Ergodicity
Why ensemble averages mislead individual investors, why the path matters as much as the destination, and why this is relevant for equity investors..
The expected value trap
Most of finance is built on expected values. We compute the probability-weighted average of possible outcomes and use that number to make decisions. This works beautifully in many contexts — but it breaks down in one critical situation: when outcomes compound multiplicatively over time.
The distinction matters because wealth is a multiplicative process. A 50% loss followed by a 50% gain does not return you to where you started — it leaves you at 75% of your original wealth. The order and path of returns matter, not just their average.
This is the domain of ergodicity: the question of whether the average outcome across many parallel experiences (the ensemble average) is the same as the average outcome for one individual over time (the time average). In most of finance, it is not.
A simple game that breaks intuition
Consider a coin flip: heads multiplies your wealth by 1.5, tails multiplies it by 0.6. The expected value per flip is (0.5 × 1.5) + (0.5 × 0.6) = 1.05 — a positive 5% expected return. Any rational expected-value maximiser would play this game all day.
But watch what happens when you actually play. The geometric growth rate — what matters for a single player over time — is √(1.5 × 0.6) ≈ 0.949, or −5.1% per round. The ensemble average grows because a small number of lucky players accumulate extreme wealth. But the median player, and most individual paths, converge toward zero.
Multiplicative Coin Flip Simulator
Each round, every agent flips a coin: heads multiplies wealth by 1.5 (+50%), tails by 0.6 (-40%). The expected value is positive, but most individuals go broke.
The ensemble average (dashed line) trends upward because a few lucky agents accumulate enormous wealth, dragging the mean higher. But the median agent -- and the vast majority of individuals -- lose money over time. This is the core of ergodicity economics: the time-average growth rate (geometric mean) governs individual outcomes, not the ensemble average. A game with positive expected value can still ruin most players when payoffs are multiplicative.
What this means for investors
Individual stock returns are non-ergodic. The path of one stock is not representative of the average experience of many stocks. A single company can go to zero — and once it does, no subsequent return can recover it. This is the “absorbing barrier” problem.
This has three practical implications for portfolio construction:
- •Diversification is not optional — it makes the portfolio more ergodic by diluting the impact of individual ruin paths
- •Position sizing must respect the asymmetry between gains and losses in multiplicative systems
- •Survival is the first objective — you cannot compound returns if you are wiped out along the way
Key Takeaway