One of the most popular assumptions used in investment modeling is that stock prices follow a random walk. However, a growing body of evidence shows that this is a bad assumption, and that stock prices go through periods of over-pricing and under-pricing that affect future returns. Many of the models used by financial planners are based on the random walk assumption. This naturally raises the question of whether we can usefully build investment models based on more realistic assumptions. In this article, I propose that we can build better models and provide some initial suggestions on how to do so.

Last fall I wrote two articles for *Advisor Perspectives* (see __here__ and __here__) discussing the use of stock market valuations to predict long-term stock market performance. The particular valuation measure I used was a price/earnings ratio that smoothes out cyclical earnings by including ten-year average earnings in the denominator. I referred to this measure as the Shiller PE because the main proponent of its use has been Professor Robert Shiller of Yale University who highlighted it in his book, "Irrational Exuberance," and in earlier research. This measure is also known as "Normalized Price Earnings Ratio," "PE 10," and "Cyclically Adjusted PE (or CAPE)." All references below to PE ratios are to Shiller PE's.

Recently I examined this measure more deeply with the goal of building a simple stochastic model based on the interaction of PE's and stock market rates of return. A stochastic model of this type could be useful in improving the Monte Carlo projections of investment returns, and making improvements to one of the principal tools used by financial planners.

My working hypothesis is that PE's are inversely correlated with future returns, but, on a year-to-year basis, any impact is mostly drowned out by random influences. Over time these PE effects accumulate and significantly affect performance. So I'm assuming that the stock market almost – but not completely – follows a random walk and that long-term investors can benefit by adjusting portfolios based on current levels of PE's.

I began by examining historic stock market performance data going back to 1928 alongside Professor Shiller's data on historic PE's. Tests of the relationship between beginning-of-year PE's and one-year returns produced the expected negative correlation with a coefficient of -.26 and an R-square measure of .067. A rough translation is that the beginning-of-year PE explains about 6.7% of that year's stock market return. The variation of the yearly returns around the regression line was considerable – equivalent to a standard deviation of 20%, about the same as expected overall stock market volatility. For modeling one-year stock returns as a function of the beginning-of-year PE, I approximated the relationship with the equation:

Return = (.26)*(.95)^PE + N(0, .20)

where N(0, .20) is a randomly generated normally distributed variable with 0 mean and a standard deviation of 20%. For example, at a PE of 7—the lowest experienced since 1928—the expected return is 18% plus or minus the random term. At the all-time high PE of 42, experienced just before the bursting of the dot-com bubble, the expected return is 3% plus the random term. Basically, the model reflects the data, which show that the beginning PE exerts an influence on the expected return for the year, but with considerable random noise.