60 months. The choice of 60 months is arbitrary and is used only for illustrative purposes. 2.    Construct a 60 X 4 matrix of returns, R(t), where each column of R(t) corresponds to a historical time series of returns over the prior 60 months. For example, the first column of R(t) represents the time series of mean-zero returns3 for stock 1; the second column of R(t) represents the time series of mean-zero returns for stock 2; and so on. 3.    The one-month volatility forecast, at time t, is based on the simple covariance matrix estimator4 V(t): V(t) = -R(t)TR(t) (20.1) 60 where the superscript "T" represents the transpose of the return matrix. The covariance matrix V{t) has 10 elements (6 covariances and 4 variances). In general, if our portfolio consists of N stocks, then the covariance matrix consists of N(N + l)/2 variances and covariances. Obviously, even mo derate-sized portfolios require many variance and covariance estimates. In practice, it is not uncommon to have a portfolio consist of 100 stocks, in which case we would have to estimate 5,050 parameters (100 variances and 4,950 covariances). In order to have a proper covariance matrix (i.e., positive semidefinite), this would require that we have at least 100 historical returns (i.e., about eight years of data) for each asset. However, a stable covariance matrix5 would require even more observations. Factor models are of interest not only because they offer an intuitive understanding of the sources of risk and return, but also because they provide parsimony. And in covariance matrix estimation, parsimony is a virtue. Therefore, it should not be surprising that much work has gone into developing methods that provide a good estimate of V{t) without requiring the estimation of a large number of parameters. The way that factor models provide parsimony should become clear in the following example. Consider a factor model that describes four stock returns in terms of two factors. For the time being we treat factors as an abstract concept. A standard factor model, at time t, can be written as follows: r1(t) = Bn(t-l)F1(t)+B12(t-l)F2(t) + u1(t) r2(t) = B21(t-l)F1(t) + B22(t-l)F2(t) + u2(t) (20.2) r3(t) = B31(t-l)F1(t)+B32(t-l)F2(t) + u3(t) r4(t) = B41(t-l)F1(t) + B42(t-l)F2(t) + u4(t) 2Briefly, excess returns are defined as the difference between total returns and the return on the one-month risk-free rate. 3We subtract the sample mean from these excess returns. 4We use the simple covariance matrix estimator just as an example. We could also employ an estimator of the covariance matrix that applies an exponential weighting scheme to the data. 5By "stable covariance matrix" we mean a covariance matrix with a low condition number.