**Estimating Future Stock Market Returns
Butler Philbrick & Associates
By Adam Butler and Mike Philbrick
December 14, 2011**

*"Mankind are so much the same, in all times and places, that history informs us of nothing new or strange in this particular. Its chief use is only to discover the constant and universal principles of human nature." - David Hume*

Long-time readers will know that we do not make predictions in the normal sense. That is, we endorse the decisive evidence that markets and economies are complex, dynamic systems which are not reducible to normal cause-effect analysis. However, we are willing to acknowledge the likelihood that the future is likely to rhyme with the past. Thus, we apply simple statistical models to discover mean estimates of what the future may hold over meaningful investment horizons (10+ years), while acknowledging the wide range of possibilities that exist around these averages.

There are several reasons why it may be useful to have a more robust estimate of future expected returns on stocks:

- People who are approaching retirement need to estimate probable returns in order to budget how much they need to save.
- A retiree's level of sustainable income is largely dictated by expected returns over the early years of retirement.
- Investors of all types must make an informed decision about how best to allocate their capital among various investment opportunities

Many studies have attempted to quantify the relationship between Shiller PE and future stock returns. Shiller PE smoothes away the spikes and troughs in corporate earnings which occur as a result of the business cycle by averaging inflation-adjusted earnings over rolling historical 10-year windows.

This study contributes substantially to research on smoothed earnings and Shiller PE by adding three new valuation indicators: the Q-Ratio, total market capitalization to GNP, and deviations from the long-term price trends. The Q-Ratio measures how expensive stocks are relative to the replacement value of corporate assets. Market capitalization to GNP accounts for the aggregate value of U.S. publicly traded business as a porportion of the size of the economy. In 2001, Warren Buffett wrote an article in Fortunewhere he states, "The ratio has certain limitations in telling you what you need to know. Still, it is probably the best single measure of where valuations stand at any given moment." Lastly, deviations from the long-term trend of the S&P inflation adjusted price series indicate how 'stretched' values are above or below their long-term averages.

These three measures take on further gravity when we consider that they are derived from four distinct facets of financial markets: Shiller PE focuses on the earnings statement; Q-ratio focuses on the balance sheet; market cap to GNP focuses on corporate value as a proportion of the size of the economy; and deviation from price trend focuses on a technical price series. Taken together, they capture a wide swath of information about markets.

We analyzed the power of each of these 'valuation' measures to explain __inflation-adjusted__ stock returns including reinvested dividends over subsequent multi-year periods. Our analysis provides compelling evidence that future returns will be lower when starting valuations are high, and that returns will be higher in periods where starting valuations are low.

This last point may seem obvious, but I want to emphasize a critical point about traditional wealth management of which most investors are not aware:

*Traditional investment planning does not account for whether markets are cheap or expensive. An investor who visited a traditional Investment Advisor at the peak of the technology bubble in early 2000 would, in practice, be advised to allocate the same proportion of his wealth to stocks as an investor who visited an Advisor near the bottom of the markets in early 2009. This despite the fact that the first investor would have had a valuation-based expected return on his stock portfolio from January 2000 of negative 2% per year, while the second investor would expect inflation-adjusted compound annual returns of 6.5%. For an investor with $1,000,000 to invest, this would represent a difference of more than $1.26 million in cumulative wealth over a decade.*

Said differently, traditional wealth advice is rooted in the assumption that the best estimate of future returns is the average long-term return to stocks. No matter where markets are on the continuum from very cheap to very expensive, traditional Advisors will make recommendations on the assumption that investors should expect 6.5% inflation adjusted returns on stocks over all investment horizons.

John Hussman at Hussman funds is careful to qualify the value of this analysis: "Rich valuation is strongly associated with weak subsequent returns, but only reliably so over periods of 7-10 years. In contrast, the present syndrome of overvalued, overbought, overbullish, rising-yield conditions is typically associated with abrupt and often steep losses, but is more commonly resolved over a period of months rather than years." (Hussman, Feb 14, 2011). Thus, we are not making a forecast of market returns over the next several months; in fact, markets could go substantially higher from here. However, over the next 10 to 15 years, markets are very likely to revert to average valuations, which are much lower than current levels. This study will demonstrate that investors should expect 6.5% returns to stocks only during those very rare occasions when the stock market passes through 'fair value' on its way to becoming very cheap, or very expensive. At all other periods, there is a better estimate of future returns than the long-term average, and this study will quantify that estimate.

Investors should be aware that, relative to __meaningful__ historical precedents, markets are currently expensive and overbought by all three measures, indicating a strong likelihood of low inflation-adjusted returns going forward over periods as long as 20 years.

This prediction is also supported by evidence from an analysis of corporate profit margins. In his recent book, Vitaly Katsenelson provided the following chart (Chart 1.) of long-term profit margins to U.S. companies. Companies have clearly been benefitting from a period of extraordinary profitability.

Source: Vitaly Katsenelson (2011)

The profit margin picture is critically important. Jeremy Grantham recently stated, "Profit margins are probably the most mean-reverting series in finance, and if profit margins do not mean-revert, then something has gone badly wrong with capitalism. If high profits do not attract competition, there is something wrong with the system and it is not functioning properly." On this basis, we can expect profit margins to begin to revert to more normalized ratios over coming months. If so, stocks may face a future where multiples to corporate earnings are contracting at the same time that the growth in earnings is also contracting. This double feedback mechanism may partially explain why our statistical model predicts such low real returns in coming years. Caveat Emptor.

### Modeling Across Many Horizons

Many studies have been published on the Shiller PE, and how well (or not) it estimates future returns. Almost all of these studies apply a rolling 10-year window to earnings as advocated by Dr. Shiller. But is there something magical about a 10-year earnings smoothing factor? Further, is there anything magical about a 10-year forecast horizon?

Kitces (2008) demonstrated that "the safe withdrawal rate for a 30-year retirement period has shown a 0.91 correlation to the annualized real return of the portfolio over the first 15 years of the time period". So there is clearly merit in studying a 15-year forecast horizon as well. Further, the tables below will demonstrate that statistical models have the greatest explanatory power at the 15-year horizon.

This study will attempt to address the question of 'perfect forecast horizon', perfect valuation factor, and 'perfect earnings smoothing factor', by analyzing the explanatory power of earnings, the Q-Ratio, and regressed historical stock returns, over return horizons from 1 to 30 years. We will also put all of the factors together to construct an optimized model.

Table 1. below provides a snapshot of some of the results from our analysis. The table shows estimated future returns based on several factor models over some important investment horizons. The "Best Fit Multiple Regression" is by far the most accurate model, but other results are provided for context.

Table 1. Factor Based Return Forecasts Over Important Investment Horizons

Source: Shiller (2011), DShort.com (2011), Chris Turner (2011), World Exchange Forum (2011), Federal Reserve (2011), Butler|Philbrick & Associates (2011)

You can see from the table that every single valuation factor model generates results which suggest a very low future return environment for stocks. Further, the 'Best Fit Multiple Regression', which has historically provided a surprising degree of forecast accuracy, confirms this outlook with a high degree of confidence (see explanation below). Those who are not interested in our process can skip to the bottom sections, '**Putting the Predictions to the Test'**, and '**Conclusion**'.

### Process

The following matrices show the R-Squared ratio, regression slope, regression intercept, and current predicted forecast returns for each valuation factor. The matrices are heat-mapped so that larger values are reddish, and small or negative values are blue-ish. Click on each image for a large version.

**Matrix 1. Explanatory power of valuation/future returns relationships**

*(Please click for larger image)*

Source: Shiller (2011), Doug Short (2011), Butler|Philbrick & Associates (2011)

You will note that the R-Squared (top chart), which is a measure of the explanatory power of the relationship, is highest for the Q-ratio over all forecast horizons up to 9 years, but that deviations from the real price regression are more predictive over horizons of 10 years or more. The explanatory power of smoothed earnings ratios gets better consistently as we extend the forecast horizon, with peak ratios at the 20-year range. No factors possess any material explanatory ability at forecast horizons less than 5 years, so we have omitted results for these horizons.

Many analysts quote 'Trailing 12-Months' or TTM PE ratios for the market as a tool to assess whether markets are cheap or expensive. If you hear an analyst quoting the market's PE ratio, odds are they are referring to this TTM number. Our analysis slightly modifies this measure by averaging the PE over the prior 12 months rather than using trailing cumulative earnings through the current month, but this change does not substantially alter the results. As it turns out, TTM average earnings have very mild explanatory value over periods greater than 8 years. However, the explanatory power of TTM earnings is substantially less reliable than all other factors studied in this analysis, so investors may wish to pay little heed to this indicator of whether stocks are cheap or expensive.

One interesting take-away from the analysis is that the simple real price series carries much better explanatory power than the real total-return price series, which includes reinvested dividends. The total return series reflects the actual returns to investors over any given period, while the price series ignores dividends altogether. Strangely however, the total return series does a relatively poor job of forecasting future total returns to stocks, and is, statistically speaking, a much less powerful explanatory factor than either the Q-ratio or the simple real price regression. As so often happens in complex fields, the evidence makes a mockery of the theory.

### Forecasting Expected Returns

The next matrices provide the slope and intercept coefficients for each regression. We have provided these in order to illustrate how we calculated the values for the final matrix below of predicted future returns to stocks.

**Matrix 2. Slope of regression line for each valuation factor/time horizon pair.**

*(Please click for larger image)*

Source: Shiller (2011), Doug Short (2011), Butler|Philbrick & Associates (2011)

**Matrix 3. Intercept of regression line for each valuation factor/time horizon pair.**

*(Please click for larger image)*

Source: Shiller (2011), Doug Short (2011), Butler|Philbrick & Associates (2011)

Our final matrix below shows predicted future real returns over each time horizon, as calculated from the slopes and intercepts above, by using the most recent values for each of the 13 earnings series, the Q-Ratio, and the return series as inputs. For statistical reasons which are beyond the scope of this study, we have substituted the ordinal rank for the nominal value for each factor in running our analysis. Therefore, when we solve for future returns based on current monthly data, we apply the monthly rank in the equations.

For example, the 15-year return prediction based on the current Q-Ratio can be calculated by multiplying the current ordinal rank of the Q-Ratio (798) by the slope from Matrix 2. at the intersection of 'Q-Ratio' and '15-Year Rtns' (-0.00007), and then adding the intercept at the same intersection (0.09709) from Matrix 3. The result is 0.0399, or 3.99%, as you can see in Matrix 4. below at the same intersection (Q-Ratio : 15-Year Rtns). Please click the matrix for a larger version.

**Matrix 4. Modeled forecast future returns using current valuations.**

*(Please click for larger image)*

Source: Shiller (2011), Doug Short (2011), Butler|Philbrick & Associates (2011)

Finally, at the bottom of the above matrix we show the forecast returns over each future horizon based on our best-fit multiple regression from the factors above. We began testing the multiple regression against the Q-ratio, the 15-year Shiller PE, the price regression, and the market cap to GNP as a 4 factor model. However, we discovered that the 15-year PE provided more noise than signal to the regression (that is, these factors were not statistically significant and reduced the F-score), so we narrowed the regression to include just the Q ratio, market cap/GNP, and the real price series over each forecast horizon. We provided the R-squared for each multiple regression at the bottom of each forecast horizon column in Matrix 4.; you can see that at the 15-year forecast horizon, our regression explains 82% of total returns to stocks. Further, the regression is very highly statistically significant, with a p value of effectively zero.

Chart 2. below demonstrates how closely the model tracks actual future 15-year returns. The red line tracks the model's forecast annualized real total returns over subsequent 15-year periods using the Q ratio and deviation from price regression as inputs at each period. The blue line shows the actual annualized real total returns over the same 15-year horizon.

**Chart 2. 15-Year Forecast Returns vs. 15-Year Actual Future Returns**

*(Please click for larger image)*

Source: Shiller (2011), Doug Short (2011), Butler|Philbrick & Associates (2011)

You can see that 15-year 'Regression Forecast' returns are 1.46% per year, and 10-year returns are forecast to be 3.75% per year using market valuations to December 9, 2011.

### Putting the Predictions to the Test

A model is not very interesting or useful unless it actually does a good job of predicting the future. To that end, we tested the model's predictive capacity at some key turning points in markets over the past century or more to see how well it predicted future inflation-adjusted returns.

**Table 2. Comparing Long-term average forecasts with model forecasts**

Source: Shiller (2011), Doug Short (2011), Butler|Philbrick & Associates (2011)

You can see we tested against periods during the Great Depression, the 1970s inflationary bear market, the 1982 bottom, and the middle of the 1990s technology bubble in 1995. The table also shows expected 15-year returns given market valuations at the 2009 bottom, and current levels. These are shaded green because we do not have 15-year future returns from these periods yet. Note real total return forecasts of 6.7% annualized from the bottom of the market in February 2009. This suggests that prices just approached *fair value* at the market's bottom, but they were nowhere near the level of cheapness that markets achieved at bottoms in 1932 or 1982. As of the end of May 2011, expected future returns over the next 15 years are almost exactly 0 percent.

We compared the forecasts from our model with what would be expected from using just the long-term average real returns of 6.5% as a constant forecast, and demonstrated that estimates form long-term average returns yield over __450% more error__ than estimations from our model over these 15-year forecast horizons (1.23% annualized return error from our model vs 5.55% using the long-term average). Clearly the model offers substantially more insight into future return expectations than simple long-term averages, especially near valuation extremes.

### Conclusions

The 'Regression Forecast' return predictions along the bottom of Matrix 4. are robust predictions for future stock returns, as they account for over 100 different cuts of the data, using 3 distinct valuation techniques, and utilize the most explanatory statistical relationships. The models explain up to 82% of future returns based on R-Squared, and are statistically significant at p~0. It is worth noting, however, that even this model has very little explanatory power over horizons less than 6 or 7 years, so almost anything is possible in the short-term.

Returns in the reddish row labeled "PE1" in Matrix 4 were forecast using just the most recent 12 months of earnings data, and correlate strongly with common "Trailing 12-Month" PE ratios cited in the media. Note that our Matrix 1. proves that this trailing 12 month measure is not worth very much as a measure for forecasting future returns over any horizon. However, the more constructive results from this metrix probably helps to explain the general consensus among sell-side market strategists that markets will do just fine over coming years. Just remember that these analysts have no proven ability whatsoever in predicting market returns (see here, here, and here). Further, it can be argued that their firms have a substantial incentive to keep their clients invested in stocks.

Investors would do much better to heed the results of robust statistical analyses of actual market history, and play to the relative odds. This analysis suggests that markets are currently expensive, and asserts a very high probability of low returns to stocks (and possibly other asset classes) in the future. Remember, any returns earned above the average are necessarily earned at someone else's expense, so it will likely be necessary to do something radically different than everyone else to capture excess returns going forward.

*Adam Butler and Mike Philbrick are Portfolio Managers with Butler|Philbrick & Associates at Macquarie Private Wealth in Toronto, Canada.*

(c) Butler|Philbrick & Associates, 2011

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