Anybody who has analysed financial market

data will have stumbled across what look

like sure ways of making money. Historical

data bristle with apparently exploitable correlations. On the basis of such correlations, it is obvious to many actuaries that markets are profoundly inefficient.

In this article, we shall review some of the traditional statistical tests which supposedly disprove market efficiency. Secondly, we shall construct a simple, efficient market model and show that the same statistical tests apparently disprove market efficiency even for this model that is known, by its construction, to be efficient. In a second article, we shall investigate what was wrong with the traditional analysis and why the statistical tests failed.

Regression analysis

Figure 1 shows data for the FTA all-share index, based at the middle of the month, for 1965 to 2001. We have calculated the dividend yield and the real return during the following year. These data are plotted with a best-fit straight line. Does the positive slope of this straight line suggests that dividend yields may be able to predict subsequent market returns?

The momentous effects of 197375 have a large influence, so we have calculated results including and excluding these periods. To avoid problems of overlapping periods, we have used annual data for these calculations, based on mid-September.

Regression analysis gives the following results:

Period covered 19652001 19762001

Intercept -29% -21%

Slope 8.3 7.5

Correlatio– 50% 49%

The slope parameter of about 8 suggests that investing when retrospective dividend yields are 1% higher, results in an expected return around 8% higher. Broadly similar results hold for many different indices over many time periods and across many major world stockmarkets.

On the basis of these results, we conclude that dividend yields predict equity returns. Therefore markets are inefficient, and prices have a tendency to revert to some multiple of dividends. Therefore the random walk model, or any other efficient market model, is an inappropriate way to model equity markets.

An efficient market model

We now build a simple random walk model, where, by construction, dividend yields cannot predict returns. But examining randomly generated ‘history’ indicates apparent predictability: the standard statistical tests are consistently fooled by this model.

Here’s the idea. We simulate first a set of independent, identically distributed lognormal total returns for the next 30 years, with an underlying arithmetic mean return of 10% and standard deviation of 20%. We then assume that the dividend during each year is 5% of the moving average of the share price over the previous 10 years, subject to a maximum dividend yield of 30%. These assumptions broadly reflect the real data set analysed earlier by regression. However, since the total returns are mutually independent, we know here that the total return in a year does not depend on the dividend yield at the start of the year.

On each simulation, we calculate the regression slope of the return against the yield at the year start. Since we know that in fact the return each year is independent of the yield at the year start, we might expect, on average, to see zero slope.

Figure 2 shows the actual distributions, obtained by Monte Carlo simulation, for a variety of different time horizons. We have plotted the 5th, 25th, 50th, 75th, and 95th percentiles of the regression coefficients.

It is clear from the picture that this model has a tendency to produce positive slopes and correlations. Indeed, for every period we look at, there is at least a 75% chance of finding a positive slope.

So what is going on?

So far we have carried out some standard statistical tests. We have seen that standard tests for market efficiency will often reject efficient markets, even when the data has been generated from a manifestly efficient market model. A positive correlation or slope parameter from a finite set of consecutive observations cannot imply that future returns depend on current yields. o

Next article: we shall explore why this happens, and what might be done to improve our statistical tests.

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