Identifying Broad and Narrow Financial Risk Factors with Convex Optimization

27 Pages Posted: 25 Jun 2016 Last revised: 21 Aug 2016

See all articles by Alexander Shkolnik

Alexander Shkolnik

University of California, Santa Barbara (UCSB); University of California at Berkeley

Lisa R. Goldberg

University of California, Berkeley; Aperio Group

Jeffrey Bohn

University of California, Berkeley - Center for Risk Management Research; State Street Corporate

Date Written: August 20, 2016

Abstract

Factor analysis of security returns aims to decompose a return covariance matrix into systematic and specific risk components. To date, most commercially successful factor analysis has been based on fundamental models, although there is a large academic literature on statistical models. While successful in many respects, traditional statistical approaches like principal component analysis and maximum likelihood suffer from several drawbacks. These include a lack of robustness, strict assumptions on the underlying model of returns, and insensitivity to narrow factors such as industries and currencies, which affect only a small number of securities, but in an important way.

We apply convex optimization methods to decompose a security return covariance matrix into low rank and sparse parts. The low rank component includes the market and other broad factors that affect most securities. The sparse component includes narrow factors and security specific effects.

We measure the variance forecasting accuracy of a low rank plus sparse covariance matrix estimator on an equally weighted portfolio of 100 securities simulated from a model with two broad factors and 21 narrow factors. We find that the low rank plus sparse estimators are more accurate than estimates made with classical principal component analysis, in particular, at forecasting risk due to narrow factors. Finally, we illustrate a low rank plus sparse decomposition of an empirical covariance matrix of 100 equities drawn from 21 countries.

Keywords: financial risk factors, broad, narrow, convex optimization, low rank plus sparse decomposition, principal component analysis

JEL Classification: C44, G10

Suggested Citation

Shkolnik, Alexander and Goldberg, Lisa R. and Bohn, Jeffrey, Identifying Broad and Narrow Financial Risk Factors with Convex Optimization (August 20, 2016). Available at SSRN: https://ssrn.com/abstract=2800237 or http://dx.doi.org/10.2139/ssrn.2800237

Alexander Shkolnik

University of California, Santa Barbara (UCSB) ( email )

5501 South Hall
Santa Barbara, CA 93106
United States

University of California at Berkeley ( email )

Berkeley, CA 94720
United States

Lisa R. Goldberg (Contact Author)

University of California, Berkeley ( email )

Department of Statistics
367 Evans Hall
Berkeley, CA 94720-3860
United States

Aperio Group ( email )

3 Harbor Drive
Suite 315
Sausalito, CA 94965
United States

Jeffrey Bohn

University of California, Berkeley - Center for Risk Management Research ( email )

581 Evans Hall
Berkely, CA
United States

State Street Corporate ( email )

1 Lincoln Street
Boston, MA 02111
United States

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