Chapter 6
Means Move—Analyze the Averages
Calculating an average is probably the most widely used statistical tool. You can look at average goals per game, grade point average, the average amount of time spent on a website. Averages pop up everywhere. There are several different types of averages that are appropriate for different types of situations, which are worth reviewing. It is also important to discuss how averages are misused. The simple fact that averages are dynamic and change often gets ignored. When there is more change, averages may become a less valuable reference point.
The most common types of calculations referred to as averages are the mode, the median, and the mean. The mode is the figure that appears most often in a data set; there can be more than one mode. The median is the figure in a data set that separates the upper half of the data set from the lower half. The mean is by far the most commonly used of these calculations. The mean is the sum of the numbers in a data set divided by the number of figures in the data set. When the word average is used in this book (and generally in conversation) it is assumed to be referring to the mean. Examples are shown in Figure 6.1.
While the mean is the most common to use there are instances where the mode and the median can be more helpful. For example, if you want to find out the most common salary in a group of workers you would use the mode, as the foreman’s salary might skew the mean. The median can be good when most numbers in the data set are clustered close together but there are one or two extreme outliers that really skew the mean.
Some other averages are quite common when running investment analysis. These include the geometric mean, a weighted average and a moving average.
Geometric means are good when each figure in the data set is dependent on the others. An example of this would be if you were tracking a company’s revenue growth from one year to another. The geometric mean multiplies all of the numbers in the data set and then takes that product to the “n” root, “n” standing for the number of figures in the data set. This type of mean is often used to examine investment returns. Typically, when using a geometric mean all of the numbers in the data set need to be positive, but you can manage around this by adding one to each figure in the data set. If the data set included annual investment returns (0.2, 0.1, 0.2, -0.1, 0.1), you could add one to each data point (1.2, 1.1, 1.2, 0.9, 1.1) then take the product of these figures to the 1/5 power and subtract one to get the geometric mean (the other option is to just never have any negative returns).
Weighted averages are commonly used in indexes, portfolio, analysis and probability theory to develop expected values. Investors often use indexes that use weighted averages. For example, an index may track the stock prices of the 100 largest public companies. The index could weight each stock the same and calculate the average price of the index. Getting this figure at the end of the day on a percentage basis would show you how much the average stock in the index moved. However, it would not show you how the value of the overall index moved. A common way of showing you the move in the value of the index is to “weight” each stock in the index based on what the equity market capitalization is (equity market capitalization = (number of shares outstanding * stock price).
In mid-2017 on a market weighted basis just four companies accounted for 10.6% of the S&P 500 U.S. stock index; these top four accounted for more of the index than the bottom 200 companies in the index combined.31 It is pretty easy to see how in this scenario the market weighted performance of the S&P 500 can look very different than the performance for the majority of the companies in the index. With the incredible proliferation of the use of indexes these types of distortions in indexes are important to understand. It is not a bad idea to occasionally look at an even weighted version of these indexes to get a different understanding of what is happening within the indexes.
Moving averages are typically used with time series studies. In a time series an investor may want to look at how an index or a stock has performed over time. Perhaps they want to look at how two different investments performed during a recession or a period of rising interest rates. Part of the value in using a moving average is that it can eliminate the impact of a one-day aberration in the price of a security.
Moving averages are often used when examining stock price performance. To calculate a moving average, you could take the trading history of the stock price of company A, and calculate a 200-day moving average. If the most recent date that you had stock price data for was November 25, the data set would consist of the average price of stock A for the 200 days prior, so from November 25 to May 9.32 On November 26, to do the same calculation the oldest day of the series would move to May 10 and May 9 data would be dropped so that there would still be 200 data points.
You might argue that deciding on 200-day old data is not as valuable as the most recent day’s data, and you would be right. However, looking at the most recent day’s data in isolation is not as valuable as looking at a long-term trend. Therefore, you might want to give more weight to the most recent data and use a weighted moving average.
One way of calculating a weighted moving average would be to use the price of the stock for each day of the period you are measuring and apply a weight to that day (usually in percentage terms). More recent days would get a larger weighting. If there are 5 days of data you are measuring the weighting of the first period could be (5/(5+4+3+2+1)) = 0.33 and then you would multiply the stock price from that first day by 0.33. The second period weighting would be 4/15 or 0.27 and so on. The weighted moving average can be customized as well, you could choose to make the weightings more subjective (e.g., maybe 10 for the most recent data and then 7 for the second oldest, etc.) or even overweight older data points rather than newer ones if you can come up with a logical reason to do so.
A more common way of increasing the weighting of the most recent data points versus older trading points is to use an exponential moving average. This formula makes the most recent price more heavily weighted than in the simple weighted average, sometimes this is called “reducing the lag.” This weighting bias will be greater in small data sets. The formula for this calculation using daily stock prices would be:
(Closing price – previous days exponential moving average (EMA) price) x (2/(N+1)) + the previous days exponential moving average price.
N is equal to the number of periods in the data set, for a 200-day moving average N would be equal to 200. To start the series the first data point can either be a simple moving average price or just the price on that date.
Investors will often use these types of moving averages as part of technical analysis to find entry and exit points for investments that are liquidly traded. They will often compare short-term trends to long-term trends. As an example, if a 50-day moving average price of a stock crosses above a 200-day moving average it may be a buy signal to a trader, as shown in Figure 6.2. This is a signal that there is a divergence in more recent trading patterns than long-term trading patterns. Technical traders may also look at the angle of the moving average, if it has minimal slope it may imply the stock is range-bound and is not showing an upward or downward trend. There are many other technical analysis tools and a large body of research in the field.
If you are looking at investments in stocks or currencies, there are several programs that prepare charts with various types of moving averages and data on volumes and more. Many stock brokerage firms offer technical analysis software on their websites for clients. Additionally, while technical analysis can be utilized when investing in many asset classes, it is most useful for highly liquid assets that have good disclosure about prices and trading volumes such as stocks and currencies.
Averages are made up of old data points. Comparing the trend lines of moving averages over different time periods can be a valuable tool to identify a divergence in newer data versus older data. However, there are problems with using averages and mistakes are constantly made because investors do not realize that an average may not be telling them the same thing today as it was 200 days ago.
One commonly used investment refrain is that, “returns tend toward the mean.” This is also known as “mean reversion.” This implies that if the price of an asset is either well above its historical average or well below it you can make a profit by assuming that the price will return to its historical average.
It is always an interesting phenomenon to explore when asset valuation is very different from its historical average and it should not be ignored. Mean reversion has worked as a strategy in many cases, especially when there is a temporary aberration in valuations. It is important to recognize when it is not going to work. This might be when the components in the “average” are going through considerable change. There are frequently times when the components (or the characteristics) of some average that is being used in a time series has changed so much that historical data that is driving the mean is just not applicable to the present. This will often make mean reversion somewhat irrelevant. Consider the earlier example of the S&P 500 Index. The types of companies that dominate the index have changed. The four largest companies in the S&P 500 Index in 2017 were Google, Amazon, Netflix, and Facebook. All internet-based companies, two of which are clearly pure service providers. Ten years earlier the top four components of this same index were Exxon Mobil, General Electric, Microsoft, and AT&T,33 all but AT&T sells products. The profit profile is very different between the top companies in 2017 and 2007. Google in 2017 converted about 33% of its revenue into cash flow, while in 2010 Exxon converted only 9%.34 If you are comparing the historical averages of the S&P 500 stocks over the last ten years, you are looking at an index that has had a significant change in its largest components, it is logical that certain data in the current index will look very different than the ten year average data. When people wave mean reversion in your face, think of the underlying components that make up that mean and apply some common sense.
Too much blind acceptance of an average as a static target will result in bad decisions. You must do some analysis of the data to understand if it is still meaningful to the current environment and apply some logic.