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We begin by applying R/S analysis to the S&P 500, for monthly data over a 38-year period, from January 1930 to July 1988. Figure 8.1 shows the log/log plot using the method described above. A long memory process is at work for N, for less than approximately 48 months. After that point, the graph begins to follow the random walk line of H - 0.50. Returns that are more than 48 months apart have little meas^ra^le correlation left, on the average. Figure 8.2 shows H values ariculitrfby running the regressions for N less than or equal to 3, 3.5, 4, 4.5, arid f years. The peak clearly occurs at N - 4 years, with H - 0.78, which we can say is the estimate for the Hurst exponent for the S&P 500. This high value for H shows that the stock market is clearly fractal, and not a random walk. It is, instead, a biased random walk, with an anomalous value of H - 0.78. Figure 8.1 graphs H - 0.78 and H - 0.50. Table 8.1 shows the results of the regression, using N less than or equal to 48 months. Regression results, using

FIGURE 8.2 R/S analysis: Estimating the cycle length; S&P 500 monthly returns, January 1950-July 1988.

FIGURE 8.2 R/S analysis: Estimating the cycle length; S&P 500 monthly returns, January 1950-July 1988.

NO JCF VEMtS

NO JCF VEMtS

Tabic 8.1 R/S Analysis of Stock Returns, January 1950-July 1988

Unscrambled Scrambled

Tabic 8.1 R/S Analysis of Stock Returns, January 1950-July 1988

Unscrambled Scrambled

Constant |
-0.32471 |
-0.04544 |

Standard error of Y (estimated) |
0.01290 |
0.02005 |

R2 |
0.99559 |
0.98564 |

X coefficient (H) |
0.778 |
0.508 |

Standard error of X |
0.008 |
0.004 |

Nst48 months, are H-0.S2±0.02, confirming that the average cycle length, jcx period, of the S&P 500 is 48 months.

We can now apply the scrambling test to the series of monthly returns. Figure 8.3 shows the log/log plot of the scrambled and unscrambled series. The scrambled series, clearly different, gives H -0.51. Scrambling destroyed the long memory structure of the original series and turned it into an independent series. There is also ho drop in slope after 48 months, as there is in the original; the series continues to scale as a random walk.

FIGUtE S3 Scrambling test: S&P 500 monthly returns, January 1950-July 1988. Unscrambled H - 0.78; scrambled H - 0.51.

FIGUtE S3 Scrambling test: S&P 500 monthly returns, January 1950-July 1988. Unscrambled H - 0.78; scrambled H - 0.51.

The sequence of price changes is important in preserving the scaling feature of the series. Changing the sequence of returns by scrambling has changed the character of the time series.

These results are inconsistent with the Efficient Market Hypothesis. Roberts (1964/1959) (as discussed in Chapter 2) described the market mechanism as a roulette wheel and asserted that "this roulette wheel has no memory." R/S analysis shows that the independence assumption, particularly regarding long i fl\i|H|f WM'|l'l1 seriously flawed.

Market returns are persistent time seriesnit£l^uiderlying fractal probability distribution, and they follow a bias®<f random walk, as described by Hurst. The market exhibits trend-reinforcing behavior, not mean-reverting behavior. Because the system is persistent, it has cycles and trends with an average cycle length of 48 months. This length is average because the system is nonperiodic and fractal.

Figure 8.4 shows the log/log graphs of four representative stocks: IBM, Mobil, Coca-Cola, and Niagara Mohawk. Values of H are persistent, and

FIGURE 8.4a R/S analysis of individual stocks: Monthly returns, January 1963-December 1989. IBM: Estimated H-0.72.

FIGURE 8.4a R/S analysis of individual stocks: Monthly returns, January 1963-December 1989. IBM: Estimated H-0.72.

The Stock Market 87

The Stock Market 87

FIGURE 8.4b R/S analysis of individual stocks: Monthly returns, January 1963-December 1989. Mobil Oil: Estimated H -0.72.

FIGURE 8.4b R/S analysis of individual stocks: Monthly returns, January 1963-December 1989. Mobil Oil: Estimated H -0.72.

cycles are of various lengths. Table 8.2 shows the results for the S&P 500 and some individual stocks. In this limited study, stocks grouped by industry tend to have similar values of H and similar cycle lengths. Industries with high levels of innovation, such as the technology industry, tend to have high levels of H and short cycle lengths. In contrast, utilities, which have a low level of innovation, have lower levels of H and very long periods. The joker shows up less often for utilities than it does for technology stocks.

These results raise an interesting question about accepted definitions of risk. According to the Capital Asset Pricing Model (CAPM), a higher-beta stock, relative to the market index, is riskier than a lower-beta stock, because the volatility as measured by the standard deviation of returns is higher for high values of beta. Apple Computer, with its beta of 1.2 relative to the S&P 500, is riskier than Consolidated Edison (ConEd) with its beta of 0.60.

The Hurst exponent (H) measures how jagged the time series is. The lower the value of H, the more noise there is in the system and the more

Table 8.2 |
R/S Analysis of Individual Stocks | |

Hurst Exponent (H) |
Cycle (Months) | |

S&P 500 |
0.78 |
48 |

IBM |
0.72 |
18 |

Xerox |
0.73 |
18 |

Apple Computer |
0.75 |
18 |

Coca-Cola |
0.70 |
42 |

Anheuser-Busch |
0.64 |
48 |

McDonald's |
0.65 |
42 |

Niagara Mohawk |
0.69 |
72 |

Texas State Utilities |
0.54 |
90 |

Consolidated Edison |
0.68 |
90 |

FIGURE 8.4d R/S analysis of individual stocks: Monthly returns, January 1963-December 1989. Niagara Mohawk: Estimated H -0.69.

FIGURE 8.4d R/S analysis of individual stocks: Monthly returns, January 1963-December 1989. Niagara Mohawk: Estimated H -0.69.

random-like the series is. (Figure 7.1 and, particularly, Figure 7.2, the cumulative graph, illustrate the difference.) Apple Computer has an H value of 0.68; for ConEd, H - 0.58. ConEd's time series is less persistent and more jagged than Apple's time series. Which stock is riskier?

Because both stocks have H values greater than 0.5, they are both fractal, and application of standard statistical analysis becomes of questionable value. Variances are undefined, or infinite, which makes volatility a useless and possibly misleading estimate of risk. A high H value shows less noise, more persistence, and clearer trends than do lower values. I suggest that higher values of H mean less risk, because there is less noise in the data. This means that Apple Computer is less risky than ConEd, despite their betas. High H stocks do have a higher risk of abrupt changes, however.

A final observation is that the S&P 500 has a higher value of H than any of the individual stocks in Table 8.2. This higher value shows that

0.4 0.« 0.* t 1.2 1.4 1.» 1.8 2 2.2 U0Q(J OF HONIM)

FIGURE 8.5a R/S analysis of international stocks: Monthly returns, January 1959-February 1990. MSCI U.K. index: Estimated H -0.69.

0.4 0.« 0.* t 1.2 1.4 1.» 1.8 2 2.2 U0Q(J OF HONIM)

FIGURE 8.5a R/S analysis of international stocks: Monthly returns, January 1959-February 1990. MSCI U.K. index: Estimated H -0.69.

diversification in a portfolio reduces risk, by decreasing the noise factor and increasing the value of H.

International markets also exhibit Hurst statistics. Figure 8.5 shows log/log plots for the U.K., Japan, and Germany, as represented by each stock market's Morgan Stanley Capital International (MSCI) index. The MSCI data used were from January 1939 to February 1990. Table 8.3 lists therewith ' ■ •

Table CJ' R/S Analysis of International Stock Indices

Hurst Expcwfjit(H) Cycle (Months)

MSCI Germany 0.72 60

MSCI Japan 0.66 48

FIGURE 8.5b R/S analysis of international stocks: Monthly returns, January 1959-February 1990. MSCI Japan index: Estimated H -0.68.

FIGURE 8.5b R/S analysis of international stocks: Monthly returns, January 1959-February 1990. MSCI Japan index: Estimated H -0.68.

If we include the S&P 500 as representative of the United States, all four countries have different H values and cycle lengths. The U.K. has the longest cycle (eight years). Germany has a six-year cycle, and the United States and Japan have four-year cycles. These cycle lengths are probably tied to economy cycles. We will examine this possibility later, for the U.S. market

' Market efficiency can be judged by the amount of noise in the data. Because the United States has the highest H value, it is the most "efficient" market: it has less noise than the others. It is followed by Germany, the UK., and Japan.

THE BOND MARKET

R/S analysis of changes in 30-year Treasury Bond (T-Bond) yields also exhibits Hunt statistics. Bond yields were examined monthly from January

FIGURE 8.5c R/S analysis of international stocks: Monthly returns, January 1959-February 1990. MSCI German index: Estimated H»0.72.

FIGURE 8.5c R/S analysis of international stocks: Monthly returns, January 1959-February 1990. MSCI German index: Estimated H»0.72.

1950 through December 1989. Therestih was H - 0.68 with a cycle length of five yearv which coincideswith' the cycle length of U.S. industrial production, as we shall see later. Figure 8 6 shows this relatioiiAip. '

A similar study was done for an average of 3-, 6-, and 12-month Treasury Bill (T-Bill) yields, as a proxyTor the short eod of the yield curve. Again, Hunt statistics result, with H - 0.65, slightly more noisy than the long bond. (See Figure 8.T.) Interestin^y,! no cyde length is apparent in the log/log plot Either there are not eootigh data; 'br, perhaps,' there is no cycle length anST-Bills do Wkle forever. Because T-Bill yields are an exception (the other series lave eyelet of fo^r to five years), itia difficult to draw any conclusions.

R/S analysis of selected currency rates also yields Hurst statistics. For this study, I ban« used currency exchange rates between the U.S. dollar and

FIGURE 8.6 R/S analysis of 30-year Treasury Bond yields: Monthly, January 1950-December 1989. Estimated H -0.68.

FIGURE 8.6 R/S analysis of 30-year Treasury Bond yields: Monthly, January 1950-December 1989. Estimated H -0.68.

the Japanese yen, British pound, German mark, and Singapore dollar. The first three exchange rates exhibit high levels of persistence. With the U.S. dollar/Singapore dollar exchange rate, we encounter our first truly tandom series.

Figure 8.8 shows the log/log plots for the three primary currencies. All three exchange rates have Hurst exponents at approximately 0.60. The currency markets are not random walks, either. Table 8.4 summarizes tiie results.

These results will come as no surprise to currency traders. Currency markets are characterized by abrupt changes traceable to central bank interventions—attempts by the governments to control the value of each respective currency, contrary to natural market forces. Currencies have a reputation as "momentum trading" vehicles in which technical analysis has more validity than usual. R/S analysis bears out the market lore that currencies have trends, but the levels of the Hurst exponent for these currencies show that they are not exceptionally persistent, when compared to equity markets.

FIGURE 8.7 R/S analysis of Treasury Bill yields: Average of 3-, 6-, and 12-month T-Bill yields, January 1950-December 1969. Estimated H - 0.65.

FIGURE 8.7 R/S analysis of Treasury Bill yields: Average of 3-, 6-, and 12-month T-Bill yields, January 1950-December 1969. Estimated H - 0.65.

This study was done on daily data from January 1973 through October 1990, almost 18 years' worth of daily observations. However, the natural cycle length is not apparent from the examination of any of the log/log plots. A flattening of the slope at the extreme end (about N - 100 months) could come from the sparseness of the data at that end. Apparently, 18 years* data do not cover enough cycles to make the cycle length visible. As

HMrst Ej^ponent (H) |
Cycle | |

Japanese yen " |
J 0.64 •• « |
Unknown |

German mark |
0.64 |
6yrsf |

U.K. pound |
0.61 |
6 yrsf |

Singapore dollar |
0.50 |
None |

we shall soon see, 30 years' stock market data are necessary for a well-defined period. Unfortunately, the United States did not go off the gold standard until 1973, so exchange rates prior to 1973 reflect an environment different from the current one. We may need another ten years' experience, to gather enough data to do a thorough analysis of the currency markets. As we shall see at the end of the chapter, more data points are not needed; tick-by-tick data will not yield more information. We need a longer time period. For that, we will have to wait.

The Singapore dollar is offered as an example of a capital market time series that does not exhibit Hurst statistics. The Singapore dollar/U.S. dollar exchange rate is a true random variable. This will be good news to the Singapore government, because the Singapore dollar is managed purposefully to track the U.S. dollar. Because of this conscious effort, any fluctuation in the exchange rate is due to random fluctuations in the timing of trades to fix the exchange rate.

1.9 | |||

1.8 | |||

1.7 |
- | ||

1.8 |
- | ||

1.9 |
- | ||

1.4 |
- | ||

1.3 |
- | ||

1.2 |
- | ||

1.1 |
- | ||

K |
1 |
- | |

f |
0.9 |
- | |

3 |
0.8 |
- | |

0.7 |
- | ||

0.8 |
- | ||

0.S |
y | ||

0.4 | |||

0.3 |
. H «0.61—3^ | ||

0.2 | |||

0.1 |
I- / | ||

0 |
iiii |
i l i l i l 1 1 . , 1 |

FIGURE 8.8b R/S analysis of currency exchange rates. U.K. pound/dollar exchange rate: Daily rate, January 1973-December 1989. Estimated H- 0.61.

Figure 8.9 is the log/log plot that shows H - 0.50 for this time series. The Singapore bank appears to be doing its job. In other currencies, where the free market determines the exchange rate, persistent values of H continue to be found. Their presence confirms that currency markets also have a fractal structure.

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