> to ' ^ ' where rn{t) = Excess return on the Kth stock at time t B , , , = nth stock's loading on the default premium n,default premium " r F,!h (t) = Default premium at time t default premium^ ' i un{t) = nxh stock's specific return Numerical Example Suppose that the current (time t) one-period return on a stock is 3.0 percent. If the current default premium is 1.5 percent (i-e., .Fdefault remium(^) = 1.5%) and the factor's sensitivity or loading to this stock is 0.5 (i.e., Bndefault remium = 0.5), then the stock's implied return due to the default premium is 0.75 percent (0.5 X 1.5%). The stock-specific return is 2.25 percent. Now, suppose that spreads are expected to widen over the forthcoming month by 50 basis points. Assuming a constant factor exposure, the expected change in the stock's return is 0.25 percent (0.5% X 0.5 = 0.25%). (We take the expected specific return to be zero.) This simple example shows how, by using factor models, practitioners can address questions about the movement of different stocks by considering a change in the factor's return and exposure. Unfortunately, however, we do not always observe a time series of factor returns and, therefore, may be required to first estimate these returns. Suppose that instead of using a macroeconomic factor we use a fundamental factor such as value. A common measure of a stock's exposure to the value factor is its ratio of net earnings to share price (E/P). In this case, we observe each stock's exposure to the value factor but not the factor itself-that is, we do not know the factor return. This is the complete opposite of the situation where we knew the default premium but not the exposure of each stock to the default premium. Mathematically, this translates into observing each stock's factor loadings, Bn{t - 1), but not the factor return F{t); that is, we know the value of the loading but not the factor return. Since we do not observe the factor return and we have information on a cross section of stocks, we estimate the return to the exposure to the value factor at a particular point in time, using a regression of N excess stock returns on N earnings-to-price exposures. Each time this regression is run, it produces one estimate of the value factor return. If we conduct these regressions over a period of time, say 60 consecutive months, then we can construct a time series of value factor returns. Once we have estimates of these factors, we can estimate the factor and stock-specific covariance matrix as described earlier. Note that the fundamental approach (value factor) is more computationally intensive than the time series method (macroeconomic factor) since we must first estimate the factor returns from a series of cross-sectional regressions. We conclude this section by expounding on the notion of factor returns and exposures. The values of F(t) in the cross-sectional approach are often referred to as factor returns. We offer two examples to help explain why the F(t)s in (20.2) are referred to as factor returns.