**2. Literature review**

#### **2.1 Systematic risk**

Lakonishok and Shapiro [4] conclude that neither the traditional measure of risk (beta) nor the alternative risk measures (variance or residual standard deviation) can explain the cross-sectional variation in returns; only size seems to matter. Gencay et al. [5] propose a new approach to estimating systematic risk (the beta of an asset) and find that the relationship between the return of a portfolio and its beta becomes stronger as the wavelet scale increases. Campbell et al. [6] state that the systematic risks of individual stocks with similar accounting characteristics are primarily driven by the systematic risks of their fundamentals. Xing and Yan [7] indicate that improving accounting information quality causes the systematic risk to decrease, thus having important implications for disclosure decisions, portfolio management, and asset pricing.

#### **2.2 PCA and the stock market**

Liu and Wand [8] study the Chinese stock market and find that the performance of the BP model integrating PCA is closer to that of the proposed model in a relatively large sample. Hargreaves and Mani [9], using PCA through a perceptual map, provide a clear picture of the winning stocks that should be selected for trading. Wang et al. [10] achieve a good level of fitness, using two-directional two-dimensional PCA and Radial Basis Functional Neural Networks (RBFNN) in the Shangai stock market. Zahedi and Rounaghi [11], studying the Tehran stock exchange, through the usage of artificial neural network models and PCA method, note that prices have been accurately predicted and modeled in the form of a new pattern consisting of all variables. Noby and Lee [12] analyze global financial indices in the years 1998–2012 and indicate that the dynamics of individual indices within the group increase in similarity with time, and the dynamics of indices are more similar during crises. Gao et al. [13] experiment the prediction of the closing price of the stock market with two-dimensional PCA and deep belief networks (DBNs).

Waqar et al. [14] analyze three stock exchanges and show how PCA can help to improve the predictive performance of machine learning methods while reducing the redundancy among the data. Zhing and Enke [15] forecast the daily direction of the S&P 500 Index ETF (SPY) return and show that DNNs using two PCA-represented datasets give slightly higher classification accuracy than the entire untransformed dataset. Nahil and Lyhyaoui [16] show that the structure of the investment decision system can be simplified through the application of kernel PCA. Berradi and Lazaar [17], using both PCA and recurrent neural network model, reduce the number of features from eight to six, giving a good prediction of total Maroc stock price. Cao and Wang [18] compare the performance of both PCA and backpropagation (BP) neural network algorithms and find that the latter has the highest prediction accuracy.

*Measuring the Systematic Risk of Sectors within the US Market via Principal Components… DOI: http://dx.doi.org/10.5772/intechopen.101860*

More recently, Wen et al. [19] demonstrate how both PCA and LTSM can accurately predict the stock price fluctuation trend of Pingon Bank. According to Liang et al. [20], using volatility information of grains and softs through PCA and FA, find significant predictive ability in forecasting the RV of the S&P 500. Xu et al. [21], through the use of PCA, investigate the Chinese A-shares market over the 2013–2019 period and find that no matter investor sentiment, stock prices react significantly to rumors as well as when the rumor goes public. Yaojie et al. [22], using PCA and other methods, show the significant ability of the combined international volatility to predict US stock volatility. The literature review shows how PCA has been useful in dimensionality reduction, predicting prices, and other features of the stock market, in particular, this paper applies this mathematical technique in an innovative way, namely measuring the systematic risk in various sectors of the US stock market.

## **3. Methods and data**

According to Ross et al. [23], systematic risk is the one that influences a large number of assets, thus having market-wide effects. On the other hand, unsystematic risk is the one that affects a single asset or a group of assets. Since the former cannot be eliminated through diversification is called non-diversifiable risk, whereas the latter is called diversifiable risk because it can be eliminated through portfolio diversification.

#### **3.1 Principal Component Analysis**

According to [24], PCA is a technique that may be useful where explanatory variables are closely related. In specific, if there are *k* explanatory variables in the regression model, PCA will transform them into *k* uncorrelated new variables. To explain, suppose that the original explanatory variables are denoted *x*1, *x*2, …, *x*k*,* and denote the principal components by *p*1, *p*2, …, *p*k*.* These principal components are independent linear combinations of the original data

$$\begin{aligned} p\_1 &= \alpha\_{11}\mathbf{x}\_1 + \alpha\_{12}\mathbf{x}\_2 + \dots + \alpha\_{1k}\mathbf{x}\_k\\ p\_2 &= \alpha\_{21}\mathbf{x}\_1 + \alpha\_{22}\mathbf{x}\_2 + \dots + \alpha\_{2k}\mathbf{x}\_k\\ \vdots &= \dots + \dots + \dots + \dots + \dots + \dots \dots \dots\\ p\_3 &= \alpha\_{k1}\mathbf{x}\_1 + \alpha\_{k2}\mathbf{x}\_2 + \dots + \alpha\_{kk}\mathbf{x}\_k \end{aligned} \tag{1}$$

Where *α*ij are coefficients to be calculated, representing the coefficient on the *j*th explanatory variable in the principal component. These coefficients are also known as factor loadings. The principal components are derived in such a way that they are in descending order of importance. In particular, for this study, we take the first component as a representative of systematic risk, that is, the risk that affects the whole sector and cannot be diversified in a stock portfolio. For this analysis we write a script in Python, particularly we use *sklearn* library to compute the principal components.

We gather all data from yahoo finance, where we include 10 sectors of the US stock market, choosing the biggest five companies per stock by market capitalization (**Table 1**), taking daily log returns of stock prices, and dividing the periods of study into two—the pre-COVID-19 era—January 10 to May 10, 2021.


*Measuring the Systematic Risk of Sectors within the US Market via Principal Components… DOI: http://dx.doi.org/10.5772/intechopen.101860*


**Table 1.**

*List of sectors/companies.*
