Example 1 Suppose that we have 100 stocks whose returns we want to explain in terms of one factor-the value factor. For a particular month, we collect returns for all 100 stocks as well as each stock's exposure to the value factor. Using these data, we estimate the factor return F(t), which turns out to be 5 percent.7 Suppose that, for a particular stock, its exposure to the value factor is measured to be two standard deviations8 (2 std) above the mean of all stocks in some predefined universe of stocks; that is, B(t - 1)= 2 std for this stock. Using F(t), we can determine the change in the average or expected stock return, given a change in exposure. In other words, we can address the question, what is the return to an increased exposure to stocks with high earnings-to-price values? It follows from Equation (20.2) that AE[f(t)] = AB(t - l)E[F(t)] where AE[r(t)] and E[F(t)] represent the expected change in stock return and the expected value of the factor return, respectively. If we expect a particular stock's exposure to the value factor to increase, say, 0.5 std-that is, the stock becomes more of a value play-then the expected change in its stock return, given this change, is

AE[r(t)] = AB(t-l)F(t) = 0.5 std x 0.05 (20.6) = 250 basis points

In this example, F(t) represents the return from an increase in the exposure to value stocks. To see this, we can rewrite (20.6):

AE[m_ AB(t-l)

So, F(t) represents the change in the average excess return for an increase (decrease) in exposure to stocks with high earnings-to-price levels.

Example 2 Factor returns are sometimes defined by first constructing a so-called factor-mimicking portfolio (FMP). Simply put, an FMP is a portfolio whose returns mimic the behavior of some underlying factor. There is a variety of techniques available to construct FMPs. A simple way9 to build a portfolio that mimics the behavior of, say, the value factor return works as follows.

1.    First, sort all assets in your portfolio according to their E/P.

2.    Split the sorted assets into two groups. The first group contains assets that fall in the top half of assets ranked by E/P. We refer to these assets collectively as

7An explanation of the factor return estimation procedures is provided in the section on cross-sectional regressions later in the chapter.

Exposures are sometimes normalized so that they are comparable. This normalization process will be discussed in more detail in the section on standardizing exposures later in the chapter.

sThe academic literature is replete with better ways to construct a factor-mimicking portfolio. Here, the example we provide is for expositional purposes only.