number 1: ensemble vs. time

general introduction to the series

One of the pleasures of setting up The Idler is that of enumerating and organizing the various principles that guide it. This exercise reveals to the writer (me) how and why the ideas connect with each other (sometimes in a sequence, other times in self-reinforcing webs and loops) and, wonderfully, how and why they were never confined to being an investment philosophy, but part of everyday life.

the expected return riddle

When discussing this idler experiment over drinks with friends and traveling from foundational principles to mathematics to tangible portfolio construction, the question that invariably comes up is "what is the expected return?"

This is a question I struggle with because it has several layers that must be worked through methodically, and the conclusion may well be that the question is in itself invalid. There are two reasons: the expected return is the wrong object (an average no investor experiences) and, in the real world, an unmeasurable quantity.

So let's start with the first element, the object: ensemble vs. time, or ergodic and non-ergodic systems.

ensemble vs. time / arithmetic vs. geometric averages

Expected return is typically the language of the group (the ensemble). It is an arithmetic average taken across all possible worlds, weighted by frequency, as though an investor could stand outside space-time and collect the mean of every branch at once. This is not possible in the real world. In this world, a portfolio is funded with one person's capital, run by one person, exposed to one person's decisions, and it lives a single path through time.

The relevant average for an investor, therefore, is not taken across worlds (or other people) at one instant. It is taken across one world, instant after instant, multiplicatively, with no resets (in this case, a geometric average).

These two averages are not approximations of each other.

the ole peters gamble

A great example to illustrate this difference is the gamble proposed by Ole Peters in his paper on ergodicity^[1]. You start with a dollar. A fair coin is flipped. Heads, your wealth grows by 50%. Tails, it shrinks by 40%.

Compute the expected outcome of one flip. With probability 50% you hold $1.50, and with probability 50% you hold $0.60. The average is $1.05. The gamble has a positive expected return of five percent per flip. This is the logic of the ensemble and, by that logic, you should take it as often as offered and, even better, lever up if you can^[2].

Except that you will go broke with probability 100%.

Now stop averaging across imagined coins and follow one player through time. Heads and tails arrive in roughly equal number, and what matters is not the sum of the outcomes but their product. One head and one tail, in either order, multiply your wealth by 1.5 × 0.6 = 0.9. Every matched pair of flips destroys ten percent of your capital.

Taking the root to express this per flip: √(1.5 × 0.6) ≈ 0.95.

That is roughly negative five percent per flip. The same gamble that the ensemble calls attractive is, for the individual player, certain ruin.

ergodic and non-ergodic systems

This is the keystone of this first element: for multiplicative processes, the average of the paths is not the path of the average. The arithmetic mean describes a population of gamblers at a moment. The geometric mean describes one gambler over time. A process for which the two coincide is called ergodic. Wealth, compounded, is not ergodic. This single fact reorganizes everything that follows from it.

It also exposes a sleight of hand the investment industry performs without quite noticing. A fund's track record, an index's long-run return, the historical equity premium: these are quoted as though they were the experience of an investor, but they are often computed and presented far closer to the ensemble average.

The number that gets repeated is the one that survived to be repeated, averaged in a way no single account holder lived. When a fact sheet reports a return it is answering an ensemble question. When you ask what a fund will do for you, over your life, you are asking a time question.

The Idler is built to answer the time question. It does not (and cannot) maximize expected return; it seeks to ensure survival first and then maximize the time-average growth rate of its portfolio. This is achieved by losing less rather than by earning more.

The inevitable consequence is: in any single period, and most reliably in the short run, The Idler will look like a fund that underperforms a conventional equity book. This is the price of optimizing the average you will actually experience instead of the one that reads well on a page.

The articles that follow will continue this attempt to disentangle the riddle of the expected return. Why, paradoxically, the underperformance is structural, why the long-run case rests on survival rather than selection, why even a well-posed forward estimate is unknowable in practice, and the specific world in which the whole enterprise is wrong.