Economies, and industrial sectors, suffered suddenly in the past from economic turbulences from time to time. In particular in the shipping sector, there were periods when ship owners made great fortunes and others when they became bankrupt [2] .

The frequency of turbulences is related to the ability, which managers have now to take faster decisions, using more rapid computers and multi-service mobile phones. In addition, millions of people work now in the financial sector. After all, we all live in interconnected and globalized world-being citizens of one and the same village: the Globe.

In order not to be misled, computer models, are not yet in a position to cope with the rapidity that real economic life changes, as manifested in the situations when stock exchanges collapsed faster than computers, contrary to what was supposed by theory.

The “Economics of Turbulence” is now at the agenda given the meltdown¹ in end-2008, which reminded us of both the 1929 “Great Depression²” and the 1987 Black Monday, just ~21 years ago. USA’s annual real GDP during 1929-1934 fell from $1000 b (constant 2005$) to $700 b (−30%) and recovered by 1938-1939 to $1000 b again (1910 = $550 b). Additional depressions have been recorded in 1819, in 1873-1896, in 1907, and in 1910-1911. Another Black Monday was the 09/05/1873 for Vienna stock exchange, and the “Long Depression” that followed.

Should the models based on normal distribution forecast turbulence? Normal distribution―as shown in Figure 1 excludes… turbulences. Turbulence is the manifestation of forces, which cause standard deviation-σ-to escape beyond ±3 from its mean... People are by now familiar with the fact that plethora of σ’s occurred beyond ±3… (Figure 13). Moreover, evidence is accumulated by now against “random walk hypothesis” [3] [4] .

As shown, 68.2% of all outcomes are within ±1σ from mean μ, 95.4% are within ±2σ and 99.6% are within ±3σ. Beyond ±3σ, the probability of an outcome to be there is between ±0.2% and ±0.4%: points at which distribution curve touches zero.

In addition to the reliance on normal distribution, science has relied on models of 1) “classical linear regression”, 2) “ARMA”(autoregressive moving average models) and 3) “VAR” (multivariate time series with lagged values on right hand side), which are linear in nature, i.e. linear in parameters: Y = Xβ + u_t, where u_t~ N(0, σ²), where “errors” follow normal distribution with mean 0 and variance σ² [1] .

Many relationships in finance, and shipping, are intrinsically non-linear [4] . Linear structural models and time series are unable to explain a number of important features, common to financial, and shipping, data: 1) leptokurtosis, meaning fat tails, and excess peaks at the mean; 2) volatility clustering/volatility pooling, meaning that volatility appears in bunches: i.e. large returns follow large returns, and small returns follow small returns. Some relate this to the way information arrives; 3) the leverage effects, meaning that volatility rises more following a large price fall, than a rise of the same magnitude [1] .

This paper aims at looking on “economic turbulence”, which shipping and financial time series exhibit at occasions. Also, a method to forecast turbulence in the shipping sector for 20 years to come is used, outside the sample, i.e. for 2016-2035 using the nonlinear method―the “Kernel density estimation”.

The paper is organized as follows: next, is a literature review, followed by methodology. Then, the concepts of volatility, risk and uncertainty are presented. Next, the more suitable method to model economic turbulences is indicated. The “maritime economics freight index” is then presented. Next, the phenomenon of power-laws in stock exchanges is analyzed, and the meaning of coefficient alpha is given. Finally, we forecast the turbulences that we expect to appear in shipping sector till 2035, and finally conclude.

Ruelle and Takens [5] defined mathematically turbulence (in physical systems) as follows: let a physical system consisting of a viscous fluid, and rigid bodies, to be subject to a zero external action―and let this action measured by a parameter μ. If μ = 0, the system now tends to equilibrium. Next, submit the system to a “steady, light, positive, action” μ > 0, thus obtaining a steady state (the physical parameters of the fluid are constant); this fluid is in dis-equilibrium. Let assume now μ > 0, and increasing: then 1) the fluid will have a change in its symmetry pattern, 2) a periodicity in time, and 3) given a sufficiently large μ, its motion will become very complicated, irregular and chaotic: this is turbulence.

Feynman [6] has described turbulence as the most important, but unsolved problem of classical physics. Turbulence has the following features: irregularity (highly)―with a lengthy scale; chaotic always; diffusive; rotational; dissipative; energy cascading; with integral length scales and with Kolmogorov ones as well Taylor’s microscales…

Mello [7] argued that “natural catastrophes” are extreme events, possessing―(at times)―an infinitesimal likelihood to occur... In statistics, extreme events occur on “the tails” of a probability distribution. He suggests the use of (fat-tailed) statistical methods coming under the broad scope of “Levy³ processes” [4] [8] .

The scientific community, Mello [7] argued, came to rely over-abundantly on Normal distribution. There is an urgent necessity to explore the notion that other processes may be better suited for determining probability estimates for extreme events (i.e. events that result to turbulence) (italics and bold added). He showed that H (the Hurst exponent defined in methodology) falls between 0 and 1, and alpha = 1/H is >2; the fractal dimension of the path cannot, however, exceed its Euclidean dimension 2.

Anonymous [9] introduced a statistical model based on the class of “symmetrica-alpha stable distributions”, which are well-suited for describing signals that are impulsive in nature: φ ( ω ) = exp ( j δ ω − γ | ω | α ) , where alpha is the characteristic exponent restricted to 0 < alpha ≤ 2, delta (−∞ < δ < ∞) is the location parameter and γ (>0) stands for variance. When α is in the interval [1-2], then delta = the mean of the distribution; when 0 < alpha ≤ 1, delta = median, as the mean is undefined. The shape of one “standard alpha-stable density function” is shown in Figure 2; we have chosen there alpha = ½ on purpose to show the high peak and the long tails involved.

As shown, the peak of the distribution (alpha = 1/2) is, at f(x) = 0.65, at a higher height than if alpha was = 1, 1.5 or 2. This is the impact of alpha on the peak of the distribution, producing also fat tails, or long tails, beyond ±5σ.

Mandelbrot and Hudson [4] argued that financial turbulence is not rare. Turbulence is at the heart of markets, where the key phenomena are: wild price swings, business failures, and windfall trading profits. Mitigation is needed with focus on the concentrated bursts of action and the discontinuities in prices,

events that common economic wisdom says should not happen and calls them “statistical outliers”. The assumptions underlying models are wrong. Standard financial models focus on a typical market behavior, i.e. modest price changes, real or seeming trends, a risky, but ultimately closely manageable world. Risk = ±3σ. For shipping this means ±308 units of the shipping index (Figure 12; 1947 = 100), where σ = ~103. The index had 42 units in 1972 min. and 795 in 1918 maximum.

Trevethan and Chanson [10] argued that turbulence is not Gaussian, particularly in Nature. Any turbulent flow is often dominated by coherent structure activities and turbulent events, meaning a series of fluctuations that contain more energy than average ones.

Juarez [11] used a “complex”⁴ model, applied in “weather forecasting” by the meteorologist Lorenz [12] in 1963 (Figure 3), known as the “wings of the butterfly” or “the eyes of owl”, to determine the relationship among: cash flow, profit & loss and assets of 70 companies in the crude oil mining and natural gas in Colombia.

When Juarez used conventional methods the explained variance was 6%, but after certain dynamic transformations he made, he succeeded to increase that to 73%, fitting a linear regression (Figure 4).

As shown, the dependent variable is the “profit & loss X”, and the independent variables are: “cash flow Y” and “assets Z”. This model resulted in a better prediction, as argued.

Zou et al. [14] argued that the financial time series of prices, deviated significantly from “standard normal”, and had nonlinear data characteristics. Skewness of −0.0273 and kurtosis of 7.1647 for 1790 days⁵ (02/01/2002-13/02/2009)― deviated from “normal” levels (kurtosis = 3; skewness 0). This indicates that the market exhibits significant abnormal return changes. The null hypothesis of “Jarque-Bera” “test of normality” and “BDS” “test of independence”, were both

rejected. This means that market returns contain unknown nonlinear dynamics, not easily captured by traditional linear models (italics added).

In summary, we showed, among other things, that turbulences are considered in a probabilistic manner, as rare economic phenomena. This, however, we doubt; looking at Black Monday, 1987, and at the Great Recession, in end-2008, there were only 21 years apart... Turbulences are also always chaotic and are expressed exclusively by the “Levy stable distributions”. Alpha < 2 creates fat tails and high peaks, a serious number of outliers and thus unexpected outcomes (turbulences).

First, we distinguish volatility/turbulence in 3 states: 1) when volatility is mild, and modelled by Normal distribution, and σ ≤ ±3. 2) When turbulence is modelled by an “in -between” distribution, and turbulence = alpha = 1.5 ≤ 1.70 and σ > ±3 ≤ ±12.99. This is a representation of the shipping sector. 3) When turbulence is modelled by a Cauchy distribution, being wild, and alpha = 1 and σ ≥ ±13.

We assume also that shipping time series behave like financial ones [15] . This assumption entitles us here to construct bridges between finance and shipping. For shipping a cause of turbulence is a serious dis-equilibrium between demand and supply, as argued by Koopmans [16] .

The relationship between Hurst exponent and alpha coefficient: Let R_n be the sum of a stable variable in a particular interval n, and R₁ be its initial value, then [8] :

R n = R 1 ∗ n 1 / alpha (1)

means that the sum of n values scales by n^1/alpha times their initial value. Taking logs:

alpha = log ( n ) / log ( R n ) − log ( R 1 ) = log ( n ) / o g ( R / S ) l = 1 / H (2)

where

H = log ( c ) + log ( R / S n ) / log n (3)

where log(c) = 0 being constant.

logR_n - R₁ ~ R/S and thus alpha = 1/H (4)

Equation (3) has been developed by Hurst [17] to determine long-memory effects and a fractional Brownian motion, using the range of a time series divided by local standard deviation (R/S_n); also

(R/S)_n = c × n^H (5)

(Rescaled) range is a method dealing with volatility rescaled to make it independent of time. It is a generalized method to measure the speed of a time series including the case of Einstein 1905 [18] as a special one. Einstein [18] proved that a particle (and assume also a time series) moves at the square root of time n, i.e.

Distance = n (6)

n stands for the number of observations or of time steps. With H = 0.69 (rounded) there is “black noise” in shipping markets, which is characterized by catastrophes and discontinuities, which are abrupt up and down, indicating high peaks at the mean and fat tails…

In forecasting, we used the “Kernel Density Estimation” method―KDE. It is slightly simpler than the other local linear forecasting methods, being in fact a weighted average. We used this method successfully in other occasions [19] [20] . The forecasts are given by

X ( N + 1 ) prediction = ∑ i = 1 k W j X j + p + ( m − 1 ) T (7)

where―if possible―

∑W_i = 1. W_i = f(R_i)/∑f(R_j) (8)

where j = 1 to k. R_j are the distances to which a weight f(R_i) is

attached = exp {−R²_i/c²} (9)

where c is the mean of R_i. For the last point X_T (bolds stand for vectors), we select the k nearest neighbors X_j, where

j = 1(1)k (10)

k. m stands for embedding dimension, the time delay is T, k is the number of the nearest neighbors and p = T − k. This method is due to Sugihara and May [21] with k = m + 1 and x_τ are included in x_j. Forecasting is based on the mean of the relocated points weighted, as far as the initial distance is concerned.

Business life―blackboard theorizing, and uncountable conferences―became familiar with the concepts of risk, volatility and uncertainty. Companies need help and effective ways of their protection from these three enemies (i.e. use of hedging). As a result, managers may easily become victims, where the “Sirens” are in the form of all kinds of derivatives.

Is volatility = turbulence?

The σ² measures the spread of data round mean:

S ( n ) = { 1 / n ∑ n = 0 n - 1 ( u n - ( u ) n ) 2 } 1 / 2 , (11)

where u_t stands for discrete time series, (u)_n stands for the mean over time lag n; σ 2 = σ [22] . Moreover, adaptive models for (mild) volatility have already created by econometricians: ARCH, GARCH 1986, EGARCH, and GJR⁶ 1993 etc. Volatility (mild) in addition is not disastrous. e.g. an earthquake at a low point on Richter’s climax causes volatility, but not devastation. Tsunamis, hurricanes, the Katrina, the Caroline, are different. Volatility⁷ is like a severe fever… to put it simply, and a broader concept of turbulence.

For econometricians, volatility is the degree to which a time series varies over time. It is usually measured by variance⁸, denoted by the Greek letter σ². In finance, σ² is often also used as a measure of risk⁹. Risk¹⁰ is very important for companies; firms wish to manage, avoid or profit from it. Moreover, σ² appears in the equation of normal distribution, and the probability of the risk of a decision is locked in ±3σ from mean… maximum, as mentioned.

Let us present volatility in shipping markets (Figure 5 and Figure 6).

As shown (left), all 3 main ship sizes and markets had strong variations in their earnings since 2011; par excellence Capes¹¹ (blue line) had for the first time in 2012 hires (=time charters), which fell from >$42,350 per day to $2,350 (!), as well in 2016. The “BDI-Baltic Dry Index” (right) fell from a high of ~11,800 units in 2008 to ~400 by 2015. Capes were heavily involved in China’s imports in serious volumes.

Shipping and Finance cannot be modeled by Normal distribution... due to the alpha they have [20] . They are better modelled by a type of distribution, where alpha is 1.50, called “Fractal or Pareto”. What is alpha? Alpha is the new indicator of risk, of uncertainty and volatility for leptokurtic distributions. Figure 7 shows the values of “shipping” alpha―starting with 1.70 and ending to 1.47 for shipping dry cargo markets from 2003 to 2015. Alpha = 1.70 as the upper benchmark of a strong market variation as indirectly “defined” by late Mandelbrot [4] .

Figure 8 presents the 3 known graphs of 3 well known distributions: normal, Cauchy and Financial/Shipping. The only difference is the value of alpha, which creates longer tails and higher peaks as the case may be. Let us see the main distributions closer.

Normal distribution

The equation of normal¹² distribution (Figure 9) is:

f ( x ) = 1 / σ 2 π ∗ e − ( ( x − μ ) / σ ) 2 / 2 (12)

where x is the level of a specific variable under study, μ is the average value of all x’s round average, σ stands for standard deviation―and e is a constant¹³.

As shown, the tails approach zero at ±3.5σ from mean¹⁴ (X = 0).

The time series of “maritime economics freight index”, 1741-2007 (extended to 2015) [2] is shown (Figure 11), with its “normal distribution” (Figure 12).

As shown, the shipping distribution―fitted to 266 years―exhibits a positive skew, and is not normal. The skew is positive and greater than zero¹⁸. More important is that the fitted distribution indicates a long tail on the right, and peak at ~144 units, which are measured by alpha―mentioned below―equal to 1.46 < 2. The shipping standard deviation went away from its mean by 6.52σ in 2008…

Turbulences/volatility in shipping historically is as follows: In class I, recorded none. In class II recorded none too. In class III were recorded 14 cases: in 1916 3.8; 1917 7.1; 1918 7.7σ; 1919 5; 1920 3.8σ; 2004 3.4σ; 2007 5.7; 2008 6.5σ; 2009 3.6; 2010 4.5; 2011 3.9; 2012 3.4; 2013 3.3 and in 2014σ = 3.1.

The dry cargo market held up till 1974 and for small bulk carriers into 1975. There was stock building in the world economy due to commodity price inflation and the heavy congestion in Middle East and Nigeria due to increased oil revenue [2] . The spot market moved into recession in 1975 till 1978. The sector entered into a deep depression in 2^nd half in 1981 till 1^st half in 1987. Stock markets are, however, wilder.

Figure 13 gives a concise picture of turbulences since 1885. It shows that serious turbulences in stock prices are discontinuous. Obviously, a depression is the main cause of an economic turbulence. Τhe four most serious turbulences, since 1885, were in: 1) 1894, 2) 1929, 3) 1987 and 4) end-2008, measured by changes in σ of monthly stock returns. These reached the maximum change in σ of ~+25%.

Figure 13 records ~48,000 days-1572 months of stock returns (1885-2016). Stock market exhibited about 15 incidents of serious economic turbulence, i.e. above 13% change in σ, since 1885. Deviations above 7.5% changes in σ were fewer―about 45―while hundreds can be found between changes in σ of 1% and up to3% (not shown)―the benchmark of normal distribution―and fewer between 3% changes in σ and 7.49%. The great number of turbulences 54.5%, indicated by changes in σ ranged from 1% to 8%; about 1/3 = 34%= of the 132 or

so years (~45 years) showed turbulences from 9% to 14% and the rest 15 years (12.5%), showed turbulences from 15% to ~25%.

Turbulence, though studied for more than 100 years, is only partly understood by theory [4] . Why are markets turbulent? Exogenous factors are to blame [4] ¹⁹, as called by economists and econometricians. Also, the size of firms and industry’s concentration influence profits, and in turn profits determine stock prices.

The key traits of turbulence are: “scaling”-γ and “long-term dependence”―or a Hurst exponent H > 0.5 ≤ 1. Also, “self-similarity” is a property of invariance against changes in scale or size. Small parts of an object are qualitatively the same―or similar―to the whole [8] . This underlies fractals, chaos and power laws. This further means a decisive symmetry. Symmetry means that matters are the same, i.e. they are invariant in change: or something stays the same, in spite of alterations [24] . The characteristics of economic turbulence are: abrupt lurches between wild motion and quite activity; discontinuities; intermittency; and uncertainty about major events in time [4] .

Economic history is also full of “near”²⁰―turbulent events―like that of October 27^th, 1997. The Dow then lost a 7.2%; cascades of selling occurred across exchange―forcing authorities to halt trading in 2 times [4] , affecting 1 billion shares. This turmoil spread round the globe stock exchanges: Hong Kong―14%; London―9%. The value of USA business fell at a rate of $100 m per second! The study of roughness, of the irregular and of jagged (turbulence) is needed.

The place where chaos²¹ reigns over charts is Wall Street. In stock and commodity exchanges, “self-similarity” weighs in on many scales. The paradox here is that the chart of minute-by-minute stock averages looks much alike the daily averages and so on in weekly, and in monthly prices… There is an uncharacteristic jump in the data―as in October 1987 as shown in Figure 13 [24] . Such price jumps are also known as “innovation processes”.

The “linear” (normality) assumption is that in stock averages the actual prices are generated by independent increments. This has a power spectrum that is proportional to the inverse square of the frequency, called “brown noise”, as an allusion to “Brownian motion”. The name of this motion is after the Scottish botanist Robert Brown (1773-1858), who in 1827 saw floating dust specks to “move” in a microscope. The innovation process consists of the independent “kicks” given to the suspended particles by the molecules of the liquid (in which they float/hover).

Pareto (1848-1923) was an industrialist, economist and sociologist. The field of economics evolved by his efforts from a branch of social philosophy―as practiced by Adam Smith (1723-1790)―into a data―intensive field of scientific research and mathematical equations. He attracted by the way “power” and “wealth” is distributed. He gathered extensive data from 1471 to 1512, and plotted them (in 1909) on a graph (Figure 14): income level is measured on vertical axis, and the number of people having that income, on horizontal. The result was that very far in the bottom was the mass of people and very thin at the top was the wealthy elite. Pareto proved that few people are extremely rich and the great masses are poorer.

As shown, the curve is not symmetrical. The slope of the Pareto curve― alpha―is equal to −3/2 = −1.5 (power law)… Money begets money, power creates power. “The first 1/2 a million $ is difficult, while the $1 m is easier thereafter” or “increasing returns to scale”, which economists ignored²²... [25] . Pareto’s formula was

P(u) = (u/m)^−α (14)

where P is the proportion of people earning more than some level of income u, and m is the minimum income²³.

Alpha is a characteristic exponent restricted to values [0-2] or rather [1-2]― where 0 < alpha ≤ 2―shapes the distribution it belongs to. The smaller alpha, the heavier (fatter) the tails of the (stable density) distribution are. Several outliers (extreme outcomes) will be observed.

As shown, alpha wanders round 2, meaning: a normal distribution since 1750 and till 1975 (225^th year). From 1975 and till 2015, however, alpha gradually fell from 2 to 1.4561976 = 1.46 (rounded) for n = 10 on 266 yearly observations. As argued by Peters [8] turbulence²⁴ is related to the velocity of a fluid and not to fluid’s movement, entailing 0 ≤ H < 0.50. As alpha decreases, both the occurrence rate and the strength of outliers increase, resulting to very impulsive processes. In Figure 15 the alpha coefficient of the “shipping index of dry cargo”

is plotted against time (years) since 1741 and till 2015^25,26.

Financial (and shipping) time series are known to be non-stationary, where the statistical calculations for σ etc., are valid only for stationary series. Here we have applied 1^st logarithmic differences to obtain stationarity. This transformation is due to Box and Jenkins [26] .

The tendency towards catastrophes (turbulences) has been called by Mandelbrot [27] “Noah effect” or the ∞ variance syndrome [8] . Fat tails are caused by crashes and stampedes, which tend to be abrupt and discontinuous. So, turbulence is related to small alphas ≤ 2 and ≥1 and especially ≤1.70. There are 13 years that alpha varies from 1.70 to 1.46 and 17 years that alpha is less than 2 (since 1975) as shown in Figure 15. The Noah effect depends on the relative size of one event [4] .

Equation: H = log ( c ) + log ( R / S n ) / log n (15)

will be used, where c is a constant, R is the range, n is a time index and it equals the number of observations, S is the local standard deviation and H is the Hurst exponent. This is based on Hurst’s equation

log ( R / σ ) = K log ( N / 2 ) (16)

or R = σ ( Ν / 2 ) Κ (17)

where K = H. Hurst [17] found K = 0.73; we found H = 0.69. The R/S or rescaled range statistic is widely used by now for testing whether long-term dependence is present in time series.

It has the advantage to be non-parametric―ignoring the organization of the original data [4] , suitable for stock prices. The R/S formula measures whether― over varying periods of time-the amount by which the data vary from maximum to minimum is > or < than what one would expect, if each data point were independent of the last (normality).

The “fractional Brownian motion”, as a generalization to processes which grow at different rates

t^H is: ([X_H(t) - X_H(0)]²)^1/2 with a linear | t | H (18)

where 0 < H < 1 is the Hurst exponent. Einstein [18] defined Brownian motion; his model became the main model for random walk in the study of statistics [22] . Einstein discovered that the distance covered by a random particle undergoing random collisions from all sides is directly related to the square root of time:

R = kT1/2 (19)

where R is the distance covered, k a constant and T the time index or n.

This, however, can be generalized to:

R/S = k T^H (20)

where T = the time index or n, or N. R/S has no dimension; the power value is equal to H. There is no matter if periods are separated by many years; and there is no need for time series to have a characteristic scale. Equation (6) has a characteristic of fractal geometry, as it is scaling in accordance with a power law― the power of H. As a result a time series may move covering time at the square root of time-this characterized the time series which are random and H = 1/2 = 0.50 (normality).

The most interesting case is the one where time series move at H > 1/2 ≤ 1, called black noise, persistent, as well possessing the Joseph and the Noah effects [22] . These time series―like shipping time series, with H = 0.69²⁷ as mentioned―are faster, and cover more distance than random walk-the system increasing in one period is more likely to keep increasing in the immediately following period. More important is that these time series has the potential of sudden catastrophes (the Noah effect) or of turbulences! This also allows for short term turbulences, say of 7 years down (according to Joseph effect)…

Forecasting turbulences outside the sample

For forecasting turbulences, we will use 1/H = alpha for 2016-2035 using NLTSA (2000) computer program due to Syriopoulos and Leontitsis [28] and the forecasting method “kernel density function” (presented briefly in methodology), which has been proved more reliable [29] . The forecast alpha is presented in Figure 16.

Figure 16 indicates the crisis periods where alpha fell below 2 (3 areas of time), with more serious the one after 1985 and 1989. Persistence for 2035 remained at a higher H = 0.70, against 0.68, and n = 277, after the 20 forecast values entered into the sample. The new alpha is lower and equal to 1.4286 against 1.47, still <1.70. The period of turbulence now transferred from 1975 to 1983 and then to 1989 till 2035. The index forecast to fall in 2016 from 242 units to 136 in 2035 (216 actual in 2015). Changing two parameters: embedding dimension to 8 from 5 and the number of the nearest neighbors from 16 to 20, alpha fell by 0.04. The new forecast values of the index will be from 128 to 121 (1947 = 100).

As a result, the forecasts indicate that turbulences in shipping dry cargo market will continue and will be intensified by and including 2035; we have 18 years still to wait to see reality if still alive…

Turbulence is an extreme economic phenomenon caused by forces, which should have a certain degree of intensity/energy, exhibiting always chaotic behavior. More important, turbulences cannot be captured by a probability density distribution like the Gaussian, because by definition it does not allow for long tails and high peaks―fact indicated by alpha < 2. Alpha < 2 ≥ 1 shows turbulences in the market to which time series belongs to.

The problem of turbulence is one of methodology: markets behave in one of three states: slow, mild and wild. Economists created only one (convenient) method to model all three: “normal distribution”. Moreover, the great majority of econometricians do not admit that normal distribution is unable to model “wild” circumstances, with ∞ variance or undefined mean, proposing²⁸ linear, models like GARCH.

GARCH in particular, when volatility jumps, it plugs-in new parameters to make the bell curve grow... and vice versa. The bell distribution is thus treated like a balloon to cope with actual volatility. Moreover, shipping sector and finance are special cases, because their time series―returns, indices or time series―do not follow either Normal distribution or Cauchy distribution…

Very astonishing is the fact that shipping sector since 1741 had time series of dry cargo freight rates, distributed following normal distribution till 1975, as indicated by the prevailing alphas. Turbulences in the sector occurred indeed after 1975: one depression appeared in 1981 till 1987, one in 2009―due to the end-2008 meltdown, and the recent depression―after China’s slowdown―in 2011-2018_.

Economists indeed accommodated 3 important concepts: “volatility, uncertainty and risk” into one, i.e. the σ of the bell curve, without reference to the distribution it belongs to. For Cauchy distribution―even if we wanted to define σ―we cannot do it… because variance is ∞ or undefined...

In addition, normality is a phenomenon of…the long run, it appears when recessions, depressions and turbulences, calm/boil down or even disappears from time series! This denies what Keynes wrote in 1936 that ‘in the long run, we are all dead’; because in the long run everything in the economy is more peaceful, as shown by normal distribution… At short time-frames prices vary wildly, and at longer time frames they settle down.

Despite the above, the yearly observations we used for shipping, detected wilderness in the last 13 years (2003-2015) pushing alpha from 2 in the interval [1.70, 1.43]. Shipping companies (and maritime economists) thus may be misled about reality according to the time-frame of data they look at… Minute by minute, hour by hour, day by day, week by week are time-frames that reveal the true wild face of markets; it is then also that decisions are taken… long runs are suitable for plans, but days are only suitable for decisions. Decisions transformed in actions are the blood and the energy for life in the economic organisms like markets. It is this energy that causes turbulences according to its energy, power and intensity.

Goulielmos, A.M. (2018) The Nature of Economic Turbulence: The Power Destructing Economies, with Application to Shipping. Modern Economy, 9, 1023-1044. https://doi.org/10.4236/me.2018.95066