MOTIVATION

In this section, we discuss the motivation behind the multifactor equity risk models. Let's assume that a portfolio manager wants to estimate and analyze the volatility of a large portfolio of stocks. A straightforward idea would be to compute the volatility of the historical returns of the portfolio and use this measure to forecast future volatility. However this framework does not provide any insight into the relationships between different securities in the portfolio or the major sources of risk. For instance it does not assist a portfolio manager interested in diversifying her portfolio or constructing a portfolio that has better risk adjusted performance.

Instead of estimating the portfolio volatility using historical portfolio returns, one could utilize a different strategy. The portfolio return is a function of stock returns and the market weights of these stocks in the portfolio. Using this, the forecasted volatility of the portfolio (σ_P) can be computed as a function of the weights (w) and the covariance matrix (Σ_s) of stock returns in the portfolio:

This covariance matrix can be decomposed into individual stock volatilities and the correlations between stock returns. Volatilities measure the riskiness of individual stock returns and correlations represent the relationships between the returns of different stocks. Looking into these correlations and volatilities, the portfolio manager can gain insight into her portfolio, namely the riskiness of different parts of the portfolio or how the portfolio can be diversified. As we outlined above, to estimate the portfolio volatility we need to estimate the correlation between each pair of stocks. Unfortunately, this means that the number of parameters to be estimated grows quadratically with the number of stocks in the portfolio.² For most practical portfolios, the relatively large number of stocks makes it difficult to estimate the relationship between stock returns in a robust way. Moreover, this framework uses the history of individual stock returns to forecast future stock volatility. However stocks characteristics are dynamic and hence using returns from different time periods may not produce good forecasts.³ Finally, the analysis does not provide much insight regarding the broad factors influencing the portfolio. These drawbacks constitute the motivation for the multifactor risk models, detailed in this chapter.

One of the major goals of multifactor risk models is to describe the return of a portfolio using a smaller set of variables, called factors. These factors should be designed to capture broad (systematic) market fluctuations, but should also be able to capture specific nuances of individual portfolios. For instance, a broad U.S. market factor would capture the general movement in the equity market, but not the varying behavior across industries. If our portfolio is heavily biased toward particular industries, the broad U.S. market factor may not allow for a good representation of our portfolio's return.

In the context of factor models, the total return of a stock is decomposed into a systematic and an idiosyncratic component. Systematic return is the component of total return due to movements in common risk factors, such as industry or size. On the other hand, idiosyncratic return can be described as the residual component that cannot be explained by the systematic factors. Under these models, the idiosyncratic return is uncorrelated across issuers. Therefore, correlations across securities are driven by their exposures to the systematic risk factors and the correlation between those factors.

The following equation demonstrates the systematic and the idiosyncratic components of total stock return:

The systematic return for security s is the product of the loadings of that security (L_s, also called sensitivities) to the systematic risk factors and the returns of these factors (F). The idiosyncratic return is given by _s. Under these models, the portfolio volatility can be estimated as

Models represented by equations of this form are called linear factor models. Here L_p represents the loadings of the portfolio to the risk factors (determined as the weighted average of individual stock loadings) and ∑_F is the covariance matrix of factor returns. w is the vector of security weights in the portfolio and Ω is the covariance matrix of idiosyncratic stock returns. Due to the uncorrelated nature of these returns, this covariance matrix is diagonal, with all elements outside its diagonal being zero. As a result, the idiosyncratic risk of the portfolio is diversified away as the number of securities in the portfolio increases. This is the diversification benefit attained when combining uncorrelated exposures.

For most practical portfolios, the number of factors is significantly smaller than the number of stocks in the portfolio. Therefore, the number of parameters in ∑_F is much smaller than in Σ_S, leading to a generally more robust estimation. Moreover, the factors can be designed in a way that they are relatively more stable than individual stock returns, leading to models with potentially better predictability.

Another important advantage of using linear factor models is the detailed insight they provide into the structure and properties of portfolios. These models characterize stock returns in terms of systematic factors that (can) have intuitive economic interpretations. Linear factor models can provide important insights regarding the major systematic and idiosyncratic sources of risk and return. This analysis can help managers to better understand their portfolios and can guide them through the different tasks they perform, such as rebalancing, hedging or the tilting of their portfolios. The Barclays Capital Global Risk Model—the model used for illustration throughout this chapter—is an example of such a linear factor model.