Factor Models

Abstract

In the Security Market Line theorem, the Tangent Portfolio happens to be a single factor, which explains alone all the excess expected returns of all the assets to the risk-free rate, and incidentally explains a portion of their risks, which is called the systematic risk. If now the aim is to explain the risk, i.e. the standard deviation of the returns of all the assets, then the Tangent Portfolio may not be the best instrument to consider, as that is not the specific purpose of this factor. In this chapter, we study techniques to find the best factors to explain the risks, and do not limit ourselves to searching for a single factor. Ideally, the set of factors identified should explain most of the risky assets’ variances and correlations, and potentially leave the residual unexplained variations as independent “noises”. Two types of factors can be considered: endogenous factors, which are statistically derived from the observed variables, i.e. from the observed returns of the assets, and exogenous factors, which are explanatory variables added to the model, such as inflation or macro-economic indicators. The normal distribution assumption is maintained here, keeping us in the Markowitz framework. When a factor model follows an additional condition, called the APT condition, it is called an APT model . For these models the fundamental APT theorem links each factor to a risk premium. In APT models the factors explain all the common sources of risks of the risky assets, which was the primary objective, but also the expected excess returns of the risky assets to the risk-free rate.