The solution to the New York Times Wordle puzzle on 16 September 2022 left a bad taste in the mouths of the online word game's enthusiasts. Many of these players lost their streaks because the solution, PARER, proved to be the hardest puzzle the game has yet presented.

We'll let Cracking the Cryptic's Mark Goodliffe demonstrate how hard it was by going through his thought process in tackling it:

Goodliffe's method is similar to that of many players. He starts with a test word, one that includes some of the more frequently occurring letters in the English language. He then takes the clues he gets from the game to choose the word he uses on the next line, and the process repeats until the sixth and final line is completed, if needed. Most players can solve the daily puzzle within four lines/guesses.

But a word like parer doesn't make it easy to use that process. There were too many viable options that could legitimately work, without even getting to the duplicate letter option.

So what would be a better approach?

If you recognize that the right way to measure your success at Wordle is not how many lines or guesses it takes you to reach the solution, but rather the amount of time you take to reach it, you can take advantage of a process of elimination. By using a series of test words that include the most frequently used letters in English words that are five letters long, you can extract a lot of information about the correct solution.

Recovering Physicist tapped a Scrabble-legal database of five-letter words to identify the letters most frequently used in them. Here's the list, from most common to least:

SEAOR ILTNU DCPYM HGBKF WVZXJ Q

Using this information, we identified some 277 sets of three five-letter words using the first 15 characters without repeating any characters. We then scored the words in each group according to the rank of their letters, where words using the most frequently used letters would get the lowest scores. For example, the score for a word like Goodliffe's test word of GRIME would be found by adding the ranks of its letters: G(17) + R(5) + I(6) + M(15) + E(2) = 45. Surely there must be a better first choice!

If you just went by the first five most frequently used letters, that word will be AROSE (15), but you won't have any test words to follow it if that guess doesn't provide you with enough clues. To have that option, from our 277 sets of words, the set with the lowest first word score is AISLE (19), MOUND (49), and CRYPT (52).

At that point, it occurred to us that it might be better to stack as many frequently used letters as possible into the first two words, ranking them by their combined score. For AISLE and MOUND for example, the combined score is 67.

Going through our list, we found the lowest combined scores, then filtered for the lowest first word score. Following that process, we think the optimal test words for Wordle are....

ARSON, CITED, and LUMPY

ARSON (or SONAR if you prefer) has a score of 22. CITED (or EDICT) would score 39, with a combined score of 61.

Between ARSON and CITED, you should be able to identify enough clues to solve most easier puzzles. If you need to use all three of these test words to get traction for your solution, you will at least be able to confirm what vowels are being used.

In the case of PARER, you could have used all three of these test words and with a bit of luck, have solved the puzzle in the fourth line. You would have identified A, R, E, and P as used letters, and knowing the location of E, you could set up PARE_ as the solution. You could reasonably identify D, R, or S as the last letter, but because the three test words eliminated D and S, you would select R and be rewarded.

But if you set up PA_ER, which would also be reasonable, you would have to consider the letters C, G, L, P, R, V, and Y as creating valid words, before eliminating C, L, and Y because you've already found they're not used. From here, your optimal choice would be to enter a new test word using letters you haven't previously used (G and V). If you used the word GLOVE, you would be down to just two possible options, P and R, with one line to go.

Your odds of winning at that final line then would have been 1 in 2, coming down to your selection of which letter between P and R to drop into that third space. If a 50% chance of winning seems too low, remember that if you're one of the majority of people who lost their Wordle streak on 16 September 2022, your chance of winning that day's game were never that high. The method we've described would have given you a fighting chance, while also helping you work out the solution faster.

P.S.: According to Recovering Physicist, the most frequently used letters found by those who extracted all the words from Wordle's database are:

EAROL TSINU CYDHP GMBFK WVXZQ J

We'll leave it as an exerise to you to identify your optimal test words based on this information!

What does the future hold for the dividends of the S&P 500 (Index: SPX) now that we're reaching the end of the third quarter of 2022?

We're now in the gap between when the index' dividend futures contracts for 2022-Q3 have expired and the actual end of the calendar quarter. We find the outlook for the S&P 500's quarterly dividends per share improved since we checked them near the end of 2022-Q2. What's more, we also have a first look for dividend futures data extending through the end of 2023. The following chart reveals those expectations before the start of trading on Monday, 26 September 2022:

Here's how the dividend futures forecast has changed for each quarter for which we presented data at the end of 2022-Q2:

• 2022-Q3: Up $0.29 per share.
• 2022-Q4: Up $0.63 per share.
• 2023-Q1: Up $0.73 per share.
• 2023-Q2: Up $0.40 per share.

As interest rates rise and recessionary pressures increase, we're starting to see firms like Fedex dial back their earnings forecasts, though not yet their dividends. It's an open question of how long that state of affairs can continue.

About Dividend Futures

Dividend futures indicate the amount of dividends per share to be paid out over the period covered by each quarters dividend futures contracts, which start on the day after the preceding quarter's dividend futures contracts expire and end on the third Friday of the month ending the indicated quarter. So for example, as determined by dividend futures contracts, the "current" quarter of 2022-Q3 began on Saturday, 18 March 2022 and will end on Friday, 16 September 2022.

That makes these figures different from the quarterly dividends per share figures reported by Standard and Poor, who reports the amount of dividends per share paid out during regular calendar quarters after the end of each quarter. This term mismatch accounts for the differences in dividends reported by both sources, with the biggest differences between the two typically seen in the first and fourth quarters of each year.

Reference

The past and projected data shown in this chart is from the CME Group's S&P 500 quarterly dividend index futures. The past data reflects the values reported by CME Group on the date the associated dividend futures contract expired, while the projected data reflects the values reported on 27 June 2022.

U.S. new home sales unexpectedly rose in August 2022. Here's how Reuters covered the most important numbers in the story:

New home sales surged 28.8% to a seasonally adjusted annual rate of 685,000 units last month, the Commerce Department said on Tuesday. July's sales pace was revised higher to 532,000 units from the previously reported 511,000 units....

The median new house price in August was $436,800, an 8.04% increase from a year ago.

What Reuters didn't say is that August 2022's new home prices fell $29,500 (-6.3%) from its revised level of $466,300 in July 2022. That previous month's median new home sale price was itself revised 6.1% higher from the $439,400 it was initially reported to be a month earlier.

At the same time, mortgage rates dipped slightly from July to August 2022, falling from 5.41% to 5.22%. The unexpectedly high number of new home sales would appear to be the result of homebuyers recognizing that respite would likely be very short lived, who then scrambled to lock in the month's combination of lower mortgage rates and lower sale prices.

We'll take a closer look at what those changes mean for new home affordability next week, but for now, we'll look at what the combination of revisions and new information involving new home sales means for the U.S. new home market. Political Calculations' initial estimate of the trailing twelve month average market capitalization for new homes in the U.S. in August 2022 is $29.81 billion, down 5.1% from our initial estimate of $31.40 billion for July 2022.

Meanwhile, our revised estimate for July 2022 is $31.47 billion, a 0.2% increase from our initial estimate, confirming this month represents the peak for this measure in 2022. The large, upward revision in new home sale prices almost fully accounts for that outcome. The latest update of our chart shows U.S.' new home market cap's shifted trailing twelve month average.

The next two charts show the latest changes in the trends for new home sales and prices:

Sales continued trending downward:

Upward price trend is slowing:

We've included the raw monthly data for new home sales and their average monthly sale prices in these latter two charts to illustrate the month-to-month noise in the data. The data for new home sales is typically finalized some three months after it is first reported.

Looking forward, we anticipate September 2022 will see both fewer sales and lesser affordability given the spike in mortgage rates during the month. The next data release for new home sales will come on 26 October 2022.

References

U.S. Census Bureau. New Residential Sales Historical Data. Houses Sold. [Excel Spreadsheet]. Accessed 27 September 2022. 

U.S. Census Bureau. New Residential Sales Historical Data. Median and Average Sale Price of Houses Sold. [Excel Spreadsheet]. Accessed 27 September 2022. 

The U.S. Census Bureau has released its annual estimate of the median income earned by the United States 131,202,000 households in 2021. At $70,784, the annual income earned by a typical American household increased by $2,774 (or 4.1%) from 2020 to 2021. Adjusted for President Biden's inflation however, real median household income fell by $402 year-over-year from 2020 in terms of constant 2021 U.S. dollars.

We've added the U.S. Census Bureau's annual data to our chart comparing the various available sources for U.S. median household income data that track this demographic characteristic on a monthly basis. The updated chart compares their nominal, noninflation-adjusted estimates over the period they overlap from January 2006 through July 2022.

In incorporating the U.S. Census Bureau's annual data in this chart, we've annotated the anomaly for 2019's median household estimate. Here, the Census Bureau's median household income estimate for 2019 was greatly impacted by the arrival of the coronavirus pandemic in March 2020, coinciding with the period in which it surveyed U.S. households about the income they earned in 2019. Lockdown measures imposed by several state goverments, including the high population states of California and New York, blocked the Census Bureau from successfully collecting as much data as they intended, particularly from lower-income earning households. In a working paper, Census Bureau analysts confirm the coronavirus pandemic reduced the number of responses to the Annual Social and Economic Survey used to collect household income data, which they believe biased the median household income figure upward by 2.8%. Our chart shows both the Census Bureau's reported figure of $68,703 and indicates the adjusted figure of $66,779 for that year.

Comparing the Census Bureau's annual estimates with the monthly estimates, we find that all these estimates range within a few percent of each other from January 2006 through October 2018. After that point, there's a major divergence between the Atlanta Fed's estimates and the others, where we continue to note the following three differences:

1. The Atlanta Fed's estimates understate the rising trend for median household income observed in 2019.
2. They also completely miss the impact and recovery from the coronavirus recession in 2020.
3. They significantly understate the robust growth in median household income we've observed since March 2021.

That difference, particularly with the U.S. Census Bureau's annual data, is growing much more substantial as time passes.

Political Calculations' estimates of median household income generally tracks with Sentier Research's estimates up through the period where they terminate in December 2019. The analysts who founded Sentier Research after retiring from the U.S. Census Bureau went on to permanently retire in 2020. Sentier Research is no longer an operating entity.

The Bureau of Economic Analysis will soon be releasing a major revision of its historic income data. Since Political Calculations uses this data in generating its monthly household income estimates, the chart above represents a snapshot of our estimates compared with other sources "before" the revised data becomes available.

References

Federal Reserve Bank of Atlanta. Home Ownership Affordability Monitor (U.S. Census Bureau American Community Survey One-Year Estimates of Median Household Income, Projected to Indicated Month by Atlanta Fed Staff using additional data produced by the Current Population Survey and Decennial Census). [Online Database]. Accessed 23 September 2022.

Sentier Research. Household Income Trends: January 2000 through December 2019.  [Excel Spreadsheet with Nominal Median Household Incomes for January 2000 through January 2013 courtesy of Doug Short]. [PDF Document]. Accessed 6 February 2020. [Note: We've converted all data to be in terms of current (nominal) U.S. dollars.] Note: Sentier Research is no longer an operating entity, we've linked to the Internet Archive's copy of this final report.

Political Calculations. Median Household Income in July 2022. [Online Article]. 1 September 2022.

U.S. Census Bureau. Historical Income Tables: Households. Table H-5. Race and Hispanic Origin of Householder -- Households by Median and Mean Income. [Excel Spreadsheet]. 13 September 2022.

Jonathan Rothbaum and Adam Bee. Coronavirus Infects Surveys, Too: Survey Nonresponse Bias and the Coronavirus Pandemic. U.S. Census Bureau Working Paper Number SEHSD WP2020-10. [PDF Document]. 30 March 2021.

The S&P 500 (Index: SPX) continued falling in the third full trading week of September 2022, closing the week down 4.65% from the previous week's close, and a full 23.0% below its 3 January 2022 peak.

This decline is consistent with how the dividend futures-based model would set stock prices if the model's basic multiplier was suddenly reset from the value of -2.5 that has applied since 16 June 2021 to be +2.0 as of 13 September 2022, assuming investors are also focusing their attention on the distant future quarter of 2023-Q2:

We're surprised by how closely the trajectory of the S&P 500 is tracking along with the model's projection of the index' trajectory associated with investors focusing their attention on 2023-Q2. That's because the model has also entered a period when we expect its projections are being affected by the past volatility of the historic stock prices it uses as the base reference points for projecting the index' future potential trajectories based on how far into the future investors are looking. That's something we'll revisit next week, when we'll estimate how much of the decline in stock prices is attributable to noise and how much is signal based on how investors have changed their view of the market.

Speaking of which, we find other analysts are picking up on just how different today's market is from that of just two weeks ago. Here's how Reuters covered that development on Friday, September 23, 2022:

Across Wall Street, banks are scrambling to adjust their forecasts to account for a Federal Reserve that shows no evidence of letting up in its fight against inflation after delivering another market-bruising rate hike this week and signaling more severe monetary policy tightening ahead.

Once-reliable technical indicators are falling by the wayside. The S&P dipped below its mid-June low of 3,666 on Friday afternoon, erasing a sharp summer rebound in U.S. stocks – the first time in history the index breached a new low after erasing more than half of its losses.

A rout in bond markets added to the pressure on stocks — yields on the benchmark 10-year Treasury, which move inversely to prices, recently stood at 3.67%, their highest level since 2010.

“These are uncharted waters,” said Sam Stovall, chief investment strategist at CFRA Research. “The market right now is going through a crisis of confidence.”

Or rather, the market has entered a new regime. Here are the market-moving headlines from the week that was:

Monday, 19 September 2022

• Signs and portents for the U.S. economy:
• Oil edges up as supply woes outweigh demand and rate hike worries
• U.S. road travel fell 3.3% in July as gas prices remain high
• U.S. home builder sentiment falls for ninth straight month in September
• Bigger stimulus developing in China:
• China to accelerate projects, boost consumption to spur recovery
• China central bank cuts 14-day reverse repo rate, steps up cash injections
• BOJ minions starting to think inflation in Japan may not be temporary:
• 'Temporary' disappears as BOJ contemplates accelerating inflation
• ECB minions thinking about looking at data before next rate hikes:
• Further interest rate hikes will depend on data, ECB's de Guindos says
• Wall Street ends higher after choppy trading ahead of Fed

Tuesday, 20 September 2022

• Signs and portents for the U.S. economy:
• U.S. two-year yield at almost 15-year high before Fed meeting
• Oil prices down, investors expect big Fed rate hike
• Bigger stimulus developing in China:
• COVID-hit Shanghai unveils $257 billion in projects in infrastructure push
• Bigger trouble developing in the Eurozone:
• German producer prices post surprise record jump in August
• Central bank rate hike global mania!
• Indonesia to raise rates by another 25 bps in Sept, some call for 50 bps: Reuters poll
• Philippines central bank to go for another 50 bps rate hike on Thursday
• Australia's central bank says hikes could slow at some point
• Sweden lifts interest rates by full percentage point with more to come
• ECB's Lagarde raises prospect of rate hikes beyond neutral level
• BOJ minions facing higher-than-target inflation, Japanese government readies more stimulus:
• Japan's inflation hits near 8-year high, stays above BOJ's target
• Japan govt will spend $24 billion in budget reserves to cope with price hikes
• Japan PM Kishida faces pressure to ramp up spending as approval slides
• Wall Street falls as Fed, Ford forecasts, give fright
• Ford stock has biggest daily drop since 2011 after inflation warning

Wednesday, 21 September 2022

• Signs and portents for the U.S. economy:
• U.S. mortgage interest rates reach 6.25%, highest level since October 2008
• Printing the future: New factory tech reshapes the U.S. industrial economy
• U.S. home sales drop for 7th straight month, house price growth cooling
• U.S. rents surge, leaving behind generation of younger workers
• Oil prices slide 1% after U.S. Fed raises interest rates
• Gold price jumps as Fed’s Powell talks chance of recession and restrictive rates
• Fed minions hike rates as expected while increasingly expecting face recession:
• Fed delivers big rate hike, sees another large move higher in 2022
• Fed forecasts show fraying faith in soft landing
• The Fed's latest rate hike: five ways Americans may feel the pain
• Fed goes big again with latest rate hike; Powell vows to 'keep at it'
• BOJ minions want leader to continue never-ending stimulus policies, half of economists think BOJ minions won't stop yen drop:
• BOJ's dovish deputy Amamiya top candidate as next head - Reuters poll
• Japan unlikely to intervene to stem weak yen, half of economists say - Reuters poll
• ECB minions says more rate hikes on top of recession needed to reduce Eurozone inflation:
• Recession alone won't subdue euro zone inflation - ECB's de Guindos
• Wall Street slumps in wake of Fed rate announcement
• "Maybe 4-5% Inflation Is The New Normal" - Wall Street Reacts To Powell's Hawkish Surprise

Thursday, 22 September 2022

• Signs and portents for the U.S. economy:
• U.S. mortgage rates rise to 6.29%, highest in 14 years
• Big U.S. banks' prime rate soars to highest since 2008 financial crisis
• Oil edges higher on Russian supply concerns in volatile trade
• U.S. leading indicator falls for sixth consecutive month in August
• Bigger stimulus developing in China:
• China's largest policy bank to boost infrastructure loans to local govts
• Central banks continue rate hike mania:
• Central banks unleash 350 basis points more of rate hikes in inflation fight
• ECB must keep raising rates despite downturn, Schnabel says
• Euro zone inflation broadening and will continue to rise, ECB's Schnabel says
• Analysis-Europe says goodbye to negative rates - or just 'au revoir'?
• Indonesia central bank raises rates by 50 bps to head off inflation risks
• Swiss National Bank exits negative rates era with 0.75% hike
• Bank of England hikes rates by 50 bps as economy slows
• Norway lifts rates to 2.25%, expects smaller hikes ahead
• BOJ minions refuse to quit never-ending stimulus, have to prop up yen to stop it from collapsing:
• Bank of Japan keeps ultra-low rates, dovish policy guidance
• Japan intervenes in FX market to stem yen falls after BOJ keeps super-low rates
• Yen jumps as Japan intervenes in currency market for first time since 1998
• Wall Street ends down for third day as growth concerns weigh on tech

Friday, 23 September 2022

• Signs and portents for the U.S. economy:
• U.S. business activity remains subdued in September - S&P Global survey
• Oil plunges to eight-month low on strong dollar, recession fears
• Bigger trouble developing in the Eurozone, United Kingdom:
• Euro zone likely entering recession as price rises hit demand -PMI
• German economic downturn deepens in Sept, outlook grim -flash PMI
• UK downturn deepens, raising recession risk -flash PMI
• Bigger stimulus developing in the U.K.:
• UK's Kwarteng to scrap top rate of income tax, cut basic rate from April
• More central bank rate hikes on tap:
• BoE needs to hike rates next week to calm markets - Deutsche Bank analyst
• Swiss National Bank ready for further steps to combat inflation - Jordan
• Hungary central bank seen hiking rate by 100 bps to 12.75% - Reuters poll
• Wall St tumbles as investors fret on rate hikes and recession
• Dow poised to confirm bear market status as recession fears mount
• 'Uncharted' sell-off in U.S. stocks has investors wondering when the pain stops

After the Fed hiked the Federal Funds Rate to a target range of 3.00-3.25%, the CME Group's FedWatch Tool projects a three-quarter point rate hike when the FOMC meets in early November (2022-Q4), followed by a half point rate hike in December (2022-Q4). In 2023, investors are expected at least a quarter point rate hike in February (2023-Q1) and another in June (2023-Q2), bringing the FFR to a target range of 4.75-5.00%. After this peak in 2023-Q2, the FedWatch tool indicates the Fed will respond to developing recessionary conditions with quarter point rate cuts projected in July (2023-Q3) and December (2023-Q4).

Meanwhile, the Atlanta Fed's GDPNow tool's projection for real GDP growth in the soon-to-end quarter of 2022-Q3 dropped again for the third consecutive week, from 0.5% to 0.3%. The Bureau of Economic Analysis will provide its first official estimate of real GDP growth in 2022-Q3 at the end of October 2022. The BEA has previously indicated the first two quarters of 2022 experienced negative real growth, but that may change as the BEA's analysts release their annual revisions.