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About combination of large cycles

research 9 January 2017 5   

How combined large cycles for different assets? Which of them move synchronously, and which live exclusively their own lives? Is it true that gold always goes against the dollar and the US stock market grow only when oil is cheap? How properly build strategic portfolio and how choose really undervalued assets? How to compare assets with fundamentally different features, and whether such a comparison would be fair?

When analysts talk about comparison, it is usually implies a correlation, if used any scientific methods. For example, correlation coefficient between gold and dollar index over the past 20 years is approximately -0.4 (on the daily data) - this is clearly not enough for formation of "no risk" portfolio, but it is enough that experts saw in them complete opposite. The following example shows how dangerous such a concept.

Let’s have a look on two cyclical price series, constructed according to the principles of Hurst:

About combination of large cycles
K=1, p=3.6, a=2

As it can be seen from formula, summands with largest period have the same phase, while the rest traded precisely in antiphase. Periods of summands pairwise equal, and amplitude reduced follow up a period, although, not proportional to it. The correlation coefficient in this system equal to -0.98 (during the interval partition 0...2*Pi into 1000 elementary segments), but this is not an obstacle to assets X and X1 globally move together (figure 1), which is create a false impression about their character.

About combination of large cycles
Figure 1 – charts X and X₁ (demonstrated only three summands of the series)

Later it will be shown, that described picture is just excessively idealized, but certainly not a purely hypothetical, and this, calls into question many of well-known regularities, and at the same time undermines the foundations of the whole built on correlations portfolio theory, because according to rules of mathematical logic, a single negative result is enough to negate the whole concept. Of course, the theory is necessary to complement, but not to discard.

The amplitude and volatility.

All assets that are traded in financial markets, are slightly different. They are different in nominal cost, expected return and accepted risk measure - volatility:

About combination of large cycles
where Cj - market close price on the bar with the index j,
q – number of observations per year (q=252 for daily data).

In traditional sense, volatility is the only responsible for the market fluctuations but in reality it is an integral characteristic, which is affects global cycles because all amplitude necessarily linked. If one of them deviates from the norm, then on certain time-frames would be beneficial systematic game with or against trend, and this is a clear contradiction of the market logic, which should not provide traders with such a trivial opportunities.

Volatility is an objective characteristic which, in contrast to all amplitudes, calculated directly on the basis of price data by use simple and proven formula. In addition, from graph of pure sentiment, can be picked-up so called full amplitude A₀, namely, the difference (in logs) between the tops and bottoms of the prime cycle. Now it is possible to associate those values, by use their analogues within the same idealistic cyclical model (formula 1):

About combination of large cycles
where u – period of large Kondratieff cycle.

Let’s conduct numerical experiment, to determine how certain variables affect volatility. Next step - divide the interval 0 ... 2*Pi into 10,000 elementary sections and fix value p and I, after that, by varying the K coefficient will try to get changes of full amplitude. By plotting chart A₀(V) (Figure 2a), it can be seen linear relationship among them (A₀=U*V), and it remains so, for all values of p and a, on which depends the only factor U.

About combination of large cycles
Figure 2a - relationship between volatility and full amplitude.
Figure 2b - relationship between amplitude and period in ideal cycles;
red line - experimental, green - approximation by formula a=0.96+0.28p.

The value of coefficient U=1/8 determines based on actual price data for the six most liquid assets, assume that they all have the same characteristic period of the prime cycle u=55 years (table 1). The range of the full amplitude with respect to its theoretical value obtained quite small, and as a result, it speaks not only about stability of founded solution, but also about fractal similarity of charts for the various financial instruments.

About combination of large cycles
Table 1 – volatility and full amplitude for major assets.

Finally, consider inverse problem, for which, initiated numerical experiment - define the values of a and p, in which the coefficient U=1/8. Because in calculation scheme the unknown more than equations, will be find only dependence of a (p), the benefit that it is obtained clearly, namely one particular value p corresponds to one particular value of a. As it can be seen from Figure 2b, this relationship is almost linear, with the amplitude that decreased much slower than period.

The theory of cycles combination.

Let’s consider two elementary sinusoidal functions f1 and f2, the periods which coincide and equal to 2*Pi (Figure 3). Since the functions, in essence, are the logarithm of the price, then for proper comparison need to use the difference f3=f1-f2 (Figure 4) - on the chart it is a complete analogue of logarithm of quotes ratio of two random assets ln(A/B).

About combination of large cycles
Figure 3 - the initial sinusoidal function.
f1=3+sin(x), f2=3+sin(x+t).

About combination of large cycles
Figure 4 - subtraction functions with the same period.
a - assets f1 and f2 traded in antiphase (t=Pi),
b - general case, when phases do not match.

It is clear that period of the resulting function in this situation will always be equal to periods two initials except one single case (t=0) when f3 disintegrate into a straight line. The amplitude f3 also depends on the value of t, and reach it peak in antiphase (t=Pi). The cycles of final function symmetrical with respect to verticals, outlined through the tops, phase is equal to -t/2.

Let’s complicate the picture and imagine that each original function is the sum of two sinusoids, the smallest of whom has a realistic amplitude commensurate with its period (Figure 5).

About combination of large cycles
Figure 5 - initial functions, which consist from two sinusoids.
f1=3+sin(x)+sin(px)/a; f2=3+sin(x+t1)+sin(px+t2)/a.

About combination of large cycles
Figure 6 - subtraction functions, which consist from two sinusoids.
The parameters of calculation: p=3.6, a=2, t1=Pi/2, t2=Pi/3.

As it can be seen from Figure 6, characteristic period of resulting function f3 still equal to 2*Pi, that is, the global cyclical picture determines by ratio of first summand of the initial functions, which has the highest amplitude. All other components f1 and f2 only affect on the shape of resulting function. If minor cycles are in antiphase, then the mood swings on f3 graph obtained especially sharp. Same phase minor cycles mutually eliminate each other.

Next example, when periods of prime cycle are slightly different, but their amplitude absolutely equal (Figure 7). The periods of minor cycles are proportional to prime, the phases are irrelevant.

About combination of large cycles
Figure 7 - initial functions with different periods.
f1=3+sin(x)+sin(px)/a; f2=3+sin(qx)+sin(qpx+t)/a.

About combination of large cycles
Figure 8 – subtraction functions with different periods.
The parameters of calculation: p=3.6, a=2, q=3/4, t=Pi/2.

The Figure 8 demonstrates what happens with resulting function f3, if initial assets are not connected by any mutual obligations. In one part of the chart can be seen a wide range with fluctuations from side to side, after that will be inevitably followed by some sharp cycles, when global bottoms on one asset will coincide with the top of another. The picture will be very similar to the case of equal periods in antiphase, but it will not last long.

About combination of large cycles
Figure 9 - initial functions (prime same phase cycles).
f1=3+sin(x)+sin(px)/a; f2=3+sin(x+t1)+sin(px+t2)/a.

The last realistic situation when prime cycles of initial functions have equal period, but they phases particularly match (Figure 9). During subtraction these cycles mutually eliminate each other, and leave behind oscillations with a period 2*Pi/p (Figure 10), but when adding they are clearly visible (Figure 11). The average period still equal to 2*Pi, but full amplitude, if minor cycles in antiphase can be almost equal to amplitude in Figure 10.

About combination of large cycles
Figure 10 - subtraction functions (prime same phase cycles)
The parameters of calculation: p=3.6, a=2, t1=Pi/6, t2=-3*Pi/4.

About combination of large cycles
Figure 11 - adding functions (prime same phase cycles).
The parameters of calculation: p=3.6, a=2, t1=Pi/6, t2=-3*Pi/4.

It is clear that full independence of some assets is a purely hypothetical case, because they all connected through inflation, which is a cyclic function. It is necessary to choose - either all assets have the same characteristic period equal to period of Kondratieff cycle or one day at the peak of inflation will get a low interest rate, low-cost commodities and bubble in stock market. The logic of pricing suggest, that it needs to choose first one.

It turns out that, whole variety of options, in fact, limited to examination of phase. At the same time, even with a relatively short history (approximately 60), on the basis of the resulting function can uniquely identify all features of initials. If ration of quotes is periodic, then the initial assets have different phase, if periodic product, then phase must match. In such a way, to identify cyclical features do not need to do any calculations!

Oil, gold and index S&P 500.

As it is clear from the theory, the analysis of compatibility of cycles should be performed on the basis of price data, cleared from inflation and all other factors of growth. However, to confirm the theory itself can be used usual quotes – if the effect of these factors conservative and at the same time leads to an increase of both assets that taken in consideration, their ratio would be a normal cyclical function, even if it is much beyond the limits of horizontal channel.

About combination of large cycles
Figure 12 - ratio of the S&P 500 to gold.

Figure 12 demonstrates the ratio of the S&P500 to gold - a periodic picture that accurately repeats Kondratieff cycles of the United States market, and global waves do not contain any extraneous noise. There is no doubt that these assets are cyclically linked, moreover, they traded almost in antiphase. Prime tops of the S&P index coincide with bottoms of gold and vice versa; the time difference between turning points not greater than 5 years.

Of course, firm quotes of gold before 70s have left their mark on the chart, but their influence best seen just where the cyclic picture contains some distortion – fluctuations of 33rd year and is too shallow bottom in the early 40 - x. The data on the chart that earlier than 1920 is not indicative, as the S&P at that time did not reflect to dynamics of market capitalization. This is the case when the impact of growth factors cannot be considered conservative.

About combination of large cycles
Figure 13 – ratio of the S&P 500 to oil.

The figure 13 demonstrates ratio of the S&P 500 to oil - and cycles are also visible. They are not as irreproachable in terms of noise, but this pattern indicates only the different nature of pricing. For gold, currency and stock markets, the growth factor is money supply for oil - only inflation. These values are cyclically linked, but there is no functional relationship between them, of course, this not - leads to "shocks" on the chart.

The comparison of pure sentiments (Figure 14) allows to get a more advanced cyclical picture. As it should be, now chart develops in horizontal channel, and excessive growth in 2014 year perfectly fit into correction that needs to stretch the cycle on the right. The volatility of oil and the presence of long periods of imbalances of supply and demand still remind about themselves, but they do not change position of major tops and bottoms.

About combination of large cycles
Figure 14 – ratio of the US stock market to oil (pure sentiment).

Throughout the XX century key bottoms on the US market accurate to a few years coincided with highs for oil, if consider them in constant dollars - it was in the 1921st, 1949th, 1982nd and 2008th years. At the same time, there are enough examples in history where oil and the US stock market over the years steadily move in the same direction, but it happened only due to impact of minor cycle, which in the case of oil has exorbitant and unstable amplitude.

The years of Great Depression, clarify some features of the pricing of oil and gold. For instance, the market decline from 1929 to 1932nd years passed in a sharp increase in demand for gold, which eventually led to an increase of its official rate, while oil, followed in line with deflation, continued decline until 1933 year. However, by 1949 year, when the US market reached significant bottom, oil caught up with gold and also set the top.

About combination of large cycles
Figure 15 – ratio of gold to oil.

It should be noted that study of cyclic features requires special thoroughness and comprehensive analysis of facts. For instance, consider the chart of ratio of gold to oil (Figure 15) it can be easily deceived by assume their different frequency. The picture has cleared up only when compared with the index S&P 500 - in spite of different nature of growth factors globally oil and gold move together, as it is required by logic of inflationary cycles.

Drift and parity of volatilities.

When setting numerical experiments, deliberately used cycles with equal amplitudes, otherwise the most volatile asset simply suppresses the movement of opponent, by not let a chance to prove themselves on the chart of ratio. The small difference in the amplitudes, as in case of gold and the S&P, allows to ignore it, but in situation where opponent of gold stands dollar index or other slow asset, such an approach does not work.

In fact, the goal is to come up with an analogue of the initial chart, which will be stored all basic proportions, but volatility and, as a consequence, full amplitude of the cycle will be equal to corresponding parameters of another asset. Logically, this means that difference of quotes between two arbitrary bars on main chart and at its analogue always differ in exactly u times. Hence, the array of data for construction of chart-analogue determined by the formula:

About combination of large cycles
where Ck - initial quote on bar with index k,
d - any positive number (d>0).

The coefficient u equal to ratio of volatilities of two compared assets, in such a way, whole principle according to which one of them stretched vertically, can be considered a parity of value V. The parameter d in this case is just a number, adding to function - it determines the nominal quote, but does not affect the logarithmic ratio. The original chart for dollar index demonstrates Figure 16, and an analogue with an amplitude increased by 3 times in Figure 17.

About combination of large cycles
Figure 16 – initial chart of dollar index.

About combination of large cycles
Figure 17 – chart-analogue with enhanced amplitude (d=100).

For gold, there is another problem - the long-term uptrend, which in this case cannot be compensated by anything, as dollar index due to pricing features originally develops almost in horizontal channel. Of course, it is possible to use calculations on pure sentiment, but it would be better, do not lay one assumptions on another and resolve this issue with trend in some other way, for example, with help of conventional drift:

About combination of large cycles
where k - serial number of bar, from the first,
e – constant of drift (e>0).

Physically correction for drift means clockwise rotation of chart around bar with serial number 0 - each subsequent bar loses in value more than the previous one, but all cyclical proportions are retained. The value of e (in this case, e=0.0054) has chosen in such a way that expected bottom in 1970, and in the early 2000s were precisely on the same level. The initial chart of gold demonstrates Figure 18, and chart that take into account the drift in Figure 19.

About combination of large cycles
Figure 18 - initial chart of gold.

About combination of large cycles
Figure 19 - chart of gold adjusted for drift.

Adjustment for drift and vertical scaling on the principle of parity volatilities is a complete set of universal methods to place any initial charts to formalized form, ie to horizontal channel with defined in advance theoretical amplitude. It is natural that such a constructions only partly reflect the essence of what is going on in financial markets, but by that disadvantage, they almost do not differ from initial chart with nominal price.

Dollar and gold

Let’s turn to history to understand how dollar and gold are linked. Formally, in the 70 years dollar index remained in downward channel, but the movement in terms of cycles cannot be considered indicative, since exchange rates were only on transition to market pricing. The real picture slightly evident only USDGBP chart (Figure 20), in which taken into account parallel investment exchange rate that existed in Britain from 1950 till 1979th years.

About combination of large cycles
Figure 20 - chart of USDGBP (until 1979 year - investment rate).

The chart demonstrates that the US currency relative to sterling rose from 1962 till 1985th years, and fell from 1985 to 2008th. It appears that globally it goes precisely according to Kondratieff cycles - against own market and with gold. In order to verify this hypothesis, alternately performed multiplication and division of above quotes of gold and dollar index. The results are shown in Figures 21 and 22, respectively. Charts are normalized so that the minimum value is always equal to one.

About combination of large cycles
Figure 21 - multiplication of above quotes of gold and the dollar index.

About combination of large cycles
Figure 22 - ratio of above quotes of gold and dollar index.

To be honest, monitoring period is not enough, but it is possible to make the main conclusions - if all assets are really have the same characteristic period, dollar and gold definitely move together. This is the only assumption that explains a great cycle on first chart, and a few small on the second. Minor cycles at the same time traded clearly in antiphase, about it says the lack of long corrections in Figure 21 and imposing amplitude in Figure 22.

Such a dual picture has a great physical sense. Locally gold moves against dollar, because it is denominated in US currency, so fall of it increases the metal's appeal outside the United States. Globally, dollar and gold are two reserve currencies, which attract capital in conditions of fear and return it when economy moves to stable growth. There is no doubt, gold is money!

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