Tag: Normal distribution

The rise of the normal distribution

“We were all Gaussians now”

This post focuses on a joint paper written in 2012 by Andrew Haldane and Benjamin Nelson titled “Tails of the unexpected”. The topic is the normal distribution which is obviously a bit technical but the paper is still readable even if you are not deeply versed in statistics and financial modelling. The condensed quote below captures the central idea I took away from the paper.

“For almost a century, the world of economics and finance has been dominated by randomness … But as Nassim Taleb reminded us, it is possible to be Fooled by Randomness (Taleb (2001)). For Taleb, the origin of this mistake was the ubiquity in economics and finance of a particular way of describing the distribution of possible real world outcomes. For non-nerds, this distribution is often called the bell-curve. For nerds, it is the normal distribution. For nerds who like to show-off, the distribution is Gaussian.”

The idea that the normal distribution should be used with care, and sometimes not at all, when seeking to analyse economic and financial systems is not news. The paper’s discussion of why this is so is useful if you have not considered the issues before but probably does not offer much new insight if you have.

What I found most interesting was the back story behind the development of the normal distribution. In particular, the factors that Haldane and Nelson believe help explain why it came to be so widely used and misused. Reading the history reminds us of what a cool idea it must have been when it was first discovered and developed.

“By simply taking repeat samplings, the workings of an uncertain and mysterious world could seemingly be uncovered”.
“To scientists seeking to explain the world, the attraction of the normal curve was obvious. It provided a statistical map of a physical world which otherwise appeared un-navigable. It suggested regularities in random real-world data. Moreover, these patterns could be fully described by two simple metrics – mean and variance. A statistical window on the world had been opened.”
Haldane and Nelson highlight a semantic shift in the 1870’s where the term “normal” began to be independently applied to this statistical distribution. They argue that adopting this label helped embed the idea that the “normal distribution” was the “usual” outcome that one should expect to observe. 
“In the 18th century, normality had been formalised. In the 19th century, it was socialised.”
“Up until the late 19th century, no statistical tests of normality had been developed.
Having become an article of faith, it was deemed inappropriate to question the faith.
As Hacking put it, “thanks to superstition, laziness, equivocation, befuddlement with tables of numbers, dreams of social control, and propaganda from utilitarians, the law of large numbers became a synthetic a priori truth. We were all Gaussians now.”

Notwithstanding its widespread use today, in Haldane and Nelson’s account, economics and finance were not early adopters of the statistical approach to analysis but eventually become enthusiastic converts. The influence of physics on the analytical approaches employed in economics is widely recognised and Haldane cites the rise of probability based quantum physics over old school deterministic Newtonian physics as one of the factors that prompted economists to embrace probability and the normal distribution as a key tool.

” … in the early part of the 20th century, physics was in the throes of its own intellectual revolution. The emergence of quantum physics suggested that even simple systems had an irreducible random element. In physical systems, Classical determinism was steadily replaced by statistical laws. The natural world was suddenly ruled by randomness.”
“Economics followed in these footsteps, shifting from models of Classical determinism to statistical laws.”
“Whether by accident or design, finance theorists and practitioners had by the end of the 20th century evolved into fully paid-up members of the Gaussian sect.”

Assessing the Evidence

Having outlined the story behind its development and increasingly widespread use, Haldane and Nelson then turn to the weight of evidence suggesting that normality is not a good statistical description of real-world behaviour. In its place, natural and social scientists have often unearthed behaviour consistent with an alternative distribution, the so-called power law distribution.
“In consequence, Laplace’s central limit theorem may not apply to power law-distributed variables. There can be no “regression to the mean” if the mean is ill-defined and the variance unbounded. Indeed, means and variances may then tell us rather little about the statistical future. As a window on the world, they are broken”
This section of the paper probably does not introduce anything new to people who have spent any time looking at financial models. It does however beg some interesting questions. For example, to what extent bank loan losses are better described by a power law and, if so, what does this mean for the measures of expected loss that are employed in banking and prudential capital requirements; i.e. how should banks and regulators respond if “…the means and variances … tell us rather little about the statistical future”? This is particularly relevant as banks transition to Expected Loss accounting for loan losses.
We can of course estimate the mean loss under the benign part of the credit cycle but it is much harder to estimate a “through the cycle” average (or “expected” loss) because the frequency, duration and severity of the cycle downturn is hard to pin down with any precision. We can use historical evidence to get a sense of the problem; we can for example talk about moderate downturns say every 7-10 years with more severe recessions every 25-30 years and a 75 year cycle for financial crises. However the data is obviously sparse so it does not allow the kind of precision that is part and parcel of normally distributed events.

Explaining Fat Tails

The paper identifies the following drivers behind non-normal outcomes:
  • Non- Linear dynamics
  • Self organised criticality
  • Preferential attachment
  • Highly optimised tolerance
The account of why systems do not conform to the normal distribution does not offer much new but I found reading it useful for reflecting on the practical implications. One of the items they called out is competition which is typically assumed by economists to be a wholly benign force. This is generally true but Haldane and Nelson note the capacity for competition to contribute to self-organised criticality.
Competition in finance and banking can of course lead to beneficial innovation and efficiency gains but it can also contribute to progressively increased risk taking (e.g. more lax lending standards, lower margins for tail risk) thereby setting the system up to be prone to a self organised critical state. Risk based capital requirements can also contribute to self organised criticality to the extent they facilitate increased leverage and create incentives to take on tail risk.

Where Next?

Haldane and Nelson add their voice to the idea that Knight’s distinction between risk and uncertainty is a good foundation for developing better ways of dealing with a world that does not conform to the normal distribution and note the distinguishied company that have also chosen to emphasise the importance of uncertainty and the limitations of risk.
“Many of the biggest intellectual figures in 20th century economics took this distinction seriously. Indeed, they placed uncertainty centre-stage in their policy prescriptions. Keynes in the 1930s, Hayek in the 1950s and Friedman in the 1960s all emphasised the role of uncertainty, as distinct from risk, when it came to understanding economic systems. Hayek criticised economics in general, and economic policymakers in particular, for labouring under a “pretence of knowledge.”
Assuming that the uncertainty paradigm was embraced, Haldane and Nelson consider what the practical implications would be. They have a number of proposals but I will focus on these
  • agent based modelling
  • simple rather than complex
  • don’t aim to smooth out all volatility

Agent based modelling

Haldane and Nelson note that …

In response to the crisis, there has been a groundswell of recent interest in modelling economic and financial systems as complex, adaptive networks. For many years, work on agent-based modelling and complex systems has been a niche part of the economics and finance profession. The crisis has given these models a new lease of life in helping explain the discontinuities evident over recent years (for example, Kirman (2011), Haldane and May (2011))
In these frameworks, many of the core features of existing models need to be abandoned.
  • The “representative agents” conforming to simple economic laws are replaced by more complex interactions among a larger range of agents
  • The single, stationary equilibrium gives way to Lorenz-like multiple, non-stationary equilibria.
  • Linear deterministic models are usurped by non linear tipping points and phase shifts
Haldane and Nelson note that these types of systems are already being employed by physicists, sociologists, ecologists and the like. Since the paper was written (2012) we have seen some evidence that economists are experimenting with “agent based modelling”. A paper by Richard Bookstabber offers a useful outline of his efforts to apply these models and he has also written a book (“The End of Theory”) promoting this path. There is also a Bank of England paper on ABM worth looking at.
I think there is a lot of value in agent based modelling but a few things impede their wider use. One is that the models don’t offer the kinds of precision that make the DSGE and VaR models so attractive. The other is that they require a large investment of time to build and most practitioners are fully committed just keeping the existing models going. Finding the budget to pioneer an alternative path is not easy. These are not great arguments in defence of the status quo but they do reflect certain realities of the world in which people work.

Simple can be more robust than complex

Haldane and Nelson also advocate simplicity in lieu of complexity as a general rule of thumb for dealing with an uncertain world.
The reason less can be more is that complex rules are less robust to mistakes in specification. They are inherently fragile. Harry Markowitz’s mean-variance optimal portfolio model has informed millions of investment decisions over the past 50 years – but not, interestingly, his own. In retirement, Markowitz instead used a much simpler equally-weighted asset approach. This, Markowitz believed, was a more robust way of navigating the fat-tailed uncertainties of investment returns (Benartzi and Thaler (2001)).
I am not a big fan of the Leverage Ratio they cite it as one example of regulators beginning to adopt simpler approaches but the broader principle that simple is more robust than complex does ring true.
The mainstay of regulation for the past 30 years has been more complex estimates of banks’ capital ratios. These are prone to problems of highly-optimised tolerance. In part reflecting that, regulators will in future require banks to abide by a far simpler backstop measure of the leverage ratio. Like Markowitz’s retirement portfolio, this equally-weights the assets in a bank’s portfolio. Like that portfolio, it too will hopefully be more robust to fat-tailed uncertainties.
Structural separation is another simple approach to the problem of making the system more resilient
A second type of simple, yet robust, regulatory rule is to impose structural safeguards on worst-case outcomes. Technically, this goes by the name of a “minimax” strategy (Hansen and Sargent (2011)). The firebreaks introduced into some physical systems can be thought to be playing just this role. They provide a fail-safe against the risk of critical states emerging in complex systems, either in a self-organised manner or because of man-made intervention. These firebreak-type approaches are beginning to find their way into the language and practice of regulation.
And a reminder about the dangers of over engineering
Finally, in an uncertain world, fine-tuned policy responses can sometimes come at a potentially considerable cost. Complex intervention rules may simply add to existing uncertainties in the system. This is in many ways an old Hayekian lesson about the pretence of knowledge, combined with an old Friedman lesson about the avoidance of policy harm. It has relevance to the (complex, fine-tuned) regulatory environment which has emerged over the past few years.
While we can debate the precise way to achieve simplicity, the basic idea does in my view have a lot of potential to improve the management of risk in general and bank capital in particular. Complex intervention rules may simply add to existing uncertainties in the system and the current formulation of how the Capital Conservation Ratio interacts with the Capital Conservation Buffer is a case in point. These two elements of the capital adequacy framework define what percentage of a bank’s earnings must be retained if the capital adequacy ratio is under stress.
In theory the calculation should be simple and intuitive but anyone who has had to model how these rules work under a stress scenario will know how complex and unintuitive the calculation actually is. The reasons why this is so are probably a bit too much detail for today but I will try to pick this topic up in a future post.

Don’t aim to eliminate volatility

Systems which are adapted to volatility will tend to be stronger than systems that are sheltered from it, or in the words of Haldane and Nelson …

“And the argument can be taken one step further. Attempts to fine-tune risk control may add to the probability of fat-tailed catastrophes. Constraining small bumps in the road may make a system, in particular a social system, more prone to systemic collapse. Why? Because if instead of being released in small bursts pressures are constrained and accumulate beneath the surface, they risk an eventual volcanic eruption.”

I am a big fan of this idea. Nassim Taleb makes a similar argument in his book “Antifragile” as does Greg Ip in “Foolproof”. It also reflects Nietzsche’s somewhat more poetic dictum “that which does not kills us makes us stronger”.

In conclusion

If you have read this far then thank you. I hope you found it useful and interesting. If you want to delve deeper then you can find my more detailed summary and comments on the paper here. If you think I have any of the above wrong then please let me know.