The Actuarial ‘Matrix’ and the Next Generation

Embracing complexity and moving actuaries forward to a new a paradigm Bryon Robidoux

Actuaries work at the intersection of economics, finance and accounting. Scholars have updated these disciplines to explain phenomena such as:

  • Predicting the size of the next recession
  • Accounting for climate change
  • Understanding the fundamentals of policyholder behavior
  • Improving the relationship between risk management and accounting

In my opinion, it would benefit us to follow suit and incorporate updates into actuarial science. In this article, I will show how scholars injected principles from physics (specifically, energy and complexity) into these disciplines, providing an entirely new perspective and insights. I will keep the technical details to the bare minimum. To make this light and fun, I will frame it in terms of my favorite ’90s movie, The Matrix. As Trinity said to the main hero, Neo: “It is the question that drives us, Neo. You know the questions just as I did. What is the Matrix?”

The Matrix

As the leader, Morpheus, explains: “The Matrix is a computer-simulated dream world … These rules are just rules of a computer system. Some of them can be bent. Some of them can even be broken. Understand?” The Matrix we live in as actuaries comprises neoclassical economics, finance and accounting.

Economics in the Matrix

The neoclassical economic models try to find when the economy is in a stable equilibrium and only displaced by external shocks.1 Production is assumed to come directly from labor and capital, which misplaces energy’s role in production because it considers energy external to production.2 But this is not a gift from the heavens. The economy must mine it from the sun, wind, oil or other sources. It takes energy to run factories, feed people, and transport goods and services. Ultimately, energy makes the economy possible because life is impossible without it. Energy is vital to everything in the universe.

Finance in the Matrix

Some of finance’s theories assume that decisions occur in a closed system. A closed system is one in which decisions and outcomes are small and cognitively manageable with little or no required heuristics. The energy cost of acquiring and processing knowledge is negligible, much like playing Monopoly.

The strong form of efficient market hypothesis (EMH) dictates that market prices embed all information. Implied in EMH is efficient behavior.3 Efficient behavior is knowing exactly how to use the information when it becomes available, which is only viable in a closed system.

The financial markets are an open system. Open system outcomes:

  • Are infinite and path-dependent
  • Involve decisions that are far beyond any one person’s cognitive ability
  • Require heuristics to cope with the complexity of problems
  • Provide few to no trials to retry solutions4

Finance extrapolates the closed system dynamics to an open system.5 In my opinion, it oversimplifies complexity and therefore overestimates efficiency in the market. This extrapolation diminishes its ability to make predictions.

Accounting in the Matrix

Michael Mainelli and Ian Harris explain in their paper, “Balancing the Odds: Stochastic Accounting”: “The search for a single number is intertwined with debates of historical, current or fair value. Accountancy’s theoretical framework assumes a deterministic system that outputs a single number. In bookkeeping, the focus on an exact single number is important. However, higher-level interpretations are probabilistic, i.e., inputs into a higher-level figure, such as turnover, and include many sorts of estimates. Not everything can, or will, balance.”6

Let’s pretend that International Financial Reporting Standards (IFRS) and the Financial Accounting Standards Board (FASB) create the perfect deterministic rules to allow for comparing two insurance companies. If Companies A and B have the same income statement, balance sheet, owner’s equity and product mix according to IFRS and FASB rules as of Dec. 31, XXXX, is Company A equal to Company B? No, because the expected value is insufficient to indicate how the organization will evolve. The financial statement has no confidence intervals, parameter sensitivity or prorogation of error metrics for the projected assets and liabilities. Understanding noise sources and levels is just as important as understanding the estimated parameters. Neglecting this reality can seem like an oversight and portray more certainty than actually exists.

As Morpheus described: “Let me tell you why you’re here. You’re here because you know something you can’t explain. You felt it your entire life. There is something wrong with the world. You do not know what it is, but it is a splinter in your mind driving you mad. The wool has been pulled over your eyes to blind you from the truth. Take the red pill to stay in Wonderland and see how deep it goes.”

‘Welcome to the Real World’ —Morpheus

Steven Keen in The New Economics is our Morpheus. He stated there are specific rules that social sciences cannot violate:

  • Be consistent with physics and the laws of thermodynamics.
  • Be grounded in empirical realism instead of assuming simplifying “as if” behavior.
  • Acknowledge that the economy is a complex system, not an equilibrium system.
  • Be based on the techniques of system science and related nonequilibrium analytical approaches.7


Thermodynamics dictates energy conservation, meaning energy cannot be created or destroyed. Energy can only transition from one form to another. When using energy to produce practical work, you cannot get 100% efficiency. There will always be energy that goes to waste in terms of heat.

As the famous astronomer, physicist and mathematician Sir Arthur Eddington said, “If your theory is found to be against the second law of thermodynamics, I can give you no hope; there is nothing but to collapse in deepest humiliation.”8

Why are these rules important for economics, finance and accounting? Because thermodynamics governs the laws of energy, computation and information, which are at the core of these disciplines. Therefore, theories that tie to laws of thermodynamics have a much better chance of predicting actual world events.

Equilibrium and Complexity

Keen states that economics has tended to deal with feedback effects through equilibrium. Equilibrium occurs when a system’s key variables are not changing. Economics does not concern itself with the transient nature of transactions. It assumes that there is enough time for transactions to take their full effect. The critical misconception, according to Keen, is that cyclical processes will come to a state of rest. The feedback loops are assumed to dampen oscillations.9

On the other hand, complex systems are nonlinear systems. They tend to oscillate indefinitely. Far from attaining equilibrium, two competing forces will start to converge. But conditions will change at some point, compelling the forces to diverge again. Complex systems are not guaranteed to come to rest or unite. For most complex systems, equilibrium will tell you where the states will never be.10

Reliance of Economics on Equilibrium

A real market is full of energy and far from equilibrium.11 Only systems far from equilibrium and bursting with localized energy can create physical order within pockets of the universe, such as Earth.12 More complex information creates more complex objects and firms by applying more energy to out-of-equilibrium systems. When energy is removed, entropy will turn order into disorder, items will disintegrate and the physical ordering will be lost. Systems or economies in equilibrium are dead!13

Implications of Complexity

Through a beautiful data-driven story, House of Debt by Atif Mian and Amir Sufi explained that the bankruptcy of Lehman Brothers was not the external shock that caused the 2008 global financial crisis. Internal market dynamics, fueled by excessive amounts of private debt in the economy, caused the 2008 global financial crisis.

Banks were selling mortgages to the market to offload the risk, which fueled an easy credit boom in 2003. The credit boom led to speculation in housing prices in a feedback loop. Many people began using home equity loans to fuel their consumption. Eventually, the severely overheated housing market in 2006 and 2007 caused declining house prices. The house price decline resulted in negative home equity, forcing a severe pullback in household spending. As household spending declined, layoffs started, causing a downward spiral.

It is a consistent trend that the amount of private debt accumulation at the front end of the economic cycle dictates the extent of the household spending pullback on the backend. Some view the size and severity of the recession as predictable based on the level and speed of private debt accumulation.14 These dynamics are critical to actuaries who price and manage variable annuity guarantees (VAGs). Private debt-fueled economic growth caused the guarantees to ratchet upward. One theory is that the massive decrease in spending caused a significant recession, which triggered the VAGs to explode and the cost of the variable annuities to increase.

Implications of Thermodynamics

Keen explained: “Energy can … be incorporated into models of production by simply treating both labor and capital as a means by which energy is harnessed to perform useful work. In mathematical terms, the standard economic models of production should be replaced by the “number” of machines, times the energy consumed by each machine each year, times the efficiency with which that energy is turned into useful work, and the same should be done for labor.”15

This mathematical equation links economics, thermodynamics and ecology together because the inefficiency will go out the smokestack of the factory or power plant. The waste and its energy do not disappear into thin air. They stay in the atmosphere and can fuel forest fires in California and intensify hurricanes in the Gulf of Mexico.

In Keen’s view, perpetual economic growth is impossible due to thermodynamics because keeping the party going requires more and more energy. Eventually, we will consume all the energy available on Earth. Until then, pollution caused by waste energy will drive the surface temperature to rise significantly. As temperature increases, ecosystems will collapse after hitting their tipping points. These collapses are already causing events, such as COVID-19, to impact human mortality and morbidity rates negatively. Only without directly connecting economics to thermodynamics is it possible to postulate infinite growth on a planet with finite resources! It is amazing how a change of perspective will drive better solutions.

Neoclassical economics, in my opinion, view the planet as akin to a small person looking down from atop Mount Everest in amazement at an infinite world with its boundless supply of resources. With thermodynamics, the planet and its resources should be viewed as a tiny finite bubble floating in the vastness of space:

Image: Cassini: Earth and Saturn “The Day Earth Smiled”

Finance Can Oversimplify Complexity

Corporate finance studies how financial markets consume and produce information, among other things. This information allows decisions to increase a firm’s value, manage financial resources and fight for market funding sources. In this vein, I see a direct connection between energy and information through thermodynamics and the Landauer limit. The Landauer limit states that no matter how efficient any physical device is, whether a computer or your brain, it must expend energy to acquire information.16 Therefore, finance, in my view, also would benefit from considering the laws of energy, information, and computation—especially as they relate to the human brain. As stated in “Improving Strategic Risk Management,” all decisions come to human judgment after performing sophisticated model calculations.

In an open system, energy and time constraints are of utmost importance. Applying the Landauer limit here, the market frictions involve computation complexities and acquiring information. In my opinion, the realization of market friction connects finance to behavioral finance. In a closed system, the friction is zero.

People frequently do not have the capacity or time to process problems of real-world complexity. The computational limits and small energy use of our brains put the EMH into serious doubt. I believe a better name for EMH would be “the frictionless market hypothesis” because efficiency only exists by neglecting market friction. Our cognitive biases and simplifying heuristics due to our limitations are the sources of our inefficient behavior.

Implications to Modeling Policyholder Behavior

In my opinion, understanding an open system’s properties is vital for any actuary modeling policyholder behavior for products that derive value from the market. From my perspective, we do not benefit by spending time or computational effort trying to model policyholder behavior under a paradigm that assumes they will be able to maximize their benefits. Benefit maximization is improbable, especially given the complexity of the products and underlying market dynamics involved.

I think it is more essential to model the consequences of inefficient behaviors because they seem far more likely. In my opinion, actuaries would benefit from playing what-if games to decide which counterproductive behaviors could destroy profitability or cause policyholders to behave against their self-interest.17

For example, some variable annuity policyholders may move their investments from equity mutual funds to the risk-free general account when the market drops to protect their account value. This defensive behavior would create a double whammy for the insurance company and policyholder. It makes no sense because it removes volatility and reduces the value of the put option they purchased.

Accounting Embracing Complexity

César Hidalgo said, “In our universe, there is no past, there is no future, but every instant calculates a new present.”18 Hence, the future is not preordained or deterministic. As time progresses, our organizations become even more complex. Complexity begets more complexity. As actuaries, our job is to project our assets and liabilities for the future uncertain, complex world. Complexity is both random and deterministic at the same time.19 Therefore, accounting, with its deterministic systems, falls short at tracking information and telling the story of the current state and ongoing concerns of an organization.

With the third law of thermodynamics, the physicist Ludwig Boltzmann shoved physics into probability and statistics through statistical mechanics. His peers hated his discovery because it destroyed the certainty inherent in the beautiful Newtonian laws of nature. In the same way that we eventually embraced the third law of thermodynamics, some, including me, believe that accounting would benefit by embracing uncertainty, embedding statistics and applying “stochastic” accounting.

The paper “Balancing the Odds: Stochastic Accounting” states: “Accountants need to move to a new theoretical framework where inputs are probabilities and outputs are distributions. At a very low level, book-keeping skills remain, but the interpretation and presentation of financial information need to shift to distributions.”20

Implications to Risk Management

Let us look at this another way. Assume the law of one price (LOOP) is the bedrock of accounting, leading to its reliance on point estimation. I have already shown that the world is complex, not at equilibrium. According to Barrucci and Fontana’s theories, equilibrium in the economy and the law of one price are a package deal.21 Therefore, I propose that the law of one price must fail with it. Potts believes our complex world naturally leads to price distribution or, as he calls it, the smudge.22 To transition accounting to stochastic accounting requires reframing the question. The question is not whether point-estimated assets are greater than the point-estimated liabilities but whether the assets-less-liabilities confidence interval contains zero. Even though it is messier, to me, this feels more natural. Would you agree?

Stochastic accounting would seem to apply to any industry that makes projections based on uncertain outcomes. Understanding this is important with IFRS and FASB moving toward market-consistent accounting. Markets are highly volatile, risks are not diversifiable with more volume and projections are complex and uncertain. Risk and uncertainty drive decisions, which puts the organizations on different paths—even with the same expected values.

The paper “Balancing the Odds: Stochastic Accounting” further states: “Enterprisewide risk/reward management systems provide risk (or reward) information and relate it to financial impact using probabilistic techniques. These risk/reward managers are hampered by insufficient financial systems information. … Stochastic accounting would integrate well with risk/reward management systems because the risk/reward managers would have the data they need to show the organization how their efforts improve organizational value.”23


As Neo urges in the final scene of The Matrix: “I know … you’re afraid of change. I don’t know the future. I didn’t come here to tell you how it will end. I came to tell you how it is going to begin. I will show people what you don’t want them to see. I am going to show them a world without [your] rules and controls, without [your] borders or boundaries.”

Neoclassical economics and finance do not consider limitations dictated by thermodynamics and complexity science. Furthermore, accounting does not address the stochastic nature of our world. If we ignore complexity and thermodynamics:

  • We sometimes may focus on the wrong problems.
  • We might concentrate time and energy on unnecessary calculations.
  • Our predictions may hit the dartboard in very narrow scenarios.24

At every moment, the complex world calculates its present state. There will always be epistemic and aleatory uncertainty. According to Potts, noise, friction and inefficiency are mandatory for markets to exist because they incentivize people to transact.25 These incentives drive people to question each other’s perceived values, which creates a range of valid prices. The laws of thermodynamics embed noise, friction and inefficiency into the universe. Adding thermodynamics to economics, finance and accounting can help us get better and more reliable forecasts. It can help us deal with the ever-growing complexity of the world and address our most pressing issues.

Bryon Robidoux, FSA, CERA, is a consulting actuary at Milliman in Chesterfield, Missouri.

Statements of fact and opinions expressed herein are those of the individual authors and are not necessarily those of the Society of Actuaries or the respective authors’ employers.

Copyright © 2022 by the Society of Actuaries, Schaumburg, Illinois.