Political Calculations
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24 December 2014

What was the most significant story in mathematics in 2014?

NASA's Real World: Mathematics - Source: http://www.nasa.gov/audience/foreducators/nasaeclips/

We thought we'd take on answering that question in our final post of the year. What we specifically set out to identify is the biggest story of 2014 where the application of maths either was or became the story.

Would it be one of PhysOrg's top-ranked mathematics news stories for 2014? Stories that included a novel ranking method for measuring the influence of academic papers or the resolution of a three decade-old controversy on how to determine the location of atomic and molecular resonances? Or perhaps even the reported proof of the Umbral Moonshine Conjecture, which has the potential to be applied to applications covering the gamut from number theory to quantum physics?

Actually, no, because all these things represent pretty esoteric applications of maths that would seem to have limited application to date.

So we dug deeper, seeking out stories where applied maths were making an impact in the real world.

There was the story of how three students from Yale University's computer science and math departments solved the Kadison-Singer Conjecture, which validates the use of the characteristics of a system that we can observe and measure to quantify the characteristics of a system that we are not able to observe and measure. In our imperfect world of limited abilities, that's a pretty big deal, where the applications run from how we might model a social network, like the thefacebook, to very complex economic systems or even quantum particles.

Or the story of how a 169-year-old algorithm for solving systems of linear equations, such as the kind of computational mechanics that might be used to model the flow of air over an aircraft's wing, had been revamped to be nearly 200 times faster, greatly increasing the speed at which an optimum design for such a structure can be developed.

And if you like a good detective story, how statistics was used to uncover potential criminal fraud in Florida's state lottery, where one suspiciously "lucky" man won 252 times, taking home $719,051. And what's more, there are indications that sort of highly improbable thing is happening in a lot more states that just Florida.

But ultimately, the biggest math story of the year belongs to a different kind of fraud, one that reaches all the way into the Oval Office. And as luck would have it, it can be considered to be a kind of sex scandal involving politicians!

The story has everything to do with a set of statistics that the White House first trumpeted in a January 2014 report "Rape and Sexual Assault: A Renewed Call to Action" and later again in its April 2014 report "Not Alone: The First Report of the White House Task Force to Protect Students from Sexual Assault", which was produced by a special task force established by President Barack Obama through a "Presidential Memorandum" on 22 January 2014 following the first report. The second report made the following claim, which was used to justify the need to act to effectively suspend the civil liberties of male students faced with allegations of sexual assault and to impose collective punishments upon them without due process at universities and college campuses across the United States in the name of "safety", or perhaps more accurately, for the purpose of advancing a key component of the President's political party's 2014 election campaign strategy:

One in five women is sexually assaulted in college. Most often, it’s by someone she knows – and also most often, she does not report what happened. Many survivors are left feeling isolated, ashamed or to blame. Although it happens less often, men, too, are victims of these crimes. The President created the Task Force to Protect Students From Sexual Assault to turn this tide. As the name of our new website – NotAlone.gov – indicates, we are here to tell sexual assault survivors that they are not alone. And we’re also here to help schools live up to their obligation to protect students from sexual violence.

The use of the "one in five" statistic began to unravel almost immediately, as a number of people, most notably University of Michigan economics professor Mark Perry, realized that when this statistic is combined with a second claim that only 12% of student victims of sexual assault report their allegations to law enforcement officials, that they cannot possibly both be true.

OK, let's do some math using crime data from the University of Michigan-Ann Arbor for 2012:

1. Number of female UM students: Approximately 21,000

2. Expected number of sexual assaults if one-in-five women is sexually assaulted in college: 4,200

3. Actual number of reported sexual assaults at UM in 2012: 34

4. Chances of a female UM student being sexually assaulted each year: 1-in-618.

5. Chances over four years that a student will be assaulted while attending college: 4-in-618 or 1-in-155.

That's nowhere close to 1-in-5.

If you accept the White House claim that 88% of college sexual assaults are not reported, you then get:

1. Estimated number of sexual assaults: 283 total, 34 reported (12%) and 249 unreported (88%).

2. Chances of a female student being sexually assaulted each year: 1-in-74 (or 1.35% chance).

3. Chances over four years of a female UM student being sexually assaulted: 1-in-18.5 (or 5.4% chance).

There's still nowhere close to a 1-in-5 (and 20%) chance of a female student being sexually assaulted while attending the University of Michigan, using the White House's own under-reporting statistics. See a similar analysis here for three Pittsburgh-area colleges (University of Pittsburgh, Carnegie Mellon University, and Duquesne University). I think maybe we need a "renewed call for the White House (and DOJ) to report accurate statistics on sexual assault (intimate partner violence)."

If you would like to do similar math for yourself, using the data that applies to the college campus of your choice, here's a tool that you can use (the default data applies to the University of Wisconsin at Madison, a reputed hotbed of sexual assaults among all U.S. college campuses).

Data for Sexual Assaults on Campus
Input Data Values
Number of Reported Sexual Assaults (During Four Year Period)
Estimated Percentage of Sexual Assaults That Are Reported to Authorities [%]
Estimated Percentage Involving Female Students [%]
College Campus Student Population Data
Total Student Population on Campus [in Single Year]
Percentage of Female Students [%]

Estimated Total of Sexual Assaults on Campus
Calculated Results Values
Total Sexual Assaults on Female Students (During Four Year Period)
Percentage Probability of Female Student Experiencing Sexual Assault
Odds of Female Student Experiencing Sexual Assault (1 in ...)

Support for the claim among the President's political party supporters in the media was so strong that Pulitzer Prize-winning political columnist George Will, who cited Mark Perry's math in a column attacking the validity of the "one in five" claim, had his column dropped from a major newspaper while also losing speaking engagements at college campuses.

Flashing forward to today, as updated campus sexual assault figures for the years 1995 through 2013 have been released by the U.S. Department of Justice, we find that the "one in five" statistic has been fully debunked, with the real figure being 1 in 41 for all the years covered by the report, and 1 in 52.6 for the four most recent years from 2010 through 2013.

The collapse of the "one in five" claim, and the weakness of the statistics behind it, can be seen in the actions of the President's political supporters in office and in the media, who are now trying to quietly drop their previous touting of the figure from public view.

GILLIBRAND DROPS DISPUTED SEX ASSAULT STATISTIC: Sen. Kirsten Gillibrand has dropped an increasingly disputed sexual assault statistic from her website. ChangeDetection.com shows how Gillibrand’s sexual assault resources web page [http://1.usa.gov/1m2xWoB] no longer includes [http://bit.ly/1C3Fi6M] a sentence citing the National Institute of Health Campus Sexual Assault Study, which concluded that one in five college women will be subject to rape or attempted rape. Gillibrand and others all the way up to President Barack Obama have cited that statistic in their push for colleges to better prevent sexual violence. But critics and media outlets [http://wapo.st/1AvC1JA] have noted the study’s flaws: It included only two large four-year universities and had a low rate of response, with more nuanced findings than lawmakers suggest. (H/T @mstratford)

— Gillibrand spokesperson Glen Caplin declined to tell Morning Education why the stat was removed. “There are some who attack this statistic to claim that sexual assault on college campuses is not a problem,” he said. “They need to get their head out of the sand. The problem is real and it is pervasive. Without this distraction, their argument has no merit.”

The rise and fall of the distractingly false claim that "one in five women is sexually assaulted in college" is the biggest math story of the year.

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23 December 2014

Having started December 2014 seeing something really weird for new home prices in October 2014, we anticipated that the record prices that were initially recorded for the month would be revised considerably downward.

And so they were! The median new home sale price for October 2014 was revised downward by nearly 5%, from $305,000 to $290,100, while the average new home sale price was revised downward by 6.5%, from $401,100 to $375,200.

These figures will continue to be revised over the next two months as the U.S. Census Bureau accumulates more sales data for the month.

Meanwhile, we find that the overall trend for median new home sale prices has continued to increase in 2014, doing so at an average pace of $9.77 for every $1 that median household income increases.

U.S. Median New Home Sale Prices vs Median Household Income, December 2000 through November 2014

This pace of growth is anywhere from 2-3 times the typical pace that was seen in the period from 1967 through 1999, or in the initial post-housing bubble crash recovery period from January 2011 through June 2012.

It is also considerably less than the rate of increase that was recorded during the primary inflation phases of the first and second U.S. housing bubbles, where median new home sale prices were rising at a rate of $21 to $25 for each $1 that median household income was increasing during those periods.

Our second chart below shows the longer term picture for the escalation of median new home sale prices in the U.S. since 1967.

U.S. Median New Home Sale Prices vs Median Household Income, 1967 through November 2014

We think that the big thing to watch for in 2015 is the re-emergence of the affordable home portion of the new home market. Since investors sparked the second U.S. housing bubble in July 2012, new home prices have largely escalated as U.S. home builders deliberately neglected this portion of the housing market, focusing instead on building premium homes, which they then attempt to sell for premium prices.

But that is a thin portion of the market that is getting thinner. Builders who have already begun adopting a strategy of building more affordable homes are being rewarded with higher volumes of sales.

“The mood of the industry heading into next year is extremely cautious,” said John Burns, CEO of his namesake firm based in Irvine, Calif. “Nothing has happened to cause sales to slow this much. But prices rose so fast in 2013, so that’s probably the primary culprit....”

This year has shaped up to be a giant stall for the new-home market, as sales through the first 10 months mustered only a 1% increase from the same period a year ago....

Some builders, however, are seeing signs of more activity from buyers. While many national builders reported lackluster results for their recent quarters, D.R. Horton Inc. posted a 38% gain by focusing on lower-priced homes and using more sales incentives to coax buyers into deals.

Paradoxically then, we would see business improve for new home builders as the median and average prices of the homes they build fall as they change up their sales mix to be more affordable for new home buyers.


Sentier Research. Household Income Trends: July 2014. [PDF Document]. Accessed 23 December 2014. [Note: We have converted all the older inflation-adjusted values presented in this source to be in terms of their original, nominal values (a.k.a. "current U.S. dollars") for use in our charts, which means that we have a true apples-to-apples basis for pairing this data with the median new home sale price data reported by the U.S. Census Bureau.]

U.S. Census Bureau. Median and Average Sales Prices of New Homes Sold in the United States. [Excel Spreadsheet]. Accessed 23 December 2014.

Previously on Political Calculations

We were among the first to declare that a second housing bubble was forming in the U.S. economy, and we were the first to back it up with an objective framework of analysis and data. Our ongoing analysis is chronicled below....


22 December 2014

Let's start today with an excerpt from an abstract published by Edwin Elton, Martin Gruber and Mustafa Gultekin during the dark ages of stock price theory, back in September 1981.

It is generally believed that security prices are determined by expectations concerning firm and economic variables. Despite this belief, there is very little research examining expectational data....

We're pointing this particular passage out because it summarizes how little work had been done to connect the dots between expectations and stock prices as recently as just over 30 years ago, despite the understanding by academics and market observers that such a connection actually exists. Let's next take a look at an early paper by Robert Shiller, which was almost contemporarily published in June 1981.

A simple model that is commonly used to interpret movements in corporate common stock price indexes asserts that real stock prices equal the present value of rationally expected or optimally forecasted future real dividends discounted by a constant real discount rate. This valuation model (or variations on it in which the real discount rate is not constant but fairly stable) is often used by economists and market analysts alike as a plausible model to describe the behavior of aggregate market indexes and is viewed as providing a reasonable story to tell when people ask what accounts for a sudden movement in stock price indexes. Such movements are then attributed to "new information" about future dividends. I will refer to this model as the "efficient markets model" although it should be recognized that this name as also been applied to other models.

What Shiller is specifically referring to at this point is the the work that Eugene Fama had done in the 1960s and 1970s to introduce the idea of efficient capital markets, which holds that "security prices at any time 'fully reflect' all available information."

Stock Market Chaos

The reason why Shiller was referencing the efficient markets model was because he observed that the movements of stock prices were "too volatile" to be accounted for in the subsequent changes of dividends. Or rather, as Shiller would argue, volatility in stock prices were not adequately explained by rationally expected future dividends, and because of that, investors had to be behaving irrationally for stock prices to be as volatile as they appeared.

Amusingly, despite this fundamental disagreement, both Shiller and Fama would share the 2013 Nobel Prize in economic sciences with Lars Peter Hansen for their collective work in "laying the foundation" for understanding how asset prices are set.

The truth is that they both screwed it up. Fama's efficient markets model simply isn't sophisticated enough to account for the observed volatility of stock prices and Shiller's reliance on the "irrationality" of investors to account for such volatility is simply misplaced.

Where Fama's efficient markets theory falls short is in missing is the time variance aspect of future expectations in setting stock prices. Instead of investors calculating some sort of net present value of the entire stream of future dividends they might rationally expect to earn from owning equities, we've observed that investors actually heavily weight changes in the growth rate of dividends that are expected to be earned at specific points of time in the near future.

But when investors suddenly shift their attention from one point of time in the future to a different point of time in the future, volatility in stock prices will result if there are differences between the changes expected in the growth rate of dividends for the different points of time in the future.

Shiller's fundamental error lies in attributing this volatility to investor "irrationality". Investors can have very rational reasons for suddenly shifting their attention from one point of time in the future to another, such as if firms announce changes in their dividends that will take effect in a particular future quarter or for other factors that might affect the amount of dividends they might pay in the future, such as changes in interest or tax rates.

In any case, what we just described pretty much accounts for all of the variation and volatility in stock prices in 2014.

Changes in the Growth Rate of Expected Future Trailing Year Dividends per Share with Daily and 20-Day Moving Average of S&P 500 Stock Prices, 2 January 2014 through 19 December 2014

Here's a link to the math we invented that converts these changes in growth rates for dividends into changes in the growth rate of stock prices, and the chart below shows the results we get when we applied that math in our standard model for each future quarter for which we had expected future dividend data throughout the fourth quarter of 2014.

Alternative Futures - S&P 500 - 2014Q4 - Standard Model - Snapshot on 19 December 2014

For the CBOE's dividend futures data, the value of the constant, m, in our math is approximately equal to 5. If you use Indexarb's bottoms-up dividend futures data, the value of the constant is approximately equal to 9.

Why are we telling you all this? Well, after six years of development following our initial key insight into the nature of how expectations for dividends actually drive stock prices, where the last several years were required to accumulate a sufficient amount of dividend futures data just to validate our theory that largely reconciles the discrepancies in both Fama's and Shiller's work, we're retiring this stream of analysis since it is no longer under development.

Someday, we hope this explains how some obscure blogger managed to get into the line where they hand out Nobel prizes in economics. Until then, because we did all of our development work in public, almost entirely from scratch and without a safety net, you have more than enough information to apply our theory on your own.

So go have fun! There's certainly a lot more possible today because of our work in just the last six years than there was in all the decades that preceded it.

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19 December 2014

Suppose, for a moment, that you could look up and see the Moon from anywhere in the Northern Hemisphere on every night of 2015.

Thanks to images collected by the Lunar Reconnaissance Orbiter and the talented staff at NASA's Scientific Visualizations Studio, here's what you would see as the lunar terminator sweeps across the Moon's surface as the Moon passes through all of its phases in 2015.

Identifying the major features on the lunar surface at their sunrises and sunsets is a nice touch, as is capturing the Moon's apparent libration motion in the sky.

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18 December 2014

Barry Ritholtz has a fine rant:

Those of you who continue to insist you can even remotely forecast what might happen next continue to reveal incredibly foolish, thoroughly disproved beliefs, despite an overwhelming avalanche of evidence that you haven’t the slightest idea what the fuck is going on now, much less what is going to happen next.

Once again, the markets prove that nobody knows ‘nuthin.

Carry on.

Okay. Here is the chart we posted before the market opened on Tuesday, 16 December 2014:

Alternative Futures for S&P 500 - Standard Model - 2014Q4 - Snapshot on 15 December 2014

Here is what the updated version of that chart looks like as of the market close on 17 December 2014:

Alternative Futures for S&P 500 - Standard Model - 2014Q4 - Snapshot on 17 December 2014

We do this sort of thing routinely. So much so that we don't even bother calling attention to it any more unless it is to point out an obvious gap in someone's understanding of what is possible.

Then again, it's not like we're going to be doing much of that beyond next week, as we'll no longer be routinely posting such things after that time!

Until then, welcome back to the cutting edge of what is possible today!

Update 18 December 2014: How about another go, seeing as Barry was really going on about the impossibility of predicting what the stock price futures were indicating would be happening with stock prices this morning? Here's what our stock price forecast chart looks like after the close of trading on 18 December 2014:

Alternative Futures for S&P 500 - Standard Model - 2014Q4 - Snapshot on 18 December 2014

Here's the text from the chart:

Converging Back to 2015-Q2 Following the Fed

Following the Fed's announcement and Janet Yellen's press conference on Wednesday, 17 December 2014, it would appear that the Federal Reserve has succeeded in directing the collective attention of investors to 2015-Q2, which is when it will most likely begin to hike short term interest rates. Consequently, stock prices have rocketed up during the last two days as they converge with the trajectory defined by an investor focus on that future quarter in setting today's stock prices, which is exactly how our model of how stock prices behave says they would behave under those circumstances. Very forecastable, Barry Ritholtz!

After a certain point, doing this sort of thing is a lot like shooting fish in a barrel, because what we're doing is much more science than art - it works because it's simply applied physics with direct parallels to the laws of motion. Which is why we're moving on to other challenges that might also benefit from that kind of approach.

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17 December 2014

It has often been remarked that falling oil prices are unambiguously good for Americans. But as we'll show you today, who they might be good for depends greatly upon where you live and the kind of government policies that are enforced there.

Let's start by reviewing how the price of crude oil has changed since the beginning of 2014. Our first chart below tracks the price per barrel of Brent crude oil since its price has the greatest impact upon gasoline prices at U.S. fuel pumps.

We see that the price per barrel of Brent crude hovered around $110 per barrel up until 4 July 2014, after which, its price has fallen at a steady pace to reach an average price nearly $44 per barrel lower as of this writing.

As you might imagine, the falling price of Brent crude oil has paced falling fuel prices across the entire U.S., which benefits Americans because it frees up the money they had been spending just on fuel, where that same amount of money can now be used to buy both the same amount of fuel they had been buying and additional things too.

One of the first industries to benefit from the savings that Americans were realizing from falling oil and gasoline prices was the restaurant industry, where business really began to improve just as fuel prices began to fall.

Cheaper gas, a rosier jobs picture and charbroiled burgers: this is the recipe for strong sales at CKE Restaurants, the privately held parent company of fast food chains Carl's Jr. and Hardee's.

"I won't give you exact numbers, but our sales have been very good since about the middle of June," CEO Andy Puzder told CNBC in a phone interview.

By October 2014, an increasing number of Americans seeking to dine out led to an unexpected improvement in the U.S. job market. Here, after having seen absolutely no sustained improvement in their employment numbers since October 2009, the number of employed teens between the ages of 16 and 19 in the U.S. suddenly increased by 266,000. Employers, predominantly in food and drinking service-related businesses, had finally responded to the increased demand they were seeing by adding U.S. teens in large numbers to their payrolls for the first time in years.

That improvement was not a one-time statistical outlier. The job gains of teens were sustained in November 2014.

Change in Number of Employed by Age Group Since Total Employment Peak in November 2007, through November 2014

So it would appear that falling oil prices have indeed been unambiguously good for around 266,000 U.S. teens who began collecting their first real paychecks in October 2014.

But as we noted at the very beginning of this post, where people live and the government policies that are enforced there matters quite a lot in determining who gets to benefit from something as apparently ambiguously good for Americans as falling oil prices.

That brings us to California.

According to the U.S. Census Bureau's population estimates, in 2013 there were 17,011,519 Americans between Age 16 and Age 19 in the entire United States. Of these, 2,143,455 live in California. Californians between the ages of 16 and 19 then represent 12.6% of the entire population of working age teens in the United States, or just over 1 out of every 8.

What percentage of the 266,000 American teens who benefited from becoming employed do you suppose are working in California?

If we go by simple statistics, we would expect that number would be equal to one-eighth of the entire increase in the number of working teenagers in the U.S., or 33,250.

According to the California's Employment Development Department, the actual number rounds to 3,000.

It's like the proverbial dog that didn't bark.

California Lottery - Source: http://www.calottery.ca.gov/play/second-chance/slp-second-chance

What are the odds of that?

Because the numbers exceed the capability of our own tool for doing the math, to answer that question, we turned to VassarStats' binomial probabilities calculator and entered the following values for n (266,000, the number of newly employed teens), k (3,000, the number of newly employed teens in California) and p (.126, the decimal equivalent of the percentage of the U.S. teen population represented by California's working age teens.)

The probability of counting 3,000 newly employed teens (or less) in California with respect to the actual population distribution of U.S. working age teenagers is less than 0.0001% (or <0.000001 as reported by the calculator). Or if you prefer, the odds of that outcome happening by chance are less than one in a million.

That result tells us that something has gone specifically and unambiguously wrong in California's job market for teens.

It occurs to us that we have an interesting natural experiment to consider in determining what exactly has gone specifically and unambiguously wrong for California's job-seeking teens.

Here, we know exactly when the environment that led to the large improvement in teen employment across the rest of the United States changed, and that the thing that caused the improvement has continued to act to the unambiguous benefit of all Americans across the country.

But not for teens in California. Something had to change just in California during the same period of time that teens in every other state in the nation were benefitting that would put teens in California at a relative disadvantage compared to their fellow American peers in their own state's job market.

And as it happens, we know exactly what changed to lead employers in California, and nowhere else, to bypass teens for consideration for the kind of low-wage jobs that the direct peers of these least educated, least skilled and least experienced members of the U.S. labor force were suddenly finding at the largest numbers seen in years elsewhere in the nation.

Labor Market for Teenage Californians, January 2005 through October 2014

Unlike every other state in the United States, California increased its minimum wage on 1 July 2014, just as the employment situation was about to improve across the entire country thanks to falling oil and fuel prices. No other state has likewise implemented an increase in their minimum wages during this period.

By arbitrarily increasing their minimum wage from $8.00 to $9.00 per hour in July 2014, California's politicians effectively jerked away the prospect of finding employment from its job-seeking teen population at a time when it would have its best chance at doing so in years, while also damaging their prospects for increased future earnings. All by making it too costly for the state's employers to employ them profitably.

How many of California's teens missed out on the opportunity of getting their first job during this time? After dividing 263,000 (October 2014's increase in teen employment minus the number attributed to California's teens) by .874 (87.4% is the percentage of U.S. teens who live outside of California), we find that if California had punched its own age demographic weight as did the rest of the country, teen employment in the U.S. would have increased by 300,915, as California would have added approximately 37,915 teen jobs to its employment totals.

Instead, the state came up some 34,915 jobs short. Just for its own teens.

It's not an accident that the New York Times is proclaiming that the economic recovery has finally spread to the middle class after having been historically terrible for so long. In finally providing enough "oomph" to create jobs for the least educated, least skilled and least experienced members of the U.S. civilian labor force, falling oil prices has indeed been unambiguously good for nearly all Americans, but especially the middle class families to which the vast majority of these newly working teens belong.

But not for middle class families in California, where the kind of government policies that are enforced there are keeping the very real economic recovery now occurring in every other part of the U.S. beyond their reach.

Just like jobs for California's teens. Just by keeping jobs away from California's teens.

Sad Teen Boy Is Seated with a Backpack - Source: http://www.stopbullying.gov/blog/2013/12/30/bullying-and-suicide-whats-the-connection


16 December 2014

In discussing the rise of volatility in the U.S. stock market at this time yesterday, we omitted our chart showing each of the trajectories that stock prices are likely to follow depending upon which particular quarter in the future they might be focusing their attention. If you haven't read that post yet, that is the explanation for why stock prices ran the gamut from one extreme to the other yesterday.

Alternative Futures for S&P 500 - Standard Model - 2014Q4 - Snapshot on 15 December 2014

Here's the text from the chart:

Over the last week, the rapid decline in oil prices to critical levels has led investors to consider the situation where oil industry-related companies, which includes the financial institutions that back them with development loans, would act to preserve their solvency by cutting their dividends in 2015-Q1. Meanwhile, investors are also factoring in the timing of when the Federal Reserve will begin to hike short term interest rates, which is expected in 2015-Q2. The result? Stock prices falling about halfway in between the alternative future projections for both quarters (if investors were simply focused on one or the other)!

Update 9:10 AM EST: The CBOE's dividend futures for 2014-Q4 (which extend through the end of this week) showed a boost yesterday as the dividend futures for 2015-Q1 showed a nearly equal decrease. What that most likely means is that investors have made the determination that dividends they had expected to be paid after this upcoming Friday will instead be paid out on or before that date - there really hasn't been a major change in the future expectations as of the market open on 15 December 2014! What that does change however is the relative position of stock prices between the alternative trajectories associated with the future quarters of 2015-Q2 and 2015-Q1, with investors appearing to be more heavily weighting their collective focus on 2015-Q2 in setting today's stock prices.

Update 10:30 AM EST: It does occur to us that there may be another possibility - with the apparent distress in the revenues of oil companies, investors could be weighing the possibility that the Fed might delay its hiking of interest rates into 2015-Q3. If so, we might get a confirmation of that on Wednesday, 17 December 2014 when the Federal Reserve's Open Market Committee issues a statement and Janet Yellen holds a press conference.

As a reminder, our regular analysis of the U.S. stock market will only continue through 23 December 2014. Since we've finished our development work in that area, we're moving on to other projects.

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15 December 2014

For all the noise related to falling oil prices and their impact upon stock prices during the past week, there is something that we have not yet seen that would fully confirm the fears of U.S. investors: large numbers of companies in the oil sector acting to cut their dividends.

That is not to say that action will not be eventually be forthcoming, but as yet, we have only seen a very limited number of oil industry related companies take that action since we noticed an uptick in their number in November 2014, so we would describe the current action with stock prices in the market as being primarily the result of speculative noise.

As a noise event then, what matters most in driving stock prices today is the timing of when investors are betting that these companies with increasingly distressed revenue prospects will act to cut their dividends. Here, rather than being solidly focused on a single point of time in the future, stock prices are becoming volatile because investors are shifting their attention between two different points of time in the future, which have different expectations for the growth rate of the dividends that investors can reasonably expect to earn from owning stocks associated with them.

We think that dynamic may explain an unusual pattern that we've previously observed in the months and weeks ahead of major market crashes, where large drops in stock prices that occurred as order broke down in the stock market were preceded by similarly-driven volatility.

With that in mind, our first chart below illustrates the current state of order in the U.S. stock market, as measured by the S&P 500 against its trailing year dividends per share, which has held since 4 August 2011.

S&P 500 Index Value vs Trailing Year Dividends per Share, 30 June 2011 Through 14 December 2011

So we're clear, the stock market can be said to be experiencing a period of relative order whenever stock prices are following an established power law trend with respect to their trailing year dividends per share, where the residual variation of stock prices about that trend can be described by a normal (or Gaussian) distribution. Or more accurately, when a statistical hypothesis test cannot reject the possibility that stock prices are behaving "normally".

In looking for signs of significant volatility that might precede the breaking down of an existing state of order in the stock market are changes in the 20-day (one-month) moving average of stock prices that exceed two standard deviations in magnitude.

You'll see several as you review this chart. The first appears when the S&P 500's trailing year dividends were below $25 per share, which occurred in the period from 30 June 2011 to 4 August 2011. This particular drop marks the beginning of the current period of order in the stock market, which coincided with the end of the Federal Reserve's second Quantitative Easing (QE 2.0) monetary stimulus program in June 2011.

After the present state of order became established, the next time that stock prices showed significant downward volatility were when trailing year dividends reached about $27.50 per share in June 2012 and again when they reached just over $30 per share in October 2012.

Under typical circumstances, this episodes of volatility would have signaled that order was breaking down in the U.S. stock market. And it was, as the U.S. economy was on the verge of entering into a new phase of contraction. The only thing that saved the U.S. economy and preserved order in the U.S. stock market was the Federal Reserve's launching of its third quantitative easing program (QE 3.0) in September 2012, which it subsequently expanded in December 2012 (QE 4.0).

So it's not that the signal of order breaking down in the U.S. wasn't sent or didn't work, but rather that it was heeded by the Federal Reserve, which acted to prevent it from breaking down in the stock market while also keeping the U.S. economy out of a full-fledged recession. They were successful in their effort.

The next time we see a two standard deviation shift in the value of U.S. stock prices occurred shortly after trailing year dividends per share neared $38.50 per share in mid-September 2014, shortly after the Fed confirmed that it would terminate its QE 3.0/4.0 programs at the end of 2014.

That brings us up to now, where we find that instead of being fully focused on 2015-Q2, as they seemed to be just a week ago, investors would also now appear to be paying increasingly close attention to 2015-Q1 in setting today's stock prices. Our chart showing the expectations that investors have for the future, as measured by the change in the growth rate of trailing year dividends per share for each of the future quarters for which we have data, places stock prices in between these two particular future quarters.

Change in Growth Rates of Expected Future Trailing Year Dividends per Share with Daily and 20-Day Moving Average of S&P 500 Stock Prices, Snapshot on 12 December 2014

In this chart, we observe that 2015-Q1 has recently drawn the attention of investors, which is significant because this is the quarter in which that those increasingly revenue-distressed oil companies would most likely act to cut their dividends. That action would be really similar to the dynamic that took hold in the stock market after the U.S. economy peaked in December 2007 ahead of the so-called Great Recession, which then set up falling dividends as the primary driver of stock prices throughout 2008.

Meanwhile, we believe that investors remain mostly focused on 2015-Q2, since that coincides with the period of time in which the Fed is most likely to begin hiking short term interest rates in the U.S.

It is the interplay between these two factors set to occur at two different points of time in the future that is what will make for the potential for a really rocky ride for investors.

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12 December 2014

For months now, the cost of a barrel of crude oil has been falling and now, those prices are falling rapidly.

But what does the falling price of crude oil today mean for you at the gasoline pump? How will the price of a barrel of crude oil translate into the price you pay per gallon?

Well, we have an app for that! It's based on James Hamilton's regression analysis of oil prices and U.S. gasoline prices from 2000 through the present, the main thing you need to know to predict where the price of gasoline in the U.S. is the price of a barrel of Brent crude oil, which if you're accessing our site directly, appears below (via Oil-Price.Net)!

All you need to do is enter that current price in our tool, and we'll estimate how much the average price of a gallon of gasoline will be in the U.S. within the next several weeks.

Crude Oil Price Data
Input Data Values
Price per Barrel of Brent Crude Oil

Future Price of Gasoline in U.S.
Calculated Results Values
Average U.S. Price per Gallon

This tool is an updated version of our original Where Are U.S. Gas Prices Going? tool, the math for which has become very relevant again!

Automobile Gas Tank Being Filled at Gas Pump - Source: http://mn.gov/commerce/weights-and-measures/

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11 December 2014

For the second month in a row, our alternative measure of the relative health of China's economy indicates that nation is potentially in recession.

Our alternative measure is based trade data collected by the U.S. Census Bureau, which is considerably more reliable than the GDP growth rate or other trade statistics reported by China's government. Here, we simply calculate the year-over-year growth rate of the value of goods that China imports from the United States in terms of China's currency, the yuan (or renminbi).

Doing that exercise for the trade data for October 2014, which was just reported on 5 December 2014, we find that the year over year growth rate for the value of the goods that China imported from the U.S. was negative for the second month in a row. That negative indication in consistent with near-zero growth in China's economy (at best) or mild economic contraction.

Year Over Year Growth Rate of U.S.-China Trade, January 1986 - October 2014

By contrast, the value of goods exported from China to the U.S. suggests that the U.S. economy continued to expand in October 2014.

Our next chart puts the year over year growth rate of the value of China's imports from the U.S. into better context.

Value of Exports from the U.S. to China, January 1986 - October 2014

In October 2013, the value of U.S. exports to China reached record levels thanks to a massive increase in the amount of oil seeds and oleagenous fruits purchased by Chinese entities following record U.S. harvests, which predominantly consists of soybeans. Of the $4.112 billion dollars worth of these products that were shipped from the U.S. to other nations, Chinese entities purchased about 72% of the total. At an estimated price of at least $13 per bushel in October 2013, China acquired the equivalent of as many as 227 million bushels of soybeans.

A year later, we find that of the total value of $4.102 billion of U.S. soybeans exported around the world, Chinese entities acquired some 75% of the total. At an estimated price of $10 per bushel for the month, the equivalent of 308 million bushels of soybeans has been imported by China from the U.S. in October 2014. Soybeans account for approximately one-quarter of the value of all the goods exported from the U.S. to China during the month.

Prices in 2014 have fallen with record soybean production around the globe. U.S. soybean production for 2014 is expected to top 3.958 billion bushels.

While the falling prices for soybeans is to be positive factor, allowing the China's soybean consumers to purchase higher quantities than ever before, it does add to the risk of further unleashing deflationary forces in China's economy, which is where the growing recession risk for the nation lies.

That's one reason why tracking the year-over-year growth rate of the value of goods imported by a nation is such an effective indicator of a nation's relative economic health - it is capable of exposing this kind of economic dynamic.


Board of Governors of the Federal Reserve System. China / U.S. Foreign Exchange Rate. G.5 Foreign Exchange Rates. Accessed 10 December 2014.

U.S. Census Bureau. Trade in Goods with China. Accessed 10 December 2014.

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10 December 2014

Now that we've made our Toolmaker's Tool fully functional and available to all, we thought we'd check in with the S&P 500 and the likely alternative trajectories that stock prices may follow. The chart below shows where things stand as of the close of trading on 9 December 2014.

Alternative Futures - S&P 500 - Standard Model - 2014-Q4 - Snapshot on 9 December 2014

The biggest change in the current week so far has been an uptick in the amount of dividends expected to be paid in 2015-Q1, which we see in the narrowing of the spread between the alternative future trajectory for that quarter and 2015-Q3.

At present however, we believe that stock prices are basically following the trajectory associated with the expectations for 2015-Q2, which is when the U.S. Federal Reserve is currently expected to begin hiking the short term interest rates it controls.

We've added some straight red lines to our chart, where our forecasting model is affected by minor, short duration echoes, which are the result of our incorporating historic stock price data in our model. These represent what we think are the most likely path for stock prices assuming investors maintain their forward-looking attention on the future quarters they were when the various echoes appear in the projections.

We should note that these are typically pretty minor, as stock prices typically fall within our expected range of volatility about the midpoint of our forecast trajectories.

As a final reminder to our regular readers, now that we have successfully completed this particular development work, we will no longer be sharing this kind of analysis after 23 December 2014.

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08 December 2014
Tools - Source: http://www.nyc.gov/html/dca/html/home/dca_additional_features2013.shtml

Nearly twenty years ago, we invented a small part of the Internet. Specifically, we invented a method that greatly automated the computer programming needed to put simple computational applications onto the Internet where they might be accessed by anyone with a personal computer or an Internet-capable mobile device. Today, the now much more sophisticated descendants of what we made easy to create are simply called "apps".

To make a long story short, we submitted a proposal to our employer at the time to patent our invention, which ultimately went nowhere. Not being in an Internet or computing-related business, they declined to file an application for our invention with the U.S. Patent and Trademark Office.

Ten years ago, we saw an opportunity to resuscitate our original invention and launched Political Calculations, where we would augment the analysis of current events with the kind of simple, yet specialized computational tools that our invention made easy to generate.

And that brings us to today, where we're not just celebrating our tenth anniversary, but also marking the occasion where if our employer at the time had applied for a patent for our invention, the term of the patent would now be expiring and entering into the public domain!

And, as an added bonus, today is also the day that the Khan Academy is encouraging people everywhere to experiment with creating computer code with its Hour of Code event!

So welcome to our Toolmaker's Tool - a stripped-down and simplified version of the invention that we've used frequently over the past decade to create many of the applications developed for the Political Calculations blog!

Our Toolmaker's Tool is designed to aid you in creating JavaScript-based math calculation tools. As such, you can take advantage of the math functions and built-in constants used in JavaScript. You may also take advantage of HTML code to help create your tool and its surrounding commentary.

By way of providing an example, we've entered some basic data for a tool that will calculate the future value of an investment, using the following formula:

F = P(1 + I/N)N*T

where F = Future Value, P = Starting Value, I = Interest Rate, N = Number of Compounding Period per Year and T = Time Held in Years.

Please feel free to overwrite this data to do your own calculations! Also, you are more than welcome to post any tool you create using this prototype on your own blog or web site, provided you retain the "Code created with the assistance of Political Calculations" text and link under your generated output table.

The Toolmaker's Tool is designed to make programming web-based computing applications as easy as filling out a form. Once you've organized the mathematical algorithm you'll use in generating your tool to identify your Constants (or Initial values), User Input Data, and Formulas - you'll probably find that you'll spend less time programming than you do writing the text around your tool! But, why wait to find out? Get started now....

Leading Post or Page Information
Enter the indicated information in the table below. The data entered will appear before the Input Data Table of the tool you are creating.
Post or Page Title:
Text Preceding Calculator:

Constants For Use in Math Formulas
Use this table to enter up to 4 values that will be used by your tool, which do not need to be entered by the user.
Symbol: = : Constant Value
Symbol: = : Constant Value
Symbol: = : Constant Value
Symbol: = : Constant Value

Input Table Information
Enter the indicated information in the table below. The data entered will control the appearance of your User Input Data Table.
Input Table Title:
Input Table Colors:

Data for User Input
Use this table to provide for up to 8 values that must be entered by the user.
Symbol: = : Default Value
Symbol: = : Default Value
Symbol: = : Default Value
Symbol: = : Default Value
Symbol: = : Default Value
Symbol: = : Default Value
Symbol: = : Default Value
Symbol: = : Default Value

Output Table Information
Enter the indicated information in the table below. The data entered will control the appearance of your Output Data Table, and designate a unique name for the computing function you create.
Output Table Title:
Output Table Colors:
Function Name: : Write without spaces.

In the following section, you have the ability to do more than to assign a value to a symbol (which is done using the single "=" symbol) - you can also incorporate one-line long conditional statements! Here's a simple example, which assigns the value 20 to the symbol k if the value x is less than, or equal to, 5:

Symbol: if (x <= 5) k = 20 : Formula

The most frequently used comparison operators in JavaScript are available in the drop-down box for the equality symbol.

Step-by-Step Math Formulas
Enter up to 12 formulas that your tool will use in the order the calculations will need to be done. Then, select whether the result will be displayed and how many decimal places to display. JavaScript's Math functions and built-in constants may be used (and are, in the example below):
Symbol: : Formula
Output?: If Yes, Show: Decimal Places
Symbol: : Formula
Output?: If Yes, Show: Decimal Places
Symbol: : Formula
Output?: If Yes, Show: Decimal Places
Symbol: : Formula
Output?: If Yes, Show: Decimal Places
Symbol: : Formula
Output?: If Yes, Show: Decimal Places
Symbol: : Formula
Output?: If Yes, Show: Decimal Places
Symbol: : Formula
Output?: If Yes, Show: Decimal Places
Symbol: : Formula
Output?: If Yes, Show: Decimal Places
Symbol: : Formula
Output?: If Yes, Show: Decimal Places
Symbol: : Formula
Output?: If Yes, Show: Decimal Places
Symbol: : Formula
Output?: If Yes, Show: Decimal Places
Symbol: : Formula
Output?: If Yes, Show: Decimal Places

Trailing Post or Page Information
Enter the indicated information in the table below. The data entered will appear after the Output Data Table of the tool you are creating.
Text Following Calculator:

To save your code, copy and paste the final version of your generated code from the appropriate fields below into your text editor to save as a *.html file (for the stand-alone web page option) or in your blogging template or JavaScript source file (for the generated functions) and your blog post editor (for the displayed text code).

Blog Template or JavaScript Source File Code

Copy and paste this code into your blog template or into a JavaScript source file. On a side note, you may also want to incorporate the table stylesheet used by this code generator to provide the same look-and-feel as that of the Test Drive feature at the bottom of the page.

Blogger Post Code

Copy and paste this code into your blog post editor or your preferred text editor. When editing this code, take care to not change the names of the form fields without also changing them to match in the blog template or JavaScript source file code. Note: We've optimized the code to work with the Blogger web platform, if you use other platforms, you may need to adapt it to work as needed.

Stand-alone Web Page Code

The code in this field puts everything together into one single package, which is ideal for copying and pasting into a *.htm or *.html file. The code generated for this field is the same as that used in the Test Drive feature.

Beyond the information we've provided or linked to above, we will not be providing technical support for using our Toolmaker's Tool. We have every confidence that you will be able to sort out any problem that you might encounter on your own.

Update 9 December 2014: Blogger made it a challenge, since our code ran in Blogger's draft status, but curiously, not after being published. We've solved that problem now, eliminating the last barrier between you and building your own online tools - it's all wide open now!

Update 8 December 2019: Google is making life difficult. We had previously solved the problem of making the underlying code by hosting it on Google Drive. Unfortunately, Google's evolution from Google Drive to Drive has broken the connection to make the Toolmaker's code work. We're working on a fix.

Celebrating Political Calculations' Anniversary

Our anniversary posts typically represent the biggest ideas and celebration of the original work we develop here each year. Here are our landmark posts from previous years:

  • A Year's Worth of Tools (2005) - we celebrated our first anniversary by listing all the tools we created in our first year. There were just 48 back then. Today, there are nearly 300....
  • The S&P 500 At Your Fingertips (2006) - the most popular tool we've ever created, allowing users to calculate the rate of return for investments in the S&P 500, both with and without the effects of inflation, and with and without the reinvestment of dividends, between any two months since January 1871.
  • The Sun, In the Center (2007) - we identify the primary driver of stock prices and describe a whole new way to visualize where they're going (especially in periods of order!)
  • Acceleration, Amplification and Shifting Time (2008) - we apply elements of chaos theory to describe and predict how stock prices will change, even in periods of disorder.
  • The Trigger Point for Taxes (2009) - we work out both when, and by how much, U.S. politicians are likely to change the top U.S. income tax rate. Sadly, events in recent years have proven us right.
  • The Zero Deficit Line (2010) - a whole new way to find out how much federal government spending Americans can really afford and how much Americans cannot really afford!
  • Can Increasing the Minimum Wage Boost GDP? (2011) - using data for teens and young adults spanning 1994 and 2010, not only do we demonstrate that increasing the minimum wage fails to increase GDP, we demonstrate that it reduces employment and increases income inequality as well!
  • The Discovery of the Unseen (2012) - we go where so-called experts on income inequality fear to tread and reveal that U.S. household income inequality has increased over time mostly because more Americans live alone!

We celebrated our 2013 anniversary in three parts, since we were telling a story too big to be told in a single blog post! Here they are:

  • The Major Trends in U.S. Income Inequality Since 1947 (2013, Part 1) - we revisit the U.S. Census Bureau's income inequality data for American individuals, families and households to see what it really tells us.
  • The Widows Peak (2013, Part 2) - we identify when the dramatic increase in the number of Americans living alone really occurred and identify which Americans found themselves in that situation.
  • The Men Who Weren't There (2013, Part 3) - our final anniversary post installment explores the lasting impact of the men who died in the service of their country in World War 2 and the hole in society that they left behind, which was felt decades later as the dramatic increase in income inequality for U.S. families and households.

And finally, our tenth anniversary post....

A Final Programming Note

We're finding ourselves increasingly being pulled toward new and exciting projects. Unfortunately, we only have a limited supply of time, which means that something that we're doing currently has to give, and that something is going to be our regular workday posting schedule here at Political Calculations.

We're not going away, but we won't be posting with the frequency that we've established over the past decade. To best keep up with us, we recommend subscribing to Political Calculations' free RSS news feed through your preferred news reader.


About Political Calculations

Welcome to the blogosphere's toolchest! Here, unlike other blogs dedicated to analyzing current events, we create easy-to-use, simple tools to do the math related to them so you can get in on the action too! If you would like to learn more about these tools, or if you would like to contribute ideas to develop for this blog, please e-mail us at:

ironman at politicalcalculations

Thanks in advance!

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