In his paper, De Groot examines the economic cycles of GDP growth in over two dozen countries. He found that their durations are grounded in golden ratio relationships. De Groot took an empirical approach to chart a pattern in the lengths of sub-cycles in GDP growth. Understanding the interrelationships between the length of economic sub-cycles then allows their cycles and their fluctuations to be detected more easily. With this information, the resulting signals of future changes in cyclical behavior can provide additional analytic tools to combat economic and societal distress.

Macro to micro economic Golden Ratio and Fibonacci patterns

To some degree, the appearance of the golden ratio in economic cycles should not be a surprise. The appearance of the golden ratio in the timing and price trends of financial markets has been studied and applied by investors for decades in the stock markets, commodities markets, Forex and elsewhere. Their application to the far more complex multi-structured cycles of entire economic cycles though has been much more challenging. Researchers have been seeking to understand these cycles for over one hundred years, so this is where De Groot’s work brings new insights and value.

Method of Analysis

To take on this task, De Groot used time series data of GDP growth for Europe and 25 OECD countries. De Groot relied primarily on Fourier analysis to detect the cycles. With this approach, a series of observations of a variable over a given period of time can be decomposed into cosine waves. When the data showed irregular trends though, this approach can become unusable. To overcome this, De Groot used a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model. This makes the data less irregular while still preserving the cyclic behavior of the original data. After performing these two tests, De Groot used an algorithm to determine which cycles were prominent. He then lastly estimated a trend-cycle that described the amplitudes and phases of the economic cycles.

Results of Study

The models confirm that the cycles he found strongly correlate with the time series data. Almost all the estimates were statistically significant at a level of 5%. The detected cycles from the sample can describe swings in GDP growth rates of up to 5 percentage points. The models can thus be useful to predict future turning points of an economy as well. This is done by letting the model forecast a part of the data that is left out of the sample, but that is readily available. It is then possible to assess whether the model actually forecasts what happens.

His results indicate two to five cycles are present that in each economy. Cycles with a length of 5–6 years and 9–10 years appear most frequently. De Groot’s paper reveals that “A meta-analysis on the detected cycle lengths reveals that the ratio between the lengths of the shorter to the longer cycles in consecutive cycles often closely matches the golden ratio.” Interestingly, this finding opposes several existing theories about multi-cycle structures, which imply that the lengths of shorter cycles should be integer fractions of the lengths of longer cycles. De Groot’s paper thus provides a new direction for theory development regarding economic cycles and dynamic stability.

GDP Cycle connection to the Golden Ratio

While supported by complex mathematical analysis and statistical correlation analysis, the underlying data and concept can be easily understood. In his paper, De Groot identifies the length of two consecutive economic cycles for over two dozen countries:

De Groot then performs a harmonic regression and other processes and develops a table with the ratios between the lengths of the consecutive cycles:

The average of all the ratios in the above table is 0.619. A probability value for the test of the results against the golden ratio at 0.618 is calculated to be 0.94, which means that the hypothesis cannot be rejected statistically.

De Groot’s full paper can be found here:

https://www.sciencedirect.com/science/article/pii/S0040162521002250

Broader implications of de Groot’s research

With de Groot’s research, we now have further evidence that markets often move in patterns related to the golden ratio. Technical traders have been applying the golden ratio and the related Fibonacci sequence for decades to improve success rates in trading of stocks, commodities, currencies, and indices. De Groot’s work corroborates, with the tools and rigor of academic research, that these same trends can be found on a global macro level in the Gross Domestic Product measures for economic cycles of entire countries, and for the world at large. Given that the economic cycles of countries are certainly related to the activities of the financial markets within them, this correlation should not be a surprise. It’s fascinating though to see his research corroborate the appearance of the golden ratio in such a broader yet application.

Golden Ratios in the timing movements of stock markets

Technical analysts often use patterns based on the golden ratio and Fibonacci sequence to identify price and timing movements in individual stocks and in market indices. The patterns can be shown to occur with relative frequency when looking historically. The challenge is knowing which pattern will appear in future movements. Technical analysts thus use these Golden Ratio and Fibonacci patterns as just one of a number of analytic tools to improve their overall success rates in trades. Market analyst Bill Erman likened it to going down an interstate highway. The analytics can give you an idea of when the next interchange is likely to come up, but you need other information to know whether to go north or south.

Here is an example of golden ratio patterns appearing in the timing movements of highs and lows in the Dow Jones Industrial Average (DJIA):

Just as De Groot’s work reveals GDP economic cycles over extended periods of years with durations in golden ratio proportion, the above chart shows multiple cycles with a single year. In this year, the DJAI hit a major low in May between February and October (gold grid), another low in September between June and November (aqua grid) and yet a third in September between September and November (red grid).

Golden Ratios in the price movements of stock markets

The grid below shows the DJIA for the same period, and reveals that the price movements also followed golden ratio patterns. The interim highs and lows are in golden ratio proportions from the market high in February to the market low in October. The price points that define the high and lows are referred to as resistance or inflection points.

PhiMatrix Golden Ratio Design Analysis software was used to create the golden ratio grids in the above charts, and is considered an essential tool by some technical traders and teachers of market trading.

For additional information on how the Golden Ratio and Fibonacci sequence are related to the price and timing movements of markets see these pages:

Stock Market Analysis, Phi and the Fibonacci Sequence

Foreign Exchange Markets (FOREX) and Trading

Stock market analysis with PhiMatrix software

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The golden ratio is found in many places, but here’s one you might not have expected: It can be found in football gaming strategy when making the “point after touchdown” decision as to whether to kick for 1 extra point or go for a 2 point conversion.

This applies to the situation when a team is trailing by 14 points late in a football game, and then scores a touchdown to be trailing the other team by 8 points.

So what should you do: Kick for 1 point after touchdown (PAT) or go for a 2 point conversion?

Most football teams kick the PAT in this situation, but in fact it is (almost) always better to go for 2.  The reason is related to phi.

If your odds of making a 2 point conversion are at least 38.2% (1/phi^2), you should then should go for 2.

This finding and the analysis presented below was contributed by Alexander (Alex) Leach. He is a high school football coach and physics teacher in Texas, and was using statistics to help make the decision as to whether to kick for 1 extra point or go for a 2 point conversion. In doing so he found this exact application of the golden ratio to gaming strategy for football.

Typical logic might tell you that you need a 50% chance of converting to justify going for 2 but this is inaccurate. The reason is that you can adjust your strategy after you know the outcome of your 2 point try.

Here are the relevant situations:

• Kick the PAT for 1 point (-7),  score again (-1), Kick again (tied), play in OT, versus
• Convert for 2 points, with:
  • Convert (-6), score again (tied), kick a PAT (+1 and win), or
  • Fail to convert (-8), score again (-2), attempt a 2 point (-2 or tied) to play in OT

Assuming that kicking the PAT is 100%  and the odds of winning in OT are 50%. Then it works out that the odds needed to justify going for 2 are exactly 1/Phi^2, or 38.2%

Professional football two point conversion success rates

Interesting enough, professional football teams have a much better chance of make the two point conversion than 38.2%, so this analysis is very relevant to a team’s probability of winning a game. Look at the following statistics by team and by year, and note that the overall success rate of making the two point conversion is almost 50%!

Source: www.SharpFootballAnalysis.com

Alex’s more in depth explanation below explains the exact relationship to phi, the math, the logic, and all possible outcomes.

Golden ratio in statistics of football strategy

There is an exact application of phi when looking at statistics used when deciding to go for a 2 point conversion after a touchdown.  The specific situation it applies to is when deciding what to do after scoring when down 14 points.  Most charts say to go for 1, but in fact it is usually better to go for 2 in that situation.  If you consider the odds of kicking your Point After Touchdown (PAT) to be 100%, it is still better to go for 2 as long as you have about a 38.2% chance of converting on your 2-point play. Actually the payoff of the two strategies breaks even when your odds of converting are exactly 38.2%. There are several ways to express 38.2% using Phi (1.618), including:

• ((1/phi)²)
• (2-phi), or
• (1-1/phi)

Here is the logic and implied math:

Suppose your team is down 14 points with time running out and you score a touchdown to make it down 8 point behind.  You can kick a PAT or attempt a 2-point conversion. There are only a few relevant situations that should weigh into the decision.

Here are those relevant situations and likelihood of each.

• Odds of converting a 2 point = C
• Odds of kicking a PAT = K
• Odds of winning in OT =T
• Odds of scoring again without other team scoring = A

When you choose to kick it, the odds of winning = KAKT+ (1-K)ACT

• (kick)(good)(score again)(kick)(good)(win OT): W = KAKT
• (kick)(good)(score again)(kick)(good)(loose OT): L = KAK(1-T)
• (kick)(good)(score again)(kick)(no good): L = KA(1-K)
• (kick)(no good)(score again)(go for 2)(good)(win OT): W = (1-K)ACT
• (kick)(no good)(score again)(go for 2)(good)(loose OT): L = (1-K)AC(1-T)
• (kick)(no good)(score again)(go for 2)(no good): L = (1-K)A(1-C)

When you choose to go for 2, the odds of winning = CAK+ CA(1-K)T+ (1-C)ACT

• (go for 2)(good)(score again)(kick)(good): W = CAK
• (go for 2)(good)(score again)(kick)(no good)(win OT): W = CA(1-K)T
• (go for 2)(good)(score again)(kick)(no good)(loose OT): L = CA(1-K)(1-T)
• (go for 2)(no good)(score again)(go for 2)(no good): L = (1-C)A(1-C)
• (go for 2)(no good)(score again)(go for 2)(good)(win OT): W = (1-C)ACT
• (go for 2)(no good)(score again)(go for 2)(good)(loose OT): L = (1-C)AC(1-T)

We can find the point where the payoff for the two strategies are equal by setting the formulas equal to each other. So:

• Odds kicking= KAKT+ (1-K)ACT = CAK+ CA(1-K)T+ (1-C)ACT = Odds going for 2

Dividing by A gives us (odds of scoring again should have no factor on decision because both strategies require another score):

• KKT + (1-K)CT = CK + C(1-K)T + (1-C)CT

Assuming you have a 50/50 chance of winning in OT, so T=.5, we get:

• .5(KK+(1-K)C) = CK + .5C(1-K) + .5C(1-C)

Now for simplifying: (skip to the bottom unless you want to see the math, and forgive me if I have a typo)

Solving for K first:

• Expanding:     KK/2-CK/2+C/2=CY/2+C/2+C(1-C)/2
• Move to left:   KK/2-CK-C(1-C)/2=0
• For Quadratic equation form: KK-2KC-C(1-C)=0

So we can complete the square:

• KK-2KC=C(1-C)
• Adding CC to both sides: KK-2KC+CC=C(1-C)+CC
• Simplify the right: KK-2KC+CC=C
• Writing left side as square: (K-C)(K-C)=C
• Square rooting both sides: K-C=sqrt(C)   Or  K-C = -sqrt(C)
• Solve for K:  K=C +or- sqrt(C)

Similarly we can solve for C:

• Starting from our original : .5(KK+(1-K)C) = CK + .5C(1-K) + .5C(1-C)
• In standard form: C(1-K)/2+KK/2=-CC/2+C(.5+{1-K}/2+K)
• Move to left: CC/2+C(1-K)/2-C(.5+{1-K}/2+K)+ KK/2=0
• In standard form: CC/2+ C((1-K)/2-.5+{K-1}/2-K)+ KK/2=0
• Quadratic equation: CC + 2C((1-K)/2-.5+{K-1}/2-K)+KK=0

So we can complete the square:

• CC + 2C((1-K)/2-.5+{K-1}/2-K)=-KK

Completing the square:

• CC+2C((1-K)/2-.5+{K-1}/2-K)+((1-K)/2-.5+{K-1}/2-K)^2=-KK+ ((1-K)/2-.5+{K-1}/2-K)^2
• Factor the left: (C+(1-K)/2-.5+{K-1}/2-K)^2=.25(4K+1)
• Square root: (C+(1-K)/2-.5+{K-1}/2-K)= +or- .5(4K+1)
• Solve for C: C=.5+(1-K)/2+(K-1)/2+K +or- .5(4K+1)

So by solving we get: (assuming K and C are real, positive, and from 0 to 1)(they are)

• K=C+sqrt(C), or
• C=(2K – sqrt(4K+1) +1)/2

Now if we assume K=1 and solve for C we get:

• (K=1 when odds of kicking a PAT are 100%, true odds in NFL are 99.3%*)
• C=.381966011250105

This is the point where the odds of winning are the same using either strategy. This is also one of the 2 percentages most closely related to phi.

When you divide your odds of not converting by your odds converting it equals phi exactly.

• (1-C)/C=phi

There are several other ways to relate this number to phi.  Here are a few examples:

• C=1-(1/phi)
• C=2-phi
• C=3-Phi^2
• C=1/phi^2

Verification of the results

For those of you who would like to test this out without going through all the steps, you can let the formula solver at WolframAlpha.com do the hard work for you.

Start with Alex’s equation above: KKT + (1-K)CT = CK + C(1-K)T + (1-C)CT

With K=1 and T=0.5, simplify it as: 1*1*.5 + (1-1)C*5 = C*1 + C(1-1)*.5 + (1-C)C*.5

Go to the formula solver at https://www.wolframalpha.com/examples/EquationSolving.html

Enter this as the polynomial equation: solve 1*1*.5 + (1-1)C*5 = C*1 + C(1-1)*.5 + (1-C)C*.5, or just click on this solution link.

The results will show as :

C = 1/2 (3-√(5))

C = 1/2 (3+√(5))

The first one is 0.382, which is 1 / Phi ².

The Golden Rule of Football

So that’s the golden rule for football. Do unto others as you would have them do unto you is great wisdom and the golden rule for life. In football, the golden rule is to use mathematics and the golden ratio as a strategy in making your game decisions. It can make the difference between winning and losing.

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Changes in stock prices largely reflect human opinions, valuations and expectations. A study by mathematical psychologist Vladimir Lefebvre demonstrated that humans exhibit positive and negative evaluations of the opinions they hold in a ratio that approaches phi, with 61.8% positive and 38.2% negative.

Phi and Fibonacci numbers are used to predict stocks

Phi (1.618), the Golden Mean and the numbers of the Fibonacci series (0, 1, 1, 2, 3, 5, 8, …) have been used with great success to analyze and predict stock market moves, known as retracements. Forbes ASAP featured a story on the work of scientist Stephen Wolfram in cellular automata (underlying rules that determine seemingly random phenomenon) stating “This seashell may hold the secret of stock market behavior, computers that think and the future of science.”

Markets may be as geometrically perfect as a spider’s web

Ermanometry Research shows the markets to be perfectly patterned, explaining that humans, being part of nature, create perfect geometric relationships in their behaviors, not unlike a spider spinning a geometrically perfect web with no conscious awareness of its amazing feat. Ermanometry applies the logarithmic spirals found in sea shells with dynamic ratios in 3D to relate one market move to others.

Phi, or Golden Ratio, patterns often define the timing of highs and lows and price resistance points

The golden ratio, or phi, appears frequently enough in the timing of highs and lows and price resistance points that adding this tool to technical analysis of the markets may help to identify fibonacci retracements, the key turning points in price movements. The photos below illustrate how the Golden Mean Gauge and Phi-based analysis software (PhiMatrix) can be used to identify these turns in the market. The middle arm of the gauge keeps the phi point of the outer arms as the gauge is opened and closed. The lines of the phi-based software are all in phi relationship to one another. The ratios of Fibonacci numbers, commonly used in technical market analysis, converge on phi as explained on the Fibonacci Series page.  Click on each photo to enlarge.

DJIA Daily Chart from 1/2004 through 11/04 using PhiMatrix software	DJIA Monthly Chart from 1/2000 through 6/2003 using a Golden Mean Gauge

Phi and Fibonacci numbers define the price movements of stocks in Elliott Wave Theory

Fibonacci numbers were used by W.D Gann and R.N. Elliott, pioneers in technical analysis of the stock market.  In Elliott Wave Theory, all major market moves are described by a five-wave series, adding to the potential to identify the turns described above. The classic Elliott Waveseries consists of an initial wave up, a second wave down (often retracing 61.8% of the initial move up), then the third wave (usually the largest) up again, then another retracement, and finally the fifth wave, which would exhaust the movement. In addition, each of the major waves (1, 3, and 5) could themselves be separated into subwaves, and so on, and exhibit other Fibonacci relationships.A sample stock price wave analysis could look something like this:

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It has long been known that phi and Fibonacci relationships appear in the stock markets.  The foreign exchange market, or Forex, is the largest market in the world.  The simplest definition of foreign exchange is the changing of one currency to another.  Since there are no touchable commodities affected, this is the most prominent and most liquid financial exchange anywhere.  In comparison to the daily trading volume averages of $300 billion in the U.S. Treasury Bond market and the less than $10 billion exchanged in the U.S. stock markets, the Forex market often averages $3.5 trillion exchanged daily.  The Forex Market is open 24 hours a day from 4 PM EST Sunday through 4 PM EST Friday.

Forex is not a “market” in the traditional sense.  There is no centralized location for trading activity as there is in currency futures. Trading occurs over the telephone and through computer terminals at thousands of established locations, as well as within home-based trading businesses worldwide.  Because of its size and the fact that the primary traders are bankers and the world’s largest corporations, it adheres to technical analysis better than any market in the world.  A trader may earn profits whether buying or selling within the currency exchange by taking advantage of daily market movements.

The PhiMatrix software created by the author of this site (Goldennumber.net) to unveil phi relationships in nature is now being used by a variety of Forex traders and trading systems to identify key market trading targets and opportunities.

One such example, from ProAct Traders, shown below, provides proprietary templates with the indicators designed to anticipate high probability movements, shown in the software by color-coded signals of potential moves. These signals allow traders to see the currency movement and take advantage of these movements for trading opportunities.  Similar strategies are used by Carlos Wolf of FXTecTips.  Elliott Wave International provides instruction on the application of phi and the Fibonacci series to all markets. While any investment vehicle or strategy has risks, the Forex market is reported to be lucrative by those who have mastered the tools and techniques of trading.

PhiMatrix software, shown below as the overlay of yellow grid lines, provides the phi-based pricing relationships that indicate probable stalling points and/or profit targets.

Download free e-book “Cooking in the FOREX”

For more information on this topic, site visitors to GoldenNumber.net and PhiMatrix.com can download a free e-book by Scott Barkley called “Cooking In The Forex” that explains the essentials of foreign exchange markets and Forex trading.  Scott was awarded the International Forex Development Award in 2003 for his work in the UK and was the awarded The Global Forex Award in both 2004 and 2005.

Two week trial on PhiMatrix

You can try PhiMatrix Software for free for two weeks to understand how it can be used to identify and predict timing and price movements in any kind of financial market.

Limitation of liability:  PhiPoint Solutions, LLC, (owner of PhiMatrix.com and GoldenNumber.net) is not affiliated with ProAct Traders and accepts no responsibility for results obtained using the software or services provided by any application of the PhiMatrix software to investments or trading techniques.

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In races, why does the most favored starter win less than half the time?  Why does it not win all the time?  Just what percentage of the time does it win?

We don’t all choose the favorite to win, but what success rate can we expect whatever we choose?  Surely not 100%, but not 0% either.  Is there an percent which would be the value most expected?

According to the work and research of Merv Pittman, mathematics predicts that favorites will win 38.2% of races, with an average of 38.2% of the people actually choosing that starter.  That percentage is both Phi squared and 1 – Phi.

This theory states the percentage of people who choose any starter is a very good indicator of its chances of success, and that the Golden Ratio is also involved in election results and even how often a goal kicker will be successful.

 

1		100
2		62	38
3		50	31	19
4		45	28	17	11
5		42	26	16	10	6
6		40	25	15	10	6	4
7		39	24	15	9	6	4	2
8		39	24	15	9	6	4	2	1
9		39	24	15	9	6	4	2	1	1
10		39	24	15	9	6	4	2	1	1	1
11		38	24	15	9	6	4	2	1	1	1	0
12		38	24	15	9	6	4	2	1	1	1	0	0
13		38	24	15	9	6	4	2	1	1	1	0	0	0
14		38	24	15	9	6	4	2	1	1	1	0	0	0
		1st	2nd	3rd	4th	5th	6th	7th	8th	9th	10th	11th	12th	13th

 

As the number of contenders grow each ranking approaches its limit, a definite number, no matter how large the field becomes. This fact becomes a powerful mathematical tool in predicting outcomes in sporting events or political elections.

Phi may also be a factor in political elections.

Note the percentage of the votes that were given in elections identified by Wikipedia as “landslide” elections in the popular vote of presidential elections, and their closeness to phi at 61.8% and phi squared at 38.2%:

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