by

Patrick Glenn

We have come to expect that certain asset classes will, in the long run, produce a certain rate of return. When that class fails to produce this return we are disappointed and look for answers and someone or something to blame. But what is the basis for these expectations? This article will attempt to address these issues and provide some answers.

**Total Return Profile of Large Cap Stocks**

Let’s start with the Large Cap Stock (S&P 500) asset class. We are told that the total return should be about 10%. But where does this number come from? Figure 1 shows the cumulative monthly total return (price + dividends) of the S&P 500 since 1928. Note that the portion of total return due to dividends has historically been in the range from 30% to 60% depending on holding period. Shown on a log axis, where equal percentage changes receive equal weight at all points on the curve, the data is very much visually about a straight line. The actual straight line shown through the data is a best fit exponential regression, Y = a exp (b X), which also displays as a straight line on a log scale and mathematically reports a very high Pearson Correlation Coefficient with this data of 0.99. The annualized rate of return this regression line exhibits, as measured by its slope, is 10.98%.

**Figure 1. Total Return of the S&P 500 since 1928**

Based on this you would expect that in the long run one should be able to achieve nearly 11% on an annual total return basis by investing in the S&P 500. But, as you can see, there are extended periods where the return fails to follow this benchmark. For example, from January 2000 through today (August 2017) the average annual total return in the S&P 500 has been only about 5.0%.

Figure 1 can be useful in a number of ways. First, the principle of ‘reversion to the mean’ indicates where the eventual value of this index will be at some time in the future. If the data is currently below the regression line, as it is now, then it is highly likely that it will eventually rise to cross it. If it is above the line, as it was in 2000, then it is equally likely that it will fall to the line and then perhaps cross below it. You can use this line as an aid in determining entry and exit points. If you buy when it is far above the line then you will experience diminished returns as compared to the 11% or better annual return expectation of those who buy when below the regression line and sell when the market is on or above the line.

A possible observation and interpretation from Figure 1 is that unlike the subprime mortgage crisis of 2008 which was a genuine crash the dot-com bubble burst of 2000 was merely an inevitable reversion to the mean.

**Total Return Profile of Small Cap Stocks**

A very similar result appears for the Small Cap Stock asset class. This asset class mirrors an index such as the the CRSP U.S. Small Cap Index which includes U.S. companies that fall between the bottom 2%-15% by market capitalization. [The Vanguard Small Cap Fund, VSMAX, uses this index as its benchmark.]

Figure 2 below indicates that a long-term annual total return of 14.2% can be expected by investing in a Small Cap stock index fund or similar. Again, this is given by the slope of the regression line and its Pearson Correlation Coefficient is also a robust 0.99.

**Figure 2. Total Return of the Small Cap Stock Asset Class**

Of course, your actual expected return depends on when you buy and when you sell. You will achieve the expected annual rate of return when you buy and sell directly on the regression line. Better returns can be achieved by buying when the value dips below the line and selling when it rises above it. The above chart indicates that right now is a good time to buy.

**Total Return Profile of REITs and International Stocks**

Exponential regressions can be performed for the Real- Estate Investment Trust (REIT) and International Stock asset classes. The results are shown in Figures 3 and 4 below. For REITs, the average annual rate of total return is 12.4%. For International Stocks, the value is 10.8%. The correlation coefficients for these above asset classes is 0.99 as well.

**Figure 3. Total Return of the REIT Asset Class**

**Figure 4. Total Return of the International Stock Asset Class**

**The impact of mean reversion**

These figures can be useful in a number of ways. First, the principle of “reversion to the mean“ indicates where the eventual value of an index will be at some time in the future. If the value is currently below its regression line (as it is in all of the above cases), then it is highly likely that it will rise to touch the line or cross it. Conversely, if it is above the regression line, it will eventually fall to the line or below it.

In discussing mean reversion, the real question is not will it ever revert to the mean, but when? It’s impossible to determine exactly that time in the future when it will revert to the mean (ie., the regression line)—it could be a matter of months or even decades. The real strength of this approach is you can improve your returns just by buying when the value is at or below the line, and selling when the value rises above it. On the other hand, if the value is currently well above the line, you may wish to wait until the value comes down, or at least take a smaller position so you don’t lose out on returns entirely.

From the above charts, you can see that if you had bought right before the 1929 market crash and held until now, your total returns in all of the asset classes would have suffered. The table below shows that your actual returns would be off by 1% – 2%, depending on asset class.

**Table 1. Comparison of Expected versus Actual Returns**

**Historical Variations in Expected Returns**

Mathematically astute readers will note that the regression lines shown are based on the data up to the present. What was the slope of the regression line in the past when less data was available? Looking at the data for the Large Cap Stocks in Figure 5, you can see that since the ending of the Great Depression in the early 1940’s the expected return as measured by the slope of the regression line climbed steadily until about 1970 when it leveled off. Since then it has been fairly stable around today’s value. A similar return profile holds for all of the four stock asset classes discussed above.

**Figure 5. Historical Expected Returns for the Large Stock Asset Class**

It’s interesting to look at the International Stock asset class date line (Figure 4). You will see that it appears to have taken a new and flatter path starting in 1987. Only the future will tell if this is a real effect signifying a fundamental change or whether it will revert to its current historical mean value.

**Stocks vs Bonds**

These stock asset classes seem to behave fairly well according to the regression line premise. However, the same cannot be said of the bond asset classes. They behave much differently. This is particularly true of the Treasury Bill asset class shown in Figure 6 below. As can be readily seen, the exponential regression line approach doesn’t do very well in representing bond performance.

This is to be expected as short-term interest rates are governed by external market forces (e.g., the Federal Reserve Bank) and not strictly by supply and demand. Being directly responsive to interest rate variations, bond performance has no single long-term trend but instead varies slowly at different rates over long periods of time. Low interest rates (and inflation) are seen through the 1960’s slowly transitioning to higher rates in the 1970’s, 1980’s, and 1990’s with lower rates and returns returning in the beginning of this century. For reference, the slope of the regression line for this bond asset class gives an expected annual return of 4.2%.

**Figure 6. Total Return of the Treasury Bill Asset Class**

**Summary**

Using regression analysis, the long-term expected returns of free floating asset classes (stocks) can be determined with a high degree of accuracy given a large enough data set. A plot of the regression line over actual return data can indicate to investors when it would be prudent to buy and when it would be prudent to sell in order to maximize portfolio returns.

**Note:** The graphs and charts presented here were all generated by the* Portfolio Preserver (TM*), a powerful software tool designed to maximize portfolio returns while minimizing risk. Find out more at PortfolioPreserver.com.

**About the Author:**

**Patrick Glenn** is a former physicist, computer scientist, and business owner. He specializes in mathematical modeling and algorithm development and is the author of the popular CarTest 2000 automobile acceleration simulation and FinanceMaster, a personal finance management program. Today he is an advocate of financial quantitative analysis and is the designer and architect of the **Portfolio Preserver** software, an innovative approach to portfolio management and risk minimization.