cash as well as U.S. stocks. Positions in both contribute to the portfolio's overall position in USD. Realized and Predicted Risk Calculations In the linear factor model framework, a portfolio risk statistic is a function of a forecast covariance matrix that, itself, is a function of asset exposures and factor and specific return covariance matrices. This model allows us to decompose risk into factor and specific components. Before we discuss portfolio risk calculations based on the linear factor model, it is important to note the differences between realized (ex post) and predicted (ex ante) risk calculations. The calculation of a portfolio's realized risk (or tracking error) consists of two steps. Step 1 A time series of the portfolio's actual (realized) returns is obtained. Typically, there are two sources of these actual returns. 1.    Officially reported returns as maintained by a firm's accounting systems or as computed by a custodian. These returns are usually what appear in monthly statements that report the portfolio's performance. 2.    Estimates of the officially reported returns.18 These returns are mostly used in cases where daily performance reporting is required and no official returns are available. In this case, a portfolio's return is approximated by using returns as of time t and weights as of time t - 1. For example, an estimate of a portfolio's active returns over a 20-day period are expressed as rg(t)=wa(t-l)TR(t) forf = l,. . .,20 (20.55) where ra(t) and wa(t-\) represent the active portfolio return and weights, respectively. Step 2 Compute the standard deviation, or some other risk statistic, of the time series of actual returns. For example, realized tracking error is defined as the standard deviation of actual active returns. Unlike realized risk calculations, predicted risk calculations rely only on the most recent set of portfolio holdings. For example, a predicted tracking error calculation at time t -1 for some future period would use portfolio weights as of time t -1. This is an important difference since by using only the most recent holdings we are allowed to carry out risk decompositions (explained later). Next, we discuss predictive risk calculations in the context of the linear factor model. Factor Model Framework We work with the global linear factor model presented earlier in the chapter in the section on global framework. There, the cross section of returns, expressed in some base currency, is modeled according to: :See the section on cash for more information on portfolio returns.