Modus Tollens

Mark Spitznagel recently wrote a great book called Safe Haven. It got me thinking about logic and finance in a way I hadn’t before and really helps to cut through the noise in finance. It’s a great book that combines ideas of probability and non-ergodicity with a passion for investing.

Spitz contends that if the goal of the investor is to increase wealth then they should be focused not on arithmetic returns but cost adjusted growth rate (CAGR). Doing so requires not only “setting sail” and going after returns but also mitigating risk through cost-effective insurance.

How does insurance help? If you can find a cost-effective way to protect wealth you are better off in the long run. Losses have an asymmetric effect on portfolio growth owing to the concavity of geometric averaging (think of the graph of a logarithm). As a simple example, a 50% loss requires a 100% gain to recover from.

The book has a central hypothesis:

If a strategy cost-effectively mitigates a portfolio’s risk, then adding that strategy raises the portfolio’s CAGR over time.

Spitz applies Popperian falsification to rigorously test the cost-effectiveness of risk mitigation. The essence of Popperian falsification comes down to a syllogism called modus tollens or “denying the consequent.”

The structure of modus tollens looks like “If hypothesis, then outcome. Not outcome. Therefore, not hypothesis.” Symbolically, it can be represented as the following:

The logic here can be a bit tricky, particularly when you consider the cases where the hypothesis is false (vacuous truth). Here is the truth table for the proof of modus tollens as a refresher:

In short, what modus ponens tells us is that if we have have an implication and we have a negation of the consequent then we can conclude the negation of the antecedent. So, we can conclude that our hypothesis is false.

What we cannot do, however, is prove that a hypothesis is true (though the logical argument may be valid). This is the basis of the scientific method: disproving hypotheses to get closer to the truth.

If we deny the outcome by finding that adding the risk mitigation strategy to a portfolio does not raise the CAGR of the portfolio, then we must also deny that the strategy cost-effectively mitigates the risk of the portfolio.

Spitz makes a good point about starting with hypothesis a priori, rather than building an ad hoc hypothesis that fits experimental data.

“We need logical explanations that independent and formed prior to those observations.”

This statement hits at the idea of formulating causal relationships for hypotheses — something Yudea Pearl talks about in his Book of Why. Nutrition is an example of a field with lots of data where you can apply an ad hoc hypothesis to and have no understanding of the causal relationships that may (or may not) exist underneath, and have a hard time falsifying with existing data.

Spitz also brings up logical fallacies to be avoided such as “affirming the consequent” and “denying the antecedent”. Again, I refer the reader to Sean McClure’s excellent podcast on logical fallacies for further investigation.

I’m currently doing a deep dive on cost-effective risk mitigation for portfolio management. In particular I’m looking at the feasibility of using convex products to protect either a single stock position, or a basket of stocks like the SPX.

Thinking About Probability #4


Continuing the detour here before I return to Papoulis & Pillai.

I recently came across a great explainer video on ergodicity by Alex Adamou on

Ergodicity is an incredibly important concept to understand when dealing with multiple trajectories of a stochastic process — which is simply a random process through time. Common examples of stochastic processes include games of chance (i.e. rolling a dice) and random walks (i.e. path a molecule takes as it travels in a liquid or gas).

Taking some examples from Alex, below is a graph representing a stochastic process X(t) with a set of four trajectories {xn(t)}.

To determine the finite time average we fix the trajectory and average horizontally over the time range

Finite time average

When time diverges the time average is given by

Time average

In the finite ensemble average we fix time and average over the trajectories

Finite ensemble average

When the number of systems diverge the ensemble average is given by

Ensemble average (imagine there are many more trajectories)

A stochastic process is ergodic if the time average of a single trajectory is equal to the ensemble average.

Shown graphically, here is what we are examining:

Ergodic perspective

And that’s really all there is to ergodicity! Ergodicity is a special case where the time average is equal to the time average.

Said differently, the time average and the ensemble average are only interchangeable when the process under consideration is ergodic.

This is a key point because the ensemble average is often easier to determine than the time average. Alex points to economics as an example where people often utilize expectation values to represent temporal phenomena without knowing if the system is ergodic.

In many real world scenarios, things exhibit strong path dependence meaning that they are very likely non-ergodic. The classic example is Polya’s urn where a ball is selected from an urn with balls of two different colors. The ball is then replaced and another ball of the same color that was selected is added to the urn. The first ball selected has a large impact on the subsequent makeup of the urn over multiple trials and breaks ergodicity.

I highly recommend watching Alex’ video for additional examples. I think he did a really good job of distilling the essence of ergodicity for the layman.

Now I am thinking about how you can tell if a process is stochastic or the result of deterministic chaos…