The S&P 500 (Index: SPX) closed the trading week ending Friday, 28 August 2026 at 7,711.76, up 0.4% from where it closed out the preceding week.

The biggest market moving event of the week came on Friday, as the new chair of the Federal Reserve spoke at the Fed's annual retreat in Jackson Hole, Wyoming. Warsh had one message he wanted to deliver, which was the Federal Reserve would act to hike interest rates while inflation in the U.S. exceeded the Fed's target for it.

That message affected investor outlook for rate hikes. The CME Group's FedWatch Tool moved up the timing of when it expects the Fed will hike the Federal Funds Rate since the previous edition of the S&P 500 chaos series. It now projects the Fed will hike this base interest rate by a quarter percent on 16 September (2026-Q3), three months sooner than anticipated a week earlier.

The FedWatch Tool now also gives a better than 50% chance the Fed will follow up with another quarter point rate hike after its 9 December (2026-Q4) meeting.

Looking further forward, having the two rate hikes in 2026 drops the likelihood of any additional rate hikes in 2027 from the FedWatch Tool's outlook. The latest update of the dividend futures chart shows investors maintained their forward-looking focus on the upcoming future quarter of 2027-Q1, however if just given the week's news, we think it would be very likely that investors are shifting their forward-looking attention to the nearer term future of 2026-Q4.

For the near term future trajectory of the S&P 500, that Lévy flight event would be accompanied by a relatively modest change in stock prices, assuming no other new information arrives to prompt investors to shift their investment horizon back out to a more distant future quarter. Which if it were significant enough, would be accompanied by a more significant upward movement in stock prices according to the dividend futures-based model's projections.

Such a move could be driven by something like the unexpected after-the-closing-bell news of the Trump administration's deal with Venezuela to secure control of more than 65 billion barrels of oil, for example. We'll be covering the impact of that news in next week's edition.

There's a lot that hinges on the random onset of new information that affects investor expectations for the future. Speaking of which, here's an example of what that kind of information looked like during the past week.

Monday, 24 August 2026

• Signs and portents for the U.S. economy:
• US hits Canadian goods with 50% tariffs after trade talks fail
• Canada to retaliate for US tariffs, worsening ties after talks fail
• Trump threatens 50% tariffs on all cars and trucks from Canada amid trade fight
• U.S. Trade Rep Greer blames Canada for failed tariff talks: ‘They wanted more’
• Oil drops more than $2 despite new US sanctions on Iran
• US threatens countries doing business with Iran, but holds off on penalties for now
• Fed minions not so worried about rising interest rates on U.S. Treasuries, new Chief Minion faces first big test:
• Fed's Kashkari not worried about rising US Treasury yields
• Bond market anxiety raises stakes for Warsh's debut Jackson Hole speech
• Bessent could tap near $1 trillion Treasury General Account to fund bond buybacks, sources said
• Chief ECB minion getting ready to jump ship and cash in on job she's long coveted:
• ECB chief Lagarde 'ready to serve' WEF, Swiss newspaper reports
• S&P 500, Nasdaq end down on tech stocks, investors weigh Iran moves
• Stock market ends mixed as tech stocks, yields slide
• Bitcoin jumps 3%, closing in on $80,000 as rally gathers pace
• Gold hits over three-month high on dollar weakness, Treasury bond buyback plans

Tuesday, 25 August 2026

• Signs and portents for the U.S. economy:
• Why the bond market may be resetting expectations about the US
• US Treasury to stick to debt auction schedule despite bigger buybacks, Bessent says
• US Treasury buybacks a 'mistake' costing credibility, says Druckenmiller
• Oil hits one-week low as investors shrug off US sanctions on Iran
• July new home sales miss estimates
• Fed minion wants higher interest rates in U.S.:
• US rates need to rise soon absent evidence of ongoing drop in inflation, Fed's Collins says
• Four Fed bank boards wanted rate hike, minutes show
• Bigger trouble, stimulus developing in China:
• China credit slump could hit U.S. stocks, Société Générale’s Albert Edwards warns
• US eyes China overcapacity tariffs of 7.5% before Xi-Trump talks, Bloomberg News reports
• China's $119 billion policy financing tool begins project applications, faces roll-out lag
• BOJ minions getting excited to hike Japan's interest rates again:
• BOJ to speed up its tightening campaign, raise key rate to 1.25% in September
• Japan's corporate services inflation accelerates in July
• Wall Street ended higher ahead of Nvidia earnings and key inflation data
• Wall Street ends higher as tech rebounds before Nvidia results

Wednesday, 26 August 2026

• Signs and portents for the U.S. economy:
• US inflation remains elevated as GDP growth outlook brightens
• Core PCE inflation rises as expected in July, headline PCE accelerates
• Americans’ spending outpaces income for record 25 months
• Mortgage rates hit highest level in 3 weeks, weakening demand further
• Canada slaps retaliatory tariffs on US goods as trade war intensifies
• Oil prices slide 2% on Iran-Oman talks to reopen Strait of Hormuz
• Bond rout ending? Massive bets on bond rally dominate options market
• Fed minions only a bit more likely to hike U.S. interest rates:
• Fed seen a bit more likely to hike after inflation data
• Bigger trouble developing in China:
• Bill Gates, alarmed by AI, has policy ideas he wants to discuss with China's Xi
• ECB minions want one more interest rate hike in Eurozone:
• Interest rates must rise further on inflation risks, ECB's Schnabel tells Bloomberg
• ECB set for September rate hike with no appetite to signal more, sources say
• Wall Street edged lower after inflation data and ahead of Nvidia earnings
• Wall Street stocks end tad lower after hot inflation data ahead of Nvidia earnings

Thursday, 27 August 2026

• Signs and portents for the U.S. economy:
• Nvidia forecasts 70% sales growth next year, signals AI spending boom has years left to run
• Oil gains as traders weigh prospects of Hormuz reopening
• OPEC+ loses oil market sway in Iran war as China gains influence
• New Chief Fed minion has got some problems:
• Fed's Warsh faces challenge whether inflation is a problem or not
• Bessent, Warsh diverge on who should set the price of money
• As Jackson Hole conference kicks off, three Fed officials issue inflation warnings
• Fed's Collins says latest US inflation readings are 'mixed'
• Fed's Hammack tells Fox Business inflation likely to ease slowly
• BOJ minions getting excited to hike Japan's interest rates some more:
• BOJ deputy governor Himino calls for 'timely' rate hike
• ECB minions see growth signs in Eurozone, want to hike interest rates:
• Euro zone lending growth picks up, ECB says
• ECB saw a further hike as likely at July meeting
• Wall Street ended higher after Nvidia results as eyes turn to Jackson Hole
• Nasdaq, S&P 500 lifted by Nvidia's forecast; investors eye speech by Fed's Warsh
• Nvidia ignites chip stock rally as forecast shows AI boom is intact
• Nvidia forecasts 70% sales growth next year, signals AI spending boom has years left to run

Friday, 28 August 2026

• Signs and portents for the U.S. economy:
• US employment growth only modestly lower than thought in BLS revision
• Oil prices fall for the week on U.S.-Iran talks stalemate
• New Chief Fed minion says to expect hikes if inflation stays above minions' target:
• Warsh signals Fed may need to hike rates if above-target inflation persists
• Warsh's 'quieter Fed' now includes a smidgen of guidance
• VIEW Rate-hike expectations rise on Warsh speech at Jackson Hole
• The August Labor Market Report To Cement The September Fed Hike
• Mixed growth signs, stimulus developing in China:
• China's factory activity seen contracting again in August
• China’s largest five banks report strongest first-half profit since height of property crash
• China home prices seen falling slightly less this year, property investment slump deepens - Reuters poll
• China to reform home sales system to curb delivery risks
• Shanghai plans subsidies to kick start dormant offshore debt market, sources say
• BOJ minions excited to see Japan's inflation near their target, even more excited for chance to hike interest rates again as they tally the cost of propping up the yen:
• Tokyo core inflation creeps near BOJ's target, bolsters rate-hike chance
• Japan spent record $96.5 billion to support yen over past month, ministry data shows
• ECB minions worried about missing out on blockchain:
• ECB should embrace blockchain to safeguard its role, Schnabel says
• S&P 500 falls Friday after Fed’s Warsh highlights inflation worries, but index posts positive week
• Wall Street ends lower after Fed Chair Warsh reaffirms inflation fight
• Stocks close lower as Warsh vows to bring inflation down
• S&P 500 posts weekly gains as Nvidia fuels renewed AI rally

The Atlanta Fed's GDPNow tool projects +4.6% real GDP growth for the U.S. economy in 2026-Q3, rebounding from the +4.0% annualized growth it forecast a week earlier.

Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of the new Federal Reserve chief pointing to a PowerPoint slide that shows a train labeled 'RATE HIKES' heading toward a Wall Street bull tied to railroad tracks by a villain holding a sign that says 'INFLATION ABOVE TARGET'." We had to add a follow on prompt to "make the Fed chief look like Kevin Warsh" because the AI image generator is still defaulting to a caricature of Jerome Powell.

A long time ago, before studio executives sucked all the remaining lifeforce out of it, fans of Star Wars eagerly anticipated the release of a new movie in the series. One that would finally continue the epic story of the original trilogy after years of diversions with mostly unsatisfying prequels.

The year was 2015. The movie was Star Wars: Episode VII - The Force Awakens. And though it was something of a rehash of the original trilogy, the talented visual and special effects team who made it succeeded in creating a new character that captured the fandom's imagination.

We are, of course, referring to the BB-8 astromech droid, which in retrospect, was perhaps the most realistic character introduced in the new movie because it was based on new, but existing technology. Not proto-AI, not CGI, but actual patented mechanical and electrical components. The movie's producers put a featurette together to show off how they did it.

Meanwhile, the studio executives made sure there was a BB-8 Droid toy available in stores when the movie came out as part of their vision for sucking the lifeforce out of Star Wars by hoovering up cash from the movies' fans.

Including droids as characters that could be easily turned into marketable toys has always been their purpose in the Star Wars franchise. Because when you really think about what their roles are in the Star Wars universe, they don't make much sense. For example, why wouldn't the people who make X-wing fighters just build-in whatever it is that droids do in X-wing fighters into the X-wing fighters themselves?

When you think about it that way, why on earth would anyone even try to make a robot like the BB-8 droid with the form factor of a ball with a static component that rides on top of it? Where would it ever make sense to have a technology like that?

Over ten years after BB-8's cinematic introduction, we might finally have an answer to that question that actually does make sense. Rollo Robotics has created the world's first autonomous and self-balancing monowheel robot, which in addition to being inspired in part by BB-8, has potential applications in mind for which it would be useful.

The following video introduces the company's prototype monowheel robot technology:

If it pans out, the monowheel robot's cost effective means of monitoring and patrolling open areas of ground could reshape how security is provided at facilities that require it. Beyond that, monowheel robots could find use as probes for exploring the surfaces of the moon and planets.

But as Core77's Rain Noe observed, the technology has a big challenge that would limit its adoption: stairs.

But to overcome that challenge, perhaps we can turn to another depleted sci-fi/fantasy series for the obvious solution:

We wouldn't say the Daleks' solution to their problems with stairs makes much sense either, but at least somebody is out there thinking about these things!

August 2026 was a good month overall for the Thanksgiving Leftover Stocks of 2025.

Compared to their July 2026 snapshot, both our hypothetical indices of the ten worst stocks within the S&P 500 (Index: SPX) during 2025 saw month-over-month gains. The market cap-weighted index of these stocks increased from 87.4% to 91.4% of its value on the day after Thanksgiving 2025, while the equal-weighted index grew more, rising from 88.5% to 94.7%.

Both these indices are still lagging behind the overall S&P 500 index. The benchmark index increased from 108.2% to 112.1% of its day-after-Thanksgiving Day 2025 value in the month from the July to August snapshots.

The following chart shows the performance of all three sets of stocks, with the two Thanksgiving Leftover stock indices continuing to lag behind the S&P 500 index by a wide margin.

It's worth noting why the equal-weighted version of the ten stock index is performing better than the market cap-weighted version. The largest component of the market-cap weighted index is Chipotle Mexican Grill (NYSE: CMG), which accounts for 23.5% of its value. Shortly after the July 2026 snapshot, Chipotle's stock plunged when jalapeno peppers served at the chain's restaurants in Minnesota were linked to an outbreak of salmonella.

Although Chipotle acted quickly to pull all potentially affected jalapenos from its restaurants, investors sent its stock down sharply, losing nearly 16% of its value in a week. Since then, Chipotle's stock has largely recovered to its pre-jalapeno recall level.

That recovery however lagged behind the improvement of several other Thanksgiving Leftover stocks, which gave the edge to the equal-weighted version of the index. The stock of Gartner (NYSE: IT) led the month, rising from 63.4% to 82.5% of its value on 28 November 2026 as investors shook off some of the AI disruption discounting they had earlier imposed on it. the stock price of Factset Research Systems (NYSE: FDS) also saw outsized gains for the same reason, rising from 95.6% to 106.6% of its post-Thanksgiving Day 2025 level.

Two of the individual Thanksgiving Leftover stocks lost notable value over the past month. Deckers Outdoor (NYSE: DECK) declined from 111.0% to 101.6% of its 28 November 2025 value, while the stock price of Trade Desk (NASDAQ: TTD) continued to fall through its continually lowering floor.

The spaghetti chart tracks the relative movements of 2025's ten Thanksgiving Leftover stocks during the last nine months with respect to their value on the day after 2025's Thanksgiving holiday.

Nine months after Thanksgiving 2025, five of the S&P 500's Thanksgiving Leftover stocks have risen above their 28 November 2025 level, while the other five have dropped below it.

In cased you missed it, our extended discussion of The Trade Desk's woes as the worst of the 2025's Thanksgiving Leftover Stocks is available here. We'll check back in with the Leftover Stocks near the end of September 2026.

Political Calculations' initial estimate of the total value of new home sales in the United States during July 2026 is $28.41 billion. This value is slightly lower than the initial estimate of $28.62 billion we presented in our previous update that covered new home sales data through May 2026.

Since then, the number of new home sales has been trending downward thanks largely to an increase in mortgage rates in recent months, which has made new homes relatively less affordable during this time. The number of new home sales has dropped to its lowest level since the start of the year.

One surprising development however is that the average new home sale prices has also declined in recent months. Builder incentives are making new homes less expensive than existing homes.

As far back as modern records go, newly built homes have almost always cost more than previously owned homes.

It only makes sense: New homes are expected to have far fewer maintenance issues, brand new appliances, and designs suited to contemporary tastes, plus they can be customized to suit the homebuyer's needs.

After all, it is only in very rare circumstances that a used car would cost more than a similar make and model purchased brand new at the dealer's lot.

But in recent months, that trend has been upended to an extent that has never been seen, with the typical new home selling at a sharp discount to existing homes.

In June, the $407,200 median sales price of new homes was about $28,000 less than for existing homes, a 6.5% discount and by far the biggest inversion in at least 25 years.

Last month, the gap narrowed a bit to nearly $19,000, or 4%—still significantly larger than any discount seen before this year. July also marked the fourth straight month of price inversion for new homes, the longest stretch on record.

The following charts present the U.S. new home market capitalization, the number of new home sales, and their average sale prices as measured by their time-shifted, trailing twelve month averages from January 1976 through July 2026.

New home sales trending down:

New home prices also trending down:

We'll update our measure of the relative affordability of new homes sometime in the next week.

References

U.S. Census Bureau. New Residential Sales Historical Data. Houses Sold. [Excel Spreadsheet]. Accessed 25 August 2026. 

U.S. Census Bureau. New Residential Sales Historical Data. Median and Average Sale Price of Houses Sold. [Excel Spreadsheet]. Accessed 25 August 2026. 

Image Credit: An aerial view of a lot of houses photo by Modunite Ltd on Unsplash.

We recently discussed how advances in technology are contributing to boosting the productivity of mathematicians during the last thirty years. But the chart we featured in that article depicting how the number of math papers being published each month has nearly doubled in the last three years may not do full justice to how fast the pace of new papers coming out has changed.

Another way to communicate that acceleration in output is Matt Parker's approach of focusing on a small subset of the new papers being published in the last few months to show how they represent a big increase in the capability of the AI large language models used to generate their findings. In the following video, in what is becoming an annual tradition, he interrupts his vacation to cover breaking news in the world of mathematics.

Parker makes an explicit point of noting the use of the Lean proof assistant to verify their findings, which gives greater confidence the findings in these papers will stand up to serious scrutiny.

Not all the preprint papers that have recently populated the mathematics category of the arXiv database meet that standard, as Reddit math community poster Salt_Attorney recently observed.

By contrast, Lean verification was used in the ten advances spanning several disciplines in mathematics OpenAI claims its Astra AI system has made. As such, they have a much better chance they'll be found valid when the findings are reviewed.

But the question now being raised is whether verification of findings by a proof assistant is enough to accept AI-generated results as presented. For further reading on that topic, we'll recommend Gary Marcus' discussion of OpenAI's claimed accomplishment to appreciate it may not be either as grand or the Astra AI system as capable as presented in OpenAI's announcement.

The S&P 500 (Index: SPX) dropped almost 1.4% from its previous week's close to wrap up the trading week ending on Friday, 21 August 2026 at 7,678.76.

Rising bond yields was perhaps the biggest driver of stock prices during the week, which comes as the U.S. government is increasingly having to compete with Big Tech to borrow money as the company's seeking to build out the infrastructure to support the expansion of Artificial Intelligence (AI) systems are borrowing big to do it.

One outcome of that dynamic is expectations of higher interest rates. The CME Group's FedWatch Tool projections of the expected future for how the Fed will set the Federal Funds Rate changed little in the past week. It anticipates a 60% chance the Fed will act to hike this core interest rate to a target range of 3.75-4.00% on 28 October (2026-Q4), while giving a much stronger 98% chance this rate will be in effect on 9 December (2026-Q4). Beyond that, the FedWatch Tool now anticipates another quarter point rate hike on 28 April (2027-Q2).

Meanwhile, stock prices behaved almost exactly as would be expected if investors were tightly focusing on 2027-Q1 as they set the level of the week's stock prices. The latest update of the alternative futures chart shows that outcome as the S&P 500's trajectory closely paced the dividend futures-based model's projection associated with investors fixing their attention on the distant future quarter of 2027-Q1.

Investors had quite a lot of other new information to absorb during the trading week. Here is the summary of the week's market-moving headlines:

Monday, 17 August 2026

• Signs and portents for the U.S. economy:
• 30-Year Treasury Yield hits 5.30%, its highest level since 2007
• Oil settles up over $2 as Iran war stalemate stokes supply concerns
• Brent oil rises above $90 as Iran rules out interim deal extension, threatens to escalate conflict
• Depleted strategic oil reserve nears level that raises concerns about damage to caverns, operations
• Economists predict Fed minions to leave interest rates unchanged in rest of 2026:
• Fed to hold interest rates this year, economists say, sticking to their view
• Bigger trouble, stimulus developing in China:
• China's recovery sputters as consumption, output lose steam
• China investment slump deepens as economy shows signs of weakness
• China new home prices stagnant in July as demand stays weak
• China's premier calls for stabilising external demand as growth sputters
• Bigger trouble developing in Japan, BOJ minions catch a break in effort to prop up Japan's currency:
• Japan Q2 growth misses forecasts on weaker spending, investment
• Japan second-quarter GDP grows 1.1% on an annualized basis, missing expectations
• Yen edges up as traders push back Fed rate hike bets
• Japan’s 10-year bond yield hits three-decade high
• Bigger trouble developing in Eurozone:
• Europe's heatwaves expose insurance gap as business losses mount
• China’s unplanned surpluses put EU in trade bind
• ECB minions say AI stocks are due for a fall:
• AI market correction is coming, ECB blog predicts
• Wall Street finished lower as investors look ahead to retail earnings
• Wall Street indexes slip as oil prices rise, retail results awaited

Tuesday, 18 August 2026

• Signs and portents for the U.S. economy:
• As US debt mounts, investors demand higher returns to lend
• US 30-year yields hit highest level since 2007 as war, oil worries fester
• US yields edge lower despite Iran worries, broader sell-off
• Oil closes at three-week high as hopes of US-Iran peace deal fade
• US diesel crack surpasses $100 a barrel for the first time on supply disruptions
• Fed minions say their monetary policy is "accommodative":
• One neutral rate estimate suggests Fed's policy stance is accommodative, paper says
• Bigger trouble, stimulus developing in China:
• China's easing steel output reflects divergent demand drivers
• China has one good reason for its credit slump
• China’s 10-year bond yield falls to 13-month low
• Bigger trouble developing in Japan:
• As Japanese Bond Yields Soar, Unrealized Losses At Life Insurers Hit $200 Billion
• Bigger trouble developing… everywhere:
• Global bond markets put governments on notice over fiscal, inflation risks
• Wall Street finished in the red as attention remains on the Middle East
• Tech selloff weighs down Wall Street as bond yields climb
• Explainer: Treasury yields are rising - why does it matter?

Wednesday, 19 August 2026

• Signs and portents for the U.S. economy:
• US housing market remains under pressure in July; factory output rises
• Oil extends climb on prolonged Hormuz export uncertainty
• Demand from US data-center boom radiates out through factory supply chains
• US 30-year Treasury yields drop from multi-year highs
• Treasury Secretary Bessent doubles US long-bond buybacks in the face of surging yields
• VIEW Yields fall after US Treasury says it will double some bond buybacks
• Port of Los Angeles posts high July volume, helped by retail goods, AI project parts
• Fed minions worried about inflation back in July 2026, just realized increase in productivity blunted it:
• Fed policymakers' inflation concerns increased at July meeting, minutes show
• Boston Fed paper says strong productivity blunted tariff inflation impact
• Bigger trouble developing in China:
• China seen holding loan rates steady in August despite economic weakness
• China robot makers seek to turn humanoid hype into useful work
• Bigger trouble developing in Japan:
• Japan bond yields near 3% as inflation, fiscal worries mount
• Japan has few answers as bond rout puts fiscal plans at risk
• Wall Street ended higher, and yields dipped after the latest Treasury Department announcement
• Wall St rises as yields ease, Moderna lifts healthcare stocks

Thursday, 20 August 2026

• Signs and portents for the U.S. economy:
• US Treasury buyback briefly eases bond rout, but debt worries persist
• Treasury's Bessent says upsized bond buybacks could increase further
• Oil settles up more than 2% after Trump threatens countries supporting Iran
• U.S. military says it aided passage of 660 million barrels of oil through Strait of Hormuz since May
• Fed minion says they're not sure how Fed will set U.S. interest rates in September 2026:
• Fed's Musalem says on CNBC he won't prejudge rate call view for September FOMC
• Bigger trouble developing in and out of China:
• China boosts imports of Russian crude, stymieing India's refiners
• BOJ minions / Bigger trouble developing in Japan:
• Japan's exports surge on chips demand, imports hit record on oil costs
• Bigger trouble developing in the Eurozone:
• Europe’s oil refining capacity set to shrink by 20% over next decade
• Wall Street ended lower as Walmart dragged down retail and oil climbed
• Wall Street sinks as bond yields rise, Walmart results disappoint
• Walmart sales growth slowdown tests consumer resilience, shares slide 9%

Friday, 21 August 2026

• Signs and portents for the U.S. economy:
• US service sector fuels acceleration in business activity, S&P Global says
• Oil rises as Trump threatens sanctions on Iran partners
• US Mortgage Rates Fall for Second Straight Week, to 6.65%
• Fed minions try not to upset their real boss:
• Fed officials tread carefully after Treasury's bond market intervention
• BOJ minions potentially planning rate hikes up through July 2027:
• A date on the calendar could shape the BOJ's endgame on interest rates
• Japan's core inflation accelerates in July, bolsters case for rate hike
• Bigger trouble developing in Eurozone (hawkish ECB minions):
• Traders are bracing for an increasingly hawkish ECB
• S&P 500 drops for week as Tech leads sector losses
• Dow surges 500 points Friday, but index posts back-to-back weekly losses
• Gains in AI company stakes juice second-quarter earnings for S&P 500
• US corporate AI debt surge tests investor limits as fatigue emerges
• Wall Street ended higher as gold and Bitcoin rallied

The Atlanta Fed's GDPNow tool anticipates +4.0% real GDP growth for the U.S. economy in 2026-Q3, dipping from the +4.3% annualized growth it projected a week earlier.

Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a Wall Street bull holding up a sign that says 'RISING BOND YIELDS' who is growling at scared investors"

It may not seem like it, but the millennia-old academic discipline of mathematics has been undergoing a productivity revolution during the last thirty years.

That revolution is being enabled by the adoption of several new technologies, each of which is following a S-shaped logistic growth curve model. Mike Roberts of Strategic Tool Kits describes the pattern of how technological advancements increase performance over time:

The S curve is a strategic concept that describes how the old ways mature and are superseded by new ways.

In the early days of new technology, it takes a long time to improve performance. People are working out the technology, and the applications, ironing out the flaws, and building the ecosystem. Slowly over time, performance accelerates.

After a certain time, the rate of improvement hits a peak and then starts to slow down. Easy wins have all been made and the learning curve has been fully ridden for continuous improvement. Some of the fundamental barriers of the technology are reached. Eventually, the improvement tapers off and a plateau with this technology is reached. This is the “S” shaped curve.

When a new productivity-enhancing technology comes along after the older technology reaches its mature phase, the process repeats with performance building on its old plateau and rising to new heights.

For mathematicians, the two great technological improvements of the twenty-first century have been the widespread adoption of the arXiv database for publishing preprint papers to more quickly communicate their discoveries and the more recent development of Artificial Intelligence (AI) systems paired with proof assistant systems like Lean that are helping automate large portions of their work.

ArXiv provides data on the number of new papers that are uploaded to its preprint paper database each month. The following chart reveals its data for math papers, not counting cross listings from other categories of academic papers, from January 1992 through July 2026:

The data shows adoption of the arXiv database for communicating advancements in maths went through the full S-curve advancement cycle, reaching a fully mature phase by the early 2020s with output holding fairly level for several years. But after ChatGPT (3.5) was launched at the end of November 2022, the Large Language Model (LLM) technology facilitiated a steady increase in the productive output of mathematicians for the first time in years. By making it possible to automate a portion of writing academic papers, the number of papers published to arXiv each month began to rise in a steady, linear trend.

But soon after OpenAI's o3-mini LLM-based reasoning model was released on 31 January 2025, the rate of output for publishing new papers to arXiv exploded in what appears to be the exponential growth phase of the S-shaped logistic growth pattern. The new cost efficient o3-mini reasoning model was specifically developed to automate analytical tasks in Science, Technology, Engineering, and Mathematics fields and also coding.

For mathematics, the new AI technologies make rapid advancements possible by drawing on the body of published work, like the thousands of math preprint papers documented in the arXiv database and other resources that catalogued large numbers of unresolved conjectures in digital-friendly formats, to test new possible ways of testing those conjectures. When promising proofs or disproofs of the conjectures are identified, the technology automates their verification using the proof assistant software technology that has also come into its own during this period. Using proof assistants also has the benefit of coding any new successful proofs into their proof libraries, which can then be mixed and matched as needed to test other conjectures.

When that's done, the remaining step for the mathematicians orchestrating what's effectively become a massive, automated collaboration exercise is to write up a new preprint paper and publish it to the arXiv database.

Mathematics isn't the only field experiencing a boom in preprint papers. Economics has likewise seen the number of papers published each month almost double since ChatGPT 3.5's public release. Unlike math however, economists don't have the equivalent of maths' proof assistants to verify their findings, which raises questions about the quality and validity of the flood of new papers in the field.

References

arXiv. Math Submissions. [Online Database]. Accessed 8 August 2026.

arXiv. Monthly Submissions. [Online Article and CSV Data]. Accessed 8 August 2026.

Ecology.net. Logistic Growth. [Online Article]. 15 December 2025.

Pablo Groisman. Math papers uploaded to arXiv per month, January 1992 to July 2026. [Online Article]. 3 August 2026. [Our article was inspired by Pablo's chart!]

Mike Roberts. S Curve – What is it? [Online Article]. 13 July 2024.

ScriptByAI. OpenAI & ChatGPT Timeline: GPT Release Dates to GPT-5.6 (2026). [Online Database]. Accessed 20 August 2026.

Image credit: S Curve Concept in Science by Oliver Tacke on Wikiversity. Creative Commons CC BY-SA 3.0 Attribution-ShareAlike 3.0 Unported Deed. 17 July 2012.

When we launched the S&P 500's Thanksgiving Leftover project the day after Thanksgiving 2025, we knew we were going to spend the next year following the stock prices of companies that weren't doing very well. After all, to even make the list, the ten companies whose stocks we would track ranked as the S&P 500's worst performing stocks of 2025.

In the nine months since then, a few of those stocks have outperformed the index, while the rest have lagged behind. Most of those stocks have fallen below their post-2025 Thanksgiving Day level, but not by anywhere near as much as they had fallen to qualify as one of the S&P 500's worst performing stocks in 2025.

But one stock in particular has gone on to plumb new depths. It has continued to fall so much more that it is on track to qualify as one of the S&P 500's worst performing stocks of 2026.

That stock is The Trade Desk (NASDAQ: TTD), the digital advertising firm analyst David Desjardins believes is facing an "existential crisis". Here's how he describes the company's now nearly two year long fall from grace:

After reporting highly disappointing financial results for the second quarter of 2026, shares of The Trade Desk, Inc. (TTD) declined by a whopping 21.9% last Friday, which came on top of a 6.8% decline on the prior day. Since the publication of my initiating coverage in early February 2026, TTD's stock price has basically been cut in half, from ~$27.00 per share at the time of publication to around $13.39 as of today's market close.

Relative to its all-time high of $141.53 reached in December 2024, The Trade Desk has now declined by a massive ~90%. As David Einhorn famously said, a stock down 90% is just a stock that was down 80% before being cut in half again, and this is exactly what happened to TTD since last February. The depth of TTD's sell-off is quite something, but what is even more impressive to me is its speed....

At this point, The Trade Desk has become one of the most hated stocks that I am aware of, and this is on top of being the worst-performing constituent in the S&P 500 (SPX) on a year-to-date basis. Pretty much everything said or written about the company is negative, and it is precisely why I decided to write an update today. In less than two years, TTD went from a market darling that could do no wrong at over 26.0x forward sales to being viewed as a melting ice cube changing hands at 2.3x forward sales today.

The following chart compares The Trade Desk's stock performance with the S&P 500, from 29 November 2024 (aka "the day after Thanksgiving Day 2024) through 18 August 2026:

Believe it or not, despite the company's continued misfortune, Desjardins views the company's low stock price as a speculative strong buy opportunity, where he makes the argument that the company has some potential for a turnaround based on its available cash balance, lack of debt, and cash flow.

We disagree, because we think The Trade Desk has further to fall before it might reach that point.

Here's why. According to SlickCharts, The Trade Desk's market cap has fallen to where the company now ranks 502 out of the 503 stocks that make up the S&P 500 index. Because it has, and because its fall is continuing, the company's stock is verging on the point where S&P will act to remove it from the index. If and when that happens, as increasingly seems likely, its stock price will experience the opposite of what happens when a company's stock is included in the index, which is to say its stock price will fall even further.

A deeper decline is almost ensured given the negative outlook CEO Jeffrey Green communicated during the company's 2026-Q2 earnings call. Gytis Zizys, who formerly held a buy rating for the company on the hope it will see a turnaround, reacted to that development:

The Trade Desk, Inc. (TTD) provided one of the worst guidances I’ve seen in recent months, which put the last nail in the coffin for many shareholders who were still clinging to the idea of a turnaround. It seems I was prematurely too bullish on the turnaround as well, and this report is forcing me to downgrade it to a hold. I don’t think there’s a point in selling at these low prices, unless you want to harvest some tax losses. If it gets to under $10 per share, I will be jumping in to see what happens over the next couple of years. It’ll either recover, or my investment will go to zero.

The only problem with this investing strategy is we can argue that the bar for being able to beat TTD's stock performance is very low. It's so low that investing almost anywhere else or just parking the money in a cash savings account would be more advantageous.

This article is a standalone feature in our ongoing Thanksgiving Leftover series, which will continue with its regular monthly installment later this month. The ongoing tragedy of the performance of The Trade Desk's stock demanded a special edition.

Something new is coming to the world of sports: real futures trading.

On 11 August 2026, the world's largest futures and options trading house, CME Group (NASDAQ: CME) announced it would partner with the National Hockey League and a startup called FutureSports to launch the world's first index-based hockey futures contracts.

Unlike the kind of futures you might find at multiple sports wagering outfits that might involve things like betting on which team will win the Stanley Cup next year or which goalie will win the Vezina Trophy, the kind of futures contracts now being developed for the NHL have a lot more to do with giving sports-related businesses and investors new tools to hedge their risks in addition to simple speculation.

The Chicago Tribune gives an overview of what the CME Group/FutureSports/NHL partnership is looking to bring about:

Futures contracts, tied to the value of indexes, allow investors to buy or sell an asset at an agreed price by a specific date. It also allows traders to hedge risks.

For decades, futures contracts have primarily been associated with agriculture products, like corn and soybeans, or energy commodities, such as oil and natural gas, as well as stock indexes like the S&P 500. Futures even got the Hollywood treatment with the 1983 film “Trading Places,” when Dan Aykroyd and Eddie Murphy’s characters cashed in on orange juice futures, and over 20 years later, inspired a federal provision nicknamed the Eddie Murphy Rule.

FutureSports and CME’s new offering means if the Blackhawks play poorly, for example, a season ticket holder could take a short position on the team’s index to try and recoup some of their ticket investment losses. Corporate sponsors, which spend millions annually on sports deals, could buy hockey futures to limit their exposure if a key athlete gets injured or the team just fails to meet expectations.

The companies highlighted other potential market participants like garage and parking lot operators, retailers and even the sports franchises.

There are real businesses and people who have real money on the line that depend on how well a team performs. For example, consider one of the Chicago Blackhawks' star players, Connor Bedard. One of the top offensive players in the NHL when healthy, Bedard's career has been repeatedly knocked by injuries, forcing him to miss playing in significant parts of several seasons.

Every time he's been sidelined, the Blackhawks' ability to score goals and win games has been notably reduced. While not the only reason for the team's bottom-of-their-division status in recent seasons, his absence from the ice when injured has certainly been a contributing factor.

With futures trading however, the businesses whose revenues rise and fall with the team's fortunes would have a way to cushion the losses they might otherwise face. For example, if Connor Bedard isn't playing, it's tougher to sell hockey sweaters with his name and number on them. Or to sell Connor Bedard Funko-pop figurines at Chicago's sporting goods stores. Or to fill restaurants near the United Center on the Blackhawks' game nights. Hedging using futures contracts could make a lot of sense for the owners of these businesses to offset their otherwise unmitigated loss of revenue and potentially even stabilize it in the face of an adverse event like a key player not being able to play.

Futures trading falls into the kind of higher risk investing we consider to be suitable mainly for well-established and well-funded operators. It's definitely not for the faint-of-heart or for those who have low tolerances for risk. The Chicago Tribune's article understates that aspect of the venture:

Trading futures is complex and its investors are savvy. Even retail traders, who buy for themselves and often make smaller trades, use similar financial analysis tools as institutional investors.

“Anyone considering trading them should understand the mechanics, costs and risks first and seek professional guidance as appropriate,” said Joseph Cusick, senior vice president and portfolio specialist at Calamos Investments in Naperville. “Futures traders can experience rapid gains or losses because of margin requirements.”

That's putting it mildly, especially since real futures trading might involve using leverage (or borrowing) to fund an inherently speculative investment.

It will be interesting to see how the proposed NHL futures market plays out. If all goes as planned, NHL futures will go live 28 September 2026. Right in time for the NHL's 2026-27 season to get underway.

Image Credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a Wall Street bull and bear playing hockey where the scoreboard shows 'CME Group / FutureSports / NHL Index' with value of 7500".