4. Scatterplot Chart

The chart on the right is not the same as in Diagoran® 2006. The raw return and raw risk coordinates in Diagoran are appropriate to Markowitz mixing rules and March curves connecting pairs of stocks in a portfolio, as mentioned on the Discover page, but not addressed here.

We believe that these raw return and risk variables are the most efficient, robust, and powerful numerical measurements that can be calculated from price history, so they appear in the report. However, the joint distribution of these two measurements is hard to interpret when you look at many stocks. For this screener we modify the display coordinate system, the better to reveal market structure.

Coordinates

Independent analytic transformations adjust the location of stocks along the two coordinate axes. These transformations do not change the order of stocks along either axis. As suggested in discussion of the Basel II Capital Accord or the Solvency 2 insurers accord, these transformations generate a marginal Gaussian normal distribution along each axis. The result is like the Gaussian copula but without high-order discontinuities at the ends of fitted segments. For an example, please see the Distribution section, below, or the Blu-ray animation described on the marginal distributions page and its free preview, the Act I, Scene 1 MPEG-2 file on a high resolution monitor.

qReturn

Return as shown in the Report appears to have a Cauchy distribution, with no valid mean or standard deviation. Compared to a Gaussian normal distribution, Cauchy distributions appear leptokurtic. They have a sharp central peak and long fat tails. To better centralize such a distribution, we subtract the median raw return from all the raw returns.

One way to generate a Cauchy distribution is to apply the trigonometric tangent function to a uniform distribution. Therefore, the inverse (arctangent) operating on the raw return distribution can improve its legibility on the chart. The arctangent of two arguments helps neutralize kurtosis with a careful choice of constant for the second term. This function smoothly inverts raw return to the plotted qReturn coordinate. The resulting measured histogram is hard to distinguish from a Gaussian probability density function.

The qReturn variable generally shows small steady changes, typically self-consistent from month to month (that is, qReturn has momentum), including dividends and splits.

pRisk

Risk as shown in the Report appears to have a lognormal distribution rather than a Gaussian normal distribution. You might expect this, since raw risk is root mean square departure. Therefore, it is a one-sided variable, unlike return. Taking the logarithm of the raw risk provides the plotting coordinate of pRisk. This also has a measured histogram indistinguishable from a Gaussian probability density function.

The pRisk variable changes very slowly, providing an even more stable measurement than qReturn.

Lines

The background of the chart shows three vertical and three horizontal lines. These mark the mean and the mean plus and minus one standard deviation along each coordinate axis. Anything near the middle of the chart is unexceptional. Any stock lying more than one standard deviation from the mean may be atypical, and warrant further examination.

The units and scales for these lines are appropriate to the qReturn and pRisk coordinates as plotted, not the raw return and risk used in the report. These lines assist graphical interpretation of the chart, rather than supporting the technical detail in the report.

In addition a purple curve marks the greatest concentration of stocks. A blue straight line roughly separates regions of stocks with rising returns from stocks with falling returns. For an example and more details, please see the Distribution section, below.

Markings

On each scatterplot chart, the individual stocks appear as oriented cross hairs, diagonal for cyclicals, upright for non-cyclicals. A color code denotes the recent market close: red for low, green for high, else black for on-trend.

We define these behaviors in Report.

Flags

A rectangular flag may appear at the upper left corner, depending on your interactions with the screener. A green flag means that you have changed the settings, a yellow flag means that the screener could not find your typed ticker symbol (or you clicked on an empty cell.)

Distribution

Far more equities than you normally might expect have negligible returns. A few show extraordinary gains, a few show extraordinary losses. This corresponds to the Cauchy distribution in raw return mentioned earlier. Applying the two analytic coordinate transformations, stocks appear densely scattered along a crescent or parabolic shape, with centerline shown by a faint purple curve on the scatterplot chart. That is, pRisk appears to vary with the square of qReturn, suggesting a separable sRisk, independent of Return. The crescent shape appears to be quite stable, persisting over many months despite movement of its component stocks.

Version 3 illustrates histograms observed in a historical Distribution Chart as blocky line drawings at the right and bottom margins, for qReturn and pRisk respectively. Measured mean and standard deviation (at the three vertical and horizontal lines) generate a smooth theoretical Gaussian curve shown at the same scale as each histogram. As discussed in the Coordinates section above, the histograms are hard to distinguish from the theoretical curves.

The gray-scale background of this distribution chart shows a two-dimensional histogram grid, shaded by density of stocks, lighter tones for higher concentrations. The highest concentration follows the smooth purple curve along the crescent mentioned above. This crescent shape consists of two equilibrium branches. The slowly changing upper branch corresponds to the Capital Asset Pricing Model, where large positive return matches large risk. The slowly changing lower branch illustrates the CAPM paradox. Here investors reject rapidly varying or risky stocks, driving down their price and therefore matching large negative return with large risk.

Between the horns of the crescent appear more rapidly moving non-equilibrium stocks. They mostly go up or down and seldom move sideways. That is, in this region pRisk does not change as much as does qReturn. Version 3 also shows relative average movement in a matching Change Chart, illustrating qReturn and pRisk during the two months before the given date as a grid of small arrows. Much like a tide and current chart, these arrows show the direction and speed of average local movement. To better illustrate the dependence upon location, we subtract the global average motion shown at the upper left corner, down and to the right in the example. That is, the market as a whole then had shown decreased return and increased risk during the previous two months.

Local changes generally accelerate this trend at the right and bottom, but move against trend at the top and left, shown by the orange and blue arrows respectively. These two areas are roughly separated along a straight blue line. You might therefore look for additions to a risk-reducing portfolio from the sweet spot above the blue line but below the purple curve. To emphasize the importance of the outliers, the change chart also illustrates stock density with contour lines. Much as currents move the grains composing a sand bank, the changes move the stocks around, changing the density structure.

Notice that the arrows are small or missing near the center countours: on the average, stocks from the region with the greatest density do not much move away from negligible qReturn and moderate pRisk. In the tidal current analogy, water above a shallow sand bank moves more slowly than water with greater depth. Notice also that local changes seldom move stocks toward regions of higher concentration, that is, outlier stocks seem to move around the edges of or away from the crowd, at present. If this continues, these contours could provide a sensitive illustration of any resulting change in the density structure over a month or so. Current and historical change charts appear in the Gallery for comparison. The sweet spot mentioned above goes from wide open in May 2007 to almost closed in November 2007.

Chart Click Action

The marking for a stock changes when you type in its ticker symbol or click on the marking (if you are not using Firefox). In addition to the most recent measurement of the return and risk at the location of the marking, Version 2 shows the measured values from one and two months earlier, connected by two line segments. These segments are green if the return is rising or red if the return is falling, independent of the marking color for recent close.

Since as many as three thousand stock markings are shown when you use the default settings, it may be hard to distinguish individual stocks, especially in highly concentrated areas along the equilibrium crescent. The screener therefore uses a grid system, and cycles though all the stocks in the grid cell where it detects a click on the chart. If you do not get the stock you expect when you click on the chart, click again at the same location, and you should eventually get the report for that marking.

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