# Factor Analysis

In statistics, factor analysis refers to a particular method of data analysis akin to principal component analysis. In risk management, factor analysis is the process of determining a portfolios exposure to different economic and market factors. Factors might include sector factors (energy, financials), country and region factors, style factors (value vs growth, small cap vs large cap), or macro factors (commodity prices, interest rates).

The earliest approach to factor analysis, sometimes referred to as risk taxonomy, simply categorized securities into different categories. A company was either a technology company or a consumer discretionary company. It was either Japanese or American. It was either large cap or small cap. The problem is that many securities have exposures to more than one factor (e.g. a technology company that sells consumer products and healthcare products, with operations in Europe and North America), and the binary yes/no categorization of securities cannot reflect the fact that some securities are strongly correlated to factors while others are only weakly correlated.

A more robust approach to factor analysis is based on regression analysis. In this approach, a factor exposure is defined relative to an index. For example, we could define exposure to India using the S&P India NIFTY 50 Index. The amount of exposure a country has to a factor is then its beta to the index. The factor beta is determined in the same way that we determine a portfolio’s market beta. For example, given the returns for a portfolio at time *t*, *Rt*, and the corresponding returns for a factor index, *Rf,t*, the factor exposure, *β*, could be determined by way of the following regression equation:

*R*=

_{t}*α*+

*β*

*R*+

_{f,t}*ε*

_{t}Just as with our exposure to the market, the exposure to a factor can be expressed as a percentage or in terms of dollars. If the beta of a security to commodity prices is 24% and the commodity index increased 10%, then we would expect the price of the security to increase 2.4% more than it would otherwise. If a portfolio has an exposure of $17 million to value vs growth and the value vs growth index increased by 20%, then we would expect the portfolio to earn $3.4 million more than it would otherwise.

This regression approach to factor analysis assume a linear relationship between the returns of a portfolio and the returns of a factor index. Factor based stress tests and other methods can be used to determine factor exposures in cases where the relationship may be non-linear.