Factor models

In finance, a factor models refers to statisticals model that aims to explain the behavior of stocks by breaking it down into a set of underlying factors. In particular, factor models relate the return of a security to a number of (risk) factors. The underlying factors should be chosen based on economic intuition and empirical evidence. The factors are meant to capture broad market trends, sector- and region specific risks, rewarded risk premia and other macroeconomic variables.

Generally, factor models have the following structure:

    $$r_{i,t} = \alpha_i + \beta_1 \cdot F_{1,t} +\beta_2 \cdot F_{2,t} + ... + \beta_N \cdot F_{N,t} + \epsilon_{i,t}$$

where ri,t is security i‘s return at time t, F1,t, F2,t, to FN,t are the factors, and β1, β2 to βN are the factor loadings.

The factor loadings measure the sensitivity of the security to changes in the factor. For example, suppose that one of the factors is the return on the S&P500. Now, imagine we estimate the model and obtain a β equal 2. This means that, if the S&P500 increases by 1%, then the security’s price tends to go up by 2%, on average.

The equation also contains a so-called error term (ε). The error term is necessary to be able to estimate the (regression) model. We can, however, also give an economic interpretation to this parameter. In particular, the error term represents security-specific news that becomes available to investors.

The coefficient that we are mainly interested in is the constant in the factor model. We have denoted the constant using the greek symbol α and investors will also typically refer to it as the ‘alpha’ . α measures the return of a strategy that cannot be explained by exposure to the factors in the model or firm-specific news. As such, it measures the strategy’s excess return.

Portfolio construction

How can factor models be used in the context of portfolio construction?

Basically, we want to achieve the following objectives:

  • maximize the (excess) return of the portfolio, i.e. maximize the alpha of the portfolio
  • minimize volatility of the portfolio that is due to stock market fluctuations, i.e. minimize the beta of the portfolio
  • minimize the impact of firm-specific news, i.e. being sufficiently diversified

Note that the approach only makes sense if the investors can correctly determine which stocks have a positive alpha. Thus, the investors should have stock-picking skills.

One way of solving the above problem was proposed by Treynor and Black. The approach is referred to as the Treynor-Black model.

Advantages and disadvantages of factor models

The main advantages of factor models are the following.

  1. Simple: the models are easy to use and allow us to simplify the complex behavior of financial variables by breaking them down into underlying factors. This makes it significantly easier to understand what drives returns. It also helps investors to make more informed investment decisions.
  2. Constructing portfolios: factor models are used to construct portfolios that are designed to capture specific rewarded factors, such as value or momentum. This will result in higher returns and/or lower risk and can improve strategies’ Sharpe ratio
  3. Managing risk: these kinds of models are used to assess the risk of a portfolio by estimating the exposures of a portfolio to different risk factors. This information is used to identify sources of risk and to make adjustments to the portfolio’s exposures
  4. Pricing assets: factor models are used to estimate the securities’ expected return by taking into account their exposures to the different risk factors. This allows investors to value assets.

While there are many advantages, there are also some disadvantages:

  1. Overfitting: the models are prone to overfitting, meaning that the model may fit the data too closely and may thus not generalize well to out-of-sample data. This can lead to poor out-of-sample predictions.
  2. Limited explanatory power: Factor models can only explain a limited portion of the variation in financial variables, and there may be other factors that are not captured by the model. This can result in incorrect predictions and poor investment decisions.
  3. Model risk: Factor models are based on a set of assumptions (the number of factors, the factor loadings, and the factor covariances). If the assumptions are incorrect, the model may produce incorrect results. One important assumption is that the relationship between between the returns and the factors is linear, which may not be the case.
  4. Data limitations: the models typically require a amount of sufficient high-quality data to be estimated accurately. Very often only limited historical data is available.

Summary

Factor models can be used to identify potentially interesting securities that can be added to a portfolio. In addition, the results can be used to construct better portfolios.