Updated: Nov 22
This article explores convexity. Convexity is also known as non-linear growth. Non-linear growth is what makes small investments in health, retirement, or other necessities pay off big when we need them. The dark side of non-linear growth is that it works for risks as well. NOT making those small investments or taking many small, but imprudent risks may lead to a big downside. While convexity by itself is agnostic to the goodness or badness of the outcome, the outcome of convexity can be decidedly good or bad. Understanding how to make convexity work for you is like finding a great tool you did not know you had in your toolbox.
Several practical convexity examples are provided... but there are many more. Most importantly, understanding convexity is only the first step to making productive use of it in your life. As we will explore, science teaches that people do not naturally perceive the power of non-linear growth. By attending to a consistent, repeatable decision process, we can identify and leverage non-linear growth opportunities in our lives.
About the author: Jeff Hulett is a career banker, data scientist, behavioral economist, and choice architect. Jeff has held banking and consulting leadership roles at Wells Fargo, Citibank, KPMG, and IBM. Today, Jeff is an executive with the Definitive Companies. He teaches personal finance at James Madison University and provides personal finance seminars. Check out his new book -- Making Choices, Making Money: Your Guide to Making Confident Financial Decisions -- at jeffhulett.com.
Table of Contents
The big problem - we do not attend to non-linear growth
What is non-linear growth or convexity?
Health and Longevity
Embedded losses and risk management - technical accounting
The global environment
Conclusion, Resources, and Notes
2. The big problem - we do not attend to non-linear growth
Let's start with the big problem - that is - the reason why you did not even know the convexity tool was in your cognitive toolbox. By natural design, the human brain is architected so non-linear growth easily escapes our attention. Instead, our genome intuitively interprets and forecasts data via straight lines. This is a little weird since nature is decidedly non-linear. When was the last time you saw a perfectly straight tree limb or rose petal?! Be that as it may, it doesn’t mean our non-linear attention can’t be trained. But even with training, under stress, our attention naturally returns to linearity.
This is a broad indictment of humanity, but we come by it naturally. Via natural selection, in the fight-or-flight mode of ancient times, survival was improved by our attention to the straight line. Whether fighting the rival tribe member or fleeing from the lion, the speed and directness of the straight line improved our chances of survival. Being the first to flee, the first to attack, and doing so most directly helped us succeed. Thus, our brains naturally evolved to attend to the straight line and this maximized our chance of survival. In the context of genetic mutations necessary for natural selection, those ancient humans with straight-line attending genes survived, and those with mutations varying away from the straight line were more likely to die. [i]
The challenge is that the world beyond that of linear survival is incredibly non-linear. Paradoxically, linear attention which helped us in the past, because we are surviving more today, has become an increasing liability.
3. What is non-linear growth or convexity?
First, let’s define non-linearity via an important natural and financial concept known as convexity. Convexity shows up in many non-linear settings. In its simplest form, convexity describes a mathematical function that increases at an increasing rate. This is the mathematical foundation for non-linear growth. Being convex to time simply means that:
Time is a transforming input to some natural process.
The outcome of that process, whether it be wealth, health, risks, or other outcomes increases at an increasing rate.
In general, for things we desire - like wealth, health, and a pollution-free environment - we want to be convex to time. For things we do not desire - like poverty, chronic disease, pollution, or other risks - we do NOT want to be convex to time. [ii]
A simple test to determine if a mathematical function is convex: You can test convexity by drawing (or imagine drawing) all the tangent lines along the surface of g(x). If the set of all tangents falls on or BELOW g(x), then it is convex. Conversely, if the tangent lines are all on or ABOVE g(x), then the function is concave.
The mathematical intuition of the non-linear “increasing at an increasing rate” convexity feature is necessary but not sufficient to understand the full power of convexity. There is another, very important part of convexity related to the volatile nature of life.
As we all know, life is volatile. Life decidedly does not move on a smooth line. The previous math example represents a sterile, long-term average growth path. However, how we get to this long-term average over time is decidedly volatile.
An investment example: Please consider a stock chart. While the long run maybe a 10% return for some diversified mutual funds, the path it got to 10% was volatile. Some days the mutual fund was up and some days it was down. In growth business cycles, it was more likely to be up than down. In recessionary business cycles, it was more likely to be down than up. But its volatile path ultimately led it to a 10% return. Next is a 14-year graph of the highly diversified S&P 500. Notice it was up significantly on average from the beginning of the timeline on the left to the end of the timeline on the right. For this example, the S&P was up 374% in about 14 years. But notice the very volatile path it took. Downward volatility is noted by the red arrows.
The beauty of convexity is this: As one reinforces the convexity path, upward volatility adds more to the objective than downward volatility subtracts from the objective. The math behind this statement is well-known. For example, bond traders look for bond portfolios with higher convexity to help improve returns GIVEN market interest rate volatility. [iii] Also, to achieve the returns associated with upside volatility, one must consistently provide inputs to the convex function, especially in downward volatility environments. This is emotionally very challenging for most people. [iv]
The net additive nature of convexity to the objective is true assuming you have diversified yourself to the point where the chance of ruin has been eliminated. This “no ruin” assumption deserves some explanation. [v] As a rule of thumb, the premise of this article is that the chance of ruin is being consistently reduced by improving public health services, medical care, food availability, and related. Thus, while our chance of undesired death or some related ruin may not be nil, think of this assumption as part of operating in a world geared toward a much greater chance of survival. [vi] This article assumes we wish to get "more life out of our years" than "more years out of our life." Since our public services are very focused on getting more years out of our life, then, focusing on convexity is a personal responsibility to help us get more life out of our years.
To summarize convexity:
Convexity transforms life’s inputs, like time, to deliver outcomes that increase at an increasing rate.
Convexity, relative to a time-based objective, enables downside volatility to be not as negative as upside volatility is positive.
Convexity is agnostic to the quality of the outputs. A series of good things exposed to convexity will get better. A series of bad things exposed to convexity will get worse.
4. Convexity examples
Next are examples from biology, personal finance, health, accounting, and the environment. There are certainly many more examples. These examples intend to show the diversity of convexity impact across nature and human affairs. The examples are intended to show how convexity could help or hurt you. The point is that being aware of convexity is the first step to overcoming its challenges or capitalizing on its benefits.
Iain McGilchrist is a psychiatrist and author of the book, The Master and His Emissary. He is an expert on how the brain operates to use non-linear information. This is a paraphrased biology example [vii]:
Let's take a bacterium in a petri dish. The challenge is to identify the point just prior to when the bacterium overwhelms the resources of the petri dish. The bacterium grows by a convex function. Which means the bacterium population grows at an increasing rate. Human beings don't understand exponential growth. So, if you put a modest amount of bacterium in the dish at 11 o'clock, it will double every minute. And you come back at 12 o'clock and the dish is full, having overcome its resources. When is it not full? 11:59. And when would we realize we have a problem with the bacterium in that jar? At 11:58, the jar was only a quarter full. At 11:57, it was only an eighth full. At 11:56, it was only a sixteenth full, but that was four minutes before it hit the buffers. So, we're in that kind of a world. And, there's far more to this world than just having more years.
In the context of risk, not understanding convexity is dangerous. Unfortunately, by the time our linear processing brains realize we have a problem, inputs to the problem are more likely irreversible. It is like a bamboo finger trap... easy to put your fingers in, but decidedly more challenging to pull them out. In this example, by the time it is realized the bacterium will overwhelm the dish, it is too close to the point of being overwhelmed.
b. Personal Finance:
Let's say you can consistently set aside $150 a month. This money is invested in a diversified securities portfolio that earns 10% a year over the long run. The idea is to save for your retirement. In this example, you wish to retire at age 62. The second-order questions are: "When should I start saving?" and "What is the impact of delaying savings?
The good news is that If you start at 22 and save 40 years until retirement, you will earn over $1.25mm for retirement.
But convexity is a double-edged sword.
Because the transformation function is convex to time, delay works against you in a big way. Just delaying 8 years, from 22 to 30, causes a significant decrease in potential retirement value. In this example, an 8-year delay causes you to lose almost $700k for retirement. As shown in the inset graphic, an 8-year delay from 22 to 30 years old causes you to lose over half your potential retirement savings. Note the blue line starts at 100% of the 22-year-old's $1.25mm in potential retirement value. As savings are delayed, the blue line drops to less than 50% of the potential retirement value at 30 years old. [viii]
The moral of the story is.... invest early and often! Even small amounts of savings will grow multiples over a longer time frame.
The impact of volatility - the further along the curve one gets, the less random volatility hurts you to the downside and the more it helps you to the upside. This occurs because of the dollar cost-averaging effect. By consistently investing, when the market goes down, you are buying more shares at a lower price per share. This averages down the cost basis for your total portfolio. This turns recessionary down markets into the equivalent of a spring that, while compressing, is storing energy to rocket forward when released in an expanding economy. [ix]
c. Health and Longevity
Dan Buettner is an explorer, educator, author, producer, storyteller, and public speaker. He wrote the book, The Blue Zones. [x] The book chronicles the characteristics of people and their communities with the longest lifespans. These 4 communities are spread across the earth. They are Sardinia, Italy; Okinawa, Japan; Loma Linda, California, and Nicoya, Costa Rica. Buettner identifies 9 characteristics correlated with their longevity. These are noted in the next graphic.
For our focus on convexity, think of these 9 characteristics as health investments. An important conclusion from Buettner’s research is that it is important to integrate all nine into your life habitually, so they can reinforce each other over the long term.
Peter Attia is a medical doctor specializing in oncology and longevity. He wrote the book Outlive: The Science and Art of Longevity. Attia’s book, which cites Buettner’s research, shows how the convexity principle relates to long-term health. The following graphic demonstrates Attia's central thesis that "getting more life out of our years" is better than "getting more years out of our life." Attia’s medical research focus is on the “Big 4” chronic disease categories including the heart, metabolism, cancer, and the brain. He identifies the difference between the current U.S. medical industry (Medicine 2.0) and the aspirational future medical focus, called Medicine 3.0. Much of his book identifies healthy habits to proactively prevent chronic disease to achieve Medicine 3.0. The more we can proactively prevent chronic diseases, the better we can "square the curve" and make our healthspan equivalent to our lifespan. [xi]
By the appearance, these curves do not appear traditionally convex. However, moving toward the outer Medicine 3.0 curve is the outcome of many small, cumulative health investments that conclude by allowing people to live long, healthy lives. These health investments were often made decades in advance. The contrary outcome relates to convexity as well. The lack of investment makes you more likely to fall in the bottom yellow "boot" characterized as longer life but much lower health. Both these outcomes are associated with a consistent or a lack of consistent investment in Buettner's Blue Zone "Power 9" healthy behaviors. This is the essence of convexity.
The impact of volatility - as we age, the outcomes of our health investments reveal themselves. Like in personal finance, random health volatility will occur. Think of a cold you may get or a slip and a fall. There could be many other random health challenges. The bottom line is, that if you have made a lifetime of blue zone health investments, you are more likely to minimize the impact of life's inevitable health volatility. On the flip side, for more fragile people, a small health volatility is more likely to lead to a significant health challenge. The slippery slope leading to death often starts with smaller health challenges compounded by existing fragile health. The convex idea is to build resilience to handle life's inevitable health volatility.
d. Embedded losses and risk management - technical accounting
This next example is provided by the author's discussion with his son, Josh. Josh is a CPA and works in financial reporting. He and his team make sure his company's SEC-required 10-Ks and 10-Qs are properly reported to investors. We were discussing how large language models (LLMs) like those used for ChatGPT could be helpful with technical accounting. Our discussion occurred not long after ChatGPT caught the interest of many people and companies:
Dad: Hi Josh! Is your company using Large Language Models or AI-based chat technologies?
Josh: Yes, we are at the beginning of our journey, but our leadership is committed to figuring it out.
Dad: Great - so glad. I have an idea for how your corporate accounting team could use LLMs to help improve technical accounting. Can I run it by you and do some brainstorming?
Josh: Sure Dad!
Dad: OK - so first let me start with some background. Let me know if this resonates. First, accounting is considered a cost center. Leadership knows they need accounting to be done well, but leaders generally do not want to overachieve in accounting. Basically, they only want to get to "good enough." This is unlike a business unit, where more revenue is usually better than less revenue.
Josh: I'm not sure all accounting leaders feel that way, but I take your general point about the difference between a cost center and a revenue center in a large company.
Dad: Also, technical accounting is notoriously project-based and understaffed to handle the peak demands for technical accounting needed in the company. When the business units are busy with new deals and contracts, technical accounting needs to advise the business units and accounting teams about how to book the journal entries and other accounting policies.
Josh: You are correct. Understanding new business transactions and applying the appropriate accounting treatment is challenging. Busy times for technical accounting are not easy to predict. Technical accounting resource capacity can be a challenge.
Dad: Now, let me explain my take on the "tyranny of immateriality." Declaring a business activity as 'immaterial' means that the technical accountant does not have to work on it. So in a busy time, accounting leadership may be more likely to declare a business activity as 'immaterial' as a way to focus limited accounting resources on more material items.
Josh: (a little uncomfortable) Let's be careful. While leadership would focus resources on the “more material” activities, it would be unethical to declare materiality solely based on available accounting capacity.
Dad: Fair enough. Josh - let me clarify. I am sure the accounting governing body has rules about materiality. I suspect those rules do require judgment to apply to each technical accounting project situation. So let's think of those rules as criteria to weigh for a particular project. That criteria weighing does take professional judgment.
Josh: OK, I'm with you so far.
Dad: Let's say you have 2 potential technical accounting projects that look about the same. The materiality designation for both projects is not readily obvious. At first blush, both projects could go either way. The only difference is project A is at a time when the technical accounting team is super busy. Project B is at a time when they are not busy. Is it possible accounting capacity could be like an invisible weight impacting how those materiality rules are applied? Is it possible Project A would be declared 'immaterial' and Project B would be declared 'material?' With both declarations being supported by regulatory guidance?
Josh: Yes, I suppose they could.
Dad: Another challenge is the volume of embedded losses associated with immateriality. When many small risks are embedded within a company, they tend to fester and grow exponentially. This means that as smaller risks accumulate, they tend to become a big deal all at once. Unfortunately, once they get to the point of being a big deal, they are much more challenging to handle. Thus, accumulated smaller risks become larger embedded losses. This is like the old saying "It doesn't make much sense to blow out the match after the forest fire is blazing." During the mega financial crisis, these smaller embedded risks grew to become larger embedded losses. They were like the "unblown out matches" seeds of destruction for some companies, like Lehman Brothers.
Josh: Wow, that makes sense. I can also see how this relates to the convexity concepts that you taught us kids growing up.
Dad: I'm glad you remember! Yes!
Dad: Ok Josh - please give me a second to summarize -- Embedded losses have a random nature to them. Depending on the environment, embedded losses may grow faster or slower. Embedded losses are challenging to see, the portfolio of embedded losses tends to grow, and the negative impacts will be revealed in high-stress economic environments.
I know you are coming up to speed on how LLMs could be used in your company and the accounting profession in general. Given the background, let me know what you think.
Josh: OK, here is my idea kernel. Given what you said about technical accounting, wouldn't it be great if the LLM could ingest all the current technical accounting regulatory guidance and our company's past technical accounting memos across the corporation? This way, we could create a labor-saving device that would increase the capacity of our very valuable technical accounting team and decrease the immateriality threshold. Thus, trained but non-technical accounting people could ask the LLM 'chat' to help establish the accounting treatment of a technical accounting question. Importantly, the chat language would be programmed in a way that returns technical accounting suggestions that are easier to understand for non-technical accounting people. As needed, a more senior person would approve. Also, the technical accounting team's role would evolve as a manager of the LLM and chat as a corporate resource. The technical accounting experts would only get pulled in when needed to handle a particularly complex, highly material need.
Dad: Josh, I like the way you are thinking! This could reduce the potential for embedded risks to be exposed to the bad side of convexity. It would not necessarily eliminate all the unseen embedded losses, but I could see where it could be very helpful to reduce both the size and significance of those smaller risks before they become embedded losses.
d. The global environment
According to the United Nations, "To keep global warming to no more than 1.5°C – as called for in the Paris Agreement – emissions need to be reduced by 45% by 2030 and reach net zero by 2050." Also, according to the United Nations, we are NOT on track to hit these targets, " [C]ommitments made by governments to date fall far short of what is required." [xii]
These estimates have two challenges. 1) Since people cannot see the challenge today, going back to our earlier discussion about the brain's attention challenge, people are less likely to make it a hot policy agenda item. As such, climate policy is more likely to be prioritized beneath "hotter" (i.e., more salient) policy priorities. 2) The future is uncertain, pretending like it is not with the U.N.'s seemingly precise estimates raises suspicion.
But the real challenge of doing nothing relates to the convex nature of the carbon emission risks. The industrial era, generally starting in the 18th century, has been emitting higher levels of carbon into the atmosphere and overwhelming nature's carbon filtering ability. Those carbon emissions present what economists call externalities. Carbon emission's true costs are not found in the information guiding market prices. Thus, businesses and consumers have not been constrained by an accurate price signal.
The convexity intuition is that embedded losses - like higher ocean levels and increased species extinctions - are increasing at an increasing rate. This means the marginal risk of a unit of carbon emission is growing rapidly. Whether the point estimate of 2030 or 2050 is accurate is mostly immaterial. The convexity intuition suggests society needs to reduce its carbon emissions now if we want to have a chance to blunt the effects of an exploding mathematical function.
Free market and anti-socialist economists like Adam Smith and FA Hayek absolutely believed there is a place for government regulation. It is not a matter of WHETHER the government should regulate but HOW they regulate is essential to the proper functioning of markets. Today, our environment is considered a 'market failure' in the traditional sense. Smith suggests there is a solution to handling environmental needs in the context of economic markets.
Including proper carbon costs to inform prices is necessary for the Smithean framework called "The Four Sources of Moral Approval." The 4th source is where the voice of the environment would be found in Smith's framework. As Smith points out, the 4th source is the most challenging, as it is less salient at the time of the market interaction and it is found in the more uncertain future. Thus, regulations such as carbon taxes may make sense as long as they are applied in a way that provides proper cost information to the price signal. The following citation offers considerations for appropriately providing carbon price information. [xiii]
Convexity is a big deal in the modern world. While it is agnostic to the goodness or badness of the outcome, the outcome of convexity can be decidedly good or bad. Understanding how to put convexity to work for you is critical. It is like a tool you did not know you had in your toolbox! We explored why people struggle with recognizing convexity and the power of non-linear growth. We gave 5 examples of convexity in biology, personal finance, health, accounting, and the environment. But there are so many more. Most importantly, now that you understand convexity, it is important to be on the lookout for it in your life. It is challenging to see the power of non-linear growth. Taking the extra step to identify it and leverage it can greatly improve your life.
Non-linear growth or convexity opportunities are abundant in our everyday lives. Making good decisions when our own brains do not naturally attend to these abundant opportunities is the challenge. Next, is a solution to help you develop and implement a consistent, repeatable decision process to help overcome these challenges and leverage the many non-linear growth opportunities. Check out my book at Jeffhulett.com for help using these decision tools.
Definitive Choice is an app decision solution to help you manage your convexity perception challenges. It does this in a way that leverages your highly tuned, linear decision-making capability provided by nature. It does this by managing the more challenging non-linear factors for you in the background.
It provides a straightforward user experience. The number-crunching occurs in the background by time-tested decision science algorithms. It uses a proprietary "Decision 6(tm)" approach that organizes the preference criteria (what is important to you?) and alternatives (what are the choices?) in a series of bite-size ranking decisions. Since it is on your smartphone, you can use it while you are curating data to support the decision. It is like having a decision expert in your pocket. The results dashboard provides a rank-ordered list of recommended "best choices," tailored to your preferences.
Also, Definitive Choice comes pre-loaded with many decision templates. You will want to customize your own preferences (aka criteria) and alternatives, but the preloaded templates provide a nice starting point.
Using decision process solutions enables DECISION A-C-T:
Accelerated: faster, less costly decisions. It enables a nimble decision environment.
Confidence-inspired: process causes people to be more confident in the decision, increasing buy-in, and decision up-take.
Transparency-enabled: reporting, documentation, and charts to help communicate the decision.
[i] "... the right hemisphere has a kind of sustained, broad, vigilant attention instead of this narrow, focused, piecemeal attention (found in the left hemisphere). And it sustains a sense of being, a continuous being, in the world. So, these are very different kinds of attention." - Iain McGilchrist
Dr. McGilchrist discusses the different kinds of attention emanating from different brain hemispheres. It is our left hemispheric originated "narrow, focused, piecemeal attention" on point for our evolutionary-based linear focus for survival. It is the understanding of non-linearity needing a "sustained, broad, vigilant attention" associated with our right hemispheric originated attention.
Perhaps, eventually, humans will evolve to better attend to non-linear growth. However, even our ability to evolve is dramatically slowing. It takes multiple millennia for humans to genetically evolve. Evolution is necessary to change how our brain naturally attends to input information. Given the rapid improvement of health care, food availability, public necessity services, and other improvements — humanity's ability to evolve through the natural selection mechanisms has slowed significantly. There is not much value to genetic mutations if all, except for the most extreme, individually mutated genes can survive to reproduce.
Certainly, saving people is a good thing. But there is no free lunch. There are unseen trade-offs. The "improved survival" to "decreased evolutionary speed" trade looks like this: A higher proportional survival-to-reproduce population decreases the available genetic information to the human genome because fewer people fail to survive. The genome needs some people to not survive to teach it how to update and adapt for the next generation. This means, do not expect our genome to evolve as quickly as it used to. Again, this is not such a bad thing. Increased diversity from those that may not of survived in earlier times may be net additive to our society. However, less evolutionary information means we are likely stuck with our current genomic state, like this linear attention limitation, for a while!
[ii] Taleb, Antifragile: Things That Gain from Disorder, 2012
[iii] Chen, Convexity in Bonds: Definition, Meaning, and Examples, Investopedia, 2023
[iv] Hulett, Zoom out! A time-tested approach for investing in challenging markets, The Curiosity Vine, 2022
[v] The subtle but very important differences between risk and ruin in the context of entropy and ergodicity are explored:
Hulett, The Regenerative Life: How to be an ergodic pathfinder, The Curiosity Vine, 2023
[vi] As an indicator of the increased chance of survival, in the last 200 years, the world's life expectancy has more than doubled.
Roser, Ortiz-Ospina, Ritchie, Life Expectancy, Our World in Data, accessed 2023
[vii] Roberts (Host), Iain McGilchrist on the Divided Brain and the Master and His Emissary, EconTalk, The Library of Economics and Liberty, 2018
[viii] Hulett, Achieve Personal Finance success with a little math intuition, The Curiosity Vine, 2023
[ix] Hulett, Zoom out! A time-tested approach for investing in challenging markets, The Curiosity Vine, 2022
See section 3 for Spring investing.
[xi] Attia, Outlive: The Science and Art of Longevity, 2023
For a deeper dive into Attia's framework, please see:
Hulett, Getting the Most Out of America's Sickcare System, The Curiosity Vine, 2023
[xii] Editors, For a livable climate: Net-zero commitments must be backed by credible action, The United Nations, accessed 2023
[xiii] Hulett, Market Failure - How Adam Smith would approach government and personal responsibility, The Curiosity Vine, 2023