In a recently published article (The ergodicity problem in economics | Nature Physics), Ole Peters introduces a novel approach of looking at century-old problems of decision-making under uncertainty. Peters highlights a difference between time average and ensemble average. He argues that time ensemble average is a poor measure for non-ergodic processes of assessments of future payoff/investment returns. By carefully addressing the question of ergodicity, the article attempts to resolve many puzzles besetting the current economic formalism in a natural and empirically testable way.
The author proposes to maximize expected growth rate, where growth rate is ergodic variable. He estimates that playing with an entire fortune a game described in the equation (2) of the original paper not an optimal investment decision.
The author proposes to maximize expected growth rate, where growth rate is ergodic variable. He estimates that playing with an entire fortune a game described in the equation (2) of the original paper not an optimal investment decision.
This game/investment provides investors a negative rate of growth when carried out in a multiplicative format. Therefore, the author argues, one should not accept a bet described in the equation (2) on the entire wealth. That confirms market wisdom: playing a game with a significant probability to lose practically all one’s wealth is never a good idea for a rational investor.
Ole Peters’s approach provides readers an interesting framework to analyse different investment opportunities and benefits of diversification between different strategies.
While playing a game described in (2) with one’s whole fortune does not make good sense, it does stand to reason to allocate part of a portfolio to uncorrelated strategies with the payoff described in (2). This will provide an investment return with lesser volatility.
Hedge funds and high frequency traders operate in this fashion. It is arguably difficult to find a standalone scalable strategy with a high Sharpe ratio, but it is possible to construct a portfolio with relatively high Sharpe ratios from standalone strategies carrying low Sharpe ratios.
The table below put this into perspective. It shows risk-return characteristics for game (2) and S&P 500 total return index over the last 30 years.
Ole Peters’s approach provides readers an interesting framework to analyse different investment opportunities and benefits of diversification between different strategies.
While playing a game described in (2) with one’s whole fortune does not make good sense, it does stand to reason to allocate part of a portfolio to uncorrelated strategies with the payoff described in (2). This will provide an investment return with lesser volatility.
Hedge funds and high frequency traders operate in this fashion. It is arguably difficult to find a standalone scalable strategy with a high Sharpe ratio, but it is possible to construct a portfolio with relatively high Sharpe ratios from standalone strategies carrying low Sharpe ratios.
The table below put this into perspective. It shows risk-return characteristics for game (2) and S&P 500 total return index over the last 30 years.
The game (2) displays attractive risk/return characteristics compared to the stock market.
It would be difficult to argue that people should abstain from investing in risky assets. Those who have not invested a part of their wealth and remained long cash, have typically regretted these decisions later on.
This leads to the conclusion that it does make sense to play the game (2) with proper risk scaling and portfolio diversification.
Ergodicity theory confirms this logic: a strategy with 90%/10% allocation has a positive rate of growth and is therefore a preferable choice compared to the alternative of remaining 100% in cash. The theory tells us interestingly that an optimal investment is game (2) is 25% of the fortune.
All in all, ergodicity theory confirms the benefits of portfolio diversification which was first proposed by Harry Markowitz[i].
One glaring question is: Will diversification help on the downside? Today risky assets are driven by one main factor: excess liquidity provision by Central Banks. The question is valid but it’s outside of the scope of this discussion.
When it comes down to a choice of an optimal strategy under uncertainty we are less convinced that an optimization of growth rate, which is an ergodic process, indeed provides a best possible solution among all alternatives.
In certain cases, it gives the results well in line with empirical observations (see above), but in others, the results are less convincing. A multiplicative growth process, for example, gives results, which are not necessarily consistent with investors’ preferences.
Specifically, according to the ergodic theory, an investor should prefer playing a game making 1% on a daily basis but losing 99.9% twice a year instead of playing a game as described in equation (2). It also says that, instead of staying in cash, an investor should play a game where she gets daily return of 1% but once every three years she loses 99.9% of her wealth in one day.
It is not certain that individual investors will reveal preferences in line with the predictions of ergodic theory. There are many option underwriting strategies and negative gamma programs, which indeed deliver superior returns at the risk of a 90% loss of capital. However, strategies, which lead to these loss percentages, are not recommended for retail investors.
It might be a different story for institutional investors though. The concept of “skin in the game[ii]” for some of these investors is a foreign concept, and many might happily accept such a huge tail risk playing with investors’ money instead of their own.
The author makes another bold statement claiming that the whole idea of utility function maximization is an error in the foundation of economics. Peters writes that people are not all that different, and the only thing which differentiates people are circumstances i.e., wealth.
Wealthy people subsequently maximizing their growth via an additive process would appear to be brave, whereas less well-to-do people maximizing their wealth via a multiplicative process would appear to be scared.
There are no doubts that wealth is one of the key components in both portfolio allocation and investment analysis, but it is only one of several parameters as shown by different portfolio allocations for investors similar wealth and consumption buckets. Human nature means that people are prone behave spontaneously and irrationally based upon hopes, fears, and beliefs.
Downside risk can be assessed differently by different people within the same wealth bracket. An anecdotal, and somewhat sad example can be drawn from the people’s preferences during the process of the so-called “Russian privatization” of the 1990s. During this period certain people were willing to accept a game with rather drastic extremes:
a 90% probability of centupling their wealth and
a 10% of ending up on in the graveyard.
Moreover, they played multiple rounds of this game. Some of the winners are known to us now as the “Russian oligarchs”. The second group remains silent and unknown.
One can argue that probabilities, and perception of probabilities, in this game were different for different people. While certainly true, it does not change the fact that some people in the same wealth bracket were willing to play the game with infinite downside while others were not.
On balance, it seems that ergodicity economics is definitely an interesting concept, which explains certain aspects of investors’ behaviour. The results of these predictions do not necessarily fall in line with empirical observations in certain cases.
Yet, the same can be also said about many models- not just in Economics. A less dismal science, such as Physics, can provide a fine example. Space satellite engineering and construction use principles of Lagrangian formalisms and application of Newton’s laws. However, applying the same principles to construction of a nuclear reactor will lead to catastrophic failure.
Therefore, it should be acknowledged that models in ergodicity economics, like many other models, do carry certain limitations.
[i] Harry Markowitz. Portfolio Selection. The Journal of Finance, Vol. 7, No. 1. (Mar., 1952), pp. 77-91.
[ii] Nassim Nicholas Taleb Skin in the Game: Hidden Asymmetries in Daily Life (2018)
It would be difficult to argue that people should abstain from investing in risky assets. Those who have not invested a part of their wealth and remained long cash, have typically regretted these decisions later on.
This leads to the conclusion that it does make sense to play the game (2) with proper risk scaling and portfolio diversification.
Ergodicity theory confirms this logic: a strategy with 90%/10% allocation has a positive rate of growth and is therefore a preferable choice compared to the alternative of remaining 100% in cash. The theory tells us interestingly that an optimal investment is game (2) is 25% of the fortune.
All in all, ergodicity theory confirms the benefits of portfolio diversification which was first proposed by Harry Markowitz[i].
One glaring question is: Will diversification help on the downside? Today risky assets are driven by one main factor: excess liquidity provision by Central Banks. The question is valid but it’s outside of the scope of this discussion.
When it comes down to a choice of an optimal strategy under uncertainty we are less convinced that an optimization of growth rate, which is an ergodic process, indeed provides a best possible solution among all alternatives.
In certain cases, it gives the results well in line with empirical observations (see above), but in others, the results are less convincing. A multiplicative growth process, for example, gives results, which are not necessarily consistent with investors’ preferences.
Specifically, according to the ergodic theory, an investor should prefer playing a game making 1% on a daily basis but losing 99.9% twice a year instead of playing a game as described in equation (2). It also says that, instead of staying in cash, an investor should play a game where she gets daily return of 1% but once every three years she loses 99.9% of her wealth in one day.
It is not certain that individual investors will reveal preferences in line with the predictions of ergodic theory. There are many option underwriting strategies and negative gamma programs, which indeed deliver superior returns at the risk of a 90% loss of capital. However, strategies, which lead to these loss percentages, are not recommended for retail investors.
It might be a different story for institutional investors though. The concept of “skin in the game[ii]” for some of these investors is a foreign concept, and many might happily accept such a huge tail risk playing with investors’ money instead of their own.
The author makes another bold statement claiming that the whole idea of utility function maximization is an error in the foundation of economics. Peters writes that people are not all that different, and the only thing which differentiates people are circumstances i.e., wealth.
Wealthy people subsequently maximizing their growth via an additive process would appear to be brave, whereas less well-to-do people maximizing their wealth via a multiplicative process would appear to be scared.
There are no doubts that wealth is one of the key components in both portfolio allocation and investment analysis, but it is only one of several parameters as shown by different portfolio allocations for investors similar wealth and consumption buckets. Human nature means that people are prone behave spontaneously and irrationally based upon hopes, fears, and beliefs.
Downside risk can be assessed differently by different people within the same wealth bracket. An anecdotal, and somewhat sad example can be drawn from the people’s preferences during the process of the so-called “Russian privatization” of the 1990s. During this period certain people were willing to accept a game with rather drastic extremes:
a 90% probability of centupling their wealth and
a 10% of ending up on in the graveyard.
Moreover, they played multiple rounds of this game. Some of the winners are known to us now as the “Russian oligarchs”. The second group remains silent and unknown.
One can argue that probabilities, and perception of probabilities, in this game were different for different people. While certainly true, it does not change the fact that some people in the same wealth bracket were willing to play the game with infinite downside while others were not.
On balance, it seems that ergodicity economics is definitely an interesting concept, which explains certain aspects of investors’ behaviour. The results of these predictions do not necessarily fall in line with empirical observations in certain cases.
Yet, the same can be also said about many models- not just in Economics. A less dismal science, such as Physics, can provide a fine example. Space satellite engineering and construction use principles of Lagrangian formalisms and application of Newton’s laws. However, applying the same principles to construction of a nuclear reactor will lead to catastrophic failure.
Therefore, it should be acknowledged that models in ergodicity economics, like many other models, do carry certain limitations.
[i] Harry Markowitz. Portfolio Selection. The Journal of Finance, Vol. 7, No. 1. (Mar., 1952), pp. 77-91.
[ii] Nassim Nicholas Taleb Skin in the Game: Hidden Asymmetries in Daily Life (2018)
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