The Problem With Computing "Expected" Returns In Finance

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We do not invent models, but we are not blind to traps that are intrinsic in models, approaches and even principles. And quite often we see higher level problems in the lower level mathematics (representations). To discuss and explain those model and method risks, we have established the UnRisk Academy.

Some principle problems have to do with time. This is why I frequently read a series of posts culminating in thinking again about time in Mark Buchanan's Blog. Ole Peters' paper (he refers to) is a great read.

Nothing to add, but one aspect seems to be important from the risk management point of view, so I try to compile ...

What the hell is Ergodicity?

Let's make it simple (from Peters' paper): if I roll my dice 100 times and report how many sixes I have got: it might be close to 17. If 100 people roll their dices once, they will find a similar number of sixes, around 17. As the number of trials increases the fraction of sixes will converge to 1/6.

Whether we look at the time average or the ensemble average it converges to the same and this system is "ergodic".

Ergodicity does not work in finance 

The simple ensemble average to compute expected returns in finance are, in many cases, inappropriate to making decisions in the real world.

Time matters. In risky investments with great return promises a few bad outcomes might wipe you out of the game completely (in extreme, a bet where I get 100 times my wealth if I roll a six, or lose it and future derivative income completely in the other cases).

In Geometric Brownian Motion, it is possible for the ensemble average to grow exponentially, while any individual trajectory does not on a long time scale.

Systems with multiplicative growth are non-ergodic.

At the other hand - the problem with Kelly criterion 

The theory and practical application of the Kelly criterion is straightforward when the underlying  distributions are fairly accurately known. However in investment applications this is usually not the case.

From the great summary paper (Ed Thorp ...)

In VaR of the jungle I have cited Aaron Brown:

.. the fence is built in a relatively save place, and remaining dangers inside are cleared out.

So (careful) Kelly might be applied in this regime.

In principle, risk management relying on risk preferences and not considering the effects of time is dangerous itself. The cost of risk and wealth growth are different things.