The Pearsonian Stock Market: Testing the Fama-French Model of Share Returns

Peter Nolan

Ordinary least squares is the most commonly used technique in any practical application of econometric methodology. In this paper, Peter Nolan using this technique, seeks to explain a cross-section of weekly returns quoted on the international stock exchange.

I can calculate the motion of heavenly bodies, but not the madness of men. Sir Isaac Newton


Apart from love and war, nothing arouses violent human passion like the stock market. Ever since gentleman speculators studied hand-drawn charts of stock price movements in the time of J.P. Morgan, investors have looked to scientific and economic methods in their quest for models to predict the future more accurately One of the greatest advances in finance has been the Capital Asset Pricing Model (CAPM) which postulates that the return on a share was positively related to the beta of this share is a measure of the volatility or dispersion of returns (which, in finance, is defined as risk) of that asset relative to the volatility of the entire set of assets investors can choose as investments. The expected return on an asset can be predicted using estimates of the risk-free rate of interest (Rf), of the expected return on the market (Rm) and of beta.

Ri = Rf + Bi(Rm - Rf), where

Bi = (Si,m)/(Sm^2)

Paul Samuelson has said that this is an excellent model because it explains between thirty and ninety percent of the cross-sectional variation in stock returns using a single factor Bernstein (1992). An article by Fama and French used regression to show that did not explain the observed cross-section of returns observed on US equities. The authors noticed that two other variables, the ratio of market equity (ME) to book equity (BE), and the size of the firm (which equals ME) had more explanatory power. Indeed is likely to be a proxy for ME, as it is almost perfectly negatively correlated with firm size (r^2 = 0.98). In this paper I shall use Ordinary Least Squares (OLS) regression to test whether the ratio of ME to BE, together with ME, alone explains ex ante the cross-section of weekly returns on holding each one of a sample of shares quoted on the International Stock Exchange over a particular trading week in mid-December 1994. Firstly I shall explain how these two variables could have an effect on the return of a stock. Then I shall describe how I gathered my data and the behaviour of the variables over the period. Next I report the results of my regression analysis. Finally I explain some of the possible implications of my results.

Model Specification

My objective is to use regression to capture the influence of both company-specific risk factors and investors' expectations in determining returns, unifying the two conflicting schools of financial thought, the fundamental and the technical. Fundamental analysts concentrate on finding the 'real' value of the firm using economic and financial analysis; technical analysts aim to forecast the changing sentiment, often using charts that show the past movement of prices. In an efficient stock market, the expected value of discounted future cashflows from purchasing an equity will equal the price at which it may be purchased Fama (1966). Book equity is the accounting figure for the acquisition value of the firm's assets. Hence the ratio of ME to BE will represent the expectations held by investors about the firm. It seems, although this is not stated in the Fama & French article, that this is an estimator for Tobin's Q. The higher this ratio is, the higher investors expect the firm's return on the value of its assets will be, as shown by the premium over the acquisition cost of the firm's assets they pay for a share in the firm's dividends and capital growth. Therefore I expect that there will be a positive relationship between this ratio and the observed return.


The size of the firm will also affect the return. I expect a negative relationship. Large firms are typically more diversified and would tend to exist in more mature or low-growth industries, which is confirmed by a casual examination of the larger companies in my sample - mainly made up of multinational giants (Shell, BAT) or utilities (BT, the regional electricity companies), or diversified industrial holding companies (BTR, Hanson). Hence the riskiness of their returns would most likely be low, leading to a lower return under capital market efficiency. Since seems to be a proxy for size, this conclusion is simply restating the conclusions of the CAPM, which were shown formally in the introduction. I could have used an estimate of the for each stock as a third independent variable, but I decided not to because it would introduce multicollinearity: has a high correlation with ME. Return over that predicted by the CAPM could have been used as my independent variable. I was aware of the difficulties in estimating the figures needed for such a computation, as the predictions of the CAPM are very sensitive to the market index chosen as a proxy of the theoretical most efficient portfolio out of all investment assets Malkiel (1992).

I ran regressions using the weekly return on the stocks as my dependent variable, Y. The first independent variable, X1, was the market equity at the opening of business on Monday. The second independent variable, X2, was the ratio of book to market equity at the opening of business on Monday. To minimise the effect of shifting parameters, the equity data was taken in a week in which no potentially significant new information, such as new macroeconomic data, was released. All the shares are quoted in sterling, and BE and ME are both measured in sterling, so that currency rates will have no direct effect on the variables.

Data collection was not difficult. Every transaction is recorded by the exchange clearing house. The cheap and easy method for obtaining price data is to use Datastream. The drawback of this data is that the price is based on averages of bid and offer prices prevailing at any particular time, so there may be many small errors present. It would be too expensive for me to obtain data for the actual first observed transactions. The individual stocks were chosen at random from the Financial Times (18/12/94) . Only non-financial companies were used, which seems to be conventional in financial studies; this avoids the high leverage of banks and other types of financial holding companies and the assosiated effects on risk and return of equity. The property and insurance companies I have included had low leverage ratios - less than thirty percent according to DataStream.

My dependent variable, Y, was the weekly holding return on each of a sample of fifty-nine shares expressed as a percentage rate of return. This was made up of capital gains only and excluded dividends paid during the period. I feel confident doing this because the assumption was used by Fama & French article also. Typical weekly returns varied between plus eleven and minus four percent, while the dividend yield for the week would be insignificant. At an average for the market as a whole of three percent per annum, the weekly yield was calculated as

(1.03)^1/52 = 0.00059%

I gathered figures for the independent variables using Datastreami programmes numbers 101S and 900B. The BE figures were taken from the 'total share capital and reserves' entries in the Datastream balance sheet database. I found the data for the ME/BE ratio using the analysis program number 101S. I was surprised that the actual figures were in the database of technical data, rather than my having to calculate them. This could indicate that the ME /BE ratio has already been commonly adopted as a tool by technical analysts.

From the Financial Times that week, I found out that none of these companies had their financial year-end during the week so this data would be as up to date as possible.

Econometric Analysis

In this investigation, I will reject the model as unworthy of further investigation if it explains less than thirty percent ,using Samuelson's criteria for a good financial model Bernstein (1992). Anything less than that would probably be too risky to use as a basis for investing my own money. I am trying to estimate the parameters of the equation

Yi = 0 + B1Xi + B2X2i + i


I ran the cross-sectional regressions on the HUMMER econometric package. The coefficient of correlation (r^2) was very low, less than 2%. This seems to indicate that the significance of my chosen X-variables in determining the observed return was negligible. The relationships observed seemed not to comply with my model. The coefficient for ME was approximately zero while that of the ME/BE ratio was -0.3. I then carried out a regression of one X-variable against the other. Again the r^2 was tiny, about 0.0001%. This indicates a low degree of collinearity between the independent variables. A final set of regressions showed that either ME alone or the ME-to-BE ratio alone is insufficient to explain returns, with r^2 of 0.2% and 1.13% respectively. The coefficients were approximately equal for the independent variables in the multiple regression and the single-variable regressions.


It is possible that my results are so flawed that I cannot inductively draw conclusions inductively from them. This could occur if the variable RETURN, as empirical studies suggest, cannot be approximated by a Gaussian distribution Peters (1991). This could be verified by a Q-Q test, which involves regressing the distribution of observed variables against those expected in a Gaussian distribution JP Morgan (1994). Another plausible explanation is that my model is misspecified, so that the two independent variables I chose do not apparently determine cross-sectional return. Consequently, it is possible that the result obtained by Fama & French (1992) is spurious. It is possible that this could be due to data-mining. If the CAPM is disproved, several econometric studies that contradict the Efficient Markets Hypothesis (EMH), of which Fama is a leading proponent, can be dismissed as being based on bad data. An example would be Basu's (1981) work on the 'P/E effect'. He states that stocks with low price-to-earinigs (P/E) ratios will return more than the level predicted by the CAPM.

From the fact that ME was shown to be uncorrelated with returns in this sample, one could infer that the variable, and hence the CAPM, does not produce an accurate estimate of risk or expected return.

The CAPM is used for determining the level of profits a firm should earn, given the risk of its business Carlton & Perloff (1994). The degree of market power held by the firm can then be estimated by comparing actual profits with the profits the CAPM says the firm should earn for the level of risk undertaken. Hence, if CAPM is incorrect, as my results suggest, anti-trust action based on risk-adjusted profits as a measure of market power could be either too zealous or too lax.

The function of capital markets is to provide funds for investment by firms and increase the aggregate level of risk-taking by lowering the risk to individuals. CAPM can play an important role in allowing us to tell whether this function is being carried out properly already or whether structural reform or new financial techniques are needed. CAPM is used to measure how much risk fund-managers have taken over a period, compared to the risk of an index-tracking fund, to achieve the given return. This provides a means of evaluating the skill of the fund managers in stock-picking or market-timing , in other words, how good they are at fundamental and technical analysis respectively. If this is done incorrectly, investment funds may be placed in the hands of incompetents who will then misallocate the money between good and bad investment projects, lowering the return from investment and lowering future national income.

As I stated above, CAPM is the foundation for many tests of stock market efficiency. Although the disproving of CAPM will discredit some tests that attempt to disprove the EMH, the hypothesis is threatened more by results involving the application of new techniques of investment analysis. On the other hand, perhaps the market is efficient, and some investors have taken note of the result published by Fama & French (1992), with the result that any potential profits have already been arbitraged away, thus undermining the original econometric relationship. This was the reason Fama (1966) gave for the lack of observable linear relationships in security price time-series data.

In summary, the evidence I have produced is of poor quality and conflicting conclusions can be drawn from it.



Bernstein,P (1992) : Capital Ideas, Macmillan, London,

Carlton & Perloff (1994): Modern Industrial Organisation, Scott, Foresman and Little.

Malkiel, B. (1992)A Random Walk Down Wall Street, Norton, London.

Peters, E. (1991)Chaos and Order in the Capital Markets, Wiley Finance Editions, New York.


Basu, S. (1981) : 'An Explanation of the Small Firm Effect', Journal of Finance

Fama, E. F.(1966) : 'The Behaviour of Common Stock Prices', Journal of Business.

Fama, E. F. & French, K.(1993) : 'The Cross-Section of Expected Stock Returns', Journal of Finance.

Financial Times, Dec. 12th to 18th., 1994.


Riskmetrics Technical Document, J. P. Morgan Inc., 1994.