“Time and chance happeneth to them all.” Ecclesiastes 9:11.
We pulled the above quote from a video clip by Ole Peters introducing his, and Alex Adamou’s, new book, “An Introduction to Ergodicity Economics”. Just as Ole mentions, this biblical reference ever so nicely sums up our overall philosophy on the importance of convexity in life. Clearly, we were not early to this challenge! It is a very useful little video clip and a truly wonderful book.
We highly recommend taking the time to watch this video. Ole does an excellent job of explaining the intentions behind the effort to produce the book. Hopefully, regular readers of our Updates will recognize just how acutely aligned we are with Ole’s thoughts. Just a few examples:
Complexity science is roughly emergence, and emergence is roughly the ergodicity problem” Ole Peters, November 2025.
This point gets to the issue of the individual versus the collective, the physics concept of complexity and emergence. How molecules, each individually not a thing, join together to become something.
“You deviate from Expected Utility Theory (EUT) because, if you do so, you will do better in your real life…Ergodicity Economics (EE) reveals rationality where orthodox economics cannot see that rationality.” Ole Peters, November 2025.
Ole gently ventures into what is known as the Rationality Wars. Logical Rationality (Rational Economic Man) does not explain how people behave but rather how economists think they should behave. Behavioural Scientists explain, therefore, people must be irrational. They are both wrong. Ergodicity Economics makes the simple mathematical case for Ecological/Bounded Rationality. You can see our note on the topic here – Convex Strategies | Risk Update: November 2024 – “Rationality Wars”.
“Orthodoxy underestimates both the power of circumstances, and our rationality.” Ole Peters, November 2025.
This quote gets right at our constant rant about the relevance of initial conditions, aka circumstances. Where we are at, matters.
“Linearity and optimization produce extreme outcomes.” Ole Peters, November 2025.
Maybe the key fundamental negative externality of Sharpe World mathematics. The progenitor of boom-bust financial cycles.
“When we focus on time-averages they incorporate non-linearities…These non-linearities, that we have in the theory (EE), introduce moderation.” Ole Peters, November 2025.
How one should go about investment cycles and life.
Ole produces this wonderful little grid laying out the characteristics of “Homo economicus” versus what they have dubbed “Homo ergodicus”. These are the equivalents of our own, regulatorily constructed stand-in as a real-world Homo economicus, “Rational Accounting Man”, and ergodicity aware actual skin-in-the-game participants, “Boundedly Rational Agents”, as discussed in our February 2025 Update – “Rational Accounting Man” Convex Strategies | Risk Update: February 2025 – “Rational Accounting Man”.
Figure 1: Homo economicus vs Homo ergodicus
Source: Why we wrote a textbook
If you have not yet gotten your hands on Ole’s book, we strongly recommend trying to get it squeezed onto your letter to Father Christmas.
Textbook – Ergodicity Economics
We described the book, in an exchange with Ole, thusly:
“It is a beautiful mathematical representation of what one might call ‘common sense’ or ‘tacit knowledge’ or ‘bounded rationality’.”
It should be taught in schools at the earliest stages of math, science, and economics.
The clever chaps at Veritasium, seemingly, would agree! They have, once again, produced an absolutely fantastic video covering many of the same topics, indeed topics that we have written about at length over the many years of spinning out our own Updates.
This graph will change how you see the world
We, of course, agree with the clipped titling of their video – “This graph will change how you see the world”. We show our version of “the graph that will change how you see the world” in every presentation that we do. The point being that, in a world of power laws and path dependencies, it is magnitude, not frequency, that makes the difference. In a non-ergodic, multiplicative (ie non-linear) world, we should prioritize that which matters the most, not that which happens the most.
Figure 2: Frequency vs Magnitude. Normal Distribution (black) and Shannon’s Entropy Curve (yellow). Shading by Percentile Contribution to SPX Index 40yr CAGR. Oct 1985 – Sept 2025
Source: Bloomberg, Convex Strategies
“If you are in a world where random additive variations cancel out over time, then you get a normal distribution. And in this case, it is the average performance, so consistency, which is important. But if you are in a world that is governed by a power law where your returns can multiply and they can grow over many orders of magnitude, then it might make sense to take some riskier bets in the hope that one of them pays off huge. In other words, it becomes more important to be persistent than consistent.” Veritasium, November 2025.
This wonderful video hits all the goodies. Most simply put, relevant parts of reality are not additive, not normally distributed, not ergodic, not linear. Reality is multiplicative, power law distributed, non-ergodic, non-linear. It is neither random nor predictable. It is chaotic, complex, emergent. It is self-organised criticality.
Long time readers will rightly be curious if, possibly, the Veritasium chaps are readers of the Convex Strategies pages, as we have touched on all parts of the subjects of their video over the course of our Updates. For a tour of some of the topics, see the links below to past Updates focused on these exact issues.
Convex Strategies | Risk Update: March 2019 – St. Petersburg Paradox.
Convex Strategies | Risk Update: September 2020 – Ergodicity.
Convex Strategies | Risk Update: August 2021 – Self-Organised Criticality.
Convex Strategies | Risk Update: September 2021 – The Challenge of Measurement.
Veritasium had another great video out this month that links in well with our discussions about paths through time, predictability, and optimization. Gets very specifically to our frequent refrain that “history alone is a poor measure of risk”.
Why People Are So Confident When They’re Wrong
“In a controlled environment where there are clear rules and guaranteed outcomes like in a chess match, there is clear feedback on whether decisions are good or bad…But in a noisy environment where there are rarely consistent or timely consequences for predictions, this feedback is unreliable.” Veritasium, November 2025.
This echoes precisely what Nassim Taleb terms the “ludic fallacy”, the mistake of using the mathematics of known probabilities from games, then applying such to the unknowable uncertainty of real life. Exactly as discussed by Ole, above, as to applying multiverse ensemble averages to, what is in reality, a path dependent, time-sequenced, uncertain future. This gets to Taleb’s running analogy between “Mediocristan” and “Extremistan”. To state it yet again, in Extremistan, “history alone is a poor measure of risk”.
Going back to the above linked Update – “Rationality Wars”, we showed this succinct explanation of the difference between “small worlds” and “large worlds”, provided by the leader of the Bounded Rationality army, Gerd Gigerenzer.
Figure 3: Risk, Ambiguity, Uncertainty, Intractability
Source: The rationality wars: a personal reflection | Behavioural Public Policy | Cambridge Core
In what has to count as Christmas coming early, Gerd has launched a new website.
Gerd Gigerenzer | Explore. Learn. Decide.
Freshly up on the website are both a video of a presentation, “What is bias? And why are we biased?”, and a paper, “Two kinds of bias”. Gerd gets into precisely these issues and tracks right along with Ole’s above discussions.
Gerd GIGERENZER – Paris IAS Ideas – What is bias? And why are we biased?
At the 23:40 mark in the video, Gerd lays it out as clearly as it can be said.
“In large worlds there is no best solution that we can know. There is no optimization. Optimization is a fiction… We feel obliged to talk about optimization in situations where it has no meaning.” Gerd Gigerenzer, November 2025.
We will quote from the abstract of this wonderful note.
“I distinguish two meanings of the term bias in the social sciences. In the first, biases are functional: they are necessary, and simultaneously enable and constrain perception and cognition. In the second, biases are viewed as errors and ideally should be reduced to zero. In the functional view, bias is value-neutral, neither good nor bad. This pragmatic perspective accepts that cognition must operate under conditions of uncertainty (rather than the certainty of a “small world”) and intractability (where the optimal solution cannot be calculated). Biases enable cognition to deal with these situations where probability theory offers no guidance, for instance, through intelligent heuristics. In contrast, the error view assigns a negative value to bias. It assumes that cognition deals with problems where the true state of the world is known with certainty – at least to some authority. This distinction has profound implications for research design: the two views lead not only to different answers, but also to different questions.” Gerd Gigerenzer, November 2025.
Just beautiful. This is our relentless refrain about the necessity of freeing oneself from the traditional trapping of “optimisation” models and get into the real world of learning by doing. Put good brakes on your race car and go figure out how to drive faster. You can’t optimise to the unknown future, and optimising to the historical average (i.e. Expected Returns/Utility) is meaningless. For a lengthy discussion on the topic of learning by doing, please refer to our June 2025 Update – “Just Do It” Convex Strategies | Risk Update: May2025 – “Just Do It”
This is precisely the #1 point that we make when discussing the benefits of negatively correlating asymmetry in an investment portfolio, the sort of explicit portfolio diversification benefit that is the literal key to concepts like Total Portfolio Approach. More often than not, people are asking the wrong questions. They are asking what did happen, not what could have happened, or what could happen in the future. In an insurance analogy, the questions focus on what the insurance cost and what past actual events were compensated by the insurance, i.e. the historical return path. They don’t ask the much more important questions, the ones that aren’t disclosed by the visible occurrences of past outcomes, those being how much insurance was/is there, and how much additional risk did/does that insurance allow the protected party to undertake.
As we say, you don’t insure risk you don’t want to take, you insure risk you do want to take. Get effective insurance, dynamic insurance, and take more of the risk you want. If you’ve got good effective insurance, and confidently pursuing the risk that allows, you shouldn’t complain when you haven’t needed your insurance, you should celebrate.
We will stress that one more time. The key question you should be posing as you evaluate providers of negatively correlating, asymmetric, risk mitigating strategies is – “how much additional participating risk will this allow me to take over the next two, three, four decades?”.
It seems like everybody in our camp is out talking about the same thing. Here is another video clip on very much the same concepts from the incomparable Nassim Taleb.
Nassim Taleb: My Most Valuable Trading Lesson |Nassim Nicholas Taleb LATEST Incerto Black Swan Money
Nassim gives a wonderfully clear example of something that is non-ergodic, something where sequencing matters and ensemble averages don’t equate to time-averages. He explains so simply that you won’t get the same outcome if you reverse the order of laundering, then ironing, your trousers. The average perturbation of the trousers may be the same, but the end state is significantly different.
He also has a jab at the “irrationality” camp noting that something that might be “irrational statically” may not be irrational on a sequencing dependent path through time. Again, bringing us back to Gerd’s decision heuristics in the realms of ambiguity, intractability, and uncertainty.
Nassim really gets to our point with the below quote. The more uncertainty there is about the future course of the racetrack, the more we know the importance of brakes. The more uncertainty there is, the simpler decision making becomes. Resilience dominates in the “large worlds” of what we don’t know. Prediction’s role, if at all, only exists in the “small worlds” of what we think we know.
“The more uncertainty there exists in a system, the more you need to follow a certain paranoid route, try to position yourself to have more upside than downside…The more uncertainty there is, the more we know how to act”. Nassim Taleb, November 2025.
Let’s take an example of our old friends who believe they are in the business of forecasting, central bankers. We give you another brilliant link to an interview with our good friend, Jim Grant, the legendary founder of Grant’s Interest Rate Observer.
Jim points out a particularly troubling aspect of central banker’s use of historical data in their efforts to adjudge current circumstances and forecast future outcomes.
“The trouble with administered rates is they return so little information to the administrator.” James Grant, December 2025.
They know they are manipulating key data sources and then taking that data as valuable information inputs to surmise future outcomes. Few are a better example of this than our friends at the Bank of Japan (BOJ). Most will by now be aware that the consensus has landed upon the BOJ hiking their policy rate from 0.50% to 0.75% at their upcoming Monetary Policy meeting on December 19th. This perception was most solidified as a result of a speech given by Governor Ueda on December 1st.
Japan’s Economy and Monetary Policy
“If the outlook for economic activity and prices outlined so far is realized, the Bank, in accordance with improvement in economic activity and prices, will continue to raise the policy interest rate and adjust the degree of monetary accommodation. As a premise for this policy stance, looking at Chart 8, the Bank recognizes that real interest rates — that is, the nominal policy interest rate minus the inflation rate — are at significantly low levels, considerably below the real interest rate that is neutral to economic activity and prices, which is often referred to as the natural rate of interest in economic terms. In other words, even if the policy interest rate is raised, accommodative financial conditions will be maintained. To use an analogy, raising the policy interest rate under accommodative financial conditions is about a process of easing off the accelerator as appropriate toward achieving stable economic growth and price developments, not about applying the brakes on economic activity.” BOJ Governor Kazuo Ueda, December 2025.
Below is the “Chart 8” that Ueda-san mentions in the above paragraph.
Figure 4: Policy Interest Rates in Japan, US, Euro (left panel) and Japan Nominal vs Real Rates (right panel)
Source: Bank of Japan, Ministry of Internal Affairs and Communications, Bloomberg
It is easy to see Mr. Grant’s point. The “administered” rates by the Bank of Japan, may not be providing the most useful information back to the administrator! We have long played the game with the BOJ asking them the question, “what if it worked?” What if their accommodative policies worked? As we have noted so often, their general refrain to this has been “it will not work”. As has been the case, we continue to disagree with them. We would argue that the rise of their price stability measure (CPI less Fresh Food, aka Core Inflation), the weakening of their currency, and the increasingly one-way direction of JGB yields, suggest it has and continues to work.
Figure 5: Japan Core CPI yoy% (white), 10yr JGB Yield (papaya), USD/JPY FX Rate (blue, left-scale). 2016 – Nov2025
Source: Bloomberg, Convex Strategies
We are surely biased, but we would argue that the only way to deal with Japan’s future is to focus on resilience, not on prediction based on historical lookbacks.
As we have mentioned, the challenges around uncertainty are getting ever increasing consideration in the world of investment management. The implications of the poor return paths delivered by some of the purveyors of traditional Sharpe World investment methodologies haven’t exactly gone unnoticed (see here Convex Strategies | Risk Update: July2025 – “Preservation” for some specific discussion on the topic). The hot topic aimed at addressing these challenges is, as we have mentioned, commonly referred to as Total Portfolio Approach (TPA).
In a huge step forward in the expanding sphere of TPA, the largest pension fund in the US, CalPERS, just announced their adoption of it as their core investment philosophy. This is a major feather in the TPA cap with one of the world’s largest purveyors of the dominant Sharpe World strategy amongst pension-type managers, Strategic Asset Allocation, making the shift over. This isn’t really a huge surprise as it comes in line with CalPERS recent hiring of Stephen Gilmore as their new Chief Investment Officer, an alum of early TPA adopters Future Fund and NZ Super.
CalPERS Board Adopts Streamlined Investment Approach to Seize Market Opportunities | CalPERS
“TPA encourages greater collaboration among the investment team, so that their collective wisdom is harnessed to judge investments based on their potential to benefit the entire portfolio.” Marcie Frost, CalPERS CEO, November 2025.
We shall see if they ask the right questions.
The folks we think are doing the best work on TPA and the effort to build resilience into their portfolio are the Future Fund in Australia. Here is a wonderful piece they just released discussing their ongoing process – “Position Paper – Portfolio Resilience: Part One”.
https://www.futurefund.gov.au/-/media/7648B5ACC4A44864B4EBBFEB43B3B2BC.ashx
“The New Investment Order and resultant consequences around greater uncertainty to macroeconomic dynamics, policy regimes, and asset class behaviours have however made us rethink portfolio construction and how to better achieve our mandate. A key outcome has been a prime focus on achieving portfolio resilience.”
This is a very good note. We might quibble that they somewhat overemphasise the projected future scenarios, as we would always stress it is the scenario that you didn’t foresee that gets you, but they recover it well with this highlighted quote:
“The use of scenarios is not about predicting; it is about preparing. And proper preparation achieves portfolio resilience.”
Again, quoting our friend Hari Krishnan – “Risk is about vulnerability, not predictability”. Prepare your race car for uncertainty, don’t flatter yourself that you can predict the future paths.
This particular paper from the Future Fund focuses in on their highlighted vulnerability of “inflation”.
“…a key conclusion from The New Investment Order is that inflation volatility and broader inflationary risk is to be a more permanent fixture in our investment future.”
Regular readers will be aware that we too have emphasised the risk of inflation volatility as a key structural fragility to the global economic system. We have been showing versions of this Bank of England chart of historical CPI averages and volatility since way back in our July 2022 Update – “The Pointlessness of Forecasting” Convex Strategies | Risk Update: July 2022 – The Pointlessness of Forecasting. Talk about an administered rate that hasn’t provided useful information. The ultimate example of Goodhart’s Law.
Figure 6: “Stability Begets Instability” 800 Years of CPI in the United Kingdom. 20yr CPI Blocks Average vs Volatility
Source: The Burst of High Inflation in 2021–22, Bank of England Millennium Dataset, Convex Strategies
The 1997-2016 dot, the era of modern-day central bank “Inflation Targeting”, is truly the anomaly of anomalies. How much of “what could have happened” is it masking in simplistic historical measurement of “what did happen”?
The Future Fund is right to highlight this as a core to future uncertainty and a driver around which resilience needs to be built into their portfolio. We look forward to their future papers!
We hear lots of people talking about increased levels of uncertainty. Far too few behaving like the Future Fund and proactively doing something about it. We, of course, do not know the future but do have memories of past episodes when seeming unforeseeable events punished the unprepared and rewarded the resilient.
We have gone on and on about our current feelings of Déjà vu with the market environments of 1995-1999. We wrote about it in our December 2024 Update – “Déjà vu” Convex Strategies | Risk Update: December 2024 – “Deja Vu”. Don’t take that too literally, we are well aware that any number of structural issues are fundamentally different today. Not least of which, our #1 bugaboo around demographics. Our point, as to the similarities, revolves around the efficacy of investment risk structured with positive convexity, targeted to our mantra of “Participate and Protect”, as well as recurring memories of central bank policies.
We have highlighted with shadowed boxes, first with the Nasdaq then with a group of global 10-year yields, the market behaviour in that window from the post-LTCM Fed rate cuts in late 1998. We have heard a lot, over most of this year, that policy rates can only go lower. Very reminiscent of the commentary post LTCM in late 1998. We have noted how the intermittent rate cuts, post the sharp rate hiking cycle of 1994, were undertaken as equity markets continued to make all-time highs and financial conditions maintained very loose setting. We have seen the same over the post-2022 rate hike cycle and the subsequent rate cuts of 2024-2025.
Figure 7: Nasdaq 100 Index (white, right log-scale) and Fed Funds Rate (white, left). Sept1998 – Sept2002 (white shaded). Oct1991 -Nov2025
Source: Bloomberg, Convex Strategies
Likewise, as we have noted before, in both periods of rate cutting, 1995-1998 and 2024-2025, we have generally seen longer term interest rates rise as policy rates were lowered.
Figure 8: Fed Funds Rate (blue) and 10yr Govt Bond Yield: US (white), Germany (papaya), UK (yellow). 10yr Swap Australia (purple). Sept 1998 – May 2000 (white shaded). Dec 1990 – Nov 2025
Source: Bloomberg, Convex Strategies
Read into that what you want. Maybe the market feels like central banks, who seem confident that their generally stalled progress in returning their price stability measures back to 2% will right itself even as they proceed to lower policy rates, are concerned about the seeming asymmetry in their attitude towards price stability. We don’t know.
What we do know is that, if you look at the market behaviour inside those shaded boxes, convexity in the form of dynamic insurance that allowed participation as markets continued to achieve historic upside, and then provided sensitive protection to subsequent downside, likely provided better investment outcomes than otherwise.
What do we advise folks worried about growing levels of uncertainty. We advise being “aggressively defensive”. In all honesty, we think there are always vast levels of uncertainty and that being “aggressively” defensive ought to probably be an investors natural state. Let us show you what we mean.
First off, let’s just start with the simple transformation from a traditional Balanced 60/40 portfolio (60% in S&P500 total return {SPXT Index} and 40% in US Treasury Bonds total return {LUATTRUU Index}). We then do our simple convex transformation in an Always Good Weather (AGW) portfolio, think of this as targeting the implications provided by our above Football/Soccer pitch picture (Figure 2). We get rid of the allocations to the bonds and split it equally between Nasdaq (our goal scorers) and Long Volatility (our goalkeepers). That leaves us with an AGW 60/20/20 (60% in S&P500, 20% in Nasdaq 100 total return {XNDX Index} and 20% in Long Volatility {our LongVol is a strapped together combo of the old CBOE Eurekahedge Long Vol Index and the new WITH Long Vol Index).
That gives us the below hypothetical historical results that we have shown so often. The superior convexity of the AGW delivers a positive return skew and better compounding over the period.
Figure 9: AGW 60/20/20 (blue) vs Balanced 60/40 (red). Scattergram and Return Distribution. March2009 – Oct2025
Source: Bloomberg, Convex Strategies
Figure 10: AGW 60/20/20 (blue) vs Balanced 60/40 (red). Compounding Path. March2009 – Oct 2025
Source: Bloomberg, Convex Strategies
Now, what if the respective investors in the two strategies decided they were concerned about elevated levels of uncertainty? What if they, given the current environment, wanted something more “defensive”?
Simple enough. Change the Balanced portfolio allocation from 60/40 to 40/60, a common expression of a defensive portfolio in the Sharpe World. Likewise, we can shift 20 of the S&P 500 allocation in the AGW over to the LongVol side, giving us 40/20/40. Let’s call these new hypothetical portfolio “Defensive AGW” and “Defensive Balanced”.
Figure 11: Defensive AGW 40/20/40 (blue) vs Defensive Balanced 40/60 (red). Scattergram and Return Distribution. March2009 – Oct2025
Source: Bloomberg, Convex Strategies
Figure 12: Defensive AGW 40/20/40 (blue) vs Defensive Balanced 40/60 (red). Compounding Path. March2009 – Oct2025
Source: Bloomberg, Convex Strategies
The hypothetical back tests on these “Defensive” portfolios, not surprisingly, show two things: 1) they both would have given up fairly significant returns over the chosen very strong equity market period, and 2) the Defensive AGW portfolio has even more significant positive return skew compared to the Defensive Balanced.
Now what if we went “Aggressively Defensive”? It is easy to anticipate what happens to the historical returns of continuing to reduce the equity exposure in the Defensive Balanced portfolio, so we will just leave that one as is. But we will construct a new “Aggressively Defensive AGW” portfolio with a 50% Nadsaq allocation and a 50% LongVol allocation.
Figure 13: Aggressively Defensive AGW 50/50 (blue) vs Defensive Balanced 40/60 (red). Scattergram and Return Distribution. March2009 – Oct2025
Source: Bloomberg, Convex Strategies
Figure 14: Aggressively Defensive AGW 50/50 (blue) vs Defensive Balanced 40/60 (red). Compounding Path. March2009 – Oct 2025
Source: Bloomberg, Convex Strategies
The hypothetical results on this “Aggressively Defensive AGW” are quite interesting. It has less risk, in terms of measures like Max Drawdown or Downside Vol or Downside Beta, but slightly higher returns. The more aggressive allocation to the negatively correlating asymmetry of the LongVol has allowed a shift up into our participating allocation into the higher upside beta of the Nasdaq.
We could take this another step. In what, in today’s nomenclature, is called “Portable Alpha” or “Return Stacking” we could utilize leverage and the efficiency of derivative markets to even more aggressively build our hypothetical defensiveness. Let’s dub this version “Stacked Aggressively Defensive AGW” and push up the Nasdaq weight to 60% as we 2x leverage the remaining 40% into a total 80% allocation to LongVol (we will use a short-term T-bill total return index {LD12TRUU} as a proxy for the funding cost on the additional 40%).
Figure 15: Stacked Aggressively Defensive AGW 60/80 (blue) vs Defensive Balanced 40/60 (red). Scattergram and Return Distribution. March2009 – Oct2025
Source: Bloomberg, Convex Strategies
Figure 16: Stacked Aggressively Defensive AGW 60/80 (blue) vs Defensive Balanced 40/60 (red). Compounding Path. March2009 – Oct 2025
Source: Bloomberg, Convex Strategies
The Scattergram and Return Skew images really start to jump out on these hypothetical back tests. We are creating something that is super convex. This is what being aggressively defensive allows you to do. Now we have hypothetical numbers that have even less downside risk but are adding compounded returns above the less defensive strategies. We would argue this is exactly how you deal with uncertainty; you get increasingly aggressively defensive. It is not about bullishness or bearishness. It is about convexity.
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