that they allow for the aggregation of multiple, sometimes weak models, to create a more robust model.

• The goal of the ensemble is to minimize the cost function.
• The goal of portfolio optimization is to minimize risk.

Thus, in this context, the risk is the cost function of portfolio optimization and creates a parallel objective to that of ensembles.

In the paper, the Stereoscopic Portfolio Optimization (SPO) framework was created by combining the traditional mean-variance optimization with Gaussian Mixture Models and Random Forests. K-Means Clustering was used to identify subgroups within the S&P 500.

Before we dive into applying the Stereoscopic Portfolio Optimization (SPO) Framework to an intraday strategy, let's review some of the components used.

### What are the Gaussian Mixture Models and How are They Related to K-Means Clustering?

An important part of conducting research is understanding how your data is distributed. The distribution of your data gives you insight into the probabilities surrounding seeing certain observations.

In a prior post on K-Means Clustering, we learned what K-Means is and how it can be applied to statistical arbitrage. In short, unsupervised learning methods allow us to explore our data and identify patterns or relationships.

K-Means is a machine learning technique that seeks to identify subgroups within our data. This means that we can find relationships within our stock universe and then test to see if those relationships have some statistical significance.

For a more thorough review of K-Means, visit the series entitled "Using K-Means Clustering for Pair Selection".

Gaussian Mixture Models are similar to K-Means in that they too are a clustering technique. One key difference between GMM and K-Means is that K-Means is a hard clustering method while GMMs are referred to as being a soft clustering technique.

In K-Means an observation only has a probability of 0 or 1 of being in the kth class.

Gaussian Mixture Models don't assign a hard probability of being in a specific class but rather assigns a probability between 0 and 1 that observation came from a specific class.

Recall that in K-Means we begin by randomly assigning cluster centres placing each observation into a specific cluster.

We then compute the centroid or mean of the clusters. We then reassign points to each cluster based on their distance from the mean of each cluster.

This process is repeated until there are no more cluster reassignments. Our cost function in K-Means is to minimize the within-cluster variation or in laymen's terms ensure that the observations within each cluster are highly similar.

Below is the objective function for K-Means.

Where,

mk is the Mu of the kth cluster,
x(n) are our observations,
r(n)k is the probability of the nth observation belonging to the kth class.

Essentially in our K-Means objective function, we are multiplying the probabilities by the squared distances of each observation from the mean of the kth class.

Similar to K-Means, Gaussian Mixture Models begin by assigning random values for the parameters. It then calculates the probabilities of an observation coming from each of the kth distributions.

Unlike in K-Means in which we would set the probability to 1 or 0, Gaussian Mixture Models will calculate a probability of the observation coming from each of the k distributions.

Note that the sum of these probabilities must equal 1.

Below is an equation for the Gaussian Mixture Model:

Where,
πk is the probability of an observation belonging to the kth distribution;
N(μkk) represents a Gaussian distribution with a mean of μk and sigma of σk.

Note here that σk is a matrix composed of the product of the sum of the distances and the transpose of the distances multiplied by the probability that kth Gaussian generated the nth observation.

This is then averaged across the number of distributions.

Below is the equation for the σk term:

The gamma, γ, the term is the responsibility of the kth Gaussian for generating the nth point.

By responsibility, we are referring to the extent that the kth Gaussian contributed to the creation of the nth point.

We can compute the gamma, γ, term by the following equation.

The above equation is the quotient of the kth Gaussian and all of the Gaussians, or in other words the proportion of the kth Gaussian.

### Let's Review Random Forests

Random Forests is a machine learning ensemble that combines several decision trees to make a single prediction. Recall that decision trees are known for their simplicity and ease of interpretation.

Bagging and boosting are alternatives to Random Forests that produce multiple trees. Trees can be applied to both regression and classification problems.

A decision tree divides the feature space into separate regions and every observation that falls into a specific region is given a prediction of the mean of that region.

The goal is to minimize the following cost function:

The above cost function is the Residual Sum of Squares and denotes the sum of the squared distances from the mean of the ith observation from the mean of the jth region, R.

Recursive binary splitting is conducted to minimize the RSS.

• The process considers every possible cutpoint, or point to split the space, of every feature with the aim of minimizing the cost function.
• This process is continued across the two new regions.
• These regions then undergo the same binary splitting as well.
• This is continued until a specific criterion is met.

One major consideration of decision trees is that if we grow a tree significantly large, it could overfit our data.

A smaller tree could lead to a better bias-variance tradeoff. A viable approach is to grow a large tree and then prune it, selecting only a subset of the tree. This is achieved via a process known as cost-complexity pruning.

Within cost-complexity pruning, we use a tuning parameter to select a sequence of subtrees.

Once we have a subset of trees we can then use k-fold cross-validation to choose the best value of the tuning parameter.

The tuning parameter a corresponds to a specific partition of the original tree. Thus when we perform k-fold cross-validation to find the value of a that achieves the best MSE, we can then select the corresponding subtree from our original tree.

Another key consideration of decision trees is that they can be non-robust. This means that a small change in the data can cause a large change in the tree. They can also lag in performance compared to other machine learning methods.

Random Forests is one method of improving the use of decision trees. Random Forests are similar to bagging decision trees in that they both resample the data and apply the model to the resamples and then average the results.

However, in Random Forests, when splitting is performed, a random sample of the features are selected as the split candidates. This is done to limit the influence of any one feature. So when splitting in Random Forests, we only consider m ∈ p possible features.

The reason Random Forests selects a fraction of the feature space at each split is that if there is a very strong feature in the space, it is likely to be chosen as the root node of every tree.

This would mean that each tree would likely be closely correlated and thus defeats the purpose of reducing variance.

### Problem Statement

We need to construct a portfolio of statistical arbitrage strategies in the most efficient manner possible.

We will be trading on an intraday basis in the U.S. equities market, namely across stocks within the S&P 500.

We will design multiple portfolios that use various methods of addressing the portfolio optimization problem. To assess our efforts we will create the Sharpe Ratio of each portfolio and make relative comparisons.

Each portfolio will be composed of the same relationships to ensure apples to apple comparison.

### Data

We will be using 5minute data over the period of 2018, beginning 1/4/18. Though at this point, we're unsure as to exactly which stocks we will be trading, we know that we will have the following four portfolios:

1. Equally Weighted,
2. Efficient Frontier,
3. Bottom Up Optimization and
4. Stereoscopic Portfolio Optimization (SPO) Framework.

The Equally Weighted Portfolio is the simplest to construct but because the remaining portfolios will require a training period, our test or assessment period will be the second half of 2018, indicating that our equally weighted portfolio will be constructed over the second half of our 2018 data.

This period will begin on May 1 and end June 12 2018. The remainder of our portfolios will be constructed using the first half of 2018 as a training period and the second half as the test or assessment period.

As we shall see, this is necessary to compare each portfolio over the same period. In short, in order to compose the Efficient Frontier, Bottom-Up, and SPO Portfolios, we first must have some period to train over (i.e. the first half of our 2018 data).

We need this training period so that we can make predictions over the period that we would like to compare all of the portfolios over (i.e. May 1 to June 12, 2018).

Our Equally Weighted Portfolio doesn't need a training period to be created so we will just create it over the period that will be used as the test period for the remaining portfolios.

As we have covered in prior posts, a key initial problem to solve with developing a statistical arbitrage strategy is that of pair selection. We will once again use K-Means clustering to solve this problem.

We will apply K-Means to the S&P 500, create subgroups, and then choose five tradeable relationships to construct our portfolios.

To get started, let's import our usual libraries.

#data analysis and manipulation
import numpy as np
import pandas as pd
#data visualization
import matplotlib.pyplot as plt
import seaborn as sns
sns.set_style('whitegrid')
#statistics and machine learning
from sklearn.cluster import KMeans
from sklearn.mixture import GaussianMixture as GM
from sklearn.ensemble import RandomForestClassifier as RF
from sklearn.metrics import confusion_matrix, classification_report
import warnings
warnings.simplefilter('ignore')

We're going to use the P/E, EPS, and Market Cap as our features to identify subgroups within the S&P 500.

Let's import our features data now.

#importing the Excel file that contains our features data
fundamentals=pd.ExcelFile('SPO_Data.xlsx')
#parsing the Fundamentals sheet from our Excel file of which holds our P/E, EPS, and MarketCap data
features=fundamentals.parse('Fundamentals')

Now that we have our features for our K-Means algorithm, let's check the head of our feature space.

#checking the head of our features dataframe

No.SymbolNameP/EEPSMarketCap
0MMM3M Company23.178.16112.74
1ABTAbbott Laboratories48.030.9477.76
2ABBVAbbVie17.553.63101.52
3ACNAccenture plc18.376.7677.29
4ATVIActivision Blizzard37.551.2836.13

We're now ready to create our K-Means method and find subgroups within the S&P 500.

Before we do so, let's drop the Name column from our features data.

#dropping name column
features.drop('Name',axis=1,inplace=True)
#rechecking our data

No.SymbolP/EEPSMarketCap
0MMM23.178.16112.74
1ABT48.030.9477.76
2ABBV17.553.63101.52
3ACN18.376.7677.29
4ATVI37.551.2836.13

Recall, an early consideration for implementing K-Means Clustering is determining the value of K that should be used. We'll use the elbow technique here to determine what value of K we should use. This technique will compare our value for K with the respective error.

Our goal is to choose the value for K that minimizes the error or cost function.

from scipy.spatial.distance import cdist
#creating our elbow technique method
def find_k(features):
#intializing a list to hold costs or errors
costs=[]
#iterating over possible values for k
for k in range(1,51):
model=KMeans(n_clusters=k)
model.fit(features)
costs.append(sum(np.min(cdist(features,model.cluster_centers_,'euclidean'),axis=1)))
#plotting our elbow graph
with plt.style.context(['classic','ggplot']):
plt.figure(figsize=(10,6))
plt.plot(costs)
plt.xlabel('Clusters')
plt.ylabel('Errors')
plt.title('Finding K')
plt.show()

Now that we have our method let's use it to find the optimal value for K. First, let's update our dataframe by making using our Symbols as our index.

We'll first make a copy of our original dataframe.

#making a copy of our original features dataframe
features_copy=features.copy()

Now we can reindex our features dataframe by our Symbol column.

#reindexing our features dataframe
features_copy=features_copy.reindex(index=features_copy['Symbol'],columns=features_copy.columns)
features_copy.head()

SymbolSymbolP/EEPSMarketCap
MMMNaNNaNNaNNaN
ABTNaNNaNNaNNaN
ABBVNaNNaNNaNNaN
ACNNaNNaNNaNNaN
ATVINaNNaNNaNNaN

Now that we have reindexed our features_copy dataframe, let's add back the values to our columns and drop the Symbol column.

#adding our data back to their respective columns
features_copy['P/E']=features['P/E'].values
features_copy['EPS']=features['EPS'].values
features_copy['MarketCap']=features['MarketCap'].values

Okay. Let's recheck our dataframe.

features_copy.head()

SymbolSymbolP/EEPSMarketCap
MMMNaN23.178.16112.74
ABTNaN48.030.9477.76
ABBVNaN17.553.63101.52
ACNNaN18.376.7677.29
ATVINaN37.551.2836.13

Now we're ready to drop the Symbol column from our dataframe.

#dropping symbol column
features_copy.drop('Symbol',axis=1,inplace=True)

Let's recheck our dataframe by calling the head method.

features_copy.head()

SymbolP/EEPSMarketCap
MMM23.178.16112.74
ABT48.030.9477.76
ABBV17.553.63101.52
ACN18.376.7677.29
ATVI37.551.2836.13

Now let's use our find_k method to find the optimal value for K.

#finding K
find_k(features_copy.fillna(0))

Okay, we will use 15 as our value for K. Now we're ready to implement our K-Means Clustering algorithm with K equal to 15 and look for tradable relationships.

Let's begin by initializing our model.

#initialzing K-Means algorithm
kmeans=KMeans(n_clusters=15,random_state=101)

Notice that I used random_state=101. This is so that you can recreate the same results posted here. We will now fit our K-Means Algorithm to our features data.

#fitting kmeans to our features data
kmeans.fit(features_copy.fillna(0))
KMeans(algorithm='auto', copy_x=True, init='k-means++', max_iter=300,
n_clusters=15, n_init=10, n_jobs=1, precompute_distances='auto',
random_state=101, tol=0.0001, verbose=0)

Note that we called the fillna method on our features_copy dataframe and replaced NaNs values with 0. We can now review our clusters.

Let's start by checking our labels.

#getting cluster labels
kmeans.labels_
array([ 6,  6,  6,  6, 12,  3,  0,  3, 13,  0,  3,  3, 12, 12, 12,  3,  3,
7,  3,  6, 12,  3,  3,  1,  1,  5, 14,  3,  3, 12,  6,  0, 12,  3,
3,  3,  3,  6,  3, 13,  3,  0,  3, 13,  3,  8,  0,  3, 13,  3,  3,
2, 13,  0, 13,  3,  3,  3, 13, 12,  2, 12, 13,  0,  0, 13, 13,  3,
0,  0,  3,  6,  7, 12,  7,  6,  6, 13,  3,  3, 13, 12,  0,  3,  3,
0,  0, 12,  3,  3,  6,  3,  3,  3,  3, 13,  0,  6, 13,  5, 10,  0,
3,  0, 13,  3,  3,  5,  5,  3,  3,  0,  3,  3,  5,  0,  0,  5,  3,
3, 13,  0,  3,  3,  3,  6, 13,  7,  3,  0,  3,  6,  3,  0,  3,  3,
0,  3,  0, 12, 13, 12,  3,  3,  3,  3,  3,  0,  3,  6,  3,  3,  0,
0, 12,  3,  3,  0,  0,  0,  3, 12,  3,  0, 13, 13,  0, 13, 12, 11,
13, 12,  3,  3,  0,  7,  3,  0,  3,  9,  3,  9,  3, 12,  0, 12,  3,
13, 13,  3,  3, 12,  3, 12,  3,  3,  0,  3,  3,  3, 13, 13,  3,  3,
0,  2,  3,  0,  0,  3,  6, 12,  6, 13,  3,  0,  3,  3,  3,  3,  3,
3,  0,  3, 13,  3, 13,  0, 12,  5,  6,  3,  3,  3, 12,  3, 12,  0,
12, 11,  3,  5,  3,  5,  3,  3,  3, 12, 12,  3,  7,  3,  3,  3,  9,
0,  9,  3,  3, 12,  3,  0,  3,  7,  3,  3,  6,  3,  3,  3,  3,  3,
3,  3,  7,  3,  6,  3, 12,  3,  6,  3,  6,  0,  3,  3,  3,  3, 13,
3, 12,  0, 12,  3,  6,  3,  3,  6,  0,  3,  6,  5,  7, 12, 13, 11,
13,  1, 12,  3,  3,  0,  0, 12,  7,  6,  3, 13, 12,  7, 13, 13, 12,
4, 12, 13, 13, 13, 13,  0, 12,  6,  3, 13,  3,  0,  3,  0, 13,  3,
0,  3,  0,  3, 12,  5, 12,  3,  3,  3,  0,  3,  3,  5,  3, 13,  5,
0,  5,  0,  3, 13,  0, 12, 12,  3,  0,  6,  3,  2,  3,  3,  0,  3,
0,  3,  3, 13,  6,  3,  3, 13,  0, 12, 12, 12, 12,  3, 12,  6,  3,
3,  3, 12,  3,  3,  3,  0,  4,  3,  6,  3,  3,  3,  3,  3, 13,  0,
3, 12,  3,  0,  0, 13,  3, 13,  6,  3, 12,  0,  3, 13,  3,  3,  3,
3,  3,  3, 12,  3,  6,  3,  0,  3, 12, 12, 12,  0,  5,  0,  3,  6,
0,  3, 12,  3,  3, 13, 12,  0,  0,  3,  6, 12, 12, 12, 12,  6,  3,
5,  6,  3,  6,  3,  3,  3,  3,  3,  3, 12,  3,  3,  5, 13,  3,  5,
3, 12,  5,  6,  0,  3,  3,  2,  3, 13, 12,  7,  3,  3,  3, 13, 12,
3, 12,  3, 13,  3,  3, 12,  0,  3,  7,  3, 12])

We will now add our cluster assignments back to our dataframe.

#adding cluster labels to dataframe
features_copy['Cluster']=kmeans.labels_

Let's now review our features dataframe

#reviewing features dataframe

SymbolP/EEPSMarketCapCluster
MMM23.178.16112.746
ABT48.030.9477.766
ABBV17.553.63101.526
ACN18.376.7677.296
ATVI37.551.2836.1312
#calling tail method our dataframe
features_copy.tail()

SymbolP/EEPSMarketCapCluster
YHOONaN-0.2343.740
YUM15.834.0422.653
ZBH77.531.5123.547
ZION22.751.999.173
ZTS32.161.6526.1112

Now that we have our cluster assignments, let's group our data by the cluster assignments.

#creating dataframe to hold data
clusters_df=pd.DataFrame()
#grouping our data by cluster for clusters with atleast 2 stocks in it.
clusters_df=pd.concat(i for clusters_df, i in features_copy.groupby(features_copy['Cluster']) if len(i) >1)

Let's now check of our new dataframe.

#checking the head of clusters df

SymbolP/EEPSMarketCapCluster
AET20.376.4145.930
AIGNaN-0.7862.150
ANTM18.029.2143.890
AMAT19.021.9439.920
#checking the tail of our cluster df
clusters_df.tail()

SymbolP/EEPSMarketCapCluster
RIG6.092.104.6713
VRTXNaN-0.4622.6913
WDCNaN-1.5922.1213
WMBNaN-0.5724.3013
XRXNaN-0.497.4813

Now that we have our data organized by cluster assignment, we're ready to check for tradeable relationships.

To do this, we will create every possible pair combination within a respective cluster. We can then later run the CADF test on specific pairs from our clusters.

Let's create a method that will take the symbols from a specific cluster as an input, compute the possible pair combinations of a respective cluster and store our pairs into a separate list.

#creating method to identify each possible pair
def create_pairs(symbolList):
#creating a list to hold each possible pair
pairs=[]
#initializing placeholders for the symbols in each pair
x=0
y=0
for count,symbol in enumerate(symbolList):
for nextCount,nextSymbol in enumerate(symbolList):
x=symbol
y=nextSymbol
if x !=y:
pairs.append([x,y])

return pairs

We will now create a list of symbols. Let's use the stock symbols from cluster 0.

#creating list of symbols from cluster 0
symbol_list_0=['ADBE','AET','AIG','ANTM','AMAT']

We will now use our create pairs method above to create a list of lists of the possible pair combinations.

#list of lists of pairs
all_pairs=create_pairs(symbol_list_0)

Let's check our pair combinations.

#printing list of all_pairs from cluster 0
all_pairs
[['ADBE', 'AET'],
['AET', 'AIG'],
['AET', 'ANTM'],
['AET', 'AMAT'],
['AIG', 'AET'],
['AIG', 'ANTM'],
['AIG', 'AMAT'],
['ANTM', 'AET'],
['ANTM', 'AIG'],
['ANTM', 'AMAT'],
['AMAT', 'AET'],
['AMAT', 'AIG'],
['AMAT', 'ANTM']]

Okay. Now that we have our possible pair combinations from cluster 0, let's import our intraday data for these stocks.

#initializing our stock variables
amat=pd.read_csv('AMAT_5min.csv')

Let's check our data.

#checking head of ADBE


02018/01/0408:30:00181.93183.74181.84183.13838623962294160921
12018/01/0408:35:00183.05183.48182.66182.89315932381784013753
22018/01/0408:40:00182.84183.35182.70183.22496543452696222692
32018/01/0408:45:00183.25183.73183.06183.73511083142156629542
42018/01/0408:50:00183.66184.00183.56183.90274761961528712189
#checking tail of AIG
aig.tail()


85742018/06/1214:30:0054.5454.5654.5154.55609023572010640796
85752018/06/1214:35:0054.5554.6154.5354.60744834252935445129
85762018/06/1214:40:0054.6054.6054.5554.58675394142068746852
85772018/06/1214:45:0054.5754.5854.5454.56821224392492557197
85782018/06/1214:50:0054.5654.6254.5654.601203287794634073988

We can see that we have 5 min bar data for our stocks ranging from January 3, 2018, to the first half of the session on June 11, 2018.

We can now test our pairs for cointegration. Recall that we are going to use part of our data as our assessment period so we don't want to include it in our test.

We'll use January to May as our training period and assess our portfolios over the remainder of our data. Let's create variables to hold our training period data.

We'll store use our Close column. So that we can parse out our data by our training period, we'll make our date column the index of our data.

We begin by making a copy of our data. Let's create a method to perform this across our symbols data.

We'll store our original symbols data in a dictionary and pass it into our function.

#creating list to hold original data
original_data={'ADBE':adbe,'AET':aet,'AIG':aig,'ANTM':antm,'AMAT':amat}

We can now create our function to copy our original data and create a dataframe for our training period data.

#function to parse out training period data
def get_training_data(original_data,symbol_list,start,end):
'''
PARAMETERS:

original_data - the dictionary we created that holds our dataframes
symbol_list - the list of symbols; data type are strings
start - the beginning date of our training period as a string
end - the ending date of our training period as a string
'''
#creating a dataframe to hold our parsed series
training_df=pd.DataFrame()
#iterating over our symbol list
for count, symbol in enumerate(symbol_list):

try:
#making a copy of our original data for each symbol
copy=original_data[symbol].copy()
#reindexing our copied data by Date column
copy=copy.reindex(index=copy['Date'],columns=copy.columns)
#restoring values of close column from our original data
copy[' Close']=original_data[symbol][' Close'].values
#parsing out our training period
copy=copy.loc[start:end][' Close']
training_df[str(symbol)]=copy.values

except:
print(str(symbol),'Threw an Exception')
print('Current Symbol Length:')
print(len(copy.loc[start:end]))
print("")
print('training_df Length:')
print(len(training_df))
continue

return training_df


We're now ready to create our training data.

We'll use our get_training_data function to parse out our training data and then use this dataframe to parse out the respective series for our pair combinations to perform our CADF test.

#creating our training data dataframe using our training period start and end dates
training_df=get_training_data(original_data,symbol_list_0,'2018/01/04','2018/04/30')

Let's check our training_df.

training_df.head()


0183.13183.4560.36229.6254.67
1182.89183.6860.61229.6354.51
2183.22183.8060.77229.7054.25
3183.73183.8260.81229.6554.08
4183.90183.9360.80229.7054.27

After checking our dataframe, we find that we have our training period data for all of our stocks. This means that we should have 20 possible pairs.

We can write a method to check our math.

#creating method to check possible pair combinations
def possitrainingnations(n):
#Parameters#
############
#n- represents the number of items or in our case stocks

possible_pairs=(n*(n-1))

return possible_pairs

Now let's check our possible pair combinations.

#checking possible pair combinations
possible_combinations(5) # we pass in 5 for our 5 stocks
20

We're now ready to check our pairs for cointegration.

We'll create a method that will allow us to iterate over our pairs, compute the slope and then perform the CADF test. The pairs that are cointegrated will be stored in a list.

Let's import our OLS method.

from scipy.stats import linregress

Now we'll write our method to create our cointegrated pairs.

def get_cointegrated(all_pairs,training_df):
'''
PARAMETERS
#########

all_pairs - the list of all possible pair combinations from Cluster 0
training_df - our dataframe holding our stock data for stocks in Cluster 0 over the training period

'''
#creating a list to hold cointegrated pairs
cointegrated=[]
#iterate over each pair in possible pairs list; pair is a list of our 2 stock symbols
for count, pair in enumerate(all_pairs):
try:
#getting data for each stock in pair from training_df
ols=linregress(training_df[str(pair[1])],training_df[str(pair[0])]) #note scipy's linregress takes in Y then X
#storing slope or hedge ratio in variable
slope=ols[0]
#checking to see if spread is cointegrated, if so then store pair in cointegrated list
print('Pair Cointegrated at 99% Confidence Interval')
#appending the X and Y of pair
cointegrated.append([pair[0],pair[1]])
print('Pair Cointegrated at 95% Confidence Interval')
#appending the X and Y of pair
cointegrated.append([pair[0],pair[1]])
print('Pair Cointegrated at 90% Confidence Interval')
cointegrated.append(pair[0],pair[1])
else:
print('Pair Not Cointegrated ')
continue
except:
print('Exception: Symbol not in Dataframe')
continue

return cointegrated


Let's initialize our get_cointegrated function and find our cointegrated pairs.

#getting our cointegrated pairs
cointegrated_from_cluster_0=get_cointegrated(all_pairs,training_df)
Pair Not Cointegrated
Pair Not Cointegrated
Pair Cointegrated at 95% Confidence Interval
Pair Not Cointegrated
Pair Not Cointegrated
Pair Not Cointegrated
Pair Cointegrated at 95% Confidence Interval
Pair Not Cointegrated
Pair Not Cointegrated
Pair Not Cointegrated
Pair Not Cointegrated
Pair Not Cointegrated
Pair Not Cointegrated
Pair Cointegrated at 95% Confidence Interval
Pair Not Cointegrated
Pair Not Cointegrated
Pair Not Cointegrated
Pair Not Cointegrated
Pair Not Cointegrated
Pair Not Cointegrated

Let's check our cointegrated list.

cointegrated_from_cluster_0
[['ADBE', 'ANTM'], ['AET', 'ANTM'], ['ANTM', 'AET']]

Okay. We've found that 3 of our possible 20 pairs are cointegrated each at the 90% confidence interval.

Recall that we called the head method on our clusters_df dataframe of which returns only the first five rows.

Let's use the Counter method to get an idea of how our symbols are distributed across all our clusters.

#importing the Counter method
from collections import Counter
#calling Counter method on our clusters
Counter(clusters_df['Cluster'])
Counter({0: 76,
1: 3,
2: 5,
3: 215,
4: 2,
5: 19,
6: 39,
7: 13,
9: 4,
11: 3,
12: 69,
13: 54})

We can see that Cluster 0 actually contains a total of 76 symbols. Given that we found 3 tradeable relationships, we'll use these.

However, I will illustrate how, in the event that none of the symbols in our first cluster were cointegrated, we would go about checking the symbols in a specific cluster.

We'll need to create a method that will allow us to iterate over our clusters column and return the symbols that are within a specific cluster.

To begin we'll add our symbols back to our dataframe as a column so that we can retrieve them if our cluster condition is met.

#adding Symbol Column back to cluster_df
clusters_df['Symbol']=clusters_df.index

Let's recheck our dataframe.

#checking update to cluster_df

SymbolP/EEPSMarketCapClusterSymbol
AET20.376.4145.930AET
AIGNaN-0.7862.150AIG
ANTM18.029.2143.890ANTM
AMAT19.021.9439.920AMAT

Now we're ready to create our method of retrieving and storing our symbols in a list based on our cluster.

For this task, we'll use a list comprehension. We'll apply our method to cluster 9.

symbols_cluster_9=[ clusters_df['Symbol'][count] for count,value in enumerate(clusters_df['Cluster'].values) if value == 9]

Let's check the length of our symbols_cluster_9 list to ensure that our method worked correctly.

Recall that our Counter method returned a value of 4 for the number of symbols in this cluster.

#getting the length of our cluster 9 list
len(symbols_cluster_9)
4

Great! We see that our list comprehension is working properly. Now we can use this list of symbols to create every possible pair combination for cluster 9.

Let's check what symbols are in cluster 9.

#checking the symbols in cluster 9
symbols_cluster_9
['XOM', 'FB', 'JNJ', 'JPM']

We'll now use this list to create our pair combinations for cluster 9 using our create_pairs method from earlier.

#getting pair combinations for cluster 9
cluster_9_pairs=create_pairs(symbols_cluster_9)

Let's check our pair combinations for cluster 9.

#checking cluster 9 pair combinations.
cluster_9_pairs
[['XOM', 'FB'],
['XOM', 'JNJ'],
['XOM', 'JPM'],
['FB', 'XOM'],
['FB', 'JNJ'],
['FB', 'JPM'],
['JNJ', 'XOM'],
['JNJ', 'FB'],
['JNJ', 'JPM'],
['JPM', 'XOM'],
['JPM', 'FB'],
['JPM', 'JNJ']]

Okay. We've learned that we have three tradeable relationships from Cluster 0 from the first five symbols we tested. We've also walked through how we would find other pairs in other clusters if we had not found these three pairs.

After finding the pairs above, we would apply the same method to check for cointegration as was used to identify our three pairs.

From here we are ready to begin constructing our portfolios.

### Brief Recap

Let's take a step back and review what we've accomplished thus far.

• We began by gaining an understanding of the Stereoscopic Portfolio Optimization (SPO) Framework.
• We then performed K-Means Clustering on the S&P 500.
• We found optimal value for K by creating the elbow graph using our fundamental data features.
• After finding K, we created our K-Means algorithm and selected stocks from Cluster 0 to test whether or not we could identify tradeable relationships.

Out of the first five symbols in Cluster 0, we found that 3 of the 20 possible combinations were cointegrated.

To illustrate how we would proceed had we not found any tradeable relationships, we created a method that would allow us to select all of the stocks from within a specific cluster and use these to create all possible pair combinations.

With these pairs in hand, we could then test these pairs to see if we could find tradeable relationships.

### Portfolio Construction: Equal Weighted

Now that we have found some likely tradeable relationships, we're ready to construct our portfolio. In this section, we're going to create a portfolio that equally weighs our strategy.

We'll assume that we have a portfolio value of $100k USD with 10% in cash. We will allocate$30k USD to each of our pairs. To begin we will create a class that will allow us to create our StatArb strategies.

Recall, our training period for our portfolios that will be employing the use of machine learning is 1/4/2018 to 4/30/2018, thus because this portfolio will not be employing the use of an optimization method, we will construct it over the period beginning 5/1/2018 so that it is consistent with the test period of our remaining portfolios.

Let's now create the variables of each of our symbols that will hold the data beginning 5/1/2018.

We'll make a copy of our data and set our date as the index so that we can parse out our test period.

#creating copies of our data
aet_copy=aet.copy()
antm_copy=antm.copy()
#reindexing our data
aet_copy=aet_copy.reindex(index=aet_copy['Date'],columns=aet_copy.columns)
antm_copy=antm_copy.reindex(index=antm_copy['Date'],columns=antm_copy.columns)

Okay. Let's check our dataframe.

#checking dataframe after reindexing

2018/01/04NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
2018/01/04NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
2018/01/04NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
2018/01/04NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN
2018/01/04NaNNaNNaNNaNNaNNaNNaNNaNNaNNaN

Now let's restore the values back to our columns. We'll also drop the date column.

#dropping date columns
aet_copy.drop('Date',axis=1,inplace=True)
antm_copy.drop('Date',axis=1,inplace=True)
#restoring our column values back to our data
antm_copy[[' Time',' Open',' High',' Low',' Close',' Volume',' NumberOfTrades',' BidVolume',' AskVolume']]=antm[[' Time',' Open',' High',' Low',' Close',' Volume',' NumberOfTrades',' BidVolume',' AskVolume']].values

Let's recheck our dataframe.

#rechecking our dataframe

2018/01/0408:30:00181.93183.74181.84183.13838623962294160921
2018/01/0408:35:00183.05183.48182.66182.89315932381784013753
2018/01/0408:40:00182.84183.35182.70183.22496543452696222692
2018/01/0408:45:00183.25183.73183.06183.73511083142156629542
2018/01/0408:50:00183.66184.00183.56183.90274761961528712189

Now we're ready to slice out our test period.

#creating variables to hold testing period data
aet_test=aet_copy.loc['2018/05/01':]
antm_test=antm_copy.loc['2018/05/01':]

Let's check the head and tail of our data.

#checking beginning of test period

2018/05/0108:30:00220.77221.28220.33220.88523512333990412447
2018/05/0108:35:00220.95221.34219.78220.09319692421535016619
2018/05/0108:40:00220.10220.15219.38219.64261631751462211541
2018/05/0108:45:00219.52220.55219.52220.34227561171089011866
2018/05/0108:50:00220.31220.82220.24220.512399489180445950
#checking the end of test period

2018/06/1214:30:00252.26252.35252.25252.34356442851599619648
2018/06/1214:35:00252.35252.35252.05252.27320742611474617328
2018/06/1214:40:00252.27252.32252.12252.20258092221390911900
2018/06/1214:45:00252.19252.36252.06252.36431283401968423444
2018/06/1214:50:00252.40252.53252.29252.47534804502547528005

We can now parse out our closing values from our test period dataframes.

#Closing price series for data
aet_test_price_series=np.array(aet_test[' Close'])
antm_test_price_series=np.array(antm_test[' Close'])

Let's check one of our price series. We'll use AET.

#checking head of AET price series
aet_test_price_series[0:5]
array([ 180.42, 179.7 , 179.3 , 179.53, 179.7 ])

We can now construct our StatArb class and create our individual strategies. We will then combine these individual strategies into an equally weighted portfolio.

class statarb(object):

def __init__(self,df1, df2,ma,floor, ceiling,beta_lookback,start,end,exit_zscore=0):
#setting the attributes
self.df1=df1 #array of prices for X
self.df2=df2 #array of prices for Y
self.ma=ma# the lookback period
self.floor=floor #the buy threshold for the z-score
self.ceiling=ceiling #the sell threshold for the z-score
self.Close='Close Long' #used as close signal for longs
self.Cover='Cover Short' #used as close signal for shorts
self.exit_zscore=exit_zscore #the z-score
self.beta_lookback=beta_lookback #the lookback for hedge ratio
self.start=start #the beginning of test period as a string
self.end=end # the end of test period as a string
#creating new dataframe
self.df=pd.DataFrame(index=range(0,len(self.df1)))

try:
self.df['X']=self.df1
self.df['Y']=self.df2

except:
print('Length of self.df:')
print(len(self.df))
print('')
print('Length of self.df1:')
print(len(self.df1))
print('')
print('Length of self.df2:')
print(len(self.df2))
#calculating the beta of the pairs
ols=linregress(self.df['Y'],self.df['X'])
self.df['Beta']=ols[0]

def generate_signals(self):
#creating the z-score
#prior z-score
self.df['Prior Z-Score']=self.df['Z-Score'].shift(1)
#Creating Buy and Sell Signals; when to be long, short, exit
self.df['Exit']=(np.abs(self.df['Z-Score'])<=self.exit_zscore)*1.0
#tracking positions via for loop implementation
self.df['Long_Market']=0.0
self.df['Short_Market']=0.0
#Setting Variables to track whether or not to be long while iterating over df
self.long_market=0
self.short_market=0
for i,value in enumerate(self.df.iterrows()):
#Calculate longs
if value[1]['Longs']==1.0:
self.long_market=1
if value[1]['Shorts']==1.0:
self.short_market=1
if value[1]['Exit']==1.0:
self.long_market=0
self.short_market=0
self.df.iloc[i]['Long_Market']=self.long_market
self.df.iloc[i]['Short_Market']=self.short_market

return

def create_returns(self, allocation,pair_number):
'''
PARAMETERS
##########
allocation - the amount of capital alotted for pair
pair_number - string to annotate the plots
'''
self.allocation=allocation
self.pair=pair_number

self.portfolio=pd.DataFrame(index=self.df.index)
self.portfolio['Positions']=self.df['Long_Market']-self.df['Short_Market']
self.portfolio['X']=-1.0*self.df['X']*self.portfolio['Positions']
self.portfolio['Y']=self.df['Y']*self.portfolio['Positions']
self.portfolio['Total']=self.portfolio['X']+self.portfolio['Y']
#creating a percentage return stream
self.portfolio['Returns']=self.portfolio['Total'].pct_change()
self.portfolio['Returns'].fillna(0.0,inplace=True)
self.portfolio['Returns'].replace([np.inf,-np.inf],0.0,inplace=True)
self.portfolio['Returns'].replace(-1.0,0.0,inplace=True)
#calculating metrics
self.mu=(self.portfolio['Returns'].mean())
self.sigma=(self.portfolio['Returns'].std())
self.portfolio['Win']=np.where(self.portfolio['Returns']>0,1,0)
self.portfolio['Loss']=np.where(self.portfolio['Returns']<0,1,0)
self.wins=self.portfolio['Win'].sum()
self.losses=self.portfolio['Loss'].sum()
#calculating sharpe ratio with interest rate of
#interest_rate_assumption=0.75
#self.sharp=(self.mu-interest_rate_assumption)/self.sigma
#win loss ratio;
self.win_loss_ratio=(self.wins/self.losses)
#probability of win
#probability of loss
#average return of wins
self.avg_win_return=(self.portfolio['Returns']>0).mean()
#average returns of losses
self.avg_loss_return=(self.portfolio['Returns']<0).mean()
#calculating payout ratio
self.payout_ratio=(self.avg_win_return/self.avg_loss_return)
#calculate equity curve
self.portfolio['Returns']=(self.portfolio['Returns']+1.0).cumprod()
self.portfolio['Portfolio Value']=(self.allocation*self.portfolio['Returns'])
self.portfolio['Portfolio Returns']=self.portfolio['Portfolio Value'].pct_change()
self.portfolio['Initial Value']=self.allocation

with plt.style.context(['ggplot','seaborn-paper']):
#Plotting Portfolio Value
plt.plot(self.portfolio['Portfolio Value'])
plt.plot(self.portfolio['Initial Value'])
plt.title('%s Strategy Returns '%(self.pair))
plt.legend(loc=0)
plt.show()

return


Okay! We're ready to apply our StatArb class to create our strategies. Recall that our pairs are ADBE & ANTM, ANTM & AET and AET & ANTM.

You may be wondering why we're using ANTM & AET and AET & ANTM. This is because though both pairs use the same symbols, they are in fact two completely different relationships.

In the first pair, ANTM is the X and AET is the Y. In a sense, we're asking what degree of variation in AET can be explained by a unit change in ANTM.

In contrast, in the AET & ANTM pair, we're asking what degree of variation in ANTM can be explained by a unit change in AET.

When we compute the slope or derivative, we get two completely different numbers, and thus two different relationships.

Let's now apply our StatArb class to our first pair.

#ADBE & ANTM statarb initialization
#passing in X, Y, MA, Floor, Ceiling, Beta Lookback, Start, End
adbe_antm=statarb(adbe_test_price_series,antm_test_price_series,17,-2,2,17,adbe_test.iloc[0],adbe_test.iloc[-1])

Let's now create our spread and generate our signals.

#creating spread


0220.88236.05-0.763772NaN
1220.09236.74-0.763772NaN
2219.64237.11-0.763772NaN
3220.34237.26-0.763772NaN
4220.51236.76-0.763772NaN

We can now generate our signals and compute our returns on our allocation of \$30k USD.

#generating signals
adbe_antm.generate_signals()
#creating returns and passing in our allocation amount
adbe_antm.create_returns(30000,'ADBE_ANTM')

We can repeat this process for our remaining pairs. Note, our create_spread method returned the head of our dataframe containing our spread.

We used a lookback of 17 which explains why those values are NaNs.

#repeating process for ANTM and AET
antm_aet=statarb(antm_test_price_series,aet_test_price_series,6,-2,2,6,antm_test.iloc[0],antm_test.iloc[-1])
antm_aet.generate_signals()
antm_aet.create_returns(30000,'ANTM & AET')

#repeating process for AET and ANTM
aet_antm=statarb(aet_test_price_series,antm_test_price_series,12,-2,2,12,aet_test.iloc[0],aet_test.iloc[-1])
aet_antm.generate_signals()
aet_antm.create_returns(30000,'AET & ANTM')

Okay. Not that we have created our individual StatArb implementations, we can combine these into a portfolio, calculate our portfolio returns, mu, sigma, and Sharpe ratio.

Note, to compute the Sharpe we will need to make an assumption about the level of interest rates.

In as of December 2017, the fed funds rate was 1.5%. We'll use this as our interest rate assumption. Recall that we started with a portfolio value of 100k USD.

We allocated 10k USD to cash and equally weighted our StatArb strategies.

### Creating the Equally Weighted Portfolio

We included a portfolio value variable in our statarb class. We'll use this to create our total equally weighted portfolio.

#creating dataframe for equally weighted portfolio
equally_weighted=pd.DataFrame()
equally_weighted['ANTM_AET']=antm_aet.portfolio['Portfolio Value']
equally_weighted['AET_ANTM']=aet_antm.portfolio['Portfolio Value']
equally_weighted['Cash']=10000
equally_weighted['Total Portfolio Value']=equally_weighted['ADBE_ANTM']+equally_weighted['ANTM_AET']+equally_weighted['AET_ANTM']+equally_weighted['Cash']

Let's check our equally_weighted portfolio dataframe.

equally_weighted.head()


030000.030000.030000.010000100000.0
130000.030000.030000.010000100000.0
230000.030000.030000.010000100000.0
330000.030000.030000.010000100000.0
430000.030000.030000.010000100000.0

We can now add our returns column and then use it to compute our Mu, Sigma and Sharpe ratio.

#adding returns column
equally_weighted['Returns']=np.log(equally_weighted['Total Portfolio Value']/equally_weighted['Total Portfolio Value'].shift(1))
#rechecking our dataframe
equally_weighted.head()

030000.030000.030000.010000100000.0NaN
130000.030000.030000.010000100000.00.0
230000.030000.030000.010000100000.00.0
330000.030000.030000.010000100000.00.0
430000.030000.030000.010000100000.00.0

We'll now get our Mu, Sigma, and Sharpe and store them in variables.

#initializing Equally_Weighted portfolio metrics
equally_weighted_mu=equally_weighted['Returns'].mean()
equally_weighted_sigma=equally_weighted['Returns'].std()
#initializing interest rate assumption of 1.5%
rate=0.015
#computing Sharpe
equally_weighted_Sharpe=round((equally_weighted_mu-rate)/equally_weighted_sigma,2)

Okay, let's check the Sharpe Ratio of our Equally Weighted Portfolio.

#getting Equally Weighted Portfolio Sharpe

print('Equally Weighted Portfolio Sharpe:',equally_weighted_Sharpe)
Equally Weighted Portfolio Sharpe: -2.1

Let's plot our portfolio equity curve.

#plotting Equally Weighted Equity Curve
plt.figure(figsize=(10,6))
plt.plot(equally_weighted['Total Portfolio Value'])
plt.title('Equally Weighted Portfolio Equity Curve')
plt.show()

We can see that even though our equally weighted portfolio value increased, it still had a negative Sharpe.

We can repeat this process for our remaining portfolios and compare our Sharpe ratios.

### Portfolio Construction: Efficient Frontier

Thus far we've created our Equally Weighted Portfolio and checked our Sharpe Ratio and Equity Curve.

We are now interested in what our Sharpe might be if we found the optimal weights for each of our StatArb implementations as well as the resulting effect on our equity curve.

We will now turn our attention to simulating our portfolio's mean and variance by assigning random weights to each of our pairs. We will store our portfolio's mean return and sigma and create a scatter plot that will aid us in selecting the most efficient weights for each of our strategy implementations.

To best decipher how we should weight our strategy implementations we will include logic that computes the Sharpe Ratios for each portfolio created. We can then create new instances of our StatArb implementations using updated allocations and survey the effects on our Sharpe Ratio and equity curve.

Though our Bottom Up Portfolio will be an extension of our Equally Weighted Portfolio in that it will only apply our machine learning concepts to the microstructure components, our Stereoscopic Portfolio Optimization (SPO) Framework will combine the weights found within this portfolio construction process and combine it with the process of constructing the Bottom Up Portfolio.

To begin, let's retrieve the MUs and Sigmas of our pairs. Recall that we created a method within the class that stores our mean and sigma.

We can use our strategy objects to retrieve these variables.

#initialzing Mus and Sigmas
#ANTM & AET
antm_aet_mu=antm_aet.mu
antm_aet_sigma=antm_aet.sigma
#AET & ANTM
aet_antm_mu=aet_antm.mu
aet_antm_sigma=aet_antm.sigma

The return of our portfolio can be expressed as

or as

where,

wi is the weight of the ith asset, and
ri is the return of the ith asset.

Let's create a method to compute returns.

#computing log returns for our portfolio values
returns=np.log(equally_weighted[['ADBE_ANTM','ANTM_AET','AET_ANTM']]/equally_weighted[['ADBE_ANTM','ANTM_AET','AET_ANTM']].shift(1))

Let's check our returns.

#checking returns


0NaNNaNNaN
10.00.00.0
20.00.00.0
30.00.00.0
40.00.00.0

We can now create our mean returns an annualize them.

avg_returns_252=returns.mean()*252

Let's check our average annualized returns.

#checking average annualized returns
avg_returns
[0.00045914208401620995, 3.3720231332227273e-06, 0.00016147773221808213]

The variance of our portfolio can be expressed as

where,

wi is the weight of the ith asset,
wj is the weight of the jth asset.

Now we will use our returns to create our annualized covariance matrix.

covariance_matrix=returns.cov()*252

Let's check our covariance matrix.

covariance_matrix


ANTM_AET0.0001211.834339e-05-1.338726e-07
AET_ANTM-0.002592-1.338726e-071.212121e-03

We can now create a variable to hold our weights for each strategy.

#assigning weights
weights=np.random.random(len(returns.columns))
weights/=np.sum(weights)
#reviewing weights
weights
array([ 0.22235774, 0.15788307, 0.61975919])

The following method is an adaptation from Dr. Yves J. Hilpisch's "Python for Finance".

We'll use it to plot our Efficient Frontier and find the most optimal weights for our strategies.

#importing optimization function
import scipy.optimize as sco
def efficient_frontier(returns,rate=0.015):
#creating a list to hold our portfolio returns, variance and Sharpe values
portfolio_returns=[]
portfolio_volatility=[]
p_sharpes=[]
# returns=returns_df

for i in range(500):
#assigning weights
weights=np.random.random(len(returns.columns))
weights/=np.sum(weights)
#getting returns
current_return=np.sum(returns.mean()*weights)*252
portfolio_returns.append(current_return)
#getting variances
variance=np.dot(weights.T,np.dot(returns.cov()*252,weights))
#getting volatility
volatility=np.sqrt(variance)
portfolio_volatility.append(volatility)
#getting Sharpe ratios
ratio=(current_return-rate)/volatility
#storing Sharpe in list
p_sharpes.append(ratio)

p_returns=np.array(portfolio_returns)
p_volatility=np.array(portfolio_volatility)
p_sharpes=np.array(p_sharpes)
#plotting
plt.figure(figsize=(10,6))
plt.scatter(p_volatility,p_returns,c=p_sharpes, marker='o')
plt.xlabel('Expected Volatility')
plt.ylabel('Expected Return')
plt.title('Efficient Frontier')
plt.colorbar(label='Sharpe Ratio')
plt.show()

return
efficient_frontier(returns.fillna(0))

def stats(weights,rate=0.015):
weights=np.array(weights)
p_returns=np.sum(returns.mean()*weights)*252
p_volatility=np.sqrt(np.dot(weights.T,np.dot(returns.cov()*252,weights)))
p_sharpe=(p_returns-rate)/p_volatility

return np.array([p_returns,p_volatility,p_sharpe])
#testing stats method
stats(weights)
array([ 0.0325289 , 0.08669494, 0.20219057])
#creating function for optimization
def minimize_func(weights):
return -stats(weights)[2]
#testing optimization function
minimize_func(weights)
-0.20219056844039546
def get_optimal_weights(weights):
#Finding Most Optimal Weights
#variables for optimization
constraints=({'type':'eq','fun':lambda x: np.sum(x)-1})
bounds=tuple((0,1) for x in range(len(returns.columns)))

starting_weights=len(returns.columns)*[1./len(returns.columns)]
most_optimal=sco.minimize(minimize_func,starting_weights, method='SLSQP', bounds=bounds, constraints=constraints)
best_weights=most_optimal['x'].round(3)

return best_weights, print('Weights:',best_weights)
#storing optimal weights in a variable
optimal_weights=get_optimal_weights(weights)
Weights: [ 0.022 0. 0.978]

Okay. We can see that the most optimal weights are to allocate 2% of our capital to ADBE_ANTM, 0% to ANTM_AET and 97.8% to AET_ANTM.

Recall that our method was designed to optimize our Sharpe ratio. We can now implement another instance of our strategies using these weights and create our portfolio and compute its equity curve.

Let's create our second instance of the ADBE_ANTM pair.

We'll create a total_allocation variable that we can use to compute our allocation to our strategies.

#total allocation variable
total_allocation=90000 #100k less 10k cash
#AET_ANTM Allocation
aet_antm_allocation=round(total_allocation*optimal_weights[0][2],2)
#creating 2nd instance of first pair
#notice here we're using our updated allocation
adbe_antm_2.create_returns(adbe_antm_allocation,'ADBE_ANTM_Portfolio _2')

Recall that our second pair received a 0% allocation. We will thus create our 3rd pair with the updated weight and create our new portfolio.

#AET_ANTM 2nd implementation
aet_antm_2=statarb(aet_test_price_series,antm_test_price_series,12,-2,2,12,aet_test.iloc[0],aet_test.iloc[-1])
aet_antm_2.generate_signals()
aet_antm_2.create_returns(aet_antm_allocation,'AET & ANTM')

We are now ready to compose our Efficient Frontier Portfolio.

### Creating the Efficient Frontier Portfolio

Let's create our Efficient Frontier Portfolio.

#creating dataframe for Efficient Frontier Portfolio
efficient_frontier_portfolio=pd.DataFrame()
efficient_frontier_portfolio['AET_ANTM']=aet_antm_2.portfolio['Portfolio Value']
efficient_frontier_portfolio['Cash']=10000
efficient_frontier_portfolio['Total Portfolio Value']=efficient_frontier_portfolio['ADBE_ANTM']+efficient_frontier_portfolio['AET_ANTM']+efficient_frontier_portfolio['Cash']

We can now add our returns column to our Efficient Frontier Dataframe.

#adding returns column to Efficient Frontier Dataframe
efficient_frontier_portfolio['Returns']=np.log(efficient_frontier_portfolio['Total Portfolio Value']/efficient_frontier_portfolio['Total Portfolio Value'].shift(1))

We can now check our Efficient Frontier Portfolio Dataframe.

#checking head of Efficient Frontier Portfolio dataframe


01980.088020.010000100000.0NaN
11980.088020.010000100000.00.0
21980.088020.010000100000.00.0
31980.088020.010000100000.00.0
41980.088020.010000100000.00.0

We'll now plot the equity curve for our Efficient Frontier Portfolio.

#plotting Equity Curve for Efficient Frontier Portfolio
plt.figure(figsize=(10,6))
plt.plot(efficient_frontier_portfolio['Total Portfolio Value'])
plt.title('Efficient Frontier Portfolio Equity Curve')
plt.show()

We can now store our Mu, Sigma, and Sharpe for our Efficient Frontier Portfolio in variables.

efficient_frontier_portfolio_mu=efficient_frontier_portfolio['Returns'].mean()
efficient_frontier_portfolio_sigma=efficient_frontier_portfolio['Returns'].std()
#recall that we initialized our interest assumption earlier
efficient_frontier_portfolio_sharpe=(efficient_frontier_portfolio_mu-rate)/efficient_frontier_portfolio_sigma

Okay. We've just finished our second portfolio. Two down and two to go.

We can now begin constructing our Bottom Up Optimization Portfolio.

### Portfolio Construction: Bottom Up Optimization

The Bottom Up Portfolio applies machine learning to the composition of the equally weighted portfolio.

This means that we will use equal weights but instead of optimizing using the Efficient Frontier, we will use bottom-up optimization.

• The idea is to have our GMM identify specific regimes or distributions.
• We can then use our GMM's predictions as labels for our Random Forests.
• Our signal generator will then trade based only when our strategy is within a regime in which it historically has not underperformed.

Recall that our assessment period is from 5/1/18 to 6/12/18. In our prior portfolios, we were able to create our portfolios using this period.

In this portfolio, however, we will be using machine learning. That means that we will need to create our models over the period beginning 1/4/18 and ending 4/30/18 and actually construct on portfolio using these models over our test period.

Let's begin by creating our training period variables.

Recall that we created a dataframe earlier called training_df that held our data over our training period.

We initially created this dataframe so that we could test for cointegration over our training period. We can now reuse it here.

Let's review our training_df dataframe.

training_df.head()


0183.13183.4560.36229.6254.67
1182.89183.6860.61229.6354.51
2183.22183.8060.77229.7054.25
3183.73183.8260.81229.6554.08
4183.90183.9360.80229.7054.27

We have the prices of our symbols from 1/4/18 to 4/30/18. No pair containing AMAT was found to be cointegrated at least 90% confidence interval so we won't be using it.

We can now recreate our equally weighted strategy implementations using our training period data.

Once we have the returns for our implementations we can then engineer some features to use in training our Gaussian Mixture Model. We will then use these predictions as labels for our Random Forests.

Once we've trained our Random Forests to predict the regimes found by our Gaussian Mixture Model, we can then use those predictions to augment our signal generator, avoiding troublesome regimes.

Let's begin creating our strategy implementations.

#### Step 1: Feature Engineering

We are now ready to engineer our features. In order to predict what regime our strategy is in based on its historical performance, we will need features or explanatory variables.

Market-Microstructure encompasses a vast array of components including liquidity, volatility, market depth, etc. For this illustration, we will use volatility.

We will track the volatility of each of our underlying components, our spread, and our z-score.

These features will be used within our Random Forests to predict the regimes identified by our Gaussian Mixture Model.

#### Step 2: Creating Strategies Over Historical Period

We'll begin with our first pair, ADBE_ANTM. Before we begin our implementation we need to split our historical period.

Recall that we imported intraday data from 1/4/18 to 6/12/18. Our assessment period or the period in which we are comparing our Sharpe ratios is from 5/1/18 to 6/12/18. In our first two portfolios, we simply applied our strategies to our assessment period.

However, in our later two portfolios, we need to first apply it to our historical period to train our models then apply it to our assessment period.

To assess our models before applying them to our true testing or assessment period we need to split our historical period into a historical training and testing set.

This will enable us to train our models on part of our total training data and then test it on our historical testing data. We can then apply them to our overall testing or assessment period.

For simplicity, we'll manually calculate an 80/20 train_test_split.

#checking length of training_df
len(training_df)
6240
#computing 80% of training_df length
len(training_df)*.80
4992.0
#computing 20% of training_df length
len(training_df)*-.20
-1248.0

Okay now that we have the index values that represent the 80% and 20% marks of our training_df dataframe, we can use them to slice our data.

The way that we can do this is for our 80%, or the first 4992 rows we can simply use the notation [0:4992]. For our testing period over our historical training period, we can parse the last 20% by using the notation [-1248].

These notations will be used to split the returns we generate over our entire historical period.

adbe_antm_3_historical=statarb(training_df['ADBE'],training_df['ANTM'],17,-2,2,17,adbe_test.iloc[0],adbe_test.iloc[-1])
#notice that we are equally weighing our strategy
adbe_antm_3_historical.create_returns(30000,'ADBE_ANTM_3 Over Training Period')

Okay now that we've created our strategy, let's check our returns.

adbe_antm_3_historical_rets=adbe_antm_3_historical.portfolio['Returns']
#checking head of returns
adbe_antm_3_historical_rets.head()
0 1.0
1 1.0
2 1.0
3 1.0
4 1.0
Name: Returns, dtype: float64

We will now repeat this process for our remaining pairs.

#ANTM_AET Bottom Up Historical Implementation
antm_aet_3_historical=statarb(training_df['ANTM'],training_df['AET'],6,-2,2,6,antm_test.iloc[0],antm_test.iloc[-1])
antm_aet_3_historical.generate_signals()
antm_aet_3_historical.create_returns(30000,'ANTM & AET_3 Over Hist. Train Period')

Let's store our returns from the ANTM_AET training period in a variable

antm_aet_3_historical_rets=antm_aet_3_historical.portfolio['Returns']

Okay, now we finish up with our AET_ANTM pair.

aet_antm_3_historical=statarb(training_df['AET'],training_df['ANTM'],12,-2,2,12,aet_test.iloc[0],aet_test.iloc[-1])
aet_antm_3_historical.generate_signals()
aet_antm_3_historical.create_returns(30000,'AET & ANTM')

Let's store our AET_ANTM returns in a variable.

aet_antm_3_historical_rets=aet_antm_3_historical.portfolio['Returns']
#checking AET_ANTM returns over the training period
aet_antm_3_historical_rets.head()
0 1.0
1 1.0
2 1.0
3 1.0
4 1.0
Name: Returns, dtype: float64


#### Step 3: Using Gaussian Mixture Model to Identify Historical Regimes

We now have the returns of our StatArb strategies over our historical period.

Our next task is to identify our historical regimes. Our goal is to identify the regimes in which our stratetgies have not performed well on a risk to reward basis and avoid these periods.

To achieve this, we must:

1. identify our historical regimes and
2. engineer features that can be used to predict these regimes.

We can then augment our signal generator and thus optimize our portfolio from the microstructure level up.

We'll begin by splitting our returns data into an 80/20 train test split.

#splitting historical returns data
#getting length of returns data
#ANTM_AET
#getting length of returns data
antm_aet_3_rets_len=len(antm_aet_3_historical_rets)
antm_aet_3_rets_train=antm_aet_3_historical_rets[0:4992]
antm_aet_3_rets_test=antm_aet_3_historical_rets[-1248:]
#AET_ANTM
#getting length of returns data
aet_antm_3_rets_len=len(aet_antm_3_historical_rets)
aet_antm_3_rets_train=aet_antm_3_historical_rets[0:4992]
aet_antm_3_rets_test=aet_antm_3_historical_rets[-1248:]

Now that we have created our train and test splits we're ready to identify our regimes. We'll start by training our GMM on our training period data, then we'll apply it to our testing period data.

We'll create a method that we can use for our GMM and Random Forests.

class gmm_randomForests(object):
def __init__(self,historical_rets_train,historical_rets_test,base_portfolio_rets,gmm_components,df,base_portfolio_df,
internal_test_start,internal_test_end):

'''
PARAMETERS
##########
The first 3 parameters should have already been sliced from the
entire sample; ie specific dates parsed from dataframe that contains
later in development within the object

historical_rets_train- the returns data over the historical period used to train on
this is the internal train of the internal 80/20 split;the 80%
of the total historical training set

historical_rets_test- retuns data for 20% of internal training test set

base_portfolio_rets - this is our figurative live data;ie returns; 5/01/18-6/12/18 from either
Equally Weighted or Efficient Frontier Portfolios dependent upon implementation
of Bottom Up or Stereoscopic Portfolio Optimization (SPO) Framework

ex. data over 01/04/18-04/30/18
we would first split this 80/20
the 80% is our training set
the 20% is our testing set

we would then do another split on our training set created above
this is so that if we can better understand the historical
regimes and recalibrate our models if necessary before actually
predicting our 5/1/18-6/12/18 testing set

in this ex. our gmm_training_train is 80% of the period 01/04/18-4/30/18
our gmm_training_test is 20% of the period 01/04/18-4/30/18 and our
gmm_test_actual is 05/01/18-6/12/18

gmm_components - type:int; for number of components for GMM

df - the entire dataframe containing prior trading history; the dataframe from either Equally Weighted or
Efficient Frontier Portfolios; Our Random Forests Implementation will take this dataframe created by
our statarb class(i.e. from the prior portfolios) and add our features to it. It will then use these
features to predict the regimes of our test period. Recall that our Equally Weighted and Efficient
Frontier Portfolios were constructed over our assessment period of 5/1/18 to 6/12/18. We will then
be able to store our predictions in a varible for our test period. These predictions will be passed
into a new statarb object as a parameter and be used to create the Bottom Up and SPO Framework Portfolios.

total_training_start- type: string; the beginning of the historical period for analysis; input as string
from example above would be 2012

total_training_end - type:string; the end of the historical period for analysis; input as string;
from example above would be 2017

base_portfolio_df - (i.e.adbe_antm.df,etc)Note: for the Bottom Up Implementation this df would be the Equally Weighted df but for the
SPO Framework df this would be the df from the Efficient Frontier implementation

internal_test_start- type:int; this is the testing period for the total training period; in the example,
this is the assessment period of 05/01/18-06/12/18; start thus is len(strategy_object)*.80

internal_test_end - type:int; this is the end of the assessment period; ie. the 20% testing split
of the broader split...this value is -len(strategy_object)*.20...Note this value is
is negative because we want the last 20% of the data

The last two methods are created as such because our Equally Weighted and Efficient Frontier
Portfolios are not indexed by dates but are indexed by a sequence of ints. This means
that in a later method(i.e. Random Forests) we have to pass in the int value of the index
position we want to parse
'''
self.historical_rets_train=historical_rets_train
self.historical_rets_test=historical_rets_test
self.base_portfolio_rets=base_portfolio_rets
self.gmm_components=gmm_components
self.max_iter=300
self.random_state=101
self.df=df
#self.total_training_start=total_training_start
#self.total_training_end=total_training_end
self.base_portfolio_df=base_portfolio_df
self.internal_test_start=internal_test_start
self.internal_test_end=internal_test_end
#creating volatility
self.volatility=self.historical_rets_train.rolling(window=5).std()
self.negative_volatility=np.where(self.historical_rets_train<0,self.historical_rets_train.rolling(window=5).std(),0)

def make_gmm(self):
model_kwds=dict(n_components=self.gmm_components,max_iter=self.max_iter,n_init=100,random_state=self.random_state)

gmm=GM(**model_kwds)

return gmm

def analyze_historical_regimes(self):
#Creating a Gaussian Mixture Model
self.gmm=self.make_gmm()
#instantiating the XTrain as the gmm_training_train; (i.e. the 80% of total training period)
self.gmm_XTrain=np.array(self.historical_rets_train).reshape(-1,1)
#Fitting the GMM on the Training Set(note this is the internal training set within the broader training set)
self.gmm.fit(self.gmm_XTrain.astype(int))
#Making predictions on the historical period; ie. the gmm_training_train
self.gmm_historical_predictions=self.gmm.predict(self.gmm_XTrain.astype(int))
#Making Predictions on the gmm_training_test (i.e. the 20% of total training period;)
self.gmm_XTest=np.array(self.historical_rets_test).reshape(-1,1)
self.gmm_training_test_predictions=self.gmm.predict(self.gmm_XTest.astype(int))
#Fitting the Model on ACTUAL data we want to Predict Regimes For
self.gmm_Actual=np.array(self.base_portfolio_rets).reshape(-1,1)
self.base_portfolio_predictions=self.gmm.predict(self.gmm_Actual)

return

def historical_regime_returns_volatility(self,plotTitle):
self.plotTitle=plotTitle
data=pd.DataFrame({'Volatility':self.volatility,'Regime':self.gmm_historical_predictions,'Returns':self.historical_rets_train})

with plt.style.context(['classic','seaborn-paper']):
fig,ax=plt.subplots(figsize=(15,10),nrows=1, ncols=2)

left = 0.125 # the left side of the subplots of the figure
right = 0.9 # the right side of the subplots of the figure
bottom = .125 # the bottom of the subplots of the figure
top = 0.9 # the top of the subplots of the figure
wspace = .5 # the amount of width reserved for blank space between subplots
hspace = 1.1 # the amount of height reserved for white space between subplots
# function that adjusts subplots using the above paramters
left = left,
bottom = bottom,
right = right,
top = top,
wspace = wspace,
hspace = hspace
)
# The amount of space above titles
y_title_margin = 2

plt.suptitle(self.plotTitle, y = 1, fontsize=20)

plt.subplot(121)
sns.swarmplot(x='Regime',y='Volatility',data=data)#,ax=ax[0][0])
plt.title('Regime to Volatility')

plt.subplot(122)
sns.swarmplot(x='Regime',y='Returns',data=data)#, ax=ax[0][1])
plt.title('Regime to Returns')
plt.tight_layout()
plt.show()

return

def train_random_forests(self):
#assumption is that this is the df
#over the entire period

'''CAN UNCOMMENT THE SELF.VIX CODE TO INCLUDE VIX IN ANALYSIS..NOTE CHANGES TO PANDAS_DATAREADER API'''
#getting vix to add as feature
#self.VIX=pdr.get_data_yahoo('^VIX',start=self.total_training_start,end=self.total_training_end)
#creating features
#self.df['VIX']=self.VIX['Close']
self.df['6 X Vol']=self.df['X'].rolling(window=6).std()
self.df['6 Y Vol']=self.df['Y'].rolling(window=6).std()
self.df['6 Z-Score Vol']=self.df['Z-Score'].rolling(window=6).std()

self.df['12 X Vol']=self.df['X'].rolling(window=12).std()
self.df['12 Y Vol']=self.df['Y'].rolling(window=12).std()
self.df['12 Z-Score Vol']=self.df['Z-Score'].rolling(window=12).std()

self.df['15 X Vol']=self.df['X'].rolling(window=15).std()
self.df['15 Y Vol']=self.df['Y'].rolling(window=15).std()
self.df['15 Z-Score Vol']=self.df['Z-Score'].rolling(window=15).std()
#adding features to our historical df
#self.base_portfolio_df['VIX']=self.VIX['Close']
self.base_portfolio_df['6 X Vol']=self.df['X'].rolling(window=6).std()
self.base_portfolio_df['6 Y Vol']=self.df['Y'].rolling(window=6).std()
self.base_portfolio_df['6 Z-Score Vol']=self.df['Z-Score'].rolling(window=6).std()

self.base_portfolio_df['12 X Vol']=self.df['X'].rolling(window=12).std()
self.base_portfolio_df['12 Y Vol']=self.df['Y'].rolling(window=12).std()
self.base_portfolio_df['12 Z-Score Vol']=self.df['Z-Score'].rolling(window=12).std()

self.base_portfolio_df['15 X Vol']=self.df['X'].rolling(window=15).std()
self.base_portfolio_df['15 Y Vol']=self.df['Y'].rolling(window=15).std()
self.base_portfolio_df['15 Z-Score Vol']=self.df['Z-Score'].rolling(window=15).std()
#replacing na values
self.df.fillna(0, inplace=True)
#Creating X_Train for RF over the Historical Period; Will train
#over Historical period, ie. self.historical_training_start/end
#then predict
self.RF_X_TRAIN=self.df[0:4992][['6 X Vol','6 Y Vol','6 Spread Vol','6 Z-Score Vol','12 X Vol','12 Y Vol',
'12 Spread Vol','12 Z-Score Vol','15 X Vol','15 Y Vol','15 Spread Vol','15 Z-Score Vol']]
#dropping unnecessary columns from train data
#self.RF_X_TRAIN.drop(['X','Y','Longs','Shorts','Exit','Long_Market','Short_Market'],inplace=True,axis=1)
#setting Y_Train for the RF to the predictions of GMM over historical period
self.RF_Y_TRAIN=self.gmm_historical_predictions
#Creating X_Test for the RF; ie the gmm_training_test period
#our features for RF
#that haven't been seen by the model
self.RF_X_TEST=self.base_portfolio_df[['6 X Vol','6 Y Vol','6 Spread Vol','6 Z-Score Vol','12 X Vol','12 Y Vol',
'12 Spread Vol','12 Z-Score Vol','15 X Vol','15 Y Vol','15 Spread Vol','15 Z-Score Vol']]\
#dropping unnecessary columns from train data
#self.RF_X_TEST.drop(['X','Y','Longs','Shorts','Exit','Long_Market','Short_Market'],inplace=True,axis=1)
#predictions for the x test over the internal testing period
self.RF_Y_TEST=self.base_portfolio_predictions #regime predictions for base portfolio
#build the RandomForest and check precision
self.RF_MODEL=RF(n_estimators=100) #Recall imported RandomForestsClassifier as RF
#training the random forests model on assessment period data
self.RF_MODEL.fit(self.RF_X_TRAIN.fillna(0),self.RF_Y_TRAIN)
#Making predictions for base portfolio period
self.RF_BASE_PORTFOLIO_PREDICTIONS=self.RF_MODEL.predict(self.RF_X_TEST.fillna(0))
#Checking Precision of Predictions
#print(confusion_matrix(self.RF_Y_TEST,self.RF_BASE_PORTFOLIO_PREDICTIONS))
#print('\n')
#print(classification_report(self.RF_Y_TEST,self.RF_BASE_PORTFOLIO_PREDICTIONS))

return


Okay. Though I've heavily commented on the above object, I'd like to reiterate the most important inputs to our gmm_randomForests method.

The most critical parameters are:

• gmm_training_train,
• gmm_training_test,
• gmm_test_actual, and
• df. gmm_training_train

Elaborating:

• df. gmm_training_train is the 80% split of the historical period returns we created earlier.
• gmm_training_test is the 20% slice of the historical period returns we created.
• mm_test actual is are the returns from either our Equally Weighted or Efficient Frontier Portfolios.

The Bottom Up Optimization builds on the Equally Weighted Portfolio in that though it offers an optimization technique, the allocations remain equally weighted.

This means that we can use the returns from our Equally Weighted Portfolio, predict the regimes, and create a new statarb object that uses the regimes (i.e. Bottom Up Optimization).
The same logic is true for our Efficient Frontier Portfolio.

The SPO Framework builds on the Efficient Frontier Portfolio. Thus when we design our SPO Framework Portfolio, we will get the regime predictions from our Efficient Frontier implementation.

We will then create a new statarb object that takes those regime predictions in as a parameter and use it to create our SPO Framework Portfolio.

An important thing to keep mind is that our GMM and Random Forests models are trained using our historical data. So when we provide the models with data from our Efficient Frontier and Equally Weighted Portfolios, it's data that our models have not seen.

This was the purpose of separating our data.

We'll begin with our first pair. Let's initialize our method.

#initializing GMM Random Forests method
#passing in the 1)returns for 80% of the training period(i.e.01/04/18-04/30/18),2)returns for 20% of the training period(i.e.01/04/18-04/30/18)
#3) Returns for actual or overall test period; (i.e. 05/01/18-06/12/18) for Equally Weighted, 4) count for GMMs,
#5)Dataframe for Equally Weighted, 6)
adbe_antm_3_historical.df, adbe_antm.df,1871,-468)

Now that we have initiated our object, let's use it to get the regimes of our adbe_antm pair. Note these are based on our historical data.

#calling our analyze historical regimes method
adbe_antm_gmm_rf.analyze_historical_regimes()

We can now plot our regimes using our historical_regime_returns_volatility method.

#Note this method takes in a string to title our plot
adbe_antm_gmm_rf.historical_regime_returns_volatility('ADBE_ANTM GMM Analysis')

From looking at the above plot, we can see that the regime most favourable for our strategy is Regime 1. This regime has the lowest relative volatility but also yields the highest return.

We will make a note of this and incorporate this into our updated statarb implementation.

• One thing to note is that we created these regimes over our historical or training period.
• Recall we performed an 80/20 split on our data.
• This means that any one of these regimes could be the actual regime in which our observations fall in over the period 05/01/18 to 06/21/18.

In practice, the idea would be to look at a historical period(i.e. trailing 3 months) and apply the analysis to the current one month.

We would update and perform this analysis at the end of each month. Let's do the same for the remainder of our pairs.

ANTM_AET GMM Implementation

#initializing object
antm_aet_gmm_rf=gmm_randomForests(antm_aet_3_rets_train,antm_aet_3_rets_test,antm_aet.portfolio['Returns'],5,antm_aet_3_historical.df, antm_aet.df,1871,-468)
#calling our analyze historical regimes method
antm_aet_gmm_rf.analyze_historical_regimes()
#plotting regime returns and volatility...Pass in String for Title of Plot
antm_aet_gmm_rf.historical_regime_returns_volatility('ANTM AET GMM Analysis')

Okay, we can see that Regime 1 appears to offer the best risk/reward tradeoff for our ANTM_AET relationship.

Our returns remain positive throughout the relative volatility distribution but it's more desirable to get equivalent returns with less volatility.

AET_ANTM GMM Implementation

#initializing object
aet_antm_gmm_rf=gmm_randomForests(aet_antm_3_rets_train,aet_antm_3_rets_test,aet_antm.portfolio['Returns'],5,aet_antm_3_historical.df, aet_antm.df,1871,-468)
#calling our analyze historical regimes method
aet_antm_gmm_rf.analyze_historical_regimes()
#plotting regime returns and volatility...Pass in String for Title of Plot
aet_antm_gmm_rf.historical_regime_returns_volatility('AET_ANTM GMM Analysis')

We can see once again that despite the level of volatility, our returns are still positive. However, one thing that we can note from the above plots is that for the most part, the levels of volatility are similar between Regime 0 and Regime 1.

However, we do see some bursts in volatility within Regime 1 and a significant portion of our historical returns, though they were positive, were less than those of Regime 0, of which didn't display any significant bursts.

We won't make any changes to the implementation of this strategy in this illustration but the further analysis could be conducted to determine the sign of returns given those volatility bursts and the same could be used to augment the strategy.

Let's begin using our Random Forests to predict our regimes and update our StatArb strategy by using our regime predictions to answer the above question.

#### Step 4: Training Random Forests to Predict Regimes

Now that we have completed our regime analysis we can use our Random Forests to predict the regimes of our assessment period. Recall that we passed in the dataframe from our Equally Weighted Portfolio created by our adbe_antm object.

We can call our train_random_forests method on our gmm_rf objects to train our Random Forests using the features we selected. We can then create a variable to hold our predictions.

These predictions will then be passed into our new statarb object and used to generate our Bottom Up Portfolio.

#calling the train_random_forests method on our adbe_antm_gmm_rf object
adbe_antm_gmm_rf.train_random_forests()
[[ 0 0]
[ 895 1444]]

precision  recall  f1-score  support
0     0.00     0.00     0.00         0
1     1.00     0.62     0.76      2339
avg / total     1.00     0.62     0.76      2339

Let's ensure that we understand what we've achieved thus far.

• Recall that we created variables to hold the returns of our training period 01/04/18 to 04/30/18.
• We used the training_df dataframe to provide the historical data and applied our StatArb strategy to it.
• We then split that training period returns, of which we stored in variables, into an 80% training and 20% testing subset
• The period of 05/01/18 to 06/12/18 was our hypothetical live market data and is the period in which we created our Equally Weighted and Efficient Frontier Portfolios.
• Our objective with the Bottom Up Portfolio was to build upon our Equally Weighted Portfolio by optimizing using the features we engineered to predict the regime that our strategy was in based on regimes seen historically.
• To achieve this we needed to train our Gaussian Mixture Model using our training over the period 01/04/18 to 04/30/18 and use our GMM Model to predict the regimes from our 05/01/18 to 06/12/18 period.
• We could then use our regimes from the 01/04/18 to 04/30/18 period as labels and the features that we engineered as parameters for our Random Forest training.
• Once we trained our Random Forests we were then able to pass in our features from over the 05/01/18 to 06/12/18 period and use it to predict the regimes.
• To test our accuracy we compared these results to the actual regimes created by our Gaussian Mixture Model over the 05/01/18 to 06/12/18 period.

Our results showed that our Random Forests had perfect precision. This is because, over our hypothetical live trading period, our observations came from the same distribution, which can be seen from our classification report above.

We can now create a variable to store our regime predictions from our hypothetical live trading period (i.e. 05/01/18 to 06/12/18) and feed these into an updated statarb implementation and see how our results would change.

We will tell our updated statarb strategy which regimes to avoid using the chart we created earlier showing our returns and volatility levels per regime.

Not that we've gotten a better understanding of our process, in future implementations we'll comment out the confusion matrix and classification reports so that we won't actually know which regimes our data fell into over our hypothetical live trading period.

We'll solely rely upon our analysis of the volatility and returns characteristics of our regimes and feed this information into our strategy implementation.

Let's store our regime predictions for our ADBE_ANTM pair into a variable that we can use for our updated StatArb implementation.

adbe_antm_regime_predictions=adbe_antm_gmm_rf.base_portfolio_predictions

ANTM_AET Random Forests Implementation

#calling the train_random_forests method on our antm_aet_gmm_rf object
antm_aet_gmm_rf.train_random_forests()

Okay. Let's now store our regime predictions in a variable.

antm_aet_regime_predictions=antm_aet_gmm_rf.base_portfolio_predictions

AET_ANTM Random Forests Implementation

#calling the train_random_forests method on our antm_aet_gmm_rf object
aet_antm_gmm_rf.train_random_forests()

We'll now store our predictions in a variable.

aet_antm_regime_predictions=aet_antm_gmm_rf.base_portfolio_predictions

We can now use these regime predictions to create our updated strategies and compose our Bottom Up Portfolio.

#### Step 5: Using the Model's Predictions in Strategy Development

We'll begin by creating an updated version of our statarb class that will allow us to pass in our regime predictions.

class statarb_update(object):
#np.seterr(divide='ignore',invalid='ignore')

def __init__(self,df1, df2, ptype,ma,floor, ceiling,beta_lookback,start,end,regimePredictions,p2Objective,avoid1=0,target1=0,exit_zscore=0):
#setting the attributes of the data cleaning object
self.df1=df1 #the complete dataframe of X
self.df2=df2 # the comlete dataframe of Y
self.df=pd.DataFrame(index=df1.index) #creates a new dataframe in the create_spread method
self.ptype=ptype #the portfolio type 1= standard implementation 2=machine learning implementation
self.ma=ma# the moving average period for the model
self.floor=floor #the buy threshold for the z-score
self.ceiling=ceiling #the sell threshold for the z-score
self.Close='Close Long' #used as close signal for longs
self.Cover='Cover Short' #used as close signal for shorts
self.exit_zscore=exit_zscore #the z-score
self.beta_lookback=beta_lookback #the lookback of beta for hedge ratio
self.start=start #the beginning of test period as a string
self.end=end # the end of test period as a string
self.regimePredictions=regimePredictions.reshape(-1,1) #the regime predictions from GMM for p2=2 implementation
self.avoid1=avoid1 #the regime to avoid
self.target1=target1 #the regime to target
self.p2Objective=p2Objective #type:string;#the objective of p2 implementation; can be 'Avoid','Target',or 'None';
if self.ptype==1:
#setting the new dataframe values for x and y of the closing
#prices of the two dataframes passed in
self.df['X']=self.df1[' Close']
self.df['Y']=self.df2[' Close']
#calculating the beta of the pairs
self.ols=linregress(self.df['Y'],self.df['X'])
#setting the hedge ratio
self.df['Hedge Ratio']=self.ols[0]

if self.ptype==2:
#setting the new dataframe values for x and y of the closing
#prices of the two dataframes passed in
self.df['X']=self.df1[' Close']
self.df['Y']=self.df2[' Close']
#calculating the beta of the pairs
self.ols=linregress(self.df['Y'],self.df['X'])
#setting the hedge ratio
self.df['Hedge Ratio']=self.ols[0]
#creating the z-score
#Creating the features columns
self.df['6 X Vol']=self.df['X'].rolling(window=6).std()
self.df['6 Y Vol']=self.df['Y'].rolling(window=6).std()
self.df['6 Z-Score Vol']=self.df['Z-Score'].rolling(window=6).std()

self.df['12 X Vol']=self.df['X'].rolling(window=12).std()
self.df['12 Y Vol']=self.df['Y'].rolling(window=12).std()
self.df['12 Z-Score Vol']=self.df['Z-Score'].rolling(window=12).std()

self.df['15 X Vol']=self.df['X'].rolling(window=15).std()
self.df['15 Y Vol']=self.df['Y'].rolling(window=15).std()
self.df['15 Z-Score Vol']=self.df['Z-Score'].rolling(window=15).std()
#Creating the Regime Prediction Column
self.df['Regime']=0
self.df['Regime']=self.regimePredictions.astype(int)

return

def generate_signals(self):
if self.ptype==1:
#creating the z-score
#prior z-score
self.df['Prior Z-Score']=self.df['Z-Score'].shift(1)
#Creating Buy and Sell Signals; when to be long, short, exit
#self.df['Prior Signal']=self.df['Signal'].shift(1)

self.df['Exit']=(self.df['Z-Score']<=self.exit_zscore)*1.0
#tracking positions via for loop implementation
self.df['Long_Market']=0.0
self.df['Short_Market']=0.0
#Setting Variables to track whether or not to be long while iterating over df
self.long_market=0
self.short_market=0
for i,value in enumerate(self.df.iterrows()):
#Calculate logns
if value[1]['Longs']==1.0:
self.long_market=1

if value[1]['Shorts']==1.0:
self.short_market=1

if value[1]['Exit']==1.0:

self.long_market=0
self.short_market=0

self.df.iloc[i]['Long_Market']=self.long_market
self.df.iloc[i]['Short_Market']=self.short_market

if self.ptype==2:

self.df['Exit']=(self.df['Z-Score']<=self.exit_zscore)*1.0
#tracking positions via for loop implementation
self.df['Long_Market']=0.0
self.df['Short_Market']=0.0
#Setting Variables to track whether or not to be long while iterating over df
self.long_market=0
self.short_market=0
for i,value in enumerate(self.df.iterrows()):
if self.p2Objective=='Avoid':
if value[1]['Regime']!= self.avoid1:
#Calculate longs
if value[1]['Longs']==1.0:
self.long_market=1
if value[1]['Shorts']==1.0:
self.short_market=1
if value[1]['Exit']==1.0:
self.long_market=0
self.short_market=0
self.df.iloc[i]['Long_Market']=value[1]['Longs']#self.long_market
self.df.iloc[i]['Short_Market']=value[1]['Shorts']#self.short_market

elif self.p2Objective=='Target':
if value[1]['Regime']==self.target1:
#Calculate longs
if value[1]['Longs']==1.0:
self.long_market=1
if value[1]['Shorts']==1.0:
self.short_market=1
if value[1]['Exit']==1.0:
self.long_market=0
self.short_market=0
self.df.iloc[i]['Long_Market']=value[1]['Longs']#self.long_market
self.df.iloc[i]['Short_Market']=value[1]['Shorts']#self.short_market

elif self.p2Objective=='None':
#Calculate longs
if value[1]['Longs']==1.0:
self.long_market=1
#Calculate Shorts
if value[1]['Shorts']==1.0:
self.short_market=1

if value[1]['Exit']==1.0:

self.long_market=0
self.short_market=0

self.df.iloc[i]['Long_Market']=value[1]['Longs']#self.long_market
self.df.iloc[i]['Short_Market']=value[1]['Shorts']#self.short_market

return self.df

def create_returns(self, allocation,pair_number):
if self.ptype==1:

'''
PARAMETERS
##########
allocation - the amount of capital alotted for pair
pair_number - string to annotate the plots
'''
self.allocation=allocation
self.pair=pair_number
self.portfolio=pd.DataFrame(index=self.df.index)
self.portfolio['Positions']=self.df['Long_Market']-self.df['Short_Market']
self.portfolio['X']=-1.0*self.df['X']*self.portfolio['Positions']
self.portfolio['Y']=self.df['Y']*self.portfolio['Positions']
self.portfolio['Total']=self.portfolio['X']+self.portfolio['Y']
#creating a percentage return stream
self.portfolio['Returns']=self.portfolio['Total'].pct_change()
self.portfolio['Returns'].fillna(0.0,inplace=True)
self.portfolio['Returns'].replace([np.inf,-np.inf],0.0,inplace=True)
self.portfolio['Returns'].replace(-1.0,0.0,inplace=True)
#calculating metrics
self.mu=(self.portfolio['Returns'].mean())
self.sigma=(self.portfolio['Returns'].std())
self.portfolio['Win']=np.where(self.portfolio['Returns']>0,1,0)
self.portfolio['Loss']=np.where(self.portfolio['Returns']<0,1,0)
self.wins=self.portfolio['Win'].sum()
self.losses=self.portfolio['Loss'].sum()
#calculating sharpe ratio with interest rate of
#interest_rate_assumption=0.75
#self.sharp=(self.mu-interest_rate_assumption)/self.sigma
#win loss ratio;
self.win_loss_ratio=(self.wins/self.losses)
#probability of win
#probability of loss
#average return of wins
self.avg_win_return=(self.portfolio['Returns']>0).mean()
#average returns of losses
self.avg_loss_return=(self.portfolio['Returns']<0).mean()
#calculating payout ratio
self.payout_ratio=(self.avg_win_return/self.avg_loss_return)
#calculate equity curve
self.portfolio['Returns']=(self.portfolio['Returns']+1.0).cumprod()
self.portfolio['Portfolio Value']=(self.allocation*self.portfolio['Returns'])
self.portfolio['Portfolio Returns']=self.portfolio['Portfolio Value'].pct_change()
self.portfolio['Initial Value']=self.allocation

with plt.style.context(['ggplot','seaborn-paper']):
#Plotting Portfolio Value
plt.plot(self.portfolio['Portfolio Value'])
plt.plot(self.portfolio['Initial Value'])
plt.title('%s Strategy Returns '%(self.pair))
plt.legend(loc=0)
plt.show()

if self.ptype==2:
'''
PARAMETERS
##########
allocation - the amount of capital alotted for pair
pair_number - string to annotate the plots
'''
self.allocation=allocation
self.pair=pair_number

self.portfolio=pd.DataFrame(index=self.df.index)
self.portfolio['Positions']=self.df['Longs']-self.df['Shorts']
self.portfolio['X']=-1.0*self.df['X']*self.portfolio['Positions']
self.portfolio['Y']=self.df['Y']*self.portfolio['Positions']
self.portfolio['Total']=self.portfolio['X']+self.portfolio['Y']

self.portfolio.fillna(0.0,inplace=True)
#creating a percentage return stream
self.portfolio['Returns']=self.portfolio['Total'].pct_change()
self.portfolio['Returns'].fillna(0.0,inplace=True)
self.portfolio['Returns'].replace([np.inf,-np.inf],0.0,inplace=True)
self.portfolio['Returns'].replace(-1.0,0.0,inplace=True)
#calculating metrics
self.mu=(self.portfolio['Returns'].mean())
self.sigma=(self.portfolio['Returns'].std())
self.portfolio['Win']=np.where(self.portfolio['Returns']>0,1,0)
self.portfolio['Loss']=np.where(self.portfolio['Returns']<0,1,0)
self.wins=self.portfolio['Win'].sum()
self.losses=self.portfolio['Loss'].sum()
#calculating sharpe ratio with interest rate of
#interest_rate_assumption=0.75
#self.sharp=(self.mu-interest_rate_assumption)/self.sigma
#win loss ratio;
self.win_loss_ratio=(self.wins/self.losses)
#probability of win
#probability of loss
#average return of wins
self.avg_win_return=(self.portfolio['Returns']>0).mean()
#average returns of losses
self.avg_loss_return=(self.portfolio['Returns']<0).mean()
#calculating payout ratio
self.payout_ratio=(self.avg_win_return/self.avg_loss_return)

#calculate equity curve
self.portfolio['Returns']=(self.portfolio['Returns']+1.0).cumprod()
self.portfolio['Portfolio Value']=(self.allocation*self.portfolio['Returns'])
self.portfolio['Portfolio Returns']=self.portfolio['Portfolio Value'].pct_change()
self.portfolio['Initial Value']=self.allocation

with plt.style.context(['ggplot','seaborn-paper']):
#Plotting Portfolio Value
plt.plot(self.portfolio['Portfolio Value'])
plt.plot(self.portfolio['Initial Value'])
plt.title('%s Strategy Returns '%(self.pair))
plt.legend(loc=0)
plt.show()

return #self.portfolio['Portfolio Value'].plot()

Let's now use our updated statarb class to create our strategies.

#creating instance of updated statarb strategy
adbe_antm_bottom_up=statarb_update(adbe_test, antm_test, 2,17,-2, 2,17,'05/01/18','06/12/18',adbe_antm_regime_predictions,'Target',avoid1=0,target1=1,exit_zscore=0)

In the above code block, we pass in our standard parameters with the exception of new fields, namely a 'Target', target1, avoid1, and regime predictions.

The Target parameter is our selection for our p2objective. It simply means that we are seeking to target a specific regime. If we set this parameter to 'Avoid', we would simply initialize the avoid1 parameter as the regime we wanted to avoid.

Given that we set this parameter to 'Target', we simply initialize the target 1 parameter to the regime we would like to target. Recall that we identified our historical regimes and trained our Random Forests to predict these regimes using the features we engineered.

We later applied our Random Forests model to our testing period and stored the regime predictions in a variable. We now pass those predictions in here and use them to update our signal generator based on our analysis of our historical regime.

Let's finish this implementation bny calling our remaining methods on our object.

#creating spread
#generating signals
#getting performance
#notice that we are passing in our equal weight amount here
adbe_antm_bottom_up.create_returns(30000,'ADBE_ANTM Bottom Up')

ANTM_AET Bottom Up Implementation

#creating instance of updated statarb strategy
antm_aet_bottom_up=statarb_update(antm_test, aet_test, 2,6,-2, 2,6,'05/01/18','06/12/18',antm_aet_regime_predictions,'Target',avoid1=0,target1=1,exit_zscore=0)
#creating spread
#generating signals
antm_aet_bottom_up.generate_signals()
#getting performance
#notice that we are passing in our equal weight amount here
antm_aet_bottom_up.create_returns(30000,'ANTM AET Bottom Up')

After using our analysis of the historical regimes that our strategy fell within and updating our statarb implementation to avoid Regime 0, we can see that our ANTM_AET implementation didn't trade over our test period.

This is an indication that observations fell within Regime 0 over this period.

AET_ANTM Bottom Up Implementation

#creating instance of updated statarb strategy
aet_antm_bottom_up=statarb_update(aet_test, antm_test, 1,12,-2, 2,12,'05/01/18','06/12/18',aet_antm_regime_predictions,'Avoid',avoid1=1,target1=0,exit_zscore=0)

Note that in our AET_ANTM implementation we chose not to avoid or target any specific regime.

Thus, in the third parameter, denoting portfolio type, we passed in 1 which is our original implementation.

#creating spread
#generating signals
aet_antm_bottom_up.generate_signals()
#getting performance
#notice that we are passing in our equal weight amount here
aet_antm_bottom_up.create_returns(30000,'AET_ANTM Bottom Up')

#### Step 6: Getting Performance Metrics

We can now aggregate our strategies to form our Bottom Up Portfolio and store our Sharpe Ratio in a variable for later use.

#creating dataframe for Bottom Up Portfolio
bottom_up_portfolio=pd.DataFrame()
bottom_up_portfolio['AET_ANTM']=aet_antm_bottom_up.portfolio['Portfolio Value']
bottom_up_portfolio['ANTM_AET']=antm_aet_bottom_up.portfolio['Portfolio Value']
bottom_up_portfolio['Cash']=10000
bottom_up_portfolio['Total Portfolio Value']=bottom_up_portfolio['ADBE_ANTM']+bottom_up_portfolio['ANTM_AET']+bottom_up_portfolio['AET_ANTM']+bottom_up_portfolio['Cash']

We can now add a column to hold the returns of our Bottom Up Portfolio.

#adding returns column to Bottom Up Dataframe
bottom_up_portfolio['Returns']=np.log(bottom_up_portfolio['Total Portfolio Value']/bottom_up_portfolio['Total Portfolio Value'].shift(1))

Let's now plot the equity curve of our Bottom Up Portfolio.

plt.figure(figsize=(10,6))
plt.plot(bottom_up_portfolio['Total Portfolio Value'])
plt.title('Bottom Up Portfolio')
plt.show()

Let's wrap up our Bottom Up Portfolio by creating and storing our Sharpe in a variable for later use.

bottom_up_portfolio_mu=bottom_up_portfolio['Returns'].mean()
bottom_up_portfolio_sigma=bottom_up_portfolio['Returns'].std()
#recall that we initialized our interest assumption earlier
bottom_up_portfolio_sharpe=(bottom_up_portfolio_mu-rate)/bottom_up_portfolio_sigma


### Portfolio Construction: SPO Framework

The SPO Framework combines the logic behind the Efficient Frontier, or top-down optimization, with that of the bottom up optimization portfolio.

Here we repeat the process used to create the bottom up portfolio except instead of equally weighing our strategies, we weight them according to the weights found when we created the Efficient Frontier.

Recall that our Efficient Frontier Analysis gave our ANTM_AET pair a weight of 0. This is congruent with what we saw when implementing our Bottom Up Portfolio.

Thus, we'll create our SPO Implementations on our ADBE_ANTM and AET_ANTM pairs using our Efficient Frontier Weights and regime predictions found during our Bottom Up implementation.

ADBE_ANTM Stereoscopic Portfolio Optimization (SPO) Framework Implementation

#creating instance of updated statarb strategy
adbe_antm_spo=statarb_update(adbe_test, antm_test, 2,17,-2, 2,17,'05/01/18','06/12/18',adbe_antm_regime_predictions,'Target',avoid1=0,target1=1,exit_zscore=0)
#creating spread
#generating signals
#getting performance
#notice that we are passing in our weight from our efficient frontier analysis here
adbe_antm_spo.create_returns(adbe_antm_allocation,'ADBE_ANTM SPO Framework')

Let's repeat this process for our AET_ANTM implementation.

AET_ANTM SPO Framework Implementation

#creating instance of updated statarb strategy
aet_antm_spo=statarb_update(aet_test, antm_test, 1,12,-2, 2,12,'05/01/18','06/12/18',aet_antm_regime_predictions,'Avoid',avoid1=1,target1=0,exit_zscore=0)
#creating spread
#generating signals
aet_antm_spo.generate_signals()
#getting performance
#notice that we are passing in our efficient frontier weight amount here
aet_antm_spo.create_returns(aet_antm_allocation,'AET_ANTM SPO Framework')

We Can Now Compose Our SPO Framework Portfolio

#creating dataframe for SPO Portfolio
spo_portfolio=pd.DataFrame()
spo_portfolio['AET_ANTM']=aet_antm_spo.portfolio['Portfolio Value']
spo_portfolio['Cash']=10000
spo_portfolio['Total Portfolio Value']=spo_portfolio['ADBE_ANTM']+spo_portfolio['AET_ANTM']+spo_portfolio['Cash']

Let's Now Compute the Returns of Our SPO Framework Portfolio

#adding returns column to SPO Portfolio dataframe
spo_portfolio['Returns']=np.log(spo_portfolio['Total Portfolio Value']/spo_portfolio['Total Portfolio Value'].shift(1))
spo_portfolio_mu=spo_portfolio['Returns'].mean()
spo_portfolio_sigma=spo_portfolio['Returns'].std()
#recall that we initialized our interest assumption earlier
spo_portfolio_sharpe=(spo_portfolio_mu-rate)/spo_portfolio_sigma

Let's Plot the Equity Curve of Our SPO Portfolio

plt.figure(figsize=(10,6))
plt.plot(spo_portfolio['Total Portfolio Value'])
plt.title('SPO Portfolio Equity Curve')
plt.show()

We can now compare the Sharpe Ratios of each of our portfolios.

### Relative Portfolio Performance Assessment

Whew!

We can now create a dataframe to hold our Sharpe ratios and compare our results.

Let's create the variables that we use to create our dataframe.

#list to hold portfolio names
names=['Equally Weighted','Efficient Frontier','Bottom Up','SPO Framework']
#variable to hold column name
column_name='Sharpe Ratio'
#list to hold Sharpe Ratios
sharpes=[equally_weighted_Sharpe,efficient_frontier_portfolio_sharpe,bottom_up_portfolio_sharpe,spo_portfolio_sharpe]

We'll now use our variables to create our dataframe.

#creating dataframe to compare Sharpe Ratios of Portfolios
portfolio_assessment=pd.DataFrame({column_name:sharpes},index=names)

Okay. We can now check our dataframe and compare our Sharpe Ratios.

portfolio_assessment


Sharpe Ratio
Equally Weighted-2.100000
Efficient Frontier-7.697951
Bottom Up-0.719018
SPO Framework-6.743110

Let's create a dataframe to compare the ending values of our portfolios.

#creating list to hold ending values of portfolios
#We pass in 1 into the tail method because it represents the last index position
portfolio_values=[equally_weighted['Total Portfolio Value'].tail(1).values.astype(int),efficient_frontier_portfolio['Total Portfolio Value'].tail(1).values.astype(int),bottom_up_portfolio['Total Portfolio Value'].tail(1).values.astype(int),spo_portfolio['Total Portfolio Value'].tail(1).values.astype(int)]
#creating dataframe to hold ending value of portfolios
pd.DataFrame({'Ending Portfolio Values':portfolio_values},index=names)

Ending Portfolio Values
Equally Weighted[124853]
Efficient Frontier[140402]
Bottom Up[86225]
SPO Framework[137868]

Okay. We can see that our best Sharpe Ratio was achieved within the Bottom Up Portfolio. But, the Bottom Up Portfolio yielded the worst ending portfolio value.

Recall that our Bottom Up implementation built on our equally weighted portfolio except that it, instead of solely weighing the strategies equally, optimized the portfolio by applying our GMM to historical regime detection and used Random Forests to predict which of our historical regimes our current market state fell in.

Our SPO Framework repeated this process except that it used the weights found by our Efficient Frontier analysis.

In the SPO and Efficient Frontier implementations, we zeroed out the second pair, ANTM_AET based on the results of our analysis.

### Review

We've covered an immense amount of information in this post.

• We began by learning what the SPO Framework is.
• We gained some understanding of Gaussian Mixture Models and how they relate to K-Means Clustering as well as Random Forests.
• We learned that the portfolio optimization problem is similar to the premise of ensembles of which is a key assumption of the SPO Framework.
• We reviewed how to use fundamental features and K-Means to find pairs for our StatArb strategy.
• We also reviewed how to find the most optimal value for K.
• We created four StatArb intraday portfolios with varying approaches to the portfolio optimization problem.
• Over this period, we found that the Bottom Up implementation yielded the best Sharpe Ratio, despite yielding the worst performance.
• Some key considerations are the fact that in reality, we wouldn't allocate as much capital to a single strategy as we did in the case of the AET_ANTM pair.
• This was due to the skewed results of our Efficient Frontier analysis of which yielded results based on the on our unoptimized pairs.
• Also, the regime in which we avoided in the ANTM_AET implementation in the Bottom Up Portfolio apparently accounted for all of the observations of our test period and thus zeroed out this pair over our test period in this portfolio, but this pair was removed by the Efficient Frontier analysis from our SPO Framework Portfolio.
• Thus the primary difference between these portfolios is the weights used.

A key takeaway from this introduction to the SPO Framework is that a portfolio can be viewed as the sum of 'n' market microstructures.

The Stereoscopic Portfolio Optimization approach to the portfolio optimization problem is a framework and not a hard and fast model.

As such, the Efficient Frontier, Gaussian Mixture Models, and Random Forests can be substituted with other top-down and bottom-up techniques, top-down being methods that focus on asset allocation and bottom being those that focus on applying machine learning to the market microstructure of component assets.

This microstructure component is not limited to volatility but can be liquidity, order arrival rates, etc.

### Next Steps

Portfolio managers and quants who wish to construct their portfolio quantitatively, generate returns and manage risks effectively can check out this course where you will learn different portfolio management techniques such as Factor Investing, Risk Parity and Kelly Portfolio, and Modern Portfolio Theory.

If you’re interested in learning more about Machine Learning for Trading, Quantinsti offers both a self-paced option, via our Quantra Online Learning Portal, and a comprehensive quantitative trading regimen, the Executive Program in Algorithmic Trading (EPAT®).

Disclaimer: All investments and trading in the stock market involve risk. Any decisions to place trades in the financial markets, including trading in stock or options or other financial instruments is a personal decision that should only be made after thorough research, including a personal risk and financial assessment and the engagement of professional assistance to the extent you believe necessary. The trading strategies or related information mentioned in this article is for informational purposes only.