be used to update volatility estimates that use only historical data. In fact, traditional volatility estimators may be too slow in incorporating extreme events. Once again, the covariance matrix decomposition in (16.8) provides an ideal ground to implement these variations. Factor Models Linear factor models are an appealing alternative to the risk models described so far. In addition to providing economic intuition on the forces that drive volatilities and correlations for asset returns, they simplify the estimation process when dealing with large sets of assets. For example, risk models for individual securities, which often include thousands of assets, are often specified as factor models. The basic assumption behind a factor model is that returns are driven by a number of systematic factors common to all assets in the economy, plus an idiosyncratic factor that reflects a random component specific to each asset. Formally, the return on a generic asset i can be described as follows: ri,id)=ai+fdbikfk,+Bitt (16-12) k=i The idiosyncratic term e: t has a mean of zero because, by assumption, it reflects unpredictable changes in the return on asset i. The K systematic factors reflect economic forces that are likely to affect all asset returns, and the coefficients bik, which are often referred to as factor loadings, capture the effect of the common factors on a specific asset. For example, in the case of equity markets the common factors may represent measures of economic growth for the economy, indicators of future expected inflation, measures of recent market performance, and so on. Since the idiosyncratic factor is asset specific, we assume that &: is uncorrected with the systematic factors, and with the idiosyncratic factor of any other asset. Given a set of N assets, we can stack their returns at time t in a vector Rt{ d) and rewrite the factor model in matrix form: Rt(d) = a + BFt +et (16.12') where a is a vector of constants with N elements, B is a matrix with N rows and K columns (each row corresponds to the factor loadings for a specific asset), Ft is a vector that contains the values of the K factors at time t, and ef is a vector that contains the idiosyncratic factors for the N assets. If we indicate with SR the covariance matrix for the N assets, then equation (16.12') combined with our assumptions on the lack of correlation between systematic and idiosyncratic factors implies the following covariance matrix decomposition: lR=BlFB' + le (16.13) where Sf is a K X K covariance matrix for the K factors, and Se is a diagonal matrix whose elements represent the variances of the idiosyncratic components.