Financial Markets: Non-Ergodic Systems?
Blog Post by Best Fin Investment
Table of Contents:
- Introduction
- Financial Markets and Non-Ergodicity
- How Ergodicity Redefines Expectations
- Strategies for Non-Ergodic Systems
- Conclusion
- References
- Related Articles
- Books from the References Section
- Additional Reads on Ergodic Systems
Introduction:
Ergodicity embodies the concept that a dynamic or stochastic system will, over time, traverse all states of the system's space. This suggests that the system's average behavior can be inferred from the trajectory of a representative point.
Similarly a sizable, i.e. sufficiently long, collection of random samples from a process can effectively capture the average statistical properties of an entire system [4].
In non-ergodic systems, this exploration of all possible states may not occur, thus not allowing the system's statistical properties to be deduced from the collected samples. While ergodicity is commonly observed in processes with Independent and Identically Distributed (IID) observations, proving or disproving it in systems with complex dependence properties remains challenging [1]. For example, failure of ergodicity is not uncommon in systems exhibiting long-range dependence such as in some physical systems (earthquake occurrences, river flow rates, solar activity) and in financial markets.
Financial Markets and Non-Ergodicity:
Arguably, one may characterize the economy and financial markets as being in a so-called "perpetual transient state", indicating continuous changes without settling into stable, predictable, patterns [2].
This implies that the dynamic nature and statistical properties of the economy and financial markets may not remain constant over time, making it difficult to deduce long-term trends from a limited set of observations. This is in essence analogous to the concept of "non-ergodicity" of the economy and financial markets.
How Ergodicity Redefines Expectations:
In contexts where ergodicity does not apply, the notion of "expected returns" loses its significance. Instead of focusing on average correlation, attention shifts to correlation patterns during crises or black swan events. Consider the analogy of safety exits in a theater: their size matters not on average, but rather how easily they facilitate evacuation during a fire. Vladimir Lenin's famous quote underscores this perspective "there are decades in which nothing happens and weeks in which decades happen", highlighting that while most periods may be uneventful, it's the moments of upheaval that truly test the system's ergodicity.
Strategies for Non-Ergodic Systems:
Effective diversification emerges as a strategy to make non-ergodic situations more ergodic. In the realm of finance, the "Barbell" strategy, as popularized by renowned author and Professor N. Taleb [5], is an investment approach that involves allocating a large portion of a portfolio to very safe, low-risk assets, while simultaneously placing a smaller portion in high-risk, high-reward assets. The idea is to minimize exposure to moderate risk and instead focus on two extremes: safety and potential for significant gains.
This approach provides protection against downside risks while still allowing for the possibility of substantial returns from the riskier investments. The term "Barbell" comes from the visual analogy of having two heavy weights on either end of a bar, representing the allocation to safe and risky assets, with little or nothing in the middle.
This strategy fosters "Antifragility" by allowing the portfolio to withstand and potentially benefit from market volatility. The conservative portion of the portfolio ensures stability, while the aggressive portion can capture outsized gains when market opportunities arise, leading to an antifragile outcome where the portfolio as a whole can thrive under stress.
Note that antifragility is a concept introduced by Professor N. Taleb [5], referring to systems or entities that actually benefit and grow stronger from volatility, stress, or shocks, rather than merely resisting them (as with resilience) or being harmed by them (as with fragility). In finance, an antifragile strategy would not only survive market volatility and uncertainty but could actually thrive in such conditions.
Another often cited approach is the so-called "Kelly Criterion".
The Kelly Criterion is a formula used to determine the optimal size of a series of bets or investments to maximize the long-term growth of wealth. It was introduced by John L. Kelly Jr., an American scientist, in a 1956 paper [3] while he was working at Bell Labs. The criterion is particularly popular in both gambling and investment circles because it offers a mathematically sound strategy for managing risk and maximizing growth.
Note that the Kelly Criterion maximizes the long-term growth rate of wealth and does not maximize the traditional expected wealth [3].
The Kelly Criterion assumes that markets are non-ergodic, hence aiming to balance risk and reward to achieve optimal growth over time, considering the inherent uncertainty and randomness of market dynamics.
This said, the Kelly Criterion should be used with care as it assumes that the probabilities and payoffs used in the formula are known with certainty, which is often not the case in real-world financial market scenarios.
One of the main drawbacks of the Kelly Criterion is its potential to recommend aggressive bet sizes that could lead to significant losses, especially if the aforementioned probabilities are inaccurate.
By using the so-called conservative "Half Kelly" strategy, investors reduce the bet size to half of what the Kelly Criterion suggests, hence mitigating the risk of large losses.
One last approach would be to consider very long investment horizons, as suggested by Professor N. Taleb, under the premise that "time will eliminate the annoying effects of randomness" [6].
This strategy is based on the idea that, over extended periods of time, the volatility and unpredictability of markets may diminish, allowing fundamental trends to prevail.
However, it is important to recognize that not all randomness can be smoothed out by time, and some risks may remain or evolve, requiring careful management throughout the investment process.
Conclusion:
The concept of ergodicity provides a lens through which we can understand the behavior of dynamic and stochastic systems, including financial markets. While ergodicity is commonly observed in processes with Independent and Identically Distributed (IID) observations, proving or disproving it in systems with complex dependence properties remains a challenge. For financial markets, the empirical observations suggest a non-ergodic nature, further complicating efforts to deduce long-term trends or behaviors from historical data.
To navigate non-ergodic environments, strategies like effective diversification using a "Barbell" approach or the "Kelly Criterion" are often employed. However, the latter should be approached with caution, considering its sensitivity to input assumptions and potential for recommending aggressive bet sizes.
Ultimately, understanding the implications of ergodicity (or the lack thereof) in financial markets is essential for investors seeking to navigate uncertainty and volatility while striving for long-term growth and stability.
References:
[1] Cont R., "Empirical properties of asset returns: stylized facts and statistical issues", Quantitative Finance, vol. 1, issue 2, pp. 223-236, 2001.
[2] Gualdi S., Tarzia M., Zamponi F., Bouchaud J.P., "Tipping points in macroeconomic agent-based models", Journal of Economic Dynamics and Control, vol. 50, pp. 29-61, 2015.
[3] Kelly J.L., "A new interpretation of information rate", The Bell System Technical Journal, vol. 35, issue 4, pp. 917-926, 1956.
[4] Petersen K., "Ergodic Theory", Cambridge University Press, 1983.
[5] Taleb N.N., "Antifragile: Things That Gain from Disorder", Penguin Books UK, 2013.
[6] Taleb N.N., "Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets", Penguin Books, 2004.