## Empirical Models

The CAPM and the APT by no means exhaust the models and techniques used in practice to measure the expected return on risky assets. Both the CAPM and the APT are risk-based models. They each measure the risk of a security by its beta(s) on some systematic factor(s), and they each argue that the expected excess return must be proportional to the beta(s). While we have seen that this is intuitively appealing and has a strong basis in theory, there are alternative approaches.

Most of these alternatives can be lumped under the broad heading of parametric or empirical models. The word empirical refers to the fact that these approaches are based less on some theory of how financial markets work and more on simply looking for regularities and relations in the history of market data. In these approaches the researcher specifies some parameters or attributes associated with the securities in question and then examines the data directly for a relation between these attributes and expected returns. For example, an extensive amount of research has been done on whether the expected return on a firm is related to its size. Is it true that small firms have higher average returns than large firms? Researchers have also examined a variety of accounting measures such as the ratio of the price of a stock to the accounting earnings, the P/E ratio, and the closely related ratio of the market value of the stock to the book value of the company, the M/B ratio. Here it might be argued that companies with low P/E's or low M/B's are "undervalued" and can be expected to have higher returns in the future.

To use the empirical approach to determine the expected return, we would estimate the following equation:

where Ri is the expected return of firm i, and where the k's are coefficients that we estimate from stock market data. Notice that this is the same form as equation (11.6) with the firm's attributes in place of betas and with the k's in place of the excess factor portfolio returns.

When tested with data, these parametric approaches seem to do quite well. In fact, when comparisons are made between using parameters and using betas to predict stock returns, the parameters, such as P/E and M/B, seem to work better. There are a variety of possible explanations for these results, and the issues have certainly not been settled. Critics of the empirical approach are skeptical of what they call data mining. The particular parameters that researchers work with are often chosen because they have been shown to be related to returns. For instance, suppose that you were asked to explain the change in SAT test scores over the past 40 years in some particular state. Suppose that to do this you searched through all of the data series you could find. After much searching, you might discover, for example, that the change in the scores was directly related to the jackrabbit population in Arizona. We know that any such relation is purely accidental but if you search long enough and have enough choices, you will find something even if it is not really there. It's a bit like staring at clouds. After a while you will see clouds that look like anything you want, clowns, bears, or whatever, but all you are really doing is data mining.

Needless to say, the researchers on these matters defend their work by arguing that they have not mined the data and have been very careful to avoid such traps by not snooping at the data to see what will work.

Of course, as a matter of pure theory, since anyone in the market can easily look up the P/E ratio of a firm, one would certainly not expect to find that firms with low P/E's did better than firms with high P/E's simply because they were undervalued. In an efficient market, such public measures of undervaluation would be quickly exploited and would not expect to last.

Ri = Rf + kp/E (P/E) . + kM/B (M/B). + kslze (size)

Ross-Westerfield-Jaffe: I III. Risk I 11. An Alternative View of I I © The McGraw-Hill

Corporate Finance, Sixth Risk and Return: The Companies, 2002

Edition Arbitrage Pricing Theory

Chapter 11 An Alternative View of Risk and Return: The Arbitrage Pricing Theory 301

Perhaps a better explanation for the success of empirical approaches lies in a synthesis of the risk-based approaches and the empirical methods. In an efficient market, risk and return are related, hence perhaps the parameters or attributes which appear to be related to returns are also better measures of risk. For example, if we were to find that low P/E firms outperformed high P/E firms and that this was true even for firms that had the same beta(s), then we have at least two possible explanations. First, we could simply discard the risk-based theories as incorrect. Furthermore, we could argue that markets are inefficient and that buying low P/E stocks provides us with an opportunity to make higher than predicted returns. Second, we could argue that both views of the world are correct and that the P/E is really just a better way to measure systematic risk, i.e., beta(s), than directly estimating beta from the data.

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