A beta of +1 indicates that the stock’s return rises and falls one for one with the systematic factor. This means, in our example, that because the stock has a GNP beta of 1, it experiences a 1 percent increase in return for every 1 percent surprise increase in GNP. If its GNP beta were –2, it would fall by 2 percent when there was an unanticipated increase of 1 percent in GNP, and it would rise by 2 percent if GNP experienced a surprise 1 percent decline.
Let us suppose that during the year the following events occur: Inflation rises by 7 percent, GNP rises by only 1 percent, and interest rates fall by 2 percent. Suppose we learn some good news about the company, perhaps that it is succeeding quickly with some new business strategy, and that this unanticipated development contributes 5 percent to its return. In other words: = 5% Let us assemble all of this information to find what return the stock had during the year. First, we must determine what news or surprises took place in the systematic factors. From our information we know that: and: Expected change in interest rates = 0% This means that the market had discounted these changes, and the surprises will be the difference between what actually takes place and these expectations: Similarly: and: The total effect of the systematic risks on the stock return, then, is: Combining this with the unsystematic risk portion, the total risky portion of the return on the stock is: + = 6.6% + 5% = 11.6% Last, if the expected return on the stock for the year was, say, 4 percent, the total return from all three components will be:
The model we have been looking at is called a factor model , and the systematic sources of risk, designated , are called the . To be perfectly formal, a is a model where each stock’s return is generated by: where is specific to a particular stock and uncorrelated with the term for other stocks. In our preceding example we had a three-factor model. We used inflation, GNP, and the change in interest rates as examples of systematic sources of risk, or factors. Researchers have not settled on what is the correct set of factors. Like so many other questions, this might be one of those matters that is never laid to rest. In practice, researchers frequently use a one-factor model for returns. They do not use all of the sorts of economic factors we used previously as examples; instead they use an index of stock market returns—like the S&P 500, or even a more broadly based index with more stocks in it—as the single factor. Using the single-factor model we can write returns like this: When there is only one factor (the returns on the S&P 500 portfolio index), we do not need to put a subscript on the beta. In this form (with minor modifications) the factor model is called a market model. This term is employed because the index that is used for the factor is an index of returns on the whole (stock) market. The market model is written as: where is the return on the market portfolio.