Maritime Businesses: Volatile Stock Prices and Market Valuation Inefficiencies

 

Robert F. Mulligan, Ph.D.

Department of Business Computer Information Systems & Economics

Western Carolina University

College of Business

Cullowhee, North Carolina 28723

Phone: 828-227-3329

Fax: 828-227-7414

Email: mulligan@wcu.edu

 

 

Gary A. Lombardo, Ph.D.

Professor of Maritime Business

Department of Marine Transportation

United States Merchant Marine Academy

300 Steamboat Road

Kings Point, New York 11024

Phone: 516-773-5066

Email: lombardog@usmma.edu

 

Acknowledgements

Robert F. Mulligan is associate professor of economics in the Department of Business Computer Information Systems and Economics at Western Carolina University College of Business and a research associate of the State University of New York at Binghamton. Gary A. Lombardo is professor of maritime business in the Department of Marine Transportation at the United States Merchant Marine Academy. Financial support was provided for a preliminary investigation in the form of a Visiting Research Fellowship from the American Institute for Economic Research and a summer research grant from the Western Carolina University College of Business, which are gratefully acknowledged. Institutional support provided by the U.S. Merchant Marine Academy is gratefully appreciated. The authors remain responsible for any errors or omissions.

 

Abstract

 

This paper examines twelve maritime equity price series for behavioural stability and efficient market pricing for the 1989-2002 period. Five self-affine fractal analysis techniques for estimating the Hurst exponent, Mandelbrot-Lvy characteristic exponent, and fractal dimension are employed to explore the price series fractal properties. Techniques employed are rescaled-range analysis, power-spectral density analysis, roughness-length analysis, the variogram or structure function method, and wavelet analysis. Formal hypothesis tests provide evidence of a change in market behaviour between the 1989-1994 and 1995-2002 periods. Hypothesis tests also provide evidence against efficient valuation of the maritime businesses sampled, supporting the multifractal model of asset returns (MMAR) and disconfirming the weak form of the Efficient Market Hypothesis (EMH). Strong evidence is presented for antipersistence of some maritime equities in the sample, suggesting market participants habitually overreact to new information, and never learn not to. An important implication of this finding is that financial derivatives based on the sampled equities cannot be efficiently priced.

 

Introduction

Twelve maritime businesses are examined in terms of their stock prices and market valuations. Statistical tests focusing on five alternative methods for estimating Hurst (1951) exponent, fractal dimension, and Mandelbrot-Lvy characteristic exponent (Lvy 1925) are used. An analysis of the findings reveal stock price volatility and have implications for the efficient market hypothesis (EMH) (Fama et al 1969; Malkiel 1987) and the multi-fractal model of asset returns (MMAR) of Mandelbrot, Fisher, and Calvet (1997), as well as for widely used models for pricing financial derivatives.

The paper is organized as follows. A literature review is provided in the second section. The data are documented in the third section. Methodology and results are presented in the fourth and fifth sections. The conclusions are provided in the sixth section. Additionally, the glossary and appendix have been prepared to assist the reader in understanding the uniqueness of the specialized statistical language used in this paper.

Mandelbrot's (1972a, 1975, 1977) and Mandelbrot and Wallis's (1969) R/S or rescaled range analysis characterizes time series as one of four types: 1.) dependent or autocorrelated series, 2.) persistent, trend-reinforcing series, also called biased random walks, random walks with drift, or fractional Brownian motion, 3.) random walks, or 4.) anti-persistent, ergodic, or mean-reverting series. Mandelbrot-Lvy distributions are a general class of probability distributions derived from the generalized central limit theorem, and include the normal or Gaussian and Cauchy as limiting cases (Lvy 1925; Gnedenko and Kolmolgorov 1954). They are also referred to as stable, Lvy-stable, L-stable, stable-Paretian, and Pareto-Lvy. Samuelson (1982) popularized the term Mandelbrot-Lvy, but Mandelbrot avoids this expression, perhaps out of modesty, and the other terms remain current. The reciprocal of the Mandelbrot-Lvy characteristic exponent alpha is the Hurst exponent H, and estimates of H indicate the probability distribution underlying a time series. H = 1/alpha = 1/2 for normally-distributed or Gaussian processes. H = 1 for Cauchy-distributed processes. H = 2 for the Lvy distribution governing tosses of a fair coin. H is also related to the fractal dimension D by the relationship D = 2 - H. Series with different fractal statistics exhibit different properties as described in Table 2. In fractal analysis of capital markets, H indicates the relationship between the initial investment and a constant amount which can be withdrawn, the average return over various samples, providing a steady income without ever totally depleting the portfolio, over all past observations. Note there is no guarantee against future bankruptcy.

Mandelbrot (1972b, 1974) and Mandelbrot, Fisher, and Calvet (1997) have developed the multifractal model of asset returns (MMAR), which shares the long-memory feature of the Fractional Brownian Motion (FBM) model introduced by Mandelbrot and van Ness (1968). Fractal structure in equity prices indicates traditional statistical and econometric methods are inadequate for analyzing security markets. The statistical theory necessary to identify empirical regularities and local scaling properties of MMAR processes with local Hlder exponents is developed by Calvet, Fisher, and Mandelbrot (1997) and applied by Fisher, Calvet, and Mandelbrot (1997).

<<Table 1 about here>>

Literature review

The search for long memory in capital markets has been a fixture in the literature applying fractal geometry and chaos theory to economics since Mandelbrot (1963b) shifted his attention from income distribution to speculative prices. Fractal analysis has been applied extensively to equities (Green and Fielitz 1977; Lo 1991; Barkoulas and Baum 1996; Peters 1994, 1996; Koppl et al 1997; Kraemer and Runde1997; Barkoulas and Travlos 1998; Koppl and Nardone 2001; and Mulligan 2003) interest rates (Duan and Jacobs 1996; and Barkoulas and Baum 1997a, 1997b), commodities (Barkoulas, Baum, and Ogutz 1998), exchange rates (Cheung 1993; Byers and Peel 1996; Koppl and Yeager 1996; Barkoulas and Baum 1997c; Chou and Shih 1997; Andersen and Bollerslev 1997; Koppl and Broussard 1999, and Mulligan 2000a), and derivatives (Fang, Lai, and Lai 1994; Barcoulas, Labys, and Onochie 1997; and Corazza, Malliaris, and Nardelli 1997).

Cheung and Lai (1993) suggest Heiner's (1980) and Kaen and Rosenman's (1986) competence-difficulty (C-D) gap hypothesis as a potential source of long memory in asset prices, offering a theoretical expectation of long memory. The C-D gap is a discrepancy between investors' competence to make optimal decisions and the complexity of exogenous risk, to which the maritime industry is particularly vulnerable. A wide C-D gap leads to investor dependency on deterministic rules, which can lead to persistent price movements in one direction either a crash or speculative bubble. Due to the irregular arrival of new information, Kaen and Rosenman argue persistent price movements may suddenly reverse direction, leading to non-periodic cycles.

Mussa (1984) introduced a disequilibrium-overshooting model for exchange rate determination. Disequilibrium-overshooting, where market participants overcorrect when adjusting prices and quantities toward equilibrium, would be supported for equity markets by a finding of antipersistence or ergodicity, that is, of consistent price overadjustment, often followed by overcorrection in the opposite direction.

 

<<Table 2 about here>>

Data

The data are daily closing prices reported by the exchanges for each traded equity. A sample of twelve maritime firms was selected. Table 3 lists the twelve maritime firms examined, giving the sample range, number of daily price observations, and the standard beta measure of relative volatility for the last year in the sample.

The maximum sample period is December 30, 1988 to September 27, 2002 over thirteen and one-half years of daily data, including approximately 3,500 observations on each variable. For three of the firms examined, data are only available for a subsample.

The data are adjusted for stock splits. Failure to adjust raw price series for splits would have introduced discontinuities unrelated to any fundamentals. Such shifts in the mean can introduce biases toward finding fractal dynamics (e.g., Barkoulas, Baum, and Oguz 1998; Granger and Hyung 1999; Diebold and Inoue 2000). It is essential to test for fractal behavior using split-adjusted prices to reflect the actual return on the equities and ensure the findings are not a statistical artifact due to changes in the mean of the process, but instead reflect genuine features of the underlying data generating process. Data not adjusted for stock splits would introduce nonstationarities, the property that the mean of the data changes over time.

 

Methodology

Long memory series exhibit non-periodic long cycles, or persistent dependence between observations far apart in time; i.e., observable patterns which tend to repeat. Long memory or persistent series tend to reverse themselves less often than a purely random series. Thus, they display a trend, and are also called black noise, in contrast to purely random white noise. Persistent series have long memory in that events are correlated over long time periods. In contrast, short-term dependent time series include standard autoregressive moving average and Markov processes, and have the property that observations far apart exhibit little or no statistical dependence. R/S or rescaled range analysis distinguishes random from non-random or deterministic series. The rescaled range is the range divided (rescaled) by the standard deviation. Seemingly random time series may be deterministic chaos, fractional Brownian motion (FBM), or a mixture of random and non-random components.

Conventional statistical techniques lack power to distinguish unpredictable random components from highly predictable deterministic components. R/S analysis evolved to address this difficulty. R/S analysis exploits the structure of dependence in time series irrespective of their marginal distributions, statistically identifying non-periodic cyclic long run dependence as distinguished from short dependence or Markov character and periodic variation (Mandelbot 1972a: 259-260). The difference between long-memory processes, also called non-periodic long cycles, and short-term dependence, is that each observation in long memory processes has a persistent effect, on average, on all subsequent observations, up to some horizon after which memory is lost, whereas in contrast, short-term dependent processes display little or no memory of the past, and what short-term dependence can be observed often diminishes with the square of the time elapsed. For equity prices, long memory can be observed when a stock follows a trend or repeats a cyclical movement, even though the cycles can have time-varying frequencies. Short-term dependence is indicated when there are no observable trends or patterns beyond a very short time span, and the impact of any outliers or extreme values diminishes rapidly over time.

Mandelbrot (1963a, 1963b) demonstrated all speculative prices can be categorized in accordance with their Hurst exponent H. The Hurst exponent was introduced in the hydrological study of the Nile valley and is the reciprocal of the characteristic exponent alpha (Hurst 1951). Some stock prices are persistent or black noise processes with (0.50 < H < 1.00). These less noisy series exhibit clearer trends and more persistence the closer H is to one, and investors should be able to earn positive returns. Neely, Weller, and Dittmar (1997) found technical trading rules, formalized with a genetic programming algorithm, provided significant out-of-sample excess returns for persistent processes. However, Hs very close to one indicate high risk of large, abrupt changes, e.g., H = 1.00 for the Cauchy distribution, the basis for the characteristic exponent test. Using the modified rescaled-range (R/S), which is robust against short-term dependence, Lo (1991) found no long memory in stock prices. In applying his modified R/S analysis to equity prices, Lo overturned earlier results based on classical R/S methods finding long memory, but he did not examine stocks in a specified industry. This research used the approach of estimating the Hurst exponent for each series over the whole sample period by five alternative techniques, then testing for Gaussian character, which is consistent with the Efficient Market Hypothesis, and finally testing for stability of the Hurst exponent over two subsamples to examine whether the behavior of the equities changed during the time studied.

 

Results

Many maritime security prices are anti-persistent or ergodic, mean-reverting, or pink noise processes with (0.00 < H < 0.50), indicating they are more volatile than a random walk. Pink noise processes are used to model dynamic turbulence. B&H Ocean Shippers was the only firm that did not provide evidence of ergodicity for at least part of the period 1989-2002. Ergodic or antipersistent processes reverse themselves more often than purely random series. Ergodicity, that is, H significantly below 0.50, strongly disconfirms the efficient market hypothesis, indicating market participants persistently over-react to new information, imposing more stock price volatility than would be consistent with market efficiency, and participants never learn not to over-react. This observed phenomenon is directly analogous to Mussa's (1984) disequilibrium overshooting, in which the market process of adjustment toward final equilbrium prices is unstable, and never quiets down. Hs significantly above 0.50 demonstrate stock prices are not random walks, shedding some doubt on weak market efficiency and indicating technical analysis can provide systematic returns. Any findings of non-normality or non-Gaussian character have serious implications for pricing financial derivatives. Because the Black-Scholes (1972; 1973) option pricing model assumes normally-distributed prices for underlying securities, financial derivatives based on non-normal securities prices cannot be priced efficiently.

<<Table 3 about here>>

This section discusses and interprets the results of five alternative fractal analysis methods for measuring the Hurst exponent H presented in Table 3. All data are converted to logarithmic returns, losing one observation. Standard errors are given in parentheses. H is estimated first over the 1989-2002 sample (whole range), and then the data is split between 1994 and 1995, giving separate estimates of H for the 1989-1994 (early range) and 1995-2002 (late range) periods. Mandelbrot, Fisher, and Calvet (1997) refer to H as the self-affinity index or scaling exponent.

Five techniques for estimating the Hurst exponent are reported in this paper: 1.) Mandelbrot's (1972a) AR1 rescaled-range or R/S analysis; 2.) power spectral-density analysis; 3.) roughness-length relationship analysis; 4.) variogram analysis; and 5.) wavelet analysis:

1.) Rescaled-range or R/S analysis: R/S analysis is the traditional technique introduced by Mandelbrot (1972a). Hs estimated by this method are mostly around 0.50, superficially suggesting Gaussian processes and supporting the efficient market hypothesis. However, the difference between estimated Hs and 0.50, though generally not great, is statistically significant over the whole sample range and both subsamples for Alexander & Baldwin and Maritrans. For Seacor Smit, Kirby, CSX, and OMI, H is significantly different from 0.50 for the earlier subsample, and for MC Shipping, H is significantly different from 0.50 for the later subsample. These results formally support the multifractal model of asset returns, which is more general, by disconfirming the weak form of the efficient market hypothesis for the maritime firms in those periods. Significant Hs are below 0.50, indicating ergodicity or antipersistence (e.g., negative serial correlation meaning the market price persistently overcorrects) for Alexander & Baldwin (all ranges), CSX (early range), Kirby (early range), MC Shipping (late range), OMI (early range), and Maritrans (all ranges). These results provide further support for the multifractal model of asset returns and further difficulty for weak form efficiency, the least restrictive form of the efficient market hypothesis. This measurable antipersistence or ergodicity demonstrates market participants habitually overreact to new information, and never learn not to.

Hs different from 0.50 demonstrate the return series have not been random walks, shedding significant doubt on weak market efficiency and indicating technical analysis could have provided systematic returns. Nevertheless, this finding may be due to short-term dependence still present after taking AR1 residuals, or systematic bias due to information asymmetries, or both.

2.) Power spectral density analysis: Hs estimated by this technique in the persistent range (H > 0.50) are OMI (all ranges), B&H and Kirby (early and late ranges), Seacor Smit, International Shipholding, MC Shipping, and Overseas Shipping Group (early range), Tidewater (whole and early ranges), and Mexican Maritime Transportation (late range). Note these results often flatly contradict those provided by other techniques. Spectral density does not provide a standard error for H, and thus cannot be used for formal hypothesis testing. Power-spectral density analysis indicates Alexander & Baldwin, CSX, and Maritrans to be consistently ergodic over all samples, providing further evidence of price overcorrection and market inefficiency.

3.) Roughness-length relationship method: This method provides the most consistent support for the multifractal model of asset returns. Formal hypothesis tests reject the Gaussian null for all series and all sample ranges, except B&H, International Shipholding, and Mexican Marine Transportation (whole and late ranges), though the null of normality is rejected for the early subsample for these three firms. One difficulty in applying the roughness-length method is that the standard errors are always so low the null hypothesis of H = 0.50 is nearly always rejected no matter how nearly normal the asset returns. The seemingly unambiguous rejection of weak market efficiency provided by this technique is best viewed cautiously. The rule of thumb adopted here is to look for rejection of the Gaussian null by at least one other technique.

4.) Variogram analysis: Variogram analysis supports the multifractal model of asset returns and disconfirms weak market efficiency for only two equities out of the sample of twelve: Alexander & Baldwin (late range) and Mexican Marine Transportation (early range). Viewed in isolation, the interpretation would be that most maritime equities are valued efficiently, but clearly not all.

5.) Wavelet analysis: This method was developed by Daubechies (1990), Beylkin (1992), and Coifman et al (1992). Wavelet H estimates indicate antipersistence or ergodicity (H < 0.50) for Alexander & Baldwin (all ranges), MC Shipping (late range), OMI (early range), Tidewater (late range), and Maritrans (whole range), indicating persistence (H > 0.50) elsewhere.

<<Table 4 about here>>

Hypothesis tests are constructed to test for 1.) the Gaussian character or normality of the underlying time series, 2.) Cauchy-character, and 3.) changes in price behavior between 1989-1994 and 1995-2002:

1.) Tests of Gaussian character or normality: Table 4 presents t-statistics for tests of the null hypothesis H = 0.50, along with two-tail probability levels. T-statistics are computed as the Hurst exponent divided by its standard error, and are based on the R/S, roughness-length, and variogram techniques. Degrees of freedom are the number of observations or average R/S samples in the regression used to estimate H, rather than the number of observations of the underlying security. The findings are that Alexander and Baldwin and Maritrans are not Gaussian processes, for the whole sample or either subsample, by both R/S and roughness-length. Seacor Smit, CSX, Kirby, MC Shipping, OMI, and Grupo TMM are non-Gaussian processes for one subsample (always 1989-1994 for all but MC Shipping which is non-Gaussian for 1995-2002) by two techniques.

Gaussian character is a sufficient condition for weak market efficiency, but not a necessary condition. This result is generally interpreted as support for the more general multifractal model of asset returns, and disconfirmation of the weak-form efficient market hypothesis; however, it really does not prove that the market is inefficient. More importantly, findings of H < 1 strongly reject weak market efficiency because they demonstrate ergodicity of antipersistence. These findings are absolutely fatal to the Black-Scholes [50-51] option pricing model and its underlying assumption of normally-distributed prices for the underlying securities. Financial derivatives based on non-normal securities prices cannot be priced efficiently. Thus, even if the equity markets for maritime stocks are efficient, in spite of the substantial empirical evidence against market efficiency, the derivatives markets clearly are not efficient.

2.) Tests of Cauchy character: the Mandelbrot-Lvy characteristic exponent test: Various statistics are available to test the null hypothesis of normality, but not for the Cauchy distribution, the other extreme. The Mandelbrot-Lvy characteristic exponent alpha is computed as the reciprocal of the Hurst exponent. Mulligan (2000b) provides tables of percentages of alpha generated by Monte Carlo experiments with 1,000 iterations for different sample sizes. These critical values can be used to evaluate estimated alphas for the Cauchy null; the null should be rejected if the estimated characteristic exponent lies outside the critical bounds. Dispersion of alpha around the theoretical value of 1 varies greatly with the sample size; for such large sample sizes (e.g., n = 3447), alpha is highly concentrated around 1.00. Critical values interpolated from Mulligan (2000b: 491) for a sample size of 3550 are: 1%, 0.829; 5%, 0.887; 10%, 0.909; 90%, 1.040; 95%, 1.059; 99%, 1.086. All alphas computed for this study are concentrated around 2.00, superficially suggestive of a Gaussian process. None are even close to being as low as the 99% critical value of 1.086; thus, the Cauchy null is always rejected by wide margins. Evidence of Cauchy character for any maritime equity would have been extremely surprising, as it would have indicated periodic, very large, and abrupt, changes in the stock price unpredictably move the equity from one trading regime to another, independent of changes in the firm's fundamentals.

<<Table 5 about here>>

3.) Tests of structural change: Table 5 presents t-statistics testing for significant differences among Hs estimated over the whole sample range and the two subsamples, referred to a ranges A, B, and C. The first null hypothesis tested for each equity is that the Hs for the two subsamples are equal (B=C), with degrees of freedom equal to the sum of sample average R/Ss in the two regressions estimating H for each subsample. The second and third null hypotheses are that the H for each subsample is equal to the H estimated over the whole sample (A=B and A=C) with degrees of freedom equal to the number of R/Ss in the whole-sample regression, because the standard error of the whole sample H is treated as the pooled standard error.

With time series that may be fractional Gaussian noise, that is, apparently random combinations of otherwise statistically well-behaved processes scrambled together with periodically-changing parameters and characteristics, it is not strictly correct to infer structural change in the conventional sense. For example, a random scrambling of several different finite-variance processes can result in an infinite-variance process over a larger sample range. The multifractal model of asset returns achieves extreme generality by incorporating the possibility of such finite-variance processes. The finding that H is not constant over two subsamples and the whole sample range is wholly consistent with a stable fractal process, but more importantly, it points to some difference in fundamentals, or at least in behavior of the variable studied, from one period to the other.

 

Conclusion

Equities traded in efficient markets should have Hurst exponents approximately equal to 0.50, indicating prices change in a purely random, normally-distributed manner. Securities with long-term trends and non-periodic cycles should display time persistence with H > 0.50, unless market efficiency imposes randomness and normality anyway. Five securities in this study; i.e., B&H, International Shipholding, Kirby, Overseas Shipping Group, Tidewater, and Maritrans, display H approximately equal to 0.50 by at least two of the three techniques. However, often results indicate H is approximately equal to 0.50 by one or more methods, and indicate H is not equal to 0.50 by others. This means there is considerable doubt that the series are truly normal.

All the maritime securities in this study yield strong evidence of antipersistence, ergodicity, or negative serial correlation, except B&H Ocean Carriers. This means the markets for these securities must be described by the multifractal model of asset returns and cannot be described by the efficient market hypothesis, not even in its least restrictive, weak form. Formal hypothesis tests indicate H is significantly less than 0.50, though often not by a great magnitude, for eleven of the twelve securities sampled (B&H being the exception), by at least one technique (usually roughness-length, but occasionally by R/S, variogram, or some combination). The conclusion suggested is that market participants are incapable of efficiently valuing some maritime equities and that they persistently overreact to the arrival of new information, and never learn not to overreact.

A possible scenario that renders this finding more intuitive is that information relevant to the valuation of a given maritime firm arrives frequently and seemingly at random. Market participants habitually ignore the vast majority of this information, until it accumulates a critical mass they must finally recognize. Then, perceiving they have ignored a body of relevant information which they have allowed to accumulate, they attempt to compensate for their history of informational sloth by overreacting. Furthermore, most investors are laypeople not well suited to evaluate relevant market information pertaining to a technically specialized industry like maritime shipping. Confirmation of the more general multifractal model of asset returns is an important implication of findings of ergodicity which disconfirm the weak form of the efficient market hypothesis. Disconfirmation of the efficient market hypothesis in its weak form suggests possibilities for constructing nonlinear econometric models for improved price forecasting and option valuation.

Any findings of non-normality or non-Gaussian character have serious implications for pricing financial derivatives. Because the Black-Scholes (1972; 1973) option pricing model assumes normally-distributed prices for underlying securities, financial derivatives based on non-normal securities prices cannot be priced efficiently. This finding of inefficient derivatives markets is completely independent of, and robust to, inefficient markets for equities which underlie the derivative assets.


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Mulligan, Robert F. 2000a. "A Fractal Analysis of Foreign Exchange Markets," International Advances in Economic Research, 6(1): 33-49.

 

_____. 2000b. "A Characteristic Exponent Test for the Cauchy Distribution," Atlantic Economic Journal, 28(4): 491.

 

_____. 2003. "Fractal Analysis of Highly Volatile Markets: an Application to Technology Equities," Quarterly Review of Economics and Finance, forthcoming.

 

Mussa, Michael. 1984. "The Theory of Exchange Rate Determination," in John F.O. Bilson and Richard C. Marston, eds., Exchange Rate Theory and Practice, Chicago: University of Chicago Press, 13-78.

 

Neely, Christopher; Weller, Paul; Dittmar, Robert. 1997. "Is Technical Analysis in the Foreign Exchange Market Profitable? A Genetic Programming Approach," Journal of Financial and Quantitative Analysis, 32(4): 405-426.

 

Osborne, M. 1959. "Brownian Motions in the Stock Market," Operations Research, 7: 145-173.

 

Peters, Edgar E. 1994. Fractal Market Analysis, New York: Wiley.

 

_____. 1996. Chaos and Order in the Capital Markets: a New View of Cycles, Prices, and Market Volatility, second edition, New York: Wiley.

 

_____. 1999. Complexity, Risk, and Financial Markets, New York: Wiley.

 

Samuelson, Paul A. 1982. "Paul Cootner's Reconciliation of Economic Law with Chance," in William F. Scharpe and Cathryn M. Cootner, eds., Financial Economics: Essays in Honor of Paul Cootner, Englewood Cliffs NJ: Scott, Foresman & Co., 101-117.

 


 

Glossary of Fractal Analysis Terms

Antipersistence a series that reverses itself more often than a purely random series, also called pink noise, ergodicity, 1/f noise, or negative serial correlation. (Peters 1994: 306).

Black noise a series that reverses itself less often than a purely random series, displaying trends or repetitive patterns over time, also called persistence, positive serial correlation, or autocorrelation. (Peters 1994: 183-187).

Brown noise the cumulative sum of a normally-distributed random variable, also called Brownian motion. (Peters 1994: 183-185; Osborne 1959).

Efficient Market Hypothesis the proposition that market prices fully and correctly reflect all relevant information. A market is described as efficient with respect to an information set if prices do not change when the information is revealed to all market participants. There are three levels of market efficiency: weak, semi-strong, and strong. (Fama et al 1969; Malkiel 1987).

Long memory the property that any value a series takes on generally has a long and persistent effect, e.g., extreme values that repeat at fairly regular intervals. (Peters 1994: 274).

Multifractal Model of Asset Returns (MMAR) a very general model of asset pricing behavior allowing for long-memory and fat-tailed distributions. Instead of infinite-variance distributions such as the Mandelbrot-Lvy and Cauchy distributions, the MMAR relies on fractional combinations of random variables with non-constant mean and variance, providing many of the properties of infinite-variance distributions. (Mandelbrot, Fisher, and Calvet 1997).

Non-periodic long cycles - a characteristic of long-memory processes, i.e., of statistical processes where each value has a long and persistent impact on values that follow it, that identifiable patterns tend to repeat over similar, though irregular, cycles (non-periodic cycles.) Also called the Joseph effect. (Peters 1994: 266).

Non-stationarity the property that a series has a systematically varying mean and variance. Any series with a trend, e.g., U.S. GDP, has a growing mean and therefore is non-stationary. Brown-noise processes are non-stationary, but white-noise processes are stationary. (Granger 1989: 58).

Persistence or persistent dependence a series that reverses itself less often than a purely random series, and thus tends to display a trend, also called black noise. Persistent series have long memory in that events are correlated over long time periods, and thus display non-periodic long cycles. (Peters 1994: 310).

Semi-strong-form Market Efficiency the intermediate form of the efficient market hypothesis, asserting that market prices incorporate all publicly available information, including both historical data on the prices in question, and any other relevant, publicly-available data, and thus it is impossible for any market participant to gain advantage and earn excess profits, in the absence of inside information. (Peters 1994: 308).

Short-term dependence - the property that any value a series takes on generally has a transient effect, e.g., extreme values bias the series for a certain number of observations that follow. Eventually, however, all memory of the extreme event is lost, in contrast to long-memory or the Joseph effect. Special cases include Markov processes and serial correlation. (Peters 1994: 274).

Spectral density or Power-spectral Density Analysis a fractal analysis based on the power spectra calculated through the Fourier transform of a series. (Peters 1994: 170-171).

Stationarity the property that a series has a constant mean and variance. White, pink, and black-noise processes are all stationary. Because it is the cumulative sum of a white-noise process, a brown-noise process is non-stationary. (Granger 1989: 58).

Strong-form Market Efficiency the most restrictive version of the efficient market hypothesis, asserting that all information known to any one market participant is fully reflected in the price, and thus insider information provides no speculative advantage and cannot offer above average returns. (Malkiel 1987: 120).

Weak-form Market Efficiency the least restrictive version of the efficient market hypothesis, asserting that current prices fully reflect the historical sequence of past prices. One implication is that investors cannot obtain above-average returns through analyzing patterns in historical data, i.e., through technical analysis. Also referred to as the Random Walk Hypothesis. One common way of testing for weak-form efficiency is to test price series for normality, however, normality is a sufficient rather than a necessary condition. (Malkiel 1987: 120).

White noise a perfectly random process. Normally-distributed processes are a special case. (Peters 1994: 312).

 


Appendix

Statistical methodology

 

Rescaled-range or R/S analysis: R/S analysis is the conventional method introduced by Mandelbrot (1972a). Time series are classified according to the estimated value of the Hurst exponent H, which is defined from the relationship

R/S = anH

where R is the average range of all subsamples of size n, S is the average standard deviation for all samples of size n, a is a scaling variable, and n is the size of the subsamples, which is allowed to range from an arbitrarily small value to the largest subsample the data will allow. Putting this expression in logarithms yields

log(R/S) = log(a) + H log(n)

which is used to estimate H as a regression slope. Standard errors are given in parentheses. H ranges from 1.00 to 0.50 for persistent series, is exactly equal to 0.50 for random walks, ranges from zero to 0.50 for anti-persistent series, and is greater than one for a persistent or autocorrelated series. Mandelbrot, Fisher, and Calvet (1997) refer to H as the self-affinity index or scaling exponent.

 

Power spectral density analysis: This method uses the properties of power spectra of self-affine traces, calculating the power spectrum P(k) where k = 2p/l is the wavenumber, and l is the wavelength, and plotting the logarithm of P(k) versus log(k), after applying a symmetric taper function which transforms the data smoothly to zero at both ends. If the series is self-affine, this plot follows a straight line with a negative slope b, which is estimated by regression and reported as beta, along with its standard error. This coefficient is related to the fractal dimension by: D = (5 - beta)/2. H and alpha are computed as H = 2 D, and alpha = 1/H. Power spectral density is the most common technique used to obtain the fractal dimension in the literature, although it is also highly problematic due to spectral leakage.

 

Roughness-length relationship method: This method is similar to R/S, substituting the root-mean-square (RMS) roughness s(w) and window size w for the standard deviation and range. Then H is computed by regression from a logarithmic form of the relationship s(w) = wH. As noted above, the roughness-length method provides standard errors so low the null hypothesis of H = 0.500 is nearly always rejected no matter how nearly normal the asset returns.

 

Variogram analysis: The variogram, also known as variance of the increments, or structure function, is defined as the expected value of the squared difference between two y values in a series separated by a distance w. In other words, the sample variogram V(w) of a series y(x) is measured as: V(w) = [y(x) y(x+w)]2, thus V(w) is the average value of the squared difference between pairs of points at distance w . The distance of separation w is also referred to as the lag. The Hurst exponent is estimated by regression from the relationship V(w) = w2H.

 

Wavelet analysis: Wavelet analysis exploits localized variations in power by decomposing a series into time frequency space to determine both the dominant modes of variability and how those modes vary in time. This method is appropriate for analysis of non-stationary traces such as asset prices, i.e. where the variance does not remain constant with increasing length of the data set. Fractal properties are present where the wavelet power spectrum is a power law function of frequency. The wavelet method is based on the property that wavelet transforms of the self-affine traces also have self-affine properties.

 

Consider n wavelet transforms each with a different scaling coefficient ai, where S1, S2,..., Sn are the standard deviations from zero of the scaling coefficients ai. Then define the ratio of the standard deviations G1, G2, ..., Gn-1 as: G1 = S1/S2, G2 = S2/S3, ..., Gn-1 = Sn-1/Sn. Then the average value of Gi is estimated as Gavg = (Gi)/(n 1). The estimated Hurst exponent H is computed as a heuristic function of Gavg. The Benoit software computes H based on first three dominant wavelet functions, i.e., n is allowed to vary up to 4, and i for the scaling coefficient ai is allowed to vary from i = 0, 1, 2, 3.

 

Mandelbrot-Lvy characteristic exponent test: The Mandelbrot-Lvy distributions are a family of infinite-variance distributions without explicit analytical expressions, except for special cases. Limiting distributions include the normal, with finite variance, and the Cauchy, with the most extreme platykurtosis or fat tails. Paul Lvy (1925) developed the theory of these distributions. The Hurst exponent H introduced in the hydrological study of the Nile valley is the reciprocal of the characteristic exponent alpha (Hurst 1951). The characteristic function of a Mandelbrot-Lvy random variable is:

log f(t) = i(delta)t (gamma)|t|alpha[1 + i(beta)(sign(t)(tan[(alpha)(pi/2)])],

where delta is the expectation or mean of t if alpha > 1; gamma is a scale parameter; alpha is the characteristic exponent; and i is the square root of -1. Gnedenko and Kolmogorov (1954) showed the sum of n independent and identically distributed Mandelbrot-Lvy variables is:

n log f(t) = in(delta)t n(gamma)|t|alpha[1 + i(beta)(sign(t)(tan[(alpha)(pi/2)])],

and thus the distributions exhibit stability under addition. Many applications of the central limit theorem only demonstrate Mandelbrot-Lvy character. The result of normality generally depends on an unjustified assumption of finite variance. Mandelbrot (1972a) introduced a technique for estimating alpha by regression, further refined by Lo (1991). Mulligan (2000b) estimates the distribution of alpha for Cauchy-distributed random variables. This distribution is used to test estimated alphas for technology equities against the Cauchy null.


 

Table 1

Fractal Taxonomy of Time Series

 

Term

 

'Colour'

 

Hurst exponent

 

Fractal dimension

 

Characteristic exponent

Antipersistent,

Ergodic,

Mean-reverting,

Negative serial correlation,

1/f noise

Pink noise

0 ≤ H <

1.50 < D ≤ 2.00

2.00 < a ≤ ∞

Gaussian process,

Normal distribution

White noise

H

D 1.50

a 2.00

Brownian motion, Wiener process

Brown noise

H

D 1.50

a 2.00

Persistent,

Trend-reinforcing,

Hurst process

Black noise

< H < 1

1.00 < D < 1.50

1 < a < 2.00

Cauchy process,

Cauchy distribution

Cauchy noise

H 1

D 1

a 1

Note: Brown noise or Brownian motion is the cumulative sum of a white-noise process, including a normally-distributed process. The changes in, or returns on, a Brownian motion, are white noise. The fractal statistics are the same for Brown and white noise because the brown-noise process should be differenced as part of the estimation process, yielding white noise.


 


Table 2

Maritime Equity Price Series

Firm

 

Ticker

Exchange(s)

Dates

N

beta

Alexander & Baldwin, Inc.

ALEX

NASD

12/30/1988-08/23/2002

3446 (1517+1929)

0.32

B&H Ocean Carriers, Ltd.

BHO

AMEX

12/30/1988-08/23/2002

3447 (1517+1930)

-0.91

Seacor Smit, Inc.

CKH

NYSE

12/21/1992-08/23/2002

2440

(511+1929)

1.14

CSX Corporation

CSX

NYSE

12/30/1988-08/23/2002

3447 (1517+1930)

0.41

International Shipholding Corporation

ISH

NYSE

12/30/1988-08/27/2002

3449 (1517+1932)

-0.14

Kirby Corporation

KEX

NYSE

12/30/1988-08/23/2002

3447 (1517+1930)

0.44

MC Shipping, Inc.

MCX

AMEX

05/25/1989-08/23/2002

3347 (1417+1930)

0.14

OMI Corporation

OMM

NYSE

12/30/1988-08/23/2002

3448 (1517+1931)

0.60

Overseas Shipping Group, Inc.

OSG

NYSE

12/30/1988-08/23/2002

3447 (1517+1930)

0.73

Tidewater, Inc.

TDW

NYSE

12/30/1988-08/23/2002

3446 (1516+1930)

1.25

Grupo TMM (Transportacin Martime Mexicana)

TMM

NYSE

06/10/1992-08/23/2002

2577

(647+1930)

 

Maritrans, Inc.

TUG

NYSE

12/30/1988-08/23/2002

3447 (1517+1930)

0.34

Note:

All raw price series are adjusted for stock splits and converted to logarithmic returns.

 


 


Table 3

Fractal Analyses of Maritime Securities

Estimated Hurst Exponent H, Various Methods

(Standard Errors in Parentheses)

 

Firm

 

Range

 

R/S

Power Spectrum

Roughness-length

 

Variogram

 

Wavelet

ALEX

12/30/1988-08/23/2002

0.431 (0.0282)

0.396

0.397 (0.0028)

0.382 (0.2215)

0.479

12/30/1988-12/30/1994

0.451 (0.0116)

0.465

0.390 (0.0023)

0.436 (0.0533)

0.460

01/03/1995-08/23/2002

0.383 (0.0317)

0.478

0.333 (0.0056)

0.376 (0.0665)

0.468

BHO

12/30/1988-08/23/2002

0.495 (0.0659)

0.448

0.474 (0.0178)

0.526 (0.3473)

0.644

12/30/1988-12/30/1994

0.492 (0.0322)

0.538

0.535 (0.0056)

0.477 (0.0800)

0.632

01/03/1995-08/23/2002

0.508 (0.0232)

0.552

0.505 (0.0053)

0.534 (0.1104)

0.562

CKH

12/21/1992-08/23/2002

0.517 (0.0302)

0.479

0.480 (0.0039)

0.405 (0.2034)

0.617

12/21/1992-12/30/1994

0.553 (0.0167)

0.587

0.527 (0.0033)

0.458 (0.0422)

0.686

01/03/1995-08/23/2002

0.513 (0.0192)

0.482

0.476 (0.0068)

0.404 (0.1802)

0.565

CSX

12/30/1988-08/23/2002

0.459 (0.0307)

0.496

0.424 (0.0072)

0.491 (0.1707)

0.551

12/30/1988-12/30/1994

0.427 (0.0348)

0.479

0.350 (0.0080)

0.490 (0.1429)

0.568

01/03/1995-08/23/2002

0.451 (0.0424)

0.481

0.428 (0.0054)

0.496 (0.1029)

0.574

ISH

12/30/1988-08/27/2002

0.476 (0.1182)

0.461

0.504 (0.0064)

0.350 (0.7979)

0.607

12/30/1988-12/30/1994

0.525 (0.0605)

0.551

0.468 (0.0031)

0.413 (0.1895)

0.530

01/03/1995-08/27/2002

0.476 (0.0565)

0.448

0.506 (0.0086)

0.435 (0.2343)

0.633

KEX

12/30/1988-08/23/2002

0.463 (0.0651)

0.466

0.438 (0.0069)

0.397 (0.2080)

0.545

12/30/1988-12/30/1994

0.452 (0.0218)

0.513

0.412 (0.0045)

0.458 (0.0590)

0.567

01/03/1995-08/23/2002

0.473 (0.0369)

0.548

0.447 (0.0034)

0.384 (0.4151)

0.546

MCX

12/30/1988-08/23/2002

0.498 (0.0244)

0.412

0.465 (0.0068)

0.557 (0.3552)

0.603

12/30/1988-12/30/1994

0.501 (0.0109)

0.535

0.461 (0.0019)

0.579 (0.2239)

0.625

01/03/1995-08/23/2002

0.437 (0.0250)

0.496

0.383 (0.0030)

0.460 (0.3809)

0.497

Table 3-continued

Fractal Analyses of Maritime Securities

Estimated Hurst Exponent H, Various Methods

(Standard Errors in Parentheses)

 

Firm

 

Range

 

R/S

Power Spectrum

Roughness-length

 

Variogram

 

Wavelet

OMM

12/30/1988-08/23/2002

0.482 (0.0493)

0.505

0.477 (0.0063)

0.487 (0.1216)

0.539

12/30/1988-12/30/1994

0.446 (0.0250)

0.521

0.401 (0.0034)

0.506 (0.2125)

0.493

01/03/1995-08/23/2002

0.503 (0.0254)

0.584

0.515 (0.0077)

0.485 (0.2333)

0.522

OSG

12/30/1988-08/23/2002

0.504 (0.0381)

0.465

0.462 (0.0063)

0.439 (0.1682)

0.558

12/30/1988-12/30/1994

0.503 (0.0163)

0.521

0.435 (0.0045)

0.492 (0.0775)

0.594

01/03/1995-08/23/2002

0.463 (0.0635)

0.465

0.448 (0.0033)

0.456 (0.0467)

0.520

TDW

12/30/1988-08/23/2002

0.489 (0.0299)

0.515

0.431 (0.0066)

0.464 (0.0666)

0.566

12/30/1988-12/30/1994

0.459 (0.0404)

0.422

0.446 (0.0045)

0.365 (0.2412)

0.617

01/03/1995-08/23/2002

0.476 (0.0184)

0.521

0.425 (0.0031)

0.490 (0.0297)

0.488

TMM

06/10/1992-08/23/2002

0.491 (0.1180)

0.480

0.509 (0.0108)

0.391 (1.2583)

0.599

06/10/1992-12/30/1994

0.463 (0.0231)

0.496

0.439 (0.0033)

0.431 (0.0175)

0.543

01/03/1995-08/23/2002

0.519 (0.0647)

0.529

0.509 (0.0100)

0.437 (0.8114)

0.653

TUG

12/30/1988-08/23/2002

0.451 (0.0261)

0.464

0.390 (0.0097)

0.539 (0.4529)

0.487

12/30/1988-12/30/1994

0.432 (0.0222)

0.440

0.387 (0.0031)

0.469 (0.1096)

0.520

01/03/1995-08/23/2002

0.436 (0.0249)

0.495

0.398 (0.0039)

0.528 (0.2402)

0.569

Note:

The Mandelbrot-Lvy characteristic exponent alpha is the reciprocal of the Hurst exponent H, thus alpha = 1/H. The fractal dimension D = 2 H.

 


 

Table 4

Hypothesis Tests for Normality or Gaussian Character (H = 0.50)

 

Firm

 

d.f.

R/S

Roughness-length

Variogram

t(R/S)

p

t(R)

p

t(V)

p

ALEX

38

2.447 **

0.019

36.786 ***

0.000

0.533

0.597

26

4.224 ***

0.000

47.826 ***

0.000

1.201

0.241

30

3.691 **

0.001

29.821 ***

0.000

1.865 *

0.072

BHO

38

0.076

0.940

1.461

0.152

0.075

0.941

26

0.248

0.806

6.250 ***

0.000

0.288

0.776

30

0.345

0.733

0.943

0.353

0.308

0.760

CKH

34

0.563

0.578

5.128 ***

0.000

0.467

0.644

30

3.174 *

0.007

8.182 ***

0.000

0.995

0.337

30

0.677

0.506

3.529 ***

0.002

0.533

0.600

CSX

30

1.336

0.190

10.556 ***

0.000

0.053

0.958

14

2.098 **

0.046

18.750 ***

0.000

0.070

0.945

20

1.156

0.257

13.333 ***

0.000

0.039

0.969

ISH

38

0.203

0.840

0.625

0.536

0.188

0.852

26

0.413

0.683

10.323 ***

0.000

0.459

0.650

30

0.425

0.674

0.698

0.491

0.277

0.783

KEX

38

0.568

0.573

8.986 ***

0.000

0.495

0.623

26

2.202 **

0.037

19.556 ***

0.000

0.712

0.483

30

0.732

0.470

15.588 ***

0.000

0.279

0.782

MCX

32

0.082

0.935

5.147 ***

0.000

0.160

0.874

26

0.092

0.928

20.526 ***

0.000

0.353

0.727

30

2.520 **

0.017

39.000 ***

0.000

0.105

0.917

OMM

38

0.365

0.717

3.651 ***

0.001

0.107

0.915

26

2.160 **

0.040

29.118 ***

0.000

0.028

0.978

30

0.118

0.907

1.948 *

0.061

0.064

0.949

OSG

38

0.105

0.917

6.032 ***

0.000

0.363

0.719

26

0.184

0.855

14.444 ***

0.000

0.103

0.919

30

0.583

0.564

15.758 ***

0.000

0.942

0.354

TDW

38

0.368

0.715

10.455 ***

0.000

0.541

0.592

26

1.015

0.320

12.000 ***

0.000

0.560

0.580

30

1.304

0.202

24.194 ***

0.000

0.337

0.739

TMM

37

0.076

0.940

0.833

0.410

0.087

0.931

15

1.602

0.130

18.485 ***

0.000

3.943 ***

0.001

30

0.294

0.771

0.900

0.375

0.078

0.939

TUG

38

1.877 *

0.068

11.340 ***

0.000

0.086

0.932

26

3.063 ***

0.005

36.452 ***

0.000

0.283

0.780

30

2.570 **

0.015

26.154 ***

0.000

0.117

0.908

Note:

Hs computed by R/S, Roughness-length, and Variogram methods are used for conventional hypothesis tests where the null hypothesis is H = 0.500, (i.e., equivalently alpha = 2, D = 1.500, or normality of the asset returns). Three independent hypothesis tests are performed for each time series. The Hurst exponent is estimated for three sample ranges A:1989-2002, B:1989-1994, and C:1995-2002. Lack of a consistent outcome with any of the three nulls for the same equity is suggestive of a structural break, i.e., a shift in H, between 1994 and 1995. Rejection at 10%, 5%, and 1% two-tail significance levels are indicated by *, **, and ***. 'd.f.' indicates degrees of freedom.


 

 

Table 5

Hypothesis Tests for Structural Stability

 

Firm

 

Null

 

d.f.

R/S

Roughness-length

Variogram

t

p (t)

T

p (t)

t

p (t)

ALEX

B=C

56

2.411 **

0.019

20.357 *

0.000

0.271

0.787

A=B

38

0.709

0.483

2.500 **

0.017

0.244

0.809

A=C

38

1.702 *

0.097

22.857***

0.000

0.027

0.979

BHO

B=C

56

0.243

0.809

1.685 *

0.097

0.164

0.870

A=B

38

0.046

0.964

3.427 ***

0.001

0.141

0.889

A=C

38

0.197

0.845

1.742 *

0.090

0.023

0.982

CKH

B=C

34

1.325

0.194

13.077

0.000

0.265

0.792

A=B

30

1.192

0.243

12.051***

0.000

0.261

0.796

A=C

30

0.132

0.896

1.026

0.313

0.005

0.996

CSX

B=C

56

0.782

0.438

10.833***

0.000

0.035

0.972

A=B

38

1.042

0.304

10.278***

0.000

0.006

0.995

A=C

38

0.261

0.796

0.556

0.582

0.029

0.977

ISH

B=C

56

0.415

0.680

5.938 ***

0.000

0.028

0.978

A=B

38

0.415

0.681

5.625 ***

0.000

0.079

0.937

A=C

38

0.000

1.000

0.313

0.756

0.107

0.916

KEX

B=C

56

0.323

0.748

5.072 ***

0.000

0.356

0.723

A=B

38

0.169

0.867

3.768 ***

0.001

0.293

0.771

A=C

38

0.154

0.879

1.304

0.200

0.063

0.950

MCX

B=C

56

2.623 **

0.011

11.471***

0.000

0.335

0.739

A=B

32

0.123

0.903

0.588

0.561

0.062

0.951

A=C

32

2.500 **

0.018

12.059***

0.000

0.273

0.787

OMM

B=C

56

1.156

0.252

18.095***

0.000

0.173

0.863

A=B

38

0.730

0.470

12.063***

0.000

0.156

0.877

A=C

38

0.426

0.673

6.032 ***

0.000

0.016

0.987

OSG

B=C

56

1.050

0.298

2.063 **

0.044

0.214

0.831

A=B

38

0.026

0.979

4.286 ***

0.000

0.315

0.754

A=C

38

1.076

0.289

2.222 **

0.032

0.101

0.920

TDW

B=C

56

0.569

0.572

3.182 ***

0.002

1.877 *

0.066

A=B

38

1.003

0.322

2.273 **

0.029

1.486

0.145

A=C

38

0.435

0.666

0.909

0.369

0.390

0.698

TMM

B=C

45

0.475

0.637

6.481 ***

0.000

0.005

0.996

A=B

37

0.237

0.814

6.481 ***

0.000

0.032

0.975

A=C

37

0.237

0.814

0.000

1.000

0.037

0.971

TUG

B=C

56

0.153

0.879

1.134

0.261

0.130

0.897

A=B

38

0.728

0.471

0.309

0.759

0.155

0.878

A=C

38

0.575

0.569

0.825

0.415

0.024

0.981

Note:

Three independent hypothesis tests are performed for each time series. The Hurst exponent is estimated for three sample ranges A:1989-2002, B:1989-1994, and C:1995-2002. The first hypothesis tested is whether the Hs estimated for ranges B and C, the split samples, are equal. The second hypothesis is whether the H estimated for range B is significantly different from that estimated for the whole sample A. The third hypothesis is whether H estimated for range C is significantly different from that for the whole sample A. Rejection of any of the three nulls indicates a structural break, i.e., a shift in H, between 1994 and 1995. Rejection at 10%, 5%, and 1% two-tail significance levels are indicated by *, **, and ***. 'd.f.' indicates degrees of freedom.