### Introduction to the BMW Method Screen

#### Screen Page Layout

There are 4 parts to a single stock's BMW Method Screen page:

• Numeric screen results
• Logarithmic price chart
• Linear price chart
• Price distribution chart (statistical info)

Each will be discussed with examples below.

#### Numeric Screen Results

This presents the numeric results of the screen over the specified time frame. An example is:

```BMW METHOD SCREEN, 30-year history:
* Using MIN RETURN FACTOR threshold of 2.000x
* Using MIN RMS BELOW AVE CAGR LINE threshold of 2.000

AVE      CUR     RETURN
TICKER   CAGR     CAGR     FACTOR      RMS   PRICE
--------------------------------------------------
IBM      6.3%     6.7%       0.91     0.23   84.85   International Business Machines Corp
```

For IBM, we see the following over the selected 30-year period:

• IBM's stock price, along a "best fit" line, achieved a 6.3% compounded annual growth rate (CAGR)
• IBM's stock price, ending at today's price, is running at a 6.7% CAGR
• If you bought IBM stock at today's price of \$84.85, and it "reverted" exactly to the mean (to where the Average CAGR line intersect's today's date), you would get back 0.91, or 91%, of your original investment. In other words, you would lose 9%.
• IBM's stock price today is 0.23 RMS above the Average CAGR line

These terms will be easier to understand and visualize once we get to the first chart, which we'll cover in quite some detail. But even based on just looking at the numbers, one can conclude that IBM stock is currently a little bit overvalued compared to its long-term historical growth rate.

#### Logarithmic Price Chart

Referring to the chart below: ##### Price Data and CAGR Lines

The blue data are the monthly closing prices for IBM for the last 30 years. The chart is logarithmic in the Y (price) axis, which is convenient because a constant growth rate shows as a straight line.

The logarithm of the data points is taken and the logarithm values are fit to a line using a standard linear regression ("best fit") routine. The resulting line represents a single compounded annual growth rate (CAGR) that best fits the data. This "best fit" line is called the Average CAGR line in BMW Method terminology.

In the chart above, the red line is the Average CAGR line. It is the statistical "best fit" line for the blue data points. Notice how the stock price varies widely, but over a long period of time it both falls below and rises above the Average CAGR line. In IBM's case, as with most stocks, the Average CAGR line shows a pretty good representation of a long term price trend.

The brown line is the Current CAGR line. It starts on the left edge of the chart at exactly the same price point as the Average CAGR line, but ends up on the right edge of the chart at today's price. It is useful just to compare to the red line for a quick visual indicator of how far from the Average CAGR line the current price is.

##### RMS, and RMS Lines

There are 4 parallel green lines on the chart, two above the red Average CAGR line and two below. These lines are also exactly parallel to the red Average CAGR line. These are very important lines because they give a statistical measure of the variation, or volatility, of the stock over the time range of the chart. Each green line represents one RMS distance away from the red Average CAGR line. The statistical "best fit" routine that generated the Average CAGR (red) line also generated an RMS value which indicates how well the price data fit the line. Statistically, about 68% of all prices should fall within 1 RMS of the Average CAGR line (in the band between 1 RMS above and 1 RMS below the Average CAGR line), and 95% should fall within 2 RMS of the Average CAGR line (again, in the band between 2 RMS above and 2 RMS below the Average CAGR line).

So stock prices should fall outside the +/-2 RMS lines only about 5% of the time, and in an ideal price distribution, exactly half, or 2.5%, of those would be above the +2 RMS line and half, or 2.5%, below the -2 RMS line. That means that over a 30-year period, the price should be below the -2 RMS line for about 9 months and above the +2 RMS line for about 9 months. As you can see in IBM's case, it actually isn't quite like that, but the general idea applies. In the mid-1992 to late 1994 range, IBM's stock was consistently under the -2 RMS line for about 1-1/2 years, and it just barely touched the +2 RMS line once.

Referring back to the Numeric Screen Results section above, one of the numbers reported for each stock is the RMS number. This is how far today's price is from the Average CAGR line in "RMS units". If the number is positive, the stock price is above the Average CAGR line; if negative, it is below. IBM's RMS today is 0.23, meaning today's price is 0.23 RMS above the Average CAGR line. Conclusion: IBM stock is not undervalued today, in fact is slightly above its Average CAGR value. It is not a stock to buy today by the BMW Method, but it certainly has been in the past.

Both statistically and from looking at the chart, a stock price going outside the +/-2 RMS lines is a fairly rare event, and is a good signal to start paying attention to the stock if you are interested in either buying or selling it. Clearly buying IBM in mid-1992 or 1993 would have been a fantastic investment decision. Selling IBM in mid 1999 was equally as good an investment decision. Both decisions would have been driven by the price going outside the +/-2 RMS band. The gain would have been about 1100% over 6 years, or a little over 50% return per year. That is an acceptable result by any definition!!

But that is in hindsight. Let's look at what you would have seen if you had generated this exact same BMW Method chart, but in October 1992 (see the chart just below). In this chart, the RMS lines are squeezed closer together because the extreme price drop of 1993 had not yet occurred; IBM's stock price had lower volatility up until then. Even still, the drop to \$20 in October 1992 would have been a -2 RMS crossing--the only one in the chart for 30 years! Had you bought at \$20 in 1992 and sold at \$120 in 1999, this still represents a 29% return per year, not at all bad. ##### Return Factor

There is one other important part of the analysis to trigger a potential buy or sell action: the Return Factor. The Return Factor is the ratio between today's Average CAGR value and today's actual price. In other words, if you were to buy IBM today at \$84.85, and it moved to be exactly on its Average CAGR line, the stock would "return" to you 0.91 of its value, in other words you would lose 9% of your investment. A value of 1.0 means you get back exactly what you invested. A value less than 1.0 means you get back less than what you invested (a loss); a value more than 1.0 means you get back more than what you invested (a gain). A positive RMS always corresponds to a Return Factor less than 1.0, since both say that the stock is currently above its Average CAGR line. For a buy point, you clearly want the price under the Average CAGR line, which means a negative RMS and a Return Factor greater than 1.0.

In back testing, there is a strong correlation between high Return Factors and good returns, even better correlation than with negative RMS values. For example, if a stock has a Return Factor of 3.0, today's price is only 1/3 the price on the Average CAGR line. In other words, if the stock price suddenly moved to be on the Average CAGR line, you would have a 3x return on your investment, or a 200% gain.

##### One Approach Using This Data

The best results, based on back testing, come from requiring both:

• A Return Factor of at least 3, meaning that simply "reverting to the mean" provides at least a 200% return (and most "reversions" overshoot the mean), and
• An RMS of -2 or below, meaning that the stock's price is in a rarely occurring low range

Using both these requirements, it turns out that there was exactly ONE DAY where IBM's stock closed below the Return Factor = 3 threshold: on Aug 16, 1993, IBM closed at \$10.25, for a Return Factor of exactly 3.00. That was the lowest close of IBM in decades. Even though this is certainly purely by chance, the general idea is well illustrated by it. During late summer in 1993 IBM was hovering around a Return Factor around 2.7 - 2.9, and that would have certainly gotten my attention if not significant investment dollars!

One final note on this chart. The legend at the top left provides the basic information for estimating CAGR and RMS quantities from the chart. For example, the bottom green key on the chart just above is "-2RMS (RF=1.67)". This says two things: the bottom green line is the -2 RMS line, and the Return Factor for any price on this line is 1.67. That provides sufficient information for estimating RMS and Return Factors at any point on the chart.

#### Linear Price Chart

The third element of the BMW Method Screen page is the Linear Price Chart. This chart presents precisely the same data as the Logarithmic Price Chart, but with a linear Y axis. The same "best fit" line from the logarithm of the monthly close prices is used as in the Logarithmic Price Chart. Only the Y axis is changed to linear for this chart. I find this chart difficult to read or to draw good conclusions from, but some people prefer it. It is easy to generate once the Logarithmic Price Chart is done.

#### Distribution Chart

The final element of the BMW Method Screen page is the Distribution Chart. Most people will not find this useful, but those who are statistically inclined may decide otherwise. In the example below, from the same IBM screen: To generate this chart, for every month's price the screen is run over, the difference between the logarithm of the actual data point (the monthly close price) and the calculated value on the Average CAGR line for the same month is calculated. This represents a "residual distribution" of the data against the "best fit" line. These residuals are binned into 100 bins ranging from -3 RMS (at the left edge of the chart) to +3 RMS (at the right edge). Green vertical lines are drawn at the same -2 RMS, -1 RMS, +1 RMS, and +2 RMS values as appear on the logarithmic and linear price charts, and at the mean residual value (which is always zero). Superimposed on this histogram in red is a normal ("bell") distribution curve of the same total area as the area under the histogram, and with standard deviation equal to the RMS value. A perfect normal distribution of residuals would exactly follow the red normal distribution.