The Rationale for Moving Averages

By guest contributor Tom Forest

Compared to Buy and Hold strategies, investing with a Simple Moving Average (SMA) system improves returns because of the effect of serial correlation in SMA signals.

Serially correlated signals are those where the value in one time interval is predicative of the value in the subsequent interval. For example, in a 10-month SMA strategy, if the one-month average is greater than the 10-month SMA (let's label this state +), then next month the one-month average is also likely to be greater than the 10-month SMA (+). Similarly, if the one-month average is less than the 10-month SMA (-), then next month the signal is also likely to be (-). In an SMA system, we don't know *when* the series will change from + to -, or - to +, but the system works because + months are more likely than average to be followed by + months, and - months by - months.

The degree of Serial Correlation in a data stream can be described mathematically by a number we can call "S" (for **S**erial Correlation). S expresses the chance that there will be a switch from + to -, or - to +, in the series (in other words, a signal to move into or out of equities). When S=0, then every + turns to a - in the next interval and every - to a +. In a world where S=0, the string would look like this: + - + - + - + - + - + - and so on to infinity. In a world where S=1, the series never changes. Depending on how it starts, it's is either - - - - - - - - - - or + + + + + + + + + + continuing forever. Investors in the first imaginary world would move in and out of the market every month. Investors in the second world would hold forever — either equities or cash — depending on the starting position.

When S=0.5, trend following is impossible because there is no way to predict next month's signal based on last month's signal. With zero trading cost and zero slippage you would break even trading using the SMA. In other words, a market where S=0.5 would be 100% efficient.

So empirically, what is S for the S&P 500? Has it moved toward 0.5 over time (become more efficient) as investors have become more sophisticated?

To answer that question, let's analyze the S&P Composite historic data from 1871 to the present and apply a 10-month SMA strategy. For the 137-year period, S averages 0.88 (which is much closer to 1 than 0.5). This finding confirms the utility of the SMA approach.

But has S changed over time? Has the US stock market become more efficient over the last century? Let's examine the rolling 10 year periods from 1871 to the present. As the table shows, S varies over time from 0.80 to 0.95. It stands at 0.88 today — squarely in the middle of the historic range.

It looks like the Animal Spirits are alive and well. The market is as inefficient today as it was in the 1870s.

*Tom Forest describes himself as "an amateur investor." However, he shares with us a professional perspective on market timing with moving averages.*

**Additional Notes**

Here's a chart of serial correlation together with the S&P composite since 1880.

The S&P Composite data used in Tom Forest's analysis is available from Robert Shiller's Yale website. The monthly prices in this dataset are the monthly averages of daily closes rather than the closing price on the last day of the month. This "data smoothing" reduces the impact of end-of-the-month outlier closes. In contrast, the moving average strategies tracked at this website are based on the closing price of the last day of the month.

An obvious question is whether the choice of monthly averages or monthly closes would change the serial correlation. To investigate this possibility, I ran my Excel 10-SMA serial correlation formula on the S&P 500 monthly closes since 1950. The result was S=0.8841. For the S&P monthly averages over the same period, S=0.8867. I'm not a professional statistician, but I think we can assume that the 0.0026 difference is statistically insignificant.

**But what about investment performance?**

For an analysis of the 10-month SMA timing strategy and its performance against buy and hold since 1972, Mebane Faber's World Beta website periodically features his thoughtful article A Quantitative Approach to Tactical Asset Allocation. See also his recent chart illustrating the relative performance of this monthly timing strategy versus buy and hold. Equally compelling are Faber's updated charts comparing timing with buy and hold.

An appropriate close to these notes is a recent comment by Mebane Faber in his review of Richard Tortoriello's book Quantitative Strategies for Achieving Alpha: "In the more than 40 single factors he tested from 1987-2006, guess which factor was most predictive? Momentum."

Momentum, of course, is the common word for high serial correlation.