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September 4, 2015
Garth Sundem's 'Your Daily Brain'

Garth Sundem has a new book out! Your Daily Brain: 24 Hours in the Life of Your Brain promises to take readers through a typical day in the life of their brains, breaking the day into 5 to 15 minute intervals to consider what is actually happening inside their heads and identifying ways to get optimal performance out of their mind's work.

That kind of structure makes it very different from his previous books, particularly those that explore how both the brain and intelligence work with greater detail. By contrast, each section of the brain's day presented in Your Daily Brain blasts through at near warp speed.

And yet, despite covering each episode of a brain's day in just a few pages, there's a wealth of information about what science has discovered about how the brain works, ranging from the practical to "really?!"

As an example of practical, just consider the exercise of trying to remember where you might have left your car keys, if you're someone who is prone to misplacing them. Remembering information like that is really a three stage process, where your brain needs to first encode the information, store it, then recall it when needed. To remember where your keys are requires you to first draw upon your brain's hippocampus when you first set them down, which does the job of encouding and setting priority tags in your head for the information, but if you really want to remember where you left them, you need to tap into your brain's emotional centers to create a stronger, easier to recall memory of where you placed them.

Which works pretty well, unless you're emotionally stressed when you're trying to remember where you left your keys. In that case, you can try a cued-retrieval approach, like going through a list, to try to prompt your brain into recalling the memory.

Then there are those "really?!" moments! Did you know that science has indicated that the color of the capsule for a placebo, or rather, a sugar pill made up to look like actual medication, can actually influence how effective it is at "treating" a given condition?

For instance, there is real research that indicates that placebos in red capsules are more effective as stimulants, while placebos in blue capsules are more effective as sleep aids. So at least Morpheus' question regarding which color pill Neo could choose from The Matrix was scientifically appropriate!

But which of these colors you should have your office painted to enhance your productivity there is something you'll have to discover in reading the book! As is the secret of controlling your co-workers minds....

Speaking of which, the book is actually co-authored/branded with retailer Marbles: The Brain Store, so if you live in a city with one of their locations, you might find it there. Or your local book store, or Amazon.

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September 3, 2015
NASA's C-OPS System in the Black Ball Covered Los Angeles Reservoir - Source: NASA - https://spinoff.nasa.gov/Spinoff2012/ee_1.html

The last time we took on the subject of balls, it was to consider the physics of New England Patriots' quarterback Tom Brady's deflated balls. Today, we're going back to the ball pit because Los Angeles' city government is spending $34.5 million to dump up to 96 million plastic balls into a water reservoir, where the apple-sized floating black spheres are intended to reduce water loss due to evaporation and to reduce the need to treat the water with chemicals to keep algae and bacteria growth at bay.

While it's an open question as to whether the black balls will be successful at their second mission, as other scientists have hypothesized that they might actually promote bacterial growth instead of hindering it, the balls should be somewhat successful in minimizing the loss of water due to evaporation by preventing direct sunlight from reaching and heating the reservoir's water surface. The question is how much might the balls succeed in reducing that evaporation.

Environmental Economics' Tim Haab worked up the basic math, which we're both immortalizing and extending today in a tool designed to answer not just that question, but also how many balls it would take if the city leaders of Los Angeles chose to substitute balls of other sizes for the apple-sized ones they are using.

In doing so, we're going to make two key assumptions. First, that the surface of the 175 acre Los Angeles Reservoir will be able to accommodate the densest packing of 2D circles in a plane, which means that a maximum of 90.7% of the water's surface will be covered by the spheres, which in turn, will represent the maximum amount by which evaporation losses will be reduced. Second, we'll also assume that the blackened spheres will not rotate to expose any wetted surface to the open air or sunlight, which would increase the evaporation rate.

Got all that? Good! Let's get to the tool.... (If you're accessing this article through a site that republishes our RSS news feed, please click through to our site to access a working version of the tool!)

Water Reservoir Shade Ball Project Data
Input Data Values
Ball Diameter [inches]
Standard Evaporation Rate for Reservoir [inches/year over 4-foot diameter area]
Surface Area of Reservoir [acres]

Shade Ball Quantity
Calculated Results Values
Maximum Number of Balls That Will Cover Reservoir in a Single Layer
Amount of Water Lost or Saved to Evaporation
Annual Water Losses If Reservoir Is Uncovered [gallons per year]
Annual Water Losses Prevented If Reservoir Is Covered by Shade Balls [gallons per year]
Annual Water Losses If Reservoir is Covered by Shade Balls [gallons per year]

That said, all the default values that appear in the tool above are those that apply to the Los Angeles Reservoir shade ball project. If you want to redo the math to consider the average annual evaporation rate in your state or to consider a project where the area being covered in balls in measured in square feet rather than acres (like a swimming pool), just enter your square footage figure and type "/43560" (without the quotation marks around it) immediately after it in the "Reservoir Area" data entry field. Doing so will convert your square footage to be in the units of acres that our tool references.

But the real fun is to consider what it would mean to substitute 1.575-inch diameter standard ping pong balls or 32-inch diameter giant rainbow beach balls, since it is California after all. Or more interestingly, the 96-inch diameter balls shown in the video below:

What could possibly go wrong?!



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September 2, 2015

We've previously written that the only time we really ever get excited about what's going on in the stock market is when it changes by 2% or more from the previous day's closing value.

That threshold is based on our statistical study of the volatility of stock prices, where we found that the percentage change of the S&P 500 from one trading day to the next fit pretty neatly inside a normal distribution, where:

  • 78.8% of all day-to-day percentage changes were within 1 standard deviations of the mean, about 10.6% higher than what would typically be expected for a perfect normal distribution.
  • 95.3% of all day-to-day percentage changes were within 2 standard deviations of the mean, almost right in line with what would typically be expected for a perfect normal distribution.
  • 98.6% of all day-to-day percentage changes were within 3 standard deviations of the mean, about 1.1% less that what would typically be expected for a perfect normal distribution, but still pretty close.

For bell curve fans, all those numbers mean two things:

  1. Small changes in stock prices are much more likely to occur than would be the case if their variation were the result of purely random factors.
  2. Big changes in stock prices from one day to the next are pretty unusual events.

So that brings us to yesterday, 1 September 2015, where for most of the day, stock prices were about 2-2.5% below where they had previously closed, before they suddenly dipped to be 3.5% lower before recovering to close at 3% lower in the last 37 minutes of trading.

And that was really pretty uninteresting because that final closing value would be exactly what our model of how stock prices work forecast it would be provided investors were setting stock prices according to the expectations they have for the current quarter, 2015-Q3.

Alternative Futures - S&P 500 - 2015Q3 - Rebaselined Model - Snapshot 1 September 2015

Now, here's the thing about how our model works. What you see in our model as day to day variation for the alternative trajectories that stock prices are likely to take when investors are focused on a particular future quarter is based upon historic stock price data. We then factor in the change in the growth rate of dividends per share for each indicated future quarter for which we have dividend futures data, which you see as the vertical separation between the various trajectories.

For our standard model, for each day's forecast value, we use the historic stock prices that were recorded 13 months earlier, 12 months earlier and 1 month earlier as our base reference points from which we project future stock prices.

But in our rebaselined model, which we use when the historic price data for our standard model contains too much volatility to provide the most accurate forecast possible, we substitute the historic stock prices from different points in time and adjust our calculations accordingly. In forecasting each day of 2015-Q3 since 28 June 2015, we've been using the historic stock prices from 25 months earlier, 24 months earlier and 1 month earlier.

S&P 500 Index Value (Historic Data Base Reference Points) Used in Rebaselined Model Forecast Projections, 2015

So when we see that today's stock prices are largely matching our forecast changes, as we basically have for all but three of the last 26 trading days, what we're seeing is that stock prices are directly echoing the events of 25 months ago, 24 months ago and 1 month ago.

Here's the multi-million dollar question: Why is that? We would only reasonably expect stock prices to fall somewhere within the range of values we forecast, where the actual trajectory of stock prices should be continually cutting across our forecast trajectories and the echoes of historic noise they capture, not paralleling them. Especially with the record levels of volatility that the market has shown recently.

But it's not, which would mean that other factors are at work. Our best guess is that those factors are somehow tied to options contracts and investments with two-year long maturities and expiration dates, where the confluence of interactions between investments initiated 25 months ago, 24 months ago and 1 month ago is compelling stock prices today to follow the course it is. Throw in a three-day long quantum shift in focus on the part of investors, and we have our recipe for explaining why stock prices have behaved as they have.

Or rather, why our model of how stock prices work has surprisingly managed to work as well as it has in the current stock market climate.

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September 1, 2015
Checkbook - Source: Virginia Department of Education - http://www.doe.virginia.gov/instruction/economics_personal_finance/

Are you tired of paying your your mobile phone bill every month? What if you could get your carrier to pay it for you?

Believe it or not, there's a way that you could actually make that happen. All you need to do is to buy the company!

Or rather, you need to buy enough shares of stock in the company to cover the cost of your monthly bills through the dividends you might earn as a partial owner of the business. But how many shares would that take?

Our latest tool answers that question! We've plugged in the quarterly dividend and share price data as it might apply to an average monthly bill for Verizon (NYSE: VZ), but you're more than welcome to substitute the data that applies to your own mobile service provider. Or for that matter, any company that bills you monthly that also pays dividends to its shareholders!

If you're reading this article on a site that republishes our RSS news feed, click here to access a working version of this tool!

Monthly Bill Data
Input Data Values
Your Monthly Bill [$USD]
Investment Data
Current Share Price [$USD per Share]
Quarterly Dividend [$USD per Share]
Dividend Income Tax Rate [%]

Shares Needed to Pay Bills with Dividends
Calculated Results Values
Annual Amount of Monthly Bills
Number of Shares Needed to Provide Enough Dividend Income to Cover Bills After Taxes
How Much Will Those Shares Cost You to Buy Today?

If you think about it, this is really the sort of thing that you're trying to accomplish for when you're not working any more through your retirement investments.

But nobody said you had to be retired before you could start working your way toward that objective.


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August 31, 2015

From time to time, we'll conclude our more remarkable posts with the phrase "Welcome back to the cutting edge!" We're going to do that again today.

The reason we'll do that today is because of a new paper that was published just 20 days ago that describes cognitive decision making as the collapse of a quantum superstate. Phys.org's Christopher Packham provides the background for the application of "Quantum Random Walks" to decision making.

Decision making in an enormous range of tasks involves the accumulation of evidence in support of different hypotheses. One of the enduring models of evidence accumulation is the Markov random walk (MRW) theory, which assigns a probability to each hypothesis. In an MRW model of decision making, when deciding between two hypotheses, the cumulative evidence for and against each hypothesis reaches different levels at different times, moving particle-like from state to state and only occupying a single definite evidence level at any given point.

But the Markov random walk theory, based in classical probability theory, runs into problems when confronted with the emerging research consensus that preferences and beliefs are constructed, rather than revealed by judgments and decisions. An international group of psychological researchers now suggests a new model called the quantum random walk (QRW) theory that specifically posits that preferences and beliefs are constructed rather than revealed by judgments and decisions, and they have published the results of an experiment that support this theory in the Proceedings of the National Academy of Sciences.

By contrast with MRW, the new theory assumes that evidence develops over time in a superposition state analogous to the wave-like state of a photon, and judgements and decisions are made when this indefinite superposition state "collapses" into a definite state of evidence.

That new theory has a direct practical application, because it describes much of the behavior we've directly observed in how investors collectively set stock prices.

To see what we mean, let's update and animate alternative futures chart, which projects the likely trajectories that stock prices will follow based upon how far forward in time investors are collectively looking when they make their current day investment decisions, where we'll pick up the action beginning one month ago.

Animation: Alternative Trajectories - S&P 500 - 2015Q3 - Rebaselined Model - 28 July 2015 through 28 August 2015

Each of the alternative future trajectories in the chart above represent a specific hypothesis, which is given by the acceleration, or change in the year over year growth rate, of the trailing year dividends per share expected to be paid out by the S&P 500 by the end of the indicated quarter.

At the beginning of the animation, we see that investors were tightly focused on 2015-Q3 on 29 July 2015, which makes sense because investors had strong reason to believe that the U.S. Federal Reserve was on track to begin hiking short term interest rates by the end of the quarter in September 2015, thanks to the Federal Open Market Committee's meeting and announcement issued that day. Thus, the trajectory indicated for 2015-Q3 represents the hypothesis that the Federal Reserve would hold to that policy.

Over the next week, the closing value of the S&P 500 closely tracked that trajectory, until moving higher on Monday, 10 August 2015, boosted by the speculative prospect that China, which announced disappointing economic data that day, would soon act to provide new stimulus measures for that nation's economy.

Stock prices remained elevated throughout the rest of that week as China made good on that speculation, as it announced the surprise devaluation of its currency on 11 August 2015 and by other measures that carried through 17 August 2015. Throughout this period, stock prices remained elevated just above the upper edge of the typical range of day-to-day volatility of stock prices that we would expect would apply for investors remaining focused on 2015-Q3.

Stock Market Crash - Source: http://www.federalreserve.gov/aboutthefed/cls-timeline/timeline/timeline_main.htm?04

That positive speculation began to deflate on 18 August 2015 as China's stock market began to decline, as China's government began to back off from enforcing its previous extraordinary measures to arrest its decline earlier in the summer.

That action also coincided with the two-year anniversary of the so-called "taper tantrum" in the U.S. stock and bond markets, which would appear to be relevant since we used historical stock prices from this period as the baseline from which to project the likely trajectories shown in our chart above. Which we began doing on 28 July 2015 because the period was much less volatile that the one-year ago period that we would normally use to forecast the future likely trajectories of stock prices in our standard model of how stock prices work.

What happened next is now the stuff of forecasting legend. Based on the echo of that two-year old event, our model anticipated that stock prices would begin falling, even if investors remained focused on 2015-Q3 in setting stock prices. And through the week ending 21 August 2015, the level of stock prices was fully consistent with investors remaining focused on 2015-Q3.

But then, on Monday, 24 August 2015, stock prices closed far lower than would be consistent with investors remaining focused on 2015-Q3 alone. Instead, stock prices plunged to a level that was much more heavily weighted toward 2016-Q1, which would be consistent with the hypothesis that the Federal Reserve would back off its plans to hike short term interest rates in the U.S. until that time at the earliest based on the continuing deterioration in China's stock markets.

On Tuesday, 26 August 2015, a sizeable rally in the U.S. stock market throughout much of that day "went up in smoke", as stock prices closed that day at a level that was fully consistent with investors having shifted their forward looking focus to 2016-Q1, in effect, collectively betting that a September 2015 rate hike in the U.S. would now be off the table. Coincidentally that day, Federal Reserve Bank of New York president William Dudley said that a September 2015 rate hike was "less compelling".

The downward swinging pendulum of investor expectations for when the Fed would seek to hike interest rates reversed on the next day with the release of positive economic news, giving more strength to the hypothesis that the Fed would hike rates in 2015-Q3. Overall however, stock prices moved to be about halfway between the levels that would be fully consistent with either 2015-Q3 and 2016-Q1, suggesting that investors gave equal weighting to the difference between these two future quarters in setting stock prices.

The positive momentum continued on 27 August 2015, with the significant upward revision of U.S. GDP recorded in the second quarter of 2015. That news strengthened the view among U.S. investors that the Fed would hold to its September rate hike plans, with the S&P 500 closing in on a level that would be more heavily weighted in favor of that hypothesis.

On Friday, 28 August 2015, there was very little movement in stock prices, but that would be expected for investors having focused once more on the likelihood that the Fed would indeed hike interest rates in September 2015, a view given great emphasis during the day by Stanley Fischer, the Number Two official at the U.S. Federal Reserve.

Random vs Quantum Walk - Source: http://rsta.royalsocietypublishing.org/content/364/1849/3407

There are four things we really need to point out about the movement of stock prices during the period from the close of trading on 21 August 2015 through 28 August 2015. First, the concept of a quantum random walk goes a very long way toward explaining why stock prices just simply don't jump straight from one quantum level to another. The uncertainty that investors have regarding the likelihood of future events can restrain the potential extent of such movements.

Second, the quantum levels themselves are not random. They're based on the real and quantifiable expectations for the change in the growth rate of dividends per share that will be paid out at specific points of time in the future, which are projected on top of where stock prices are and have been. That's important because it is the relative distance between the quantum levels that exist when investors are choosing between the hypotheses that apply at different points of time in the future that determines whether stock prices follow a true random walk-style Brownian Motion, or go into a full Lévy Flight such as we observed in the last week.

Third, as stock prices change, so does the likely trajectory of stock prices in the future. Given the math involved, we now see the echo of the China-driven stock market crash in September in our animated chart above.

However, that doesn't mean that stock prices will follow that trajectory. In general, the echoes of past volatility do not tend to affect the trajectory of current day stock prices, although in rare cases, they can. What provides the potential for determining whether they might is the amount of money that might have been tied up in investments with fixed maturity dates, such as options contracts or bonds, which would come due when such echoes appear in our model.

But whether they do have any real world impact depends on what the investment climate looks like when they do come due. It is always the prospects for the future that determine the actual trajectory of stock prices.

Fourth, the existence of the quantum random walk phenomenon helps explain why stock returns have fat tail distributions. To the extent those distributions primarily resemble bell curves is an indication of the extent to which investors collectively tend to focus on one potential future, or investment-driving hypothesis, at a time as they make investing decisions.

Welcome back to the cutting edge!

Previously on Political Calculations

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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:

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