Risk measurement, nonlinearities, and chaos.

This paper examines the efficacy of the statistical measures of risk in the light of results obtained from the analysis of stock market data using contemporary techniques of mathematical modelling of dynamical systems like the Rescaled Range Analysis and the related Hurst's Exponent, Fractal Dimensions, and the Lyapunov Exponents.

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Traditionally, the modelling of a system proceeds along one of two lines. In one approach, deterministic equations of motion are derived from first principles, initial conditions are measured, and the equations of motion are integrated forward in time. Alternatively, when a first-principles model is unavailable or intractable or when initial conditions are not accessible, then the dynamics can be modelled as a random process, using nondeterministic and stochastic, though typically linear laws of motion.

Until recently, the notions of determinism and randomness were seen as opposites and were studied as separate subjects with little or no overlap. Complicated phenomena were assumed to result from complicated interactions among many degrees of freedom, and thus were analysed as random processes. Simple systems were assumed to produce simple phenomena, so only simple phenomena were modelled deterministically.

The modus operandi for studies on stock market phenomena was no different and one could go to the extent of saying that the Efficient Market Hypothesis (Sharpe, 1970; Elton, 1981) was formulated with one primary objective--to create a scenario which would justify the use of stochastic calculus (Ross, 1999) for the modelling of capital markets.

The Efficient Market Hypothesis contemplates a market where all assets are fairly priced according to the information available and neither buyers nor sellers enjoy any advantage. Market prices reflect all public information, both fundamental and price history and prices move only as sequel to new information entering the market. Further, the presence of large number of investors ensures that all prices are fair. Memory effects, if any at all, are extremely short ranging and dissipate rapidly. Feedback effects on prices are, thus, assumed to be marginal. The investor community is assumed rational as benchmarked by the traditional concepts of risk and return.

An immediate corollary to the Efficient Market Hypothesis is the independence of single period returns, so that they may be modelled as a random walk and the defining probability distribution, in the limit of the number of observations being large, would be the normal distribution.

While several adaptations of the Random Walk EMH have been postulated with the corresponding variants of the normal distribution and martingale and submartingale approaches being used to model stock prices or returns, the variance continues to be the cardinal measure of risk, as defined by the volatility of stock prices.

Ever since the studies of Fama in 1964-65, evidence has been accumulating against the validity of the Efficient Market Hypothesis--the existence of negatively skewed observations and fat tails and distortion around the mean values are but a few.

Furthermore, the access to enhanced computing power during the last decade has enabled analysts to try refined methods like the phase space reconstruction methods for determining the Lyapunov Exponents (Wolf et al, 1985) of stock market price data, besides doing Rescaled Analysis (Mandlebrot, 1977) etc. Results of these studies have unambiguously established the existence of significant nonlinearities and chaotic behaviour in these time series.

Before we examine the implications of these studies, it is appropriate to elaborate on the concept of chaos in view of it being of nascent origin.

What is Chaos?

The flapping of butterfly wings in Rio de Janeiro could bring on a tornado in Texas several weeks later--you don't believe it? Nor did I. But that is what Edward Lorenz concluded one day when he was running a mathematical model of the weather on his computer. It so happened that in order to recheck some results on his weather forecasting model, he decided to re-input his data from the earlier printouts and run the program again. The results were quite inconceivable--although the immediate values of the variable were identical, major divergences surfaced as the run steps were extended. In fact, no significant resemblance was observed between the results of the two runs after a sufficiently long period. Thus, starting from nearly the same initial conditions, weather patterns were produced that grew further and further apart until all association disappeared. On investigation, the cause of this highly paradoxical scenario was traced to a very trivial matter--while the printer had printed data up to six decimal p oints which constituted the data fed for the second run, the computer had calculated data up to eight decimals. The data that fed for the second run were, therefore, minutely different from that used in the first run. Amazing as it may sound, it was these minute differences that manifested themselves as gigantic divergences in the output--this, indeed, is chaos. The property that manifests itself through sensitivity to initial conditions with a consequential unpredictability is generally termed as chaos. The discovery of chaos has destroyed the deterministic image of the modern world leading to new directions of research and providing a fillip to the ergodic description of systems.

The origin of chaos, in its modern form, may be traced to the work of the master French mathematician Henri Poincare in the 1890s, on the mathematical aspects of planetary motion, treating it as a three-body problem. Through the use of topological methods, he established that there is no simple solution to the three-body problem. During the course of his analysis, he realised that if one takes two different readings on the position of a planet, then, irrespective of the proximity of the two readings, the orbits of the planets might separate away from each other, after enough time. Hence, accurate prediction of the orbit of any planetary body was impossible. Chaos was, thus, born.

Chaos provides a link between deterministic systems and random processes, with both good and bad implications for the prediction problem. In a deterministic system, chaotic dynamics can amplify small differences, which in the long run produces effectively unpredictable behaviour. On the other hand, chaos implies that not all random-looking behaviour is the product of complicated interactions. Under the intoxicating influence of nonlinearity, only a few degrees of freedom are necessary to generate chaotic motion. In this case, it is possible to model the behaviour deterministically and to make short-term predictions that are far better than those that would be obtained form a linear stochastic model. Chaos is thus a double-edged sword: it implies that even approximate long-term predictions may be impossible, but that very accurate short-term predictions may be possible.

Decision-making of all kinds, including investments in the capital markets, rests on our ability to predict the future. However, business and, indeed, life in general, is not predictable. Conventionally, in decision theory, this lack of predictability is explained by factors such as lack of information or the limitations of prediction techniques. Chaos theory, however, provides a radically opposite explanation, in that it accepts unpredictability as an inherent attribute of a wide range of phenomena, so that, forecasting may be an entirely futile and wasteful exercise.

Prediction and forecasting have, hitherto, relied essentially on various linear models like regression, linear programming, capital budgeting and so on. It is, however, now established beyond doubt that all fundamental processes of nature have various degrees of non-linearity. In fact, chaos is a manifestation of the non-linearities inherent in a system in so far as such unpredictable phenomena are forbidden in linear systems by the very virtue of their linearity.

A corollary to this ubiquitous non-linearity is the high degree of approximation incumbent in all the contemporary decision-making processes. Chaos theory emphasises that because of this sensitivity to initial conditions, many events simply cannot be predicted, because it would be impossible to know and monitor all the variations that might have a significant effect on the outcomes.

A compact, concise and universally acceptable definition of chaos has, hitherto, eluded the scientific community. However, the following are conventionally accepted as the inherent characteristics of a chaotic system:

* Exponential divergence of trajectories in phase space;

* Sensitive dependence on initial conditions;

* Fractal dimensions;

* Critical levels and bifurcations;

* Time dependent feedback systems; and

* Far from equilibrium conditions.

As had happened in the case of Edward Lorenz cited above, chaotic systems are highly sensitive to initial conditions insofar as minor differences in their initial conditions tend to get magnified manifold with the evolution of the system. This is illustrated in the Henon Map, defined by the following set of simultaneous difference equations:-

[x.sub.(t+1)] = 1 + [y.sub.t] - [ax.sup.2.sub.t]

[y.sub.(t+1)] = [bx.sub.t]

While the origin of this sensitive dependence may be attributed to the existence of time dependent feedback mechanisms, the implications are devastating. Unpredictability becomes an inherent attribute and long-term forecasting becomes a futile exercise. Marginally small errors in data collection would manifest themselves manifold magnified in forecasting and, for all we know, even with the best available measurement devices error-fee measurement is impossible, a fundamental lower limit being imposed by Heisenberg's Uncertainty Principle.

The Chaos theory suggests the need to adopt a radically new perspective to forecasting. It emphasises the need to acknowledge the true dimensions of uncertainty in its absoluteness and to discard the conventional and traditional so called "rational" models. The theory recognises the existence of disorder, discontinuities and randomness as inherent properties or norms rather than as aberrations. Consequent to the acceptance of unpredictability as an inherent property, the chaos theory tends to dwell heavily on the necessity of development of adequate "fire fighting" mechanisms as an indispensable part of planning and forecasting.

Chaos and the Capital Markets

As mentioned above, several studies adopting largely diverse and independent approaches have established the existence of the following characteristics in the behaviour of stock markets:

* Long-term correlation and memory effects

* Erratic markets under certain conditions and at certain times

* Fractal time series of returns, and

* Less reliable forecasts with increase in the horizon,

thereby establishing the existence of chaotic behaviour.

Not only do the quantitative tests contradict the validity of the EMH, the assumption of "Investor Rationality" as envisaged in the EMH may also be questioned on the following grounds:

* People are not risk averse in all situations. They can be risk takers, if confronted with a situation involving perceived sure losses, for example, a trade off between a certain loss of $85,000 versus a loss of $100,000 with a probability of 0.85 and a zero loss with probability of 0.15 would generally find the investor opting for the latter;

* People are not unbiased in setting subjective probabilities They are likely to be more confident of their forecasts than is warranted by the available information. People tend to ignore new information if it does not fit in with their current forecasts of the future;

* They do not react to trends until fully established. They will not begin to extrapolate a phenomenon until it is firmly established, a factor that normally takes some time. They will then take a decision that incorporates information that they have ignored until that time. They do not react to information in a continuing fashion as and when it is received, but rather in discrete blocks and clumps in a cumulative fashion. Thus, reaction to information occurs in blocks and chunks rather than linearly. This is contrary to the behaviour of the rational investor as postulated by the Efficient Market hypothesis;

* Investors are less likely to change their forecasts unless they receive enough confirming information that the environment has changed. Once the level of information reaches a critical level, investors react to all the information received till then. Hence, memory effects subsist;

* As a corollary, markets lose their efficiency since all information is not reflected in prices. Much is ignored and reaction comes later.

Risk Measures and Chaos

Variance has been, traditionally, used in one guise or another as the statistical measure of risk. Variance measures the probability that an observation will be a certain distance from the average observation. The larger this numbers, the wider the dispersion. Wide dispersion would mean that there is a high probability of large swings in returns. The security is risky. However, the use of variance as a measure of risk inherently assumes that the underlying system is random. If the observations are correlated, then the usefulness of variance as a measure of risk is considerably weakened.

A time series will be truly random when it is influenced by a number of events that are equally likely to occur that is, the system has a large number of degrees of freedom. In a non-random series the data will clump together to reflect the correlation inherent in its influences and the time series will be a fractal.

We illustrate the point by an example. We consider two possible series of stock market returns, say A & B:

   Observation       A     B

        1            2     1
        2           -1     2
        3           -2     3
        4            2     4
        5           -1     5
        6            2     6
Standard Deviation   1.70  1.71
Fractal Dimension    1.42  1.13

A is a trendless series while B has a clear trend. Both have almost the same standard deviation. The two stocks with virtually identical risk (as measured by the standard deviation) have vastly differing return characteristics. The obvious fallacy is that both series are not normally distributed, but then the same is the case with the stock markets. As mentioned above, numerous studies have shown the non-random character of the stock market-returns, thereby questioning the usefulness of variance as a comparative measure of risk.

In contrast, the fractal dimension clearly shows that A is more jagged than B, and hence, more risky. As a corollary, the fractal dimension may, at least qualitatively, depict the risk profile of a stock more rationally than the standard deviation.

There is a simple mathematical link between the Fractal Dimension or the associated Hurst's Exponent H and the physical dimension. H=1 corresponds to a perfectly persistent series representable by a unidimensional line whereas 11=0.5 corresponds to random or Brownian motion, it is equal to a dimension of 1.5, a fractal or noninteger dimension halfway between a line and a plane. Where H=0, a perfectly antipersistent time series, the corresponding physical dimension is a plane or 2.

Temporal Nonlinearity and Risk

The efficacy or otherwise of statistical techniques for risk measurement, can be questioned at a more fundamental level arising out of the Einsteinian Relativity and the String Theory where time itself is considered nonlinear.

The stock markets are modelled as a process that happens in time. As is the case with most systems modelling, this process is treated either as a discrete static process or a continuous random process. However, neither assumption entirely gels with reality and nor extreme is a complete and sophisticated treatment of the subject. The commonality underlying both these assumptions is that they are linear, that is, either, they are always static or always random. Time either does not effect the system or does so at a uniform rate.

However, what happens if time itself is nonlinear? This is not a trivial issue. In fact, Einsteinian Relativity and its modern and perhaps upgraded counterpart, the so-called theory of everything--the Superstring Theory--does indeed postulate the existence of four dimensional manifolds consisting of the three spatial dimensions and a temporal dimension as the basic structure of our universe. Introduction of the fourth dimension results in the space-time having a curved structure like the windscreen of a car. It is, however, not static, but a dynamic structure more like a crumpled piece of paper which keeps on getting recrumpled.

Einsteinian Relativity in essence implies that space and time are part of the same stuff called space-time, and that the surface of space--time is curved. The fabric of space-time is warped and everyone's vantage point is different.

But why do we need to consider all this? The reason is simple--"risk unfolds with time". The unfolding of risk may, however, be commensurate with time or it may not. Yet we cannot dispute some association between risk and time, since most investors perceive that reward does increase with risk. In fact, this risk-reward relationship forms the basis of the traditional theory of finance. The linear school ends up assuming that they are perfectly positively related. A corollary to which is that the prices and hence return are normally distributed. In reality this is not the case.

An obvious implication of the commensurate relationship between risk and time could be that, say, 1/30 of the daily risk measurement evaporates every day all the time. This is evidenced by the fact that most models conceive of risk as a function of time, for example, the stock averaged 20 per cent volatility today. Risk can vary, but time is always assumed to flow at an even rate. However, life is not so simple. A more formal interpretation could be that when a risk event happens, time changes speed. The time value of money still holds--if there is no risk then time is slow and the return demanded by investors' approaches the risk free interest rate. As risk is encountered, time speeds up as the warped up periods of the Einsteinian space-time suggest. Risk goes multifractal in these speeded up periods of time accounting for the lumps and clumps in the volatility. In fact, if the risk is large enough, a singularity may be reached warranting the concept of infinite return.

Our fallacy is that we measure time in uniform discrete chunks called hours and days, resulting in the conclusion as to the linearity of the flow of events. The lumping of volatility as characterised by fractals suggests that this is not the case. What if the relationship itself between the observer and the observed changes with time?

However, the obvious question is--would such factors as relativity that manifest themselves visibly only at the microscopically and macroscopically extreme physical systems have a tangible impact on investor rationality and the stock markets? They may or they may not. But we could certainly, at least qualitatively, borrow the ideas at the conceptual level, should they fit our case.

The fact that the space-time fabric is warped and constantly getting recrumpled would mean not only that the vantage point of each investor is different but also that it keeps on varying with the passage of time. It might also explain why people can be rational investors and still make very different investment decisions. Everyone's perspective is different and is changing at different rates. Investors may be considered rational in that they are internally self-consistent with what they know. However, their actions might be illogical from a different informational point of view. But how can we seemingly have these different views all at once? That is, how can it be that reasonable men may differ? If they are rational, viewing the same set of facts, how can they arrive at different conclusions, other than the explanation of having different points of view? If time flowed as uniformly or smoothly according to our current assumptions and measurements then the fabric of space-time would be planar. In fact it is not so, as evidenced by relativity in physics and in financial economics by the lumping of volatility. With a uniform Newtonian clock, investors' viewpoints might still differ, but the fact that time is happening at different rates makes a uniform viewpoint unlikely, if not impossible.

The comparison of peoples' viewpoints is also uneven in the sense that perfect information is never attainable. The perceptions of providers and users of capital can never be unique because they are not the same entity. This is another way of saying that they cannot share the same place in space-time.

We have stated earlier that the Hurst Exponent provides us with a qualitative benchmark for measuring risk. To continue, if a stock exhibited perfect persistence or a straight line, it would imply that the price history is steady. In practice, the steady state of the Hurst exponent is approached by event-driven examples such as bankruptcy or merger arbitrage, where values are locked by easily identifiable circumstances. In theory, assuming a zero-valued stock could stay listed, it would be perfectly persistent and thus very predictable and not risky. It would also be dead. The Hurst exponent is also a scaling factor in time. If time does not scale for a dead company, what can we conclude? Would it be accurate to say that there is no risk?

The answer is in the negative. There is risk, It is just that risk exists which is not captured by the constant perspective of the time series itself. It is also worth stating that Brownian motion cannot scale. Randomness is a very singular view and the mistakes perpetuated by using it will persist until there is a change.

Given that volatility lumps and clumps or exhibits what we call multifractal time like a bedsheet, how can one reasonably expect any risk-management program to calculate meaningful result when the assumptions are wrong? Time moves like an ant on a string that is stretched out and the ends brought back together. This is why equilibrium-based finance will never portray the real essence of finance.

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