In 1711 Abraham De Moivre came up with the mathematical definition of risk as:

*The Risk of losing any sum is the reverse of Expectation; and the true measure of it is, the product of the Sum adventured multiplied by the Probability of the Loss.*

Abraham de Moivre, De Mensura Sortis, 1711 in the Ph. Trans. of the Royal Society

*"Applying decision theory to the qualitative and subjective estimation of risk"*

For new systems we generally do not have statistical data .... and high consequence events are (usually) quite rare leaving us with a paucity of information.

So we end up arguing our .... case using low base rate data, and in the final analysis we usually fall back on some form of subjective (and qualitative) risk assessment.

The risk matrix was developed to guide this type of risk assessments, it’s actually based on decision theory, De’Moivres definition of risk and the principles of the iso-risk contour

**Iso-risk contour**

Well, I've given you De’Moivres definition of risk in the opening to this posting. What then is an iso-risk contour?

"iso" from the Greek, meaning "equal"

"contour", typically referring to a plotted line (or curve) meaning all points on the line are equal. A common usage is 'contour map' which is a mapping of equal elevation lines.

So, iso-risk contours are lines on a risk mapping where all the risk values are the same.

**Risk matrix**

Enter: decision theorists. These guys provide the methodology for constructing the familiar risk matrix (or grid) that is dimensioned impact by probability. The decision guys recognized that unless you "zone" or compartmentalize or stratify the impacts and probabilities it's very hard to draw any conclusions or obtain guidance for management. Thus, rather than lists or other means, we have the familiar grid.

Each grid value, like High-Low, can be a point on a curve (curve is a generalization of line that has the connotation of straight line), but Low-High is also a point on the same curve. Notice we're sticking with qualitative values for now.

However, we can assign arbitrary numeric scales so long as we define the scale. The absence of definition is the achilles heel of most risk matrix presentations that purport to be quantitative. And, these are scales simply for presentation, so they are relative not absolute.So for example, we can define High as being 100 times more of an impact than Low without the hazard of an uncalibrated guess as to what the absolute impact is.

If you then plot the risk grid using Log Log scaling, the iso-contours will be straight lines. How convenient! Of course, it's been a while since I've had log log paper in my desk. Thus, the common depiction is linear scales and curved iso-lines.

Using the lines, you can make management decisions to ignore risks on one side of the line and address risks on the other.

**Common problems**

There are two common problems with risk matrix practices:

- What do you do with the so-called "bury the needle" low probability events (I didn't use 'black swan' here) that don' fit on a reasonably sized matix (who needs 10K to 1 odds on their matix?)
- How do you calibrate the thing if you wanted to?

For "2", pick a grid point, perhaps a Medium-Medium point, that is amenable to benchmarking. A credible benchmark will then "anchor" the grid. Being cautious of "anchor bias" (See: Kahneman and Tversky), one then places other risk events in context with the anchor.

If you've read this far, it's time to go.

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