“In most parts of the world, children are taught the mathematics of certainty, not uncertainty.” Gerd Gigerenzer, Summer 2025.
The above quote comes from yet another wonderful note from Gerd Gigerenzer, this time reminiscing on his relationship and intellectual battles with the recently departed Nobel Laureate Daniel Kahneman.
View of The Legacy of Daniel Kahneman:
It is a wonderful note, touching both on the personal aspects of their relationship, as well as the scientific disagreements which they very publicly debated. Those disagreements, as we have discussed often (e.g. Convex Strategies | Risk Update: November 2024 – “Rationality Wars”), came to be known as the “Rationality Wars”, a term that Gerd discloses his distaste for in this note, not wanting scientific debate to be framed in the linguistics of war.
We are, ourselves, quite outspoken about the shortcomings of exactly the sort of education that Gerd references in this quote. In last month’s Update, “Aggressively Defensive” Convex Strategies | Risk Update: November 2025 – “Aggressively Defensive”, we quoted ourselves, in public exchanges we have had with Ole, in reference to the unparalleled new textbook on the subject, Ole Peters and Alex Adamou’s “An Introduction to Ergodicity Economics”, stressing this exact point.
“It (An Introduction to Ergodicity Economics) 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.”
This is the mathematics of uncertainty. It is the mathematics that factors in that oh so tricky variable, the one that the mathematics of certainty (aka Sharpe World) makes so much effort to ignore, “time”.
Here is another great summary note from Ole Peters that wraps up much of what you can learn about in more detail in his book.
The ergodicity problem in economics
“… it turns out a surprising reframing of economic theory follows directly from asking the core ergodicity question: is the time average of an observable equal to its expectation value?… Famously, ergodicity is assumed in equilibrium statistical mechanics, which successfully describes the thermodynamic behaviour of gases. However, in a wider context, many observables don’t satisfy equation (1). And it turns out a surprising reframing of economic theory follows directly from asking the core ergodicity question: is the time average of an observable equal to its expectation value?” Ole Peters, December 2019.
For those who would really like to get their mind twisted on the topic and the complexity of time, we gift you with this thought provoking discussion from legendary physicist Richard Feynman.
Richard Feynman Explains Time Like You’ve Never Seen Before
“So, what is time? Time is the thing that stops everything from happening at once. Time is the heat death of the universe waiting to happen. Time is the curvature of space caused by matter. Most importantly, time is the mirror. It reflects the limitations of your own brain…Real time, physical time, is a wild untamed beast that moves at different speeds, stops near black holes, might not even flow at all.” Richard Feynman.
Mind-warping stuff! It is indeed complicated. We would, however, argue that it is best to try to cope with it, as opposed to simply ignoring it. This gets to our persistent rant about focusing on resilience, anti-fragility, initial conditions, not on pointless predictions of an unknowable future. Manage your car, your balance sheet, your investment portfolio, your health, so that you are resilient to unforeseen outcomes, so that divergences from expectations are a relative benefit, not a disbenefit. As we put it last month:
“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.”
Two of the most relevant purveyors of bad mathematics, those who ignore the implications of time in non-ergodic realms, are the central banking luminaries from the US, the Federal Reserve (Fed), and from Japan, the Bank of Japan (BOJ). Both made key policy decisions this month.
Let’s start with the Bank of Japan. As signalled well in advance, the BOJ hiked their policy rate another 25bp, raising it to the lofty level of 0.75%.
https://www.boj.or.jp/en/mopo/mpmdeci/mpr_2025/k251219a.pdf
They once again provided this lovely clear brochure to explain the factors behind their policy decisions.
Figure 1: Bank of Japan December 2025 Monetary Policy Meeting Explainer
Source: Bank of Japan
Nice and clear. We would point out a couple of the issues which they highlighted. First, they again referred to the mythical ‘Underlying CPI inflation”, this time in a formal official policy meeting announcement, a metric that has never been explicitly explained or numerated. Second, they are explicit that their current policy setting still entails real interest rates that are “significantly negative” steering to overall financial conditions that are still accommodative with the intent to “firmly support economic activity”. We certainly would not disagree with this perception.
Figure 2: Real Policy Rates vs CPI yoy%
Source: Bloomberg, Convex Strategies
The Bank of Japan obviously stands out above, but nobody seems to be in a particular rush to get back to their 2% inflation targets.
The implications of ongoing accommodative policies have been, at least in the short period since the policy meeting, taken to mean that the BOJ wants more of what they have been getting from their policy. A weaker currency. A stronger equity market. Higher bond yields. And, presumably, faster increases in their price stability measure. It seems to be working.
Figure 3: Japan CPI ex-Fresh Food yoy% (white). BOJ Policy Rate (blue). USD/JPY FX Rate (papaya). JGB 10yr Yield (purple). Nikkei Equity Index (yellow). 2016-2025
Source: Bloomberg, Convex Strategies
They do, however, expect further ongoing reductions in the scale of accommodation should the economy continue along its expected path, i.e. further rate hikes to come. Our guess is that they will continue with such on a very gradual path.
“As for the future conduct of monetary policy, given that real interest rates are at significantly low levels, if the outlook for economic activity and prices presented in the October Outlook Report will be realized, the Bank, in accordance with the improvement in economic activity and prices, will continue to raise the policy interest rate and adjust the degree of monetary accommodation.” Bank of Japan, December 2026. `
You can get a broader perspective on the differing views amongst the policymakers when the BOJ subsequently releases their Summary of Opinions.
Summary of Opinions at the Monetary Policy Meeting on December 18 and 19, 2025
As we saw very clearly at the October meeting, where two members dissented to the Committee’s decision to hold the policy rate unchanged, there are some differing opinions. At least one member would agree with the market’s seeming interpretation of the recent policy announcement.
“The yen’s depreciation and the rise in long-term interest rates reflect to some extent the policy interest rate being too low relative to the inflation rate. Raising the policy interest rate in a timely manner could curb future inflationary pressure and lead to holding down long-term interest rates.”
We can’t resist showing the below picture. Going back to the old musing that Japanese bank equity values and JGB yields shared the commonality of both being on their way to zero. It is pretty clear, that relationship, while reversing in direction, still holds.
Figure 4: Topix Bank Index (papaya) vs JGB 10yr Yields. 1990-2025
Source: Bloomberg
Over on the other side of the Pacific Ocean, the Fed, also trying to avoid getting back to their 2% price stability target too quickly, chose to cut the Fed Funds Rate 25bp, the third such cut since September, down to 3.75%. Again, this was very much in line with market expectations and prior signalling from key FOMC members. There was, which is becoming a bit of a theme after many years of total consensus voting, another round of dissenting votes. Two dissents to hold rates unchanged, and one to go for an even larger 50bp cut.
Federal Reserve Board – Federal Reserve issues FOMC statement
Arguably, more interesting than the widely anticipated rate cut was the introduction of a new form of asset purchases, quite explicitly framed as not Quantitative Easing (QE). In the official policy statement, they gave this little standalone sentence on the topic.
“The Committee judges that reserve balances have declined to ample levels and will initiate purchases of shorter-term Treasury securities as needed to maintain an ample supply of reserves on an ongoing basis.” FOMC statement, December 2025.
Fed Chair Jerome Powell, ever so sparsely, elaborated on this new (sort of) practice of purchasing shorter-term Treasury securities in his official statement at the subsequent press conference.
Transcript of Chair Powell’s Press Conference — December 10, 2025
“In support of our goals, and in light of the balance of risks to employment and inflation, today the Federal Open Market Committee decided to lower our policy interest rate by ¼ percentage point. As a separate matter, we also decided to initiate purchases of shorter-term Treasury securities solely for the purpose of maintaining an ample supply of reserves over time, thus supporting effective control of our policy rate.” Jerome Powell, December 2025.
After the usual mumble about views of the economy and whatnot, he eventually got back to this technical adaptation of policy. Again, stressing that it was not in any way a part of their monetary policy stance.
“Let me turn now to issues related to the implementation of monetary policy, with the reminder that these issues are separate from—and have no implications for—the stance of monetary policy.” Jerome Powell, December 2025.
He then explained the practical execution of this, that being the purchase of $40 billion per month of US Treasury bills, which was actually in a formal announcement from the Federal Reserve Bank of NY, not the FOMC. There were no specifics about possible time horizons or overall size limitations. As best we can tell, this process has been assigned to FRB NY and changes, expansions, reductions, will no longer be at the discretion of the voters on the FOMC.
Amazingly, in our opinions, the issue only came up as one follow up question during the Q&A portion of the press conference. Chair Powell brushed it aside as nothing more than a technicality to manage to their new favourite catch phrase – “ample reserves”.
“So, you know, we announced that we’re resuming reserve management purchases. That is completely separate from monetary policy. It’s just we need to keep an ample supply of reserves out there.” Jerome Powell, December 2025.
Not surprisingly, this generated quite a bit of discussion around market commentators on the concept of QE vs not-QE. When considering the early days of QE under Chair Bernanke and his persistent insistence that it would always be temporary and, as such, should not be considered “monetization”, we have decided to dub whatever this version is as “perpetual monetization”.
The Fed has ended its brief experiment in QT, thus are now reinvesting all their maturing proceeds, plus now buying the additional $40bio of bills every month. Giving us at least one answer to our oft-asked question: “Who is going to buy the bonds?” Their balance sheet will commence growing with immediate effect.
Figure 5: Federal Reserve Assets. 2000-2025
Source: Bloomberg
There are any number of interesting commentaries on this topic. This one, from Bill Nelson of the Bank Policy Institute comes with a very nice table reflecting the evolving scale of bank reserves.
https://www.linkedin.com/pulse/forward-guidance-fed-needs-seen-working-hard-get-smaller-bill-nelson-jalve/ Bill Nelson, Chief Economist and Head of Research, Bank Policy Institute
Figure 6: Fed Estimates of Minimum Required Bank Reserves
Really quite remarkable. The new policy would seem to entail that there is only one direction of movement: ever larger.
Here is a short and simple note from Thomas Hoenig, former President of the Kansas City Fed, that echoes many of our own surface-level concerns.
The Fed’s QE—and the Claim of a Technical Adjustment
“It is surprising that the media failed to ask for more detail as to why restarting large open market purchases of US Treasuries was not a restart of quantitative easing (QE), a monetary policy action that perhaps should have required a FOMC vote. The purchase of $40 billion per month of Treasury securities, if it continues through May, as Chairman Powell hinted, would be an annual increase of 7 percent in bank reserves and if continued for all of 2026, would be an increase of 16 percent. Such increases will exceed projected GDP growth over these periods and are a substantial increase in liquidity for the financial system. Such purchases through May would equal 10 percent of the government’s deficit, and if continued through year-end, would equal almost 25 percent.” Thomas Hoenig, December 2026.
Perpetual monetization.
This rolls back into past discussions, and criticisms, from the likes of Treasury Secretary Scott Bessent and future Fed Chair candidate Kevin Warsh, about the Fed’s conflicting roles. While pursuing their supposed “dual mandate” of price stability and maximum employment, it is hard to miss their ever-leaning bias towards their third mandate, that of financial stability, and the possible conflicts with their supervisory/regulatory role of financial institutions.
This is a good note on the topic from Amit Seru in Project Syndicate.
How the Fed Became a Lender of Immediate Resort by Amit Seru – Project Syndicate
“The greatest threat to independence of the US Federal Reserve does not come from President Donald Trump’s attack or a Supreme Court ruling that might expand his authority. It is the Fed’s longer-term shift from lender of last resort to lender of immediate resort. Without a clear distinction between temporary liquidity support and protection for insolvent institutions, the Fed’s independence turns into cover for ad hoc bailouts, and monetary policy becomes a hostage of weak institutions and authorities’ reluctance to admit supervisory failure.” Amit Seru, December 2025.
We have long asked the question of our various central banking friends – “What if it works?” We titled our February 2021 Update just that Convex Strategies | Risk Update: February 2021 – What if it works?. What happens if the efforts towards monetary debasement succeed? We discussed this at length in the February 2021 Update and might argue that through 2021 and 2022 the world got a good taste of some of the negative externalities. Updating one of the charts that we showed back then now looks like the below.
Figure 7: US M2 Money Supply (white) vs Copper (papaya), Corn (purple), Shiller 20 City Home Price Index (fuchsia), CPI Index (blue). Normalized. 2000-2025. Feb 2021 (white vertical)
Source: Bloomberg, Convex Strategies
It may not be fair but, given what has gone on this year, we can’t help but add to this chart two more components: gold and silver.
Figure 8: US M2 Money Supply (white) vs Copper (papaya), Corn (purple), Shiller 20 City Home Price Index (fuchsia), CPI Index (blue), Gold (yellow), Silver (light blue). Normalized. 2000-2025. Feb 2021 (white vertical)
Source: Bloomberg, Convex Strategies
Is currency debasement working? Are there signs of classical inflation, i.e. the growth of money and credit, sneaking into the repricing, in debasing currency terms, of items that may not be captured by narrow consumer pricing indices? Pretty darn difficult to argue against it.
Take for example the UK’s FTSE 100 Index which has rocketed to fresh all-time highs.
Figure 9: UK FTSE 100 Index. 1990-2025
Source: Bloomberg
Is the FTSE Index the beneficiary of currency debasement? Well, if you look at the performance of Gold relative to the currency, GBP, you would certainly come to that conclusion.
Figure 10: XAU/GBP Spot Rate. 1990-2025
Source: Bloomberg
The ratio of the FTSE 100 to Gold makes the debasement argument rather strongly.
Figure 11: UK FTSE 100 Index (white). XAU/GBP Spot Rate (papaya). Ratio of FTSE 100 Index to Gold (lower panel). 1990-2025
Source: Bloomberg
In fact, one could say, it is all debasement.
Of course, there is no need to pick on the UK. We can say the same thing about Europe.
Figure 12: Europe SX5E Index (white). XAU/EUR Spot Rate (papaya). Ratio of SX5E Index to Gold (lower panel). 1990-2025
Source: Bloomberg
We can say the same thing about Japan.
Figure 13: Japan Nikkei Index (white). XAU/JPY Spot Rate (papaya). Ratio of Nikkei Index to Gold (lower panel). 1990-2025
Source: Bloomberg
We can even say the same thing about the almighty US equity market, though it has done a lot better than most.
Figure 14: US S&P 500 Index (white). XAU/USD Spot Rate (papaya). S&P 500 Index Ratio to Gold (lower panel). 1990-2025
Source: Bloomberg
Debasement is all around.
For an absolutely exceptional discussion around the implications of accumulated imbalances/fragility in Complex Adaptive Systems, we cannot recommend highly enough a listen to our dear friend and mentor, William (Bill) White, on the popular podcast Top Traders Unplugged. Few, if any, explain it better than Bill has done over the last two or three decades.
GM93: The Calm Before a System… – Top Traders Unplugged – Apple Podcasts
One of the highlights was Bill’s uniquely clear explanation of what is popularly known as “fiscal dominance.
“All of a sudden people are starting to grasp the nature of debt dynamics. Which is, if debt levels are high enough and short enough, that interest rates only need to go up a little, relative to the nominal growth rate of the economy, to insure debt sustainability (levelling out of debt/GNE ratio) you need primary surpluses as a proportion of GDP that are so great that you look at them and you say ‘this is not going to happen’. In which case you are into a situation every year where you know the debt/GNE is going higher and higher. The ultimate end of that is governments find they cannot fund themselves at a reasonable rate in the market. So, they turn to the central banks.” William White, January 2026.
We stand firmly in the camp that we are already well into the situation Bill is describing. It is the very bifurcation, significantly influenced by the carved-in-stone underlying demographic issues, that is driving economic systems and financial markets in their critical states towards some inevitable future phase transition.
We can get some sense of the impact of this ongoing debasement by taking a look at the performance of the hypothetical “Anti-Fiat Portfolio” that we introduced back in our June 2025 Update – “I See No Ships” Convex Strategies | Risk Update: June2025 – “I See No Ships”. We constructed the Anti-Fiat Portfolio to roughly match, over the chosen lookback period, the downside volatility risk of a traditional 60/40 US Balanced Portfolio comprised of 60% S&P 500 total return (SPXT Index) and 40% US Treasury total return (LUATTRUU Index). That kicked out weights of 35% Gold (GLD US Equity), 15% Bitcoin (XBTUSD), and 50% LongVol (our bootstrapped version of the old EurekaHedge Long Volatility Index and the new WITH Long Volatility Index). For this analysis, we will cover January 2017 through December 2025 and, as always, assume annual rebalancing.
Figure 15: Hypothetical Anti-Fiat Portfolio vs 60/40 Balanced Portfolio. Scattergram and Return Distribution. 2017 -2025
Figure 16: Hypothetical Anti-Fiat Portfolio vs 60/40 Balanced Portfolio. Compounding Path. 2017 -2025
Source: Bloomberg, Convex Strategies
Hypothetically, similar risk (downside volatility and max drawdown of 6.2% and 15.3% for the Anti-Fiat vs 6.5% and 19.6% for the Balanced Portfolio), more return (CAGR of 15.8% for Anti-Fiat vs 9.2% for Balanced). In other words, Anti-Fiat has a far superior Sortino Ratio over the Balanced Portfolio. The scattergram (figure 15) shows the very compelling convexity of the Anti-Fiat Portfolio, obviously following our principles of being “Aggressively Defensive” with the heavy 50% allocation to the LongVol component.
We can, as always, look at the relationship from the other side, instead of same risk and more return, we can do same return and less risk. In this case, we can get that by comparing the same Anti-Fiat weightings against a 100% investment in S&P 500.
Figure 17: Hypothetical Anti-Fiat Portfolio vs 100% S&P 500. Scattergram and Return Distribution. 2017 -2025
Source: Bloomberg, Convex Strategies
Figure 18: Hypothetical Anti-Fiat Portfolio vs 100% S&P 500. Compounding Path. 2017 -2025
Source: Bloomberg, Convex Strategies
Now the two have more similar returns, still 15.8% for Anti-Fiat vs 14.2% for the go-go 100% long-only S&P 500 portfolio, but that portfolio has significantly more risk (downside vol of 10.6% and max drawdown of 23.9%). The Anti-Fiat, by our calculations, has a very useful Downside Beta of -0.15 and, yet hypothetically captured the equivalent returns, and more, of one of the best Beta runs ever. We will leave readers to imagine what could be done, per our “Aggressively Defensive” examples last month, were we to play around with weights, risks, stacking and the like.
Our point here is that “debasing’ is not a new thing. Indeed, you could argue it has been going on, in some form or another, since governments took control of money. We just happen to be living in a particularly large, advanced version. We agree with Mr. White: central bankers, and their central planning partners on the government side, all rely on the wrong models. If they keep following those models, in a misspent belief that they can control complex economic systems, we would expect to get more of the same. More debasement, more asset inflation, more fragility. Bill quotes one of his oldest friends, Charles Goodhart, who sums it up very nicely.
“There is absolutely nothing, in any of these models, that is of significant interest to a central banker” William White quoting Charles Goodhart, January 2026.
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