CFA Institute

!"#$%#&'(%")$%)"*'+,'%-*'.#/+%#&'0#"1*%2 3)%-4"5267'89:#"'8;')&;'D'3):;?'EFGF6?'//;'HIDHJ (et-MN),t= 1

where H equals the Hurst exponent. H can range between zero and one.6 An H equal to 0.5 implies pure random walk behavior-the absence of long-term statisticaldependence.7 But what does an H different from 0.5 imply? An H between zero and 0.5 implies antibehavior. This means that if a trend persistent has been positive in the last period, it is more likely that it will be negative than positive in the next period. Conversely, if it has been negative in the last period, it is more likely than not to be positive in the next period. An H greaterthan 0.5 but less than or equal to behavior. This means that one implies persistent if the trend has been positive in the last observed period, the chances are that it will continue to be positive in the next period. The level of persistence (hence the probability of a continuing trend) is judged by how far H is above 0.5. Hurst found that a variety of natural phenomena exhibit an H with a mean of 0.73 and a standard deviation of 0.09.8 The Biased Hurst Process Perhaps the best example of a biased Hurst process is contained in Hurst's own simulation

(1)

where

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1989 O 33

method, devised for an age before computers. Hurst's simulations used a "probabilitypack of cards." The cards contained in the pack (52 in all) were numbered +1, -1, +3, -3, +5, -5, +7, -7, in proportions to estimate a normal curve. By shuffling and cutting the deck, Hurst created a random-numbergenerator. To create a biased distribution, Hurst would cut the deck and note its number-for example, +5. Hurst would then reshuffle the deck and deal two hands, A and B, each containing 26 cards. Because the cut produced a +5, the five highest cards would be removed from hand A and placed in hand B. The five lowest cards would be removed from hand B. Hand B would now have a positive bias. A joker would then be placed in hand B, which became the probability pack of cards. This pack would be shuffled and cut, as before. When the joker was cut, however, the whole deck would be reshuffled, and a new biased hand created. Using this method, Hurst generated an H of 0.714 (plus or minus 0.9). This is consistent with his observations in nature. The biased probabilitydeck gives some idea of how Hurst statistics can arise. A phenomenon has a bias in one direction, which remains until some exogenous event (the joker)changes the bias in direction, magnitude or both. In the capital markets, the exogenous event could be the economy, which causes market sentiment to persist in one direction until an economic reversal changes that bias.

motion, or a pure random walk, as the memory effect dissipates. The regression referred to above would thus be performed on the data prior to the convergence of H to 0.5. The correlationbetween periods can be calculated as follows:12CN = 2(2Hl- 1)

(6)

where CN equals the correlationover period N. From this we see that, as expected, an H equal to 0.5 results in zero correlation. A persistent series, with an H greater than 0.5, results in positive correlation. An anti-persistent series results in negative correlation. This is not a simple matter of autocorrelation, despite the similarities. The autocorrelation function assumes Gaussian, or near-Gaussian, properties in the underlying distribution. (The distributionhas the normal bell-shaped curve.) The autocorrelationfunction works well in determining short-termdependence, but it tends to understate long-run correlation for nonGaussian series.13

Persistence in the Capital MarketsI estimated the Hurst exponent for monthly S&P 500 returnsand 30-yearTreasurybonds, as well as relative returns between the two series. The following equations were used:Sr(i+ 1) = [(Si + I + Di + 1)/(Si + Di) - 1] 100, (7)Br(i+ 1) = [(Bi + 1 + Ci + 1)/(Bi + Ci)SBr(l) = Sr(i) -Br(i), 1] 100, (8) (9)

Estimating the Hurst ExponentMandelbrothas found that:9R/S = NH (4)

whereSr = stock return, Si= S&P 500 price in month i, Di= S&P 500 dividend (paid by ex-date) in

when observed over various N. From this it follows that:H = log (R1S)/log (N). (5)

month i,Br = bond return,

H can thus be estimated by performing an ordinary-least-squaresregression between log (R/S) and log (N) for various N. This method gives a more accurateestimate of H than Hurst's EmpiricalLaw (Equation(3)) because the latter tends to overestimate H between 0.5 and 0.7 and underestimate H between 0.7 and 1.0.10 For very long N, the series can be expected to converge to the value H = 0.5.11 In other words, observations with long N can be expected to exhibit properties similar to regular brownian

Bi= 30-yearT-bond price in month i, Ci = 30-year T-bond accrued interest in month i and SBr(i) stock/bondrelativereturnin month i. = Returns were generated for the period January 1950 through June 1988, using data from The Boston Company's database. This resulted in 463 months of data. These months were then divided into various N, from one 463-month observationto 77 continguous six-month observations. The rescaledrange analysis was performedas

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1989 0 34

Table I R/S Analysis for S&P 500 Monthly Returns, 1/50-6/88N RIS Log (RIS) Log (N)

Returns, Table II R/S Analysis of 30-YearTreasury-Bond 1/50-6/88N RIS Log (RIS) Log (N)

463 230 150 116 75 52 36 25 18 13 6

31.877 22.081 16.795 12.247 12.182 10.121 7.689 6.296 4.454 3.580 2.168

1.503 1.344 1.225 1.088 1.086 1.005 0.886 0.799 0.649 0.554 0.336

2.667 2.362 2.176 2.064 1.875 1.716 1.556 1.398 1.255 1.114 0.778

463 232 150 116 75 52 36 25 18 13 6

45.050 21.587 15.720 12.805 10.248 9.290 7.711 5.449 4.193 4.471 2.110

1.654 1.334 1.196 1.107 1.011 0.968 0.887 0.736 0.622 0.650 0.324

2.666 2.362 2.176 2.064 1.875 1.716 1.556 1.398 1.255 1.114 0.778

RegressionResults:

RegressionResults:

-0.103 Constant Std. Err.of Y Est. 0.041 0.988 R-squared 0.611 X Coefficient(H) 0.027 Std. Err.of Coef. CN 0.168

Constant -0.151 Std. Err.of Y Est. 0.057 0.978 R-squared X Coefficient(H) 0.641 0.031 Std. Err.of Coef. 0.215 CN

outlined in Equations (1) and (2). The R/S used in the regression was the average R/S for each N. The lower levels of N, where there are many observations, can thus be considered more accurate than the larger N, which had few observations. Tables I, II and III show the results.

Figure A illustrates the result of the regression. The solid line shows a pure random walk, with an H equal to 0.5. After 38 years, the observationsshow no tendency to deviate from the regression line. Table II shows that the bond market has a slightly higher persistent bias than stocks, with an H of 0.64 (plus or minus 0.03). The fit is Results Table I shows the data used to estimate the good, with an R-squaredof 97.8 per cent and a Hurst exponent for the S&P 500. H was esti- standard error of 0.06. The correlation is 21.5 mated to be 0.61 (plus or minus 0.03). The high per cent. FigureB offers a visual illustration.As R-squared (98.8 per cent) and low standard with stocks, the memory effect persists even error (plus or minus 0.04) illustrate the good- after 38 years. Table III shows that stock/bond relative reness of the fit. CN is calculated to be 16.8 per turns have a Hurst exponent of 0.66 (plus or cent.R/S Analysis for the S&P 500, 1/50-6/88 Figure B R/S Analysis for 30-Year T-Bonds,

Figure A

1/50-6/88Log (R/S) 1.6 1.4 1.2 H =0.5 Fitted . . . 1 1 1 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 Log (no. of months) Log (R/S) 2 1.8 1.6 1.41.2 I 0.8H=5 0.6 0.4

0

0.8 0.60.4*

Fitteda a a I

0.7 0.9

0.7 0.9 1.1

1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 Log (no. of months)

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1989 O 35

Table III R/S Analysis of Relative Stock/Bond Returns, 1/50-6/88N RIS Log (RIS) Log (N)

463 230 150 116 75 52 36 25 18 13 6

27.977 18.806 15.161 11.275 11.626 8.790 7.014 4.958 4.444 3.459 2.209

1.447 1.274 1.181 1.052 1.065 0.944 0.846 0.695 0.648 0.539 0.344

2.666 2.362 2.176 2.064 1.875 1.716 1.556 1.398 1.255 1.114 0.778

tice, any modeling of the relationbetween stock and bonds should not use periods longer than six years. This regression run uses N less than or equal to 75 months.

ImplicationsThe higher the Hurst exponent, the strongerthe persistence and the less "white noise" there is in a time series. While the stock market shows persistent trends, the relatively low level of H also implies a good deal of noise. We can thus expect that attempts to forecastthe stock market over the short term will be difficult, given the level of short-term noise. The H of 0.61, however, shows that it may be possible. In addition, attempts to find "deterministic chaos" in the stock marketwill be difficultbecause of the level of noise in the data. The same is true of the bond market. The level of the Hurst exponent for relative stock/bond returns is closer to that found for naturalsystems, where fractalshave been somewhat successful in explaining behavior.15 It seems likely that the relation between stocks and bonds can be modeled using deterministic chaos, given the high level of persistence, relatively low level of noise and short period of persistence. This may result in an improved model for tacticalasset allocation, which is built on the premise that there is an appropriatetime to invest in one asset over the other. Finally, the results show that pure random walk theory does not apply to the capital markets. The capitalmarketsinstead follow a biased random walk. The trick is to determine the current bias as well as to anticipate when the "joker" will crop up to change the bias. The biased Hurst phenomenon shows that there is a theoreticalbasis for market timing and tactical asset allocation.

Results: Regressioni

Constant -0.185 Std. Err.of Y Est. 0.022 0.994 R-squared X Coefficient(H) 0.658 Std. Err.of Coef. 0.023 0.245 CN

minus 0.02)-an even higher level of persistence than stocks or bonds alone. Again, the high R-squared (99.4 per cent) and low standard error (plus or minus 0.02) illustrate the goodness of the fit. The higher correlation(CN = 24.5 per cent) illustrates the stronger influence of sentiment on relative stock/bond returns, as compared with stock or bond returns alone. FigureC illustratesthe findings. Note that the relation between stock and bond returnsbreaks down when N is greater than 75 months. After 75 months, the relationshipconsistently follows the H = 0.5 line.14The "memory effect" in the relation between stocks and bonds dissipates after about six years. This means that, in pracFigure C R/S Analysis for Stock/Bond Returns, 1/50-6/88

Log (R/S)1.8 1.6 1.4 1.2 1

What's Behind Persistence?What are the underlying reasons for the biased random walk? We have learned that the Hurst phenomenon indicates that capital market returns are influenced by the past. This influence goes across time scales. One sixmonth period influences all subsequent sixmonth periods. One 10-year period influences all subsequent 10-year periods. The degree of influence varies across the asset classes studied. 16 We can postulate that this effect is caused by investor bias, or market sentiment. The correla-

0.8 0.6 0.4 0.7 0.9 Fitted 1.1 1.3

H =0.5

1.5 1.7 1.9 2.1 2.3 2.5 2.7 Log (no. of months)

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1989 O 36

TheFractal Geometry Nature(New York:W. H. of tion derived from the Hurst exponent thus Freeman, 1982). becomes a measure of the impact of market sentiment (generated by past events) upon fu- 3. R/Sanalysis is robust (as shown by B. B. Mandelbrot, "StatisticalMethodology for Nonperiodic ture returns in the capital markets. The bond Cycles:FromCovarianceto R/SAnalysis," Annals and stock markets separately are influenced by and of Economic SocialMeasurement July 1972, 1, regulatory policies, as well as by specific comand Mandelbrotand J. R. Wallis, "Robustnessof the Rescaled Range R/S in the Measurement of pany events, and show a low level of carryover Noncyclic Long Run Statistical Dependence," influence from sentiment. This memory effect WaterResour.Res. 5 (1969), pp. 967-988) and is persists for a long time, however. Neither marvalid whether the underlying distribution is ket showed signs of deviating from the trend Gaussian or non-Gaussian. after even 38 years of data. 4. An explanationof fractalsand other chaos theoStock/bond returns show a much stronger ries for the layman is contained in J. Gleick, influence from investor sentiment. A preference Chaos: MakingA New Science (New York:Viking, 1987). for one asset class over the other is a purer op. measure of market sentiment. If the Hurst cor- 5. Feder (Fractals, cit.) has found this to be true. relation is a measure of market sentiment, a 6. This is demonstrated in Mandelbrot,"Statistical Methodology," op. cit relative return between asset classes is likely to 7. This is demonstratedin H. E. Hurst, "Long-term exhibit strong investor sentiment. But this carStorageof Reservoirs,"Trans.Am. Soc. Civ. Eng. ryover sentiment dissipates after six years; 116 (1951), pp. 770-808 and W. Feller, "The while the memory effect is strong, it does not AsymptoticDistributionof the Range of Sums of last as long as it does in either the stock or the Independent Variables," Ann. Math. Stat. 22 (1951),pp. 427-433. It is true whether the underbond market. In all cases, however, the sentilying random distribution is Gaussian, logment bias exists, until an economic equivalentof normal, Markovianor some other distribution. the joker changes the bias. 8. Hurst tested his empiricallaw for, among other Investor sentiment represents investors' innatural phenomena, river discharges, rainfall, terpretationof the events that influence capital sunspot numbers and tree rings. A detailed list is markets. This interpretationis not immediately availablein Feder, Fractals, cit. Much of this op. reflected in prices, as the EMH asserts. It is discussion of Hurst is taken from Feder. instead manifested as a bias in returns, one that 9. See Mandelbrot, "StatisticalMethodology," op. cit. persists for decades. This bias accounts for a 10. See Feder, Fractals, cit. op. significant portion of capital market returns. 11. Feder(Fractals, cit.) makes this case. He found op. The challenge now is to develop an asset pricing that the memory effect dissipated only after N model that takes this underlying, non-linear greaterthan 4000, using simulated data. process into account. U

12. See Mandelbrot, The FractalGeometry Nature, of op. cit. 13. This is demonstrated in Mandelbrot, "Statistical Footnotes Methodology," op. cit 1. A good summary may be found in E. F. Fama, 14. This is the convergence with the Gaussian as"EfficientCapital Markets:A Review of Theory ymptote predicted by Feder for very large N. and Empirical Work," Journalof Finance,May 15. See footnote 8, as well as the study of wave 1970. heights in Feder, Fractals, cit. op 2. More detailed descriptions of the history of 16. Mandelbrot (The FractalGeometry, cit.) has op. (New Hurst's work are found in J. Feder, Fractals shown this influence to be fractaland caused by non-lineardynamics. York:Plenum Press, 1988)and B. B. Mandelbrot,

FINANCIAL ANALYSTS JOURNAL / JULY-AUGUST 1989 O 37