We're going to play a little bit of catch up today, starting from the last time in 2015 that we discussed where the S&P 500 was going to head next.

On that day, we added an update to our original post to note where the market closed:

Update 21 December 2015, 6:38 PM EST: Here are the current expectations for the S&P 500's quarterly dividends per share for 2016, as determined from today's recorded values for the CBOE's dividend futures contracts for the S&P 500:

2016-Q1: $11.40 | 2016-Q2: $11.45 | 2016-Q3: $11.64 | 2016-Q4: $11.77

Meanwhile, here's the updated version of our alternative futures forecast chart through the close of trading on 21 December 2015:

Still focused on 2016-Q2. At least, until that changes....

We're now going to roll our chart to the left, to capture all the time from when the dividend futures contracts for 2015-Q4 expired on the third Friday of December 2015 through the end of the first quarter of 2016.

In between then and last Friday (8 January 2016), here's what really drove U.S. stock prices:

• 21 December 2015: U.S. investors focused on 2016-Q2 in setting stock prices.
• 23 December 2015: The third estimate of U.S. GDP in the third quarter of 2015 is released. The less than stellar growth that occurred going into what is widely believed to be an even slower fourth quarter leads investors to speculate that the U.S. Federal Reserve will push back the date it will next implement a short term interest rate hike. In our chart above, we see that as investors shifting their focus from 2016-Q2 to 2016-Q3. This shift in focus leads to an increase in stock prices, as the change in the year over year rate of growth of dividends in 2016-Q3 is expected to be stronger than in 2016-Q2.
• 23 December 2015 - 31 December 2015: Investors sustain their forward-looking focus on 2016-Q3, as no other news is sufficient to prompt investors to focus on other periods of time through the end of the year.
• 4 January 2015: The first day of trading in 2016 coincides with a large selloff of stocks in China, as that nation relaxes the restrictions it had imposed on selling during 2015, believing its new stock market "circuit breakers", which would halt trading if stock prices fell at least 7% during a trading day, would work. They were tested almost immediately. In the U.S., the noise from China combined with news that both manufacturing and construction slumped during 2015-Q4 to lead investors to speculate that the Fed's next rate hike might be pushed back even further, from the 2016-Q3 they were speculating it would at the end of 2015 to 2016-Q4 instead. Stock prices rise once again as the year over year rate of growth of dividends in 2016-Q4 is expected to be even stronger than in 2016-Q3.

What we just described above hopefully explains why bad economic news can produce the seemingly paradoxical result of rising stock prices. The trajectory that stock prices follow depends not just on where they just were, but also upon how far forward in time investors are focusing as they make their current day investment decisions.

But that only brings us up to the beginning of the past week. Let's resume our day-by-day play-by-play:

• 5 January 2016: U.S. GDP growth estimates for the last quarter of 2015 weaken further, suggesting that the economic headwinds for 2016 are stronger than previously anticipated. Stock prices remain consistent with investors focusing on 2016-Q4, which is consistent with the speculation that the Fed will hold off hiking interest rates again until that quarter in the face of the declining economic expectations.
• 6 January 2015: The minutes of the Federal Open Market Committee's December 2015 meeting are released, showing that the Fed is both concerned by low inflation and strongly committed to hiking interest rates several times during 2016. U.S. stock prices begin falling as investors shift their focus back to 2016-Q3 in reaction to the statements of Fed officials.
• 7 January 2016: The carnage in the stock market continues as investors continue to align their expectations for the next Fed rate hike, pulling back to focus upon 2016-Q2 as the quarter when the next rate hikes will take effect. Coincidentally, more noise emanates from China regarding the health of its economy.
• 8 January 2016: News from the December 2015 employment situation report that wage growth in the U.S. is accelerating combined with statements by San Francisco Fed President John Williams and Richmond Fed President Jeffrey Lacker combine to lead investors to split their focus between 2016-Q2 and 2016-Q1, as they begin speculate that the Fed may act to hike short term interest rates in the U.S. before the end of the first quarter of 2016.

Given all the news from China, many might be surprised that we don't consider it to have been much of a major influence over what has been happening in U.S. markets. Looking at what's happened during the past week however, we do see a repeating theme from August 2015.

Back then, the massive selloff that took place in China's markets came in response to Chinese authorities relaxing some of the restrictions they had placed on selling stocks earlier in the year.

In January 2016, after setting up their market's new circuit breakers, Chinese authorities once again relaxed their other restrictions on selling activity, believing they would be sufficient to arrest downward pressure on Chinese stock prices. Days after taking effect, the circuit breakers are being dumped as restrictions on selling stocks are now being reimposed.

Meanwhile, the same factors that initially drove such widespread selling activity are still present in China, with the pressure for selling growing because that activity has been forcefully constrained without the sufficient improvement in China's economic situation that would be needed to allow that pressure to be relaxed naturally as yet.

Speaking of which, Zero Hedge reports that Goldman Sachs is estimating that China's leaders have spent 1.8 trillion yuan to prop up Chinese stock prices by buying up the shares of Chinese firms. Funny - if they had only underwritten the dividend payments of the Chinese firms instead, decoupling them from their earnings, they could have both been more successful in propping up prices and spent quite a lot less, as that action would have relaxed a good portion of the pressure upon Chinese investors to sell.

As for what's happening in U.S. markets, the best way to describe what's happening there is that investors have been engaged in a speculative dispute with the Fed over the number of rate hikes in 2016, with investors saying fewer and the Fed saying more. Recent falling stock prices are consistent with investors shifting their forward-looking focus from 2016-Q3 to 2016-Q2 (mostly) as the likely timing for the Fed's next rate hike. The activity in China's markets is mostly coincidental to what's really been driving U.S. markets.

The big question of course is whether the Fed really wants to fully convince the market that they're solidly committed to boosting interest rates again during 2016-Q1. If they do, watch out below....

Once upon a time, the only way that sailors crossing oceans could know if they were on course to their destinations required them to know what time it was. Verna Allee described the problem in The Knowledge Evolution:

In the Age of Exploration [1300-1600 AD] competitive advantage resided with those who had the best ability to navigate the globe. Those countries that ruled the seas--ruled the world. In those days, they found the great challenge in navigation is finding a way to tell time at sea. Unless you know what time it is, you cannot determine longitude. So, you see, in order to master space and thus circumnavigate the globe, you must first find a way to master time. A time measurement allows you to understand the relationship between the point where you are and the point where you want to be. If you understand that relationship, then you can chart a course and find your way home again....

Navigation boils down to a few simple elements: a good map, something like a sextant to tell you where you are, and a timepiece to help you measure your progress. These are good things to remember when trying to navigate:

• In order to know where you are on earth, you must first look at the stars (Shift your perspective.)
• To chart a course you must know both where you are and what time it is. (Understand the relationship of time and space.)

Once you know these things, navigation becomes a simple problem in kinematics. Where you will end up depends first upon where you are (your position), and then it depends upon your direction and speed (velocity), as well as any changes in your direction and speed (acceleration).

Stock prices work almost exactly like that, with one important difference. Because the acceleration of stock prices is directly proportionate to the change in the rate of growth of their underlying dividends per share expected at specific points of time in the future, correctly anticipating where stock prices are going to go also requires investors navigating the markets to know the specific point of time in the future to which investors are collectively heading.

What makes that such a challenge to pull off successfully is that investors can collectively shift their forward-looking focus from one specific point of time in the future to another with very little warning. When they fully shift their focus from one such point of time to another, depending upon how different the expected acceleration of dividends per share is for each, the resulting change in stock prices can vary anywhere from a little to a lot. This shift in focus is not just what gives stock prices of their volatility, but also their quantum-like properties.

And because the things, or new information, that might prompt them to shift their attention from one point of time in the future to another often comes at random intervals, the effect is to give stocks a seemingly random trajectory where they are most often characterized by small movements, as investors can sustain their focus on a particular point on the future horizon for relatively long periods of time, that are then interspersed by large movements, which come as investors suddenly shift their focus to different points of time on the horizon. Stock prices can then be said to follow a quantum random walk, which really makes forecasting their trajectory a problem in quantum kinematics.

That's why competent forecasts for stock prices need to consider the potential quantum levels for stock prices based upon the rational expectations for dividends per share associated with the different points of time on the future horizon to which investors might turn their attention. Keep that in mind then when you read the following remarks, which we originally posted at 3:35 AM on Thursday, 17 December 2015 after observing that investors had just appeared to have shifted their forward-looking focus to 2016-Q3 in setting today's stock prices in response to the Fed's rate hike and Fed Chair Janet Yellen's comments the previous day:

Whenever the U.S. Federal Reserve has the full attention of investors, they give us the ability to tell how far forward investors are looking in time as they are making their current day investment decisions.

So when the Fed announced on 16 December 2015 that it would hike short term interest rates in the U.S. by a quarter percent, while also providing additional information about the likely future for additional rate hikes, we couldn't help but notice that investors would appear to have shifted their forward-looking attention to the third quarter of 2016.

For example, simply shifting their focus to 2016-Q2 before the end of 2015 would likely coincide with an approximately 3% decline in the value of the S&P 500. Meanwhile, a complete shift in forward looking focus to 2016-Q1 would correspond to a full 10% correction, with stock prices dropping by over 200 points.

But if investors stay focused on 2016-Q3, then the market is within just a percent or two of its ceiling for the remainder of the year.

After identifying the last known position of stock prices going into that trading day, we identified three very different outcomes for how stock prices would be most likely to change in the near term, all completely hinging upon whichever distinct future point of time that investors might shift their attention toward as the only differentiating factor for what the actual outcome would be.

As we expected, one of those three distinct outcomes came to pass. Conveniently for us, within two days of trading after the Fed's rate hike announcement, giving us something to write about this morning, as the value of the S&P 500 dropped by 3.3% through the close of trading on 18 December 2015....

The shift in focus to the nearer term future horizon of 2016-Q2 is more than what would be expected by just purely random chance with the stock market's typical day-to-day volatility, and yet, if we're being honest, it's not really very random at all, as perhaps the most random thing about it is the timing of its occurrence.

This will be our last post on the S&P 500 this year, where we'll wrap up 2015 sometime early in 2016, where we'll also revisit our forecasting performance during the periods of time where we relied upon our rebaselined model. Sometime later this week, perhaps as early as later today, we'll start getting dividend futures data for 2016-Q4, allowing us to consider that new point on the horizon in our forecasts.

Until then, since investors would be most likely to focus upon the three quarters preceding that distant future quarter, we've provided you with what you need to anticipate where stock prices will go next through the end of 2015, having already marked their last known position on our charts above. All you need to do now is to determine how far forward in time investors are collectively fixing their attention on the horizon to tell where stock prices will go next.

It is, after all, just a simple problem in quantum kinematics - one that anybody can solve if they understand the relationships of time and space and can shift their perspective on the future's horizons. Welcome back to the cutting edge!

Update 21 December 2015, 6:38 PM EST: Here are the current expectations for the S&P 500's quarterly dividends per share for 2016, as determined from today's recorded values for the CBOE's dividend futures contracts for the S&P 500:

2016-Q1: $11.40 | 2016-Q2: $11.45 | 2016-Q3: $11.64 | 2016-Q4: $11.77

Meanwhile, here's the updated version of our alternative futures forecast chart through the close of trading on 21 December 2015:

Still focused on 2016-Q2. At least, until that changes....

Whenever the U.S. Federal Reserve has the full attention of investors, they give us the ability to tell how far forward investors are looking in time as they are making their current day investment decisions.

So when the Fed announced on 16 December 2015 that it would hike short term interest rates in the U.S. by a quarter percent, while also providing additional information about the likely future for additional rate hikes, we couldn't help but notice that investors would appear to have shifted their forward-looking attention to the third quarter of 2016.

Which, when you see how it compares to all the alternative points of time in the future that investors might otherwise have chosen to focus upon, is a really dangerous place to be, as any news that would prompt investors to shift their focus to a less distant future would coincide with falling stock prices.

For example, simply shifting their focus to 2016-Q2 before the end of 2015 would likely coincide with an approximately 3% decline in the value of the S&P 500. Meanwhile, a complete shift in forward looking focus to 2016-Q1 would correspond to a full 10% correction, with stock prices dropping by over 200 points.

But if investors stay focused on 2016-Q3, then the market is within just a percent or two of its ceiling for the remainder of the year.

The one wild card is the expected future for 2016-Q4, for which we won't have data until sometime next week.

The one thing we absolutely know for sure is that investors will not turn their attention back to 2015-Q4. Although there are 16 days left before the calendar quarter officially ends, from the perspective of investors, 2015 is done.

The caption from the chart below:

After briefly looking at 2016-Q2 on Tuesday, 1 December 2015, after news indicating that the U.S. economy is slowing broke and investors speculated that news would delay a Fed rate hike, Fed Chair Janet Yellen effectively grabbed investors by the scruffs of their necks and forced them to focus on 2015-Q4 instead.

In terms of the alternative future trajectories that we forecast for U.S. stock prices, shown in the chart covering the fourth quarter of 2015 below, that shift in focus accounts for the changing trajectory of the S&P 500 since Monday, 30 November 2015, which went from the dotted dark orange line representing investors being focused on 2015-Q4, to the dashed purple line representing 2016-Q2, and then back to the dotted dark orange line representing 2015-Q4, as Janet Yellen showed the market who's in charge over the last two days.

The interesting thing is that the expectations for 2015-Q4's dividends per share deteriorated over the same time, falling by 18 cents per share, yet stock prices still behaved just as our model would predict as they took their quantum random walk over the last several days, almost perfectly converging with where our model's projected trajectory for 2015-Q4 moved over that same time.

That's largely because we are currently in a period where we have optimal conditions for producing this kind of forecast, with minimal noise to distract investors (indicated by the unshaded regions in our chart above).

Now, if you want to use our model to directly forecast where stock prices will go next, you really only need to anticipate the timing of news that would either prompt investors to once again shift their forward-looking focus from where it is now to a different point of time in the future, or that might alter their expectations for the fundamental driver for U.S. stock prices (their underlying dividends per share) at the future point of time they might focus upon.

If you can do that, you might be able to show Janet Yellen who's really in charge of the U.S. stock market.