The Hidden Multiplicative Nature Of Additive Processes

Some have argued mainly in social media that multiplicative processes have higher risks than additive. This is a misunderstanding: risk does not depend on the model of a phenomenon but on its impact on an observable quantity. In fact, some additive processes may have higher impact than multiplicative. After all, the relative impact depends on how things are multiplied versus how they are added. It can be shown that every additive process has an implicit multiplicative nature.

In another article I wrote why most deaths are multiplicative. The reason for that is as follows: the death of a person in reproductive age reduces future population growth in a multiplicative fashion because descendant generation is terminated. Since the average family in USA for example has about 1.9 children, then the effect on population reduction due to a death of a person in reproductive age is shadow-multiplicative according to the model in the referenced article above.

Multiplicative processes

Suppose an observable quantity has size X(0). If at the next step j the observable size X(1) is equal to F(1)X(0), then this is a multiplicative process. In general, a multiplicative process is defined as follows:

X(j) = F(j)X(j-1)

where F(j) is a random variable. This is a loose definition that lacks rigor but suffices for the purposes of this article.

Multiplicative processes are used in finance, biology, geology, computer science, ecology (population growth) and in many other fields.

Usually multiplicative processes yield a lognormal distribution but under certain conditions the distribution is a power law. These technical issues go beyond the scope of this brief article, which is to demonstrate that additive processes have a hidden multiplicative nature.

If X(0), X(1)-X(0), X(2)-X(1),…,X(j)-X(j-1) are independent random variables then the process is called additive. This is an non-formal and far from rigorous definition but provides the general idea.

The conundrum

Many think for some reason that all multiplicative processes give rise to power laws with infinite mean and can be more harmful than additive processes especially when the observable is death counts, mortality rates, etc. This is simply not so.

Let us look at an example below of world road traffic deaths with data from CDC.

Each year, 1.35 million people are killed on roadways around the world. Source: CDC

The graph above shows the linear increase in the total number of deaths (approximate) since 2006 (blue line.) The red line shows the implied multiplicative process with a constant rate of 27.56% annual growth.

How do we calculate this rate? This is easy; we do that as in every other multiplicative process, such as stock price growth, as follows:

In this case CAGR is 0.2756 or 27.56%. This means that the implied rate of growth of the additive process is 27.56%. This is equivalent to a multiplicative process with the same CAGR.

You may notice that the blue and red curves above match only at the end points since this is an implied multiplicative process with constant growth rate. In fact, the two curves can be made to match exactly if the growth rate at each step in taken into account.

So what is the difference between an additive and a multiplicative process if the former can be implicitly represented by a similar model as the latter?

Multiplicative processes scare some people because they have been associated with virus infections growth and market bubbles, just to mention two. During a pandemic for example, what happens to one individual can affect many other individuals in a multiplicative fashion even at far away distances due to travel. In road accidents, what happens to one individual will not affect other individuals except possibly in the immediate vicinity and the effect is limited.

However, there are additive processes that have the potential of inflicting much greater damage than multiplicative: wars, meteorite strikes and even climate changes.

The issue is not so much whether a process is multiplicative or additive but the multiplication vs. addition factors. One meteorite strike has the potential of exterminating the whole Earth population although the effect is additive. At the same time, in the history of the planet there have been many pandemics that are described by multiplicative processes and the human race has survived through the development of herd immunity. This of course does not imply that herd immunity should be taken for granted but this is what we know up to this point.