Factor analysis of security returns aims to decompose a covariance matrix into systematic and specific components. The systematic component is based on the common factors that drive financial risk, and the specific component is based on firm specific, or diversifiable effects. Risk factor identification by covariance matrix decomposition is essential to building portfolio-risk models used by investors. It is equally as crucial for active monitoring of systemic risk by institutions that promote financial market stability (International Monetary Fund, Office of Financial Research, Bank of International Settlements, etc).

To date, financial practitioners have relied primarily on fundamental models to decompose covariance matrices. In a fundamental model, risk factors are identified by analysts, and they are readily interpretable. However, the structure of a fundamental model–the number of factors and their identities–is static until an analyst changes it. Statistical models are an alternative to fundamental models, and they occupy a central position in the academic literature. Statistical factors are dynamic: they adapt to changing relationships among securities as well as the emergence of new industries and new risk drivers (e.g., climate change.) The innovation and instability that have dominated financial markets since the turn of the millennium underscore the importance of adaptability.

Most statistical models are based on principal component analysis (PCA), which identifies factors as eigenvectors of a sample covariance matrix. While successful in many respects, principal component analysis suffers from several drawbacks. These include a lack of robustness to noise and an insensitivity to narrow factors such as countries, industries and currencies. Narrow factors affect only a small number of securities, but in an important way. The tendency of PCA to neglect narrow factors is elemental. *PCA identifies factors as variance maximizing latent variables*. It therefore has a tendency to conflate broad and narrow risk factors leading to incorrect risk profiles.

### Sparse and low rank structure of security returns

A recent innovation in the area of statistical models is the application of convex optimization to decompose a security returns covariance matrix \(\Sigma\) into two components. The first component, \(L\), is driven by broad factors, such as the market, interest rates and creditworthiness. Each broad factor affects most securities. The second component, \(S\), is driven by narrow factors, each of which affects a small number of securities, and specific returns.

\[ \Sigma = L + S \]

Convex optimization methods for covariance decomposition operate on different set of assumptions from PCA. They presuppose the underlying data is generated by a process that possesses a sparse and low rank (SLR) structure. That is, \(S\) is sparse with the small number of nonzero entries induced by the narrow factors. And, \(L\) is low rank, that is, the number of broad factors describing the data is not too large. The fact that this structural assumption yields a unique decomposition of \(\Sigma\) under very mild conditions is owed to recent, deep results in matrix decomposition theory.

- Alex Shkolnik delivered a survey talk on some methods for factor analysis including PCA and convex optimization at the April 12th, 2016 Risk Seminar.
- Lisa Goldberg and Alex Shkolnik presented a poster illustrating an approach to identifying financial risk factors via SLR decompositions at the BIDS’ 2016 Data Science Faire.

The SLR decomposition represents an important step toward a dynamic decomposition of a covariance matrix into systematic and specific components that is sensitive to narrow factors. Recent empirical literature has confirmed statistically significant industry and country effects in the residuals to statistical factors. Moreover, correlation in global equity markets has steadily risen over the past two decades. While broad factors surely drive security returns, narrow factors can often present higher volatility risk. New industries, emerging markets, climate change, geopolitical events and many other narrow factors present sources of risk that may be impossible to uncover by PCA-based methods. SLR decomposition techniques constitute a promising alternative to risk factor identification warranting further investigation and research.

### Simulation and empirical case studies

Alex Shkolnik, Lisa Goldberg and Jeffrey Bohn investigated the effectiveness of an existing SLR method for graphical latent models in the context of risk forecasting.

- The study, Identifying Broad and Narrow Financial Risk Factors with Convex Optimization uses both simulation and empirical experiments to explore the efficacy of SLR decompositions.

Simulations were performed on \(125\) securities with the number of observations is set to roughly a year’s worth of daily data, \(250\). The algorithm can be easily scaled up to handle larger portfolios over a longer time window. Using metrics that assess the accuracy of variance forecasts, the authors find that SLR decompositions are more accurate than classical PCA decompositions.

The performance of the SLR method on empirical data was illustrated with heat maps of low rank and sparse return correlation matrices recovered from an empirical covariance matrix, \(\hat \Sigma\), a noisy approximation to the true covariance \(\Sigma\). The sample consisted of randomly selected securities from \(25\) countries. The block diagonal structure of the recovered sparse component indicates that country effects contribute to empirical correlations. Said differently, the narrow-factor effects cluster to create correlation risk in a portfolio exposed to this particular narrow factor (i.e., the country.) If the relevant narrow factors were not recovered, a portfolio-risk model that simulates factors and generates forward-looking portfolio value distributions would ignore those risks. The probability of losses associated with securities exposed to a relevant narrow factor would be underestimated. In future applications SLR methods could be utilized to detect narrow factors at various scales without significant guidance (in this study countries were manually selected to serve a narrow factors).

- This work has most recently been presented by Lisa Goldberg and Alex Shkolnik at MMDS 2016. Lisa’s talk focused on describing the SLR methodology for extracting factors and specific risks from security returns data. Alex’s talk focused on results of numerical simulations and a sketch of future work on theoretical performance guarantees for SLR-based estimators.
- An early and preliminary version of this work was presented by Lisa Goldberg at the inaugural Berkeley-Columbia Meeting in Engineering and Statistics (2016).

### Beyond simple sparsity: monitoring financial stability

Recent reports of several financial stability monitoring bodies (IMG, OFR, BIS, etc) have sounded warnings of spillovers risk from emerging markets. A recent example is the drop of the Chinese stock market in the first month of 2016 that led to adverse effects globally. Accompanying the warnings is the realization that current methods and metrics fail to indicate *contagion vulnerabilities* before they are realized. A key to tackling this problem appears to be the accurate identification of narrow factors that direct the channels of financial spillover.

In preliminary work Lisa Goldberg, Samim Ghamami and Alex Shkolnik are investigating the use of SLR decompositions to identify and characterize channels of financial spillover. Their research posits that the sparse component \(S\) in the standard SLR decomposition \(\Sigma = L + S\) is no longer sparse once securities become exposed to a larger and larger number of narrow factors. A more promising model for \(S\) is then

\[ S = X X^T + \Delta \]

where \(\Delta\) is a diagonal matrix of security specific risks and \(X\) is a matrix of narrow factor exposures. The model supposes that it is \(X\) and not \(S\) that should exhibit a structure of sparsity. The nonzero entries of \(X\) describe the dependence of each security on some narrow factor (China, a geopolitical event, etc). The techniques being developed in this research recover \(X\) via nonconvex optimization methods. This recovery supplies a characterization of how narrow factors direct financial spillover in global equity markets.

- This work was presented for the first time at Stanford’s Center for Financial Risk Analytics on October 13, 2016.