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Principal Component Analysis As A Factor Model

Introduction

Principal component analysis (PCA) is a statistical technique which enjoys applications in image processing and quantitative finance. In this article, we focus on the later application in quantitative trading, in particular using PCA as a multi-factor model of portfolio returns. We use the multi-factor model to design a momentum trading strategy, backtest it under different investment universes, and present the performance results.

The project is shared on my online repository https://github.com/DinodC/pca-factor-model.

Import packages

import numpy as np
import pandas as pd
import pickle
import statsmodels.api as sm
import matplotlib.pyplot as plt

Magic

%matplotlib inline

Data Collection

In this section, we collect S&P consittuents' historical data from a previous project https://quant-trading.blog/2019/06/24/backtesting-a-trading-strategy-part-2/.

Set S&P Index keys

keys = ['sp500',
        'sp400',
        'sp600']

Initialize S&P indices close data

close = {}

Pull S&P indices close data

for i in keys: 
    # Load OHLCV data
    with open(i + '_data.pickle', 'rb') as f:
        data = pickle.load(f)
    
    # Update close prices data
    close[i] = data.close.loc['2014-06-12':]
    
    # Close file
    f.close

Inspect S&P 500 index constituents data

close['sp500'].head()
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Symbols A AAL AAP AAPL ABBV ABC ABMD ABT ACN ADBE ... XEL XLNX XOM XRAY XRX XYL YUM ZBH ZION ZTS
date
2014-06-12 39.7726 38.2958 123.2252 84.5860 44.6031 65.9975 23.03 35.7925 74.5018 66.56 ... 25.8547 41.1019 83.8928 46.4230 28.5372 35.2780 51.2950 101.0367 27.7535 30.8448
2014-06-13 39.8407 38.4672 123.7407 83.6603 45.0187 66.2930 22.95 35.7745 74.7820 66.82 ... 25.8884 41.6091 84.7098 46.3744 28.4920 36.0532 51.5945 101.2861 27.8192 30.9702
2014-06-16 39.7181 39.1150 124.0086 84.5035 44.8857 66.0529 23.27 35.8734 74.8634 67.62 ... 26.0740 41.9028 84.9326 46.4424 28.3791 35.9318 51.5099 100.7105 27.4060 30.9798
2014-06-17 40.0927 39.8867 124.9411 84.3935 45.1351 66.2561 23.43 35.8375 74.5832 67.54 ... 26.0910 42.2409 84.5200 46.2677 28.8310 36.0346 51.7639 100.4707 27.9601 31.6355
2014-06-18 40.3651 40.6392 128.6809 84.4852 45.3595 66.5886 23.37 36.2960 74.7459 73.08 ... 26.7810 42.1964 84.7758 46.3841 28.7406 36.3521 51.8876 101.7178 27.9977 31.7126

5 rows × 505 columns

For WordPress

close['sp500'].iloc[:5, :5]
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</style>
Symbols A AAL AAP AAPL ABBV
date
2014-06-12 39.7726 38.2958 123.2252 84.5860 44.6031
2014-06-13 39.8407 38.4672 123.7407 83.6603 45.0187
2014-06-16 39.7181 39.1150 124.0086 84.5035 44.8857
2014-06-17 40.0927 39.8867 124.9411 84.3935 45.1351
2014-06-18 40.3651 40.6392 128.6809 84.4852 45.3595 
close['sp500'].tail()
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Symbols A AAL AAP AAPL ABBV ABC ABMD ABT ACN ADBE ... XEL XLNX XOM XRAY XRX XYL YUM ZBH ZION ZTS
date
2019-06-04 67.95 29.12 154.61 179.64 76.75 82.21 267.06 77.46 177.97 268.71 ... 57.78 106.90 73.59 54.40 33.30 77.40 106.97 117.41 44.40 108.12
2019-06-05 68.35 30.36 154.61 182.54 77.06 81.65 268.80 78.69 179.56 272.86 ... 59.32 105.60 72.98 55.38 33.42 78.88 107.29 118.54 44.18 108.50
2019-06-06 69.16 30.38 154.90 185.22 77.07 81.75 269.19 80.09 180.40 274.80 ... 59.80 106.01 74.31 55.63 34.03 79.15 108.42 120.31 44.24 108.89
2019-06-07 69.52 30.92 155.35 190.15 77.43 83.48 267.87 80.74 182.92 278.16 ... 59.43 107.49 74.58 55.94 34.16 79.56 109.07 120.73 43.64 110.06
2019-06-10 70.29 30.76 153.52 192.58 76.95 84.77 272.43 81.27 184.44 280.34 ... 59.26 110.88 74.91 57.10 34.69 80.38 108.65 121.71 43.84 110.22

5 rows × 505 columns

For WordPress

close['sp500'].iloc[-6:-1, :5]
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Symbols A AAL AAP AAPL ABBV
date
2019-06-03 66.99 27.20 153.17 173.30 75.70
2019-06-04 67.95 29.12 154.61 179.64 76.75
2019-06-05 68.35 30.36 154.61 182.54 77.06
2019-06-06 69.16 30.38 154.90 185.22 77.07
2019-06-07 69.52 30.92 155.35 190.15 77.43 
close['sp500'].describe()
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Symbols A AAL AAP AAPL ABBV ABC ABMD ABT ACN ADBE ... XEL XLNX XOM XRAY XRX XYL YUM ZBH ZION ZTS
count 1257.000000 1257.000000 1257.000000 1257.000000 1257.000000 1257.000000 1257.000000 1257.000000 1257.000000 1257.000000 ... 1257.000000 1257.000000 1257.000000 1257.000000 1257.000000 1257.000000 1257.000000 1257.00000 1257.000000 1257.000000
mean 52.020384 41.208459 145.434801 135.133436 66.670458 84.568308 162.852343 49.123858 118.508010 142.302737 ... 39.362328 58.865112 75.435801 53.179431 26.876008 51.309177 66.614527 111.70591 36.711273 59.821177
std 13.868035 6.368716 24.131106 38.117633 17.717831 9.473348 116.557311 12.652720 31.178171 70.409857 ... 8.317325 22.243159 4.713198 7.834761 3.222998 16.350151 15.998532 9.92955 10.738373 20.214756
min 32.258600 24.539800 79.168700 82.743800 42.066600 65.718100 22.220000 33.935700 68.852300 60.880000 ... 25.247700 32.502600 58.967500 34.178400 18.532600 28.874500 44.181800 89.23610 18.885300 30.652000
25% 39.391500 36.574400 132.526400 103.637500 53.175300 77.745400 80.700000 39.410700 92.630500 82.100000 ... 31.489100 42.116300 72.522500 48.734500 24.214700 35.032500 53.061000 103.35200 26.823800 44.771300
50% 46.123400 40.847900 150.479700 119.476200 58.475100 83.862300 118.580000 43.045400 112.149800 107.970000 ... 39.059500 52.731200 75.815100 54.533600 26.571900 48.112000 61.584700 112.77000 38.252800 51.338500
75% 65.603800 46.127100 161.921100 167.857700 82.898700 89.855500 261.940000 58.174900 149.608800 212.280000 ... 45.463000 68.820200 78.488900 59.653000 29.655300 67.341800 80.207700 119.13230 46.435300 80.866200
max 81.940000 57.586600 199.159900 229.392000 116.445400 107.649700 449.750000 81.270000 184.440000 289.250000 ... 59.800000 139.263300 86.137400 67.795300 35.000000 83.549000 109.070000 130.91280 57.139500 110.220000

8 rows × 505 columns

For WordPress

close['sp500'].describe().iloc[:, :5]
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</style>
Symbols A AAL AAP AAPL ABBV
count 1257.000000 1257.000000 1257.000000 1257.000000 1257.000000
mean 52.020384 41.208459 145.434801 135.133436 66.670458
std 13.868035 6.368716 24.131106 38.117633 17.717831
min 32.258600 24.539800 79.168700 82.743800 42.066600
25% 39.391500 36.574400 132.526400 103.637500 53.175300
50% 46.123400 40.847900 150.479700 119.476200 58.475100
75% 65.603800 46.127100 161.921100 167.857700 82.898700
max 81.940000 57.586600 199.159900 229.392000 116.445400 
close['sp500'].shape
(1257, 505)

Fill NaNs with previous observation

close['sp500'].fillna(method='ffill', inplace=True)
close['sp400'].fillna(method='ffill', inplace=True)
close['sp600'].fillna(method='ffill', inplace=True)

Initialize daily returns

returns = {}

Calculate daily returns

for i in keys:
    returns[i] = close[i].pct_change()

Momentum Strategy Implementation

In this section, we present the model (factor model) and tool (principal compenent analysis) used in implementing the momentum trading strategy.

1. Factor Models

Factor models use economic (e.g. interest rates), fundamental (e.g. price per earnings), and statistical (e.g. principal component analysis) factors to explain asset prices (and returns). Fama and French initially designed the three-factor model which extends the capital asset pricing model (CAPM) to include size and value factors. The general framework is known as the arbitrage pricing theory (APT) developed by Stephen Ross and proposes multiple factors.

2. Principal Component Analysis (PCA)

Principal component analysis is a statistical procedure for finding patterns in high dimension data. PCA allows you to compress high dimension data by reducing the number of dimensions, without losing much information. Principal component analysis has the following applications in quantitative finance: interest-rate modeling and portfolio analysis.

PCA implementation has the following steps:

  1. Pull data
  2. Adjust data by subtracting the mean
  3. Calculate the covariance matrix of the adjusted data
  4. Calculate the eigenvectors and eigenvalues of the covariance matrix
  5. Choose the most significant components which explain around 95% of the data

Note that in this article we are only interested in applying principal component analysis, for a complete illustration you can start by checking out https://en.wikipedia.org/wiki/Principal_component_analysis#Applications.

3. Momentum Trading Strategy

Momentum trading strategies profit from current market trends continuation. The momentum trading strategy proposed below assumes that factor returns have momentum. The idea is to long the winning (losing) stocks which have the highest (lowest) expected returns according to factors.

Before implementation, we set the following parameters:

  1. Lookback period
  2. Number of significant factors
  3. Number of winning and losing stocks to pick
lookback = 250
number_of_factors = 5
top_n = 50

Initialize the trading positions

positions = {}

for i in keys:
    # Update positions
    positions[i] = pd.DataFrame(np.zeros((returns[i].shape[0], returns[i].shape[1])),
                                 index=returns[i].index,
                                 columns=returns[i].columns
                                )

Implementation

for i in keys:
    for j in range(lookback + 1, len(close[i])):
        # Calculate the daily returns
        R = returns[i].iloc[j - lookback + 1:j, :]

        # Avoid daily returns with NaNs
        has_data = (R.count() == max(R.count()))
        has_data_list = list(R.columns[has_data])
        R = R.loc[:, has_data_list]

        # Calculate the mean of the daily returns
        R_mean = R.mean()

        # Calculate the adjusted daily returns
        R_adj = R.sub(R_mean)

        # Calculate the covariance matrix
        cov = R_adj.cov()

        # Calculate the eigenvalues (B) and eigenvectors (X)
        eigen = np.linalg.eig(cov)
        B = eigen[0]
        X = eigen[1]

        # Retain only a number of factors
        X = X[:, :number_of_factors]

        # OLS
        model = sm.OLS(R_adj.iloc[-1], X)
        results = model.fit()
        b = results.params

        # Calculate the expected returns
        R_exp = R_mean.add(np.matmul(X, b))

        # Momentum strategy
        shorts = R_exp.sort_values()[:top_n].index
        positions[i].iloc[j][shorts] = -1
        longs = R_exp.sort_values()[-top_n:].index
        positions[i].iloc[j][longs] = 1

Remarks:

  1. Investment universes used in backtesting are the S&P 500, S&P 400 MidCap and S&P 600 SmallCap indices.
  2. Ordinary least squares (OLS) method is used to calculate stocks' expected returns from significant factors.
  3. Only a single stock is bought for each of the winning (losing) stocks, this could be improved by adjusting the number by the rank.

Performance Analysis

In this section, we present the performance of the momentum trading strategy based on principal component analysis.

Adjust the positions because we consider close prices

for i in keys:
    positions[i] = positions[i].shift(periods=1)

Calculate the daily PnL of the momentum strategy

pnl_strat = {}
avg_pnl_strat = {}

for i in keys:
    # Daily pnl
    pnl_strat[i] = (positions[i].mul(returns[i])).sum(axis='columns')
    # Annualized average pnl of the momentum strategy
    avg_pnl_strat[i] = pnl_strat[i].mean() * 250

Average daily PnL of momentum strategy using different investment universes i.e. S&P 500, S&P 400 and S&P 600 indices

pd.DataFrame(avg_pnl_strat,
            index=['Average PnL'],
            )
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sp500 sp400 sp600
Average PnL -0.033961 -0.491665 -1.941058

Remark: the average daily pnl of the momentum strategy is negative regardless of the investment universe used.

Plot the cumulative PnL of the momentum trading strategy

# Set size
plt.figure(figsize=(20, 10))

for i in range(len(keys)):
    plt.plot(pnl_strat[keys[i]].cumsum())
    
plt.xlim(pnl_strat['sp500'].index[0], pnl_strat['sp500'].index[-1])
# plt.ylim(-10, 5)

# Set title and legend
plt.title('Cumulative PnL Of Momentum Trading Strategy')
plt.legend(keys)
<matplotlib.legend.Legend at 0x1c192b4978>

png

Remarks:

  1. From July 2015 to January 2017, the momentum trading strategy generated negative PnL.
  2. From July 2017 to January 2019, the momentum trading strategy turned it around and generated positive PnL.
  3. From January 2019, the momentum trading strategy continuted to generate positive PnL for S&P 500 and S&P 400 MidCap indices only.

Conclusion

In this article, we implemented a momentum trading strategy based on principal component analysis. The momentum trading strategy generated positive PnL from July 2017 to January 2019, and negative PnL from July 2015 to July 2017. A way to enhance the current momentum trading strategy is to include exit and entry points depending on the expected profitability of the trading system.