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 (σP) can be computed as a function of the weights (w) and the covariance matrix (Σs) 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, ...