Currencies A True Hurst Process

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As we have stated in previous chapters, currencies are often confused with securities. When traders buy and sell currencies, they do not realize an investment income on the currencies themselves. Instead, currencies are bought and sold in order to invest in short-term interest-rate securities in the selected country. Currency "value" is not necessarily related to activity in the country's underlying economy. Currencies are tied to relative interest-rate movements in the two countries. In addition, the markets themselves are manipulated by their respective governments for reasons that may not be considered "rational" in an efficient market sense. For instance, if a country wants to stimulate exports, it will allow, or even encourage, the value of its currency to drop. On the other hand, if it wishes to encourage imports and reduce its trade surplus, it would like its currency to appreciate. Both objectives could be desirable, whether the country is in recession or expansion.

There are two ways in which the central bank of a country can manipulate its currency. First, it can raise or lower interest rates, making its government securities more or less attractive to foreign investors. Because this alternative can impact the overall economic growth of a country, it is generally considered a last resort, even though it has the most long-lasting effects.

The second method is more direct and usually occurs when the currency has reached a level considered acceptable by the central bank. Central banks typically buy or sell in massive quantities, to manipulate the value of the currency. At certain times, the largest trader in the currency market can be the central bank, which does not have a profit maximization objective in mind.

Because of these two factors, currency markets are different from other traded markets. For instance, they are not really a "capital market" because the objective of trading currency is not to raise capital, but to create the ability to trade in stocks and bonds, which are real markets for raising capital. Currencies are "pure" trading markets, because they are truly a zero sum game. In the stock market, asset values will rise and fall with the economy. Interest rates also rise and fall, in an inverse relationship with the economy. Both relationships are remarkably stable. However, currencies have no stable relationship with the economy. As a pure trading market, currencies are more inclined to follow fads and fashions. In short, currencies follow crowd behavior in a way that is assumed for stock and bond markets.

So far, we have examined markets that have some tie to economic activity. Stocks, bonds, and (probably) gold have nonperiodic cycles that have an average length. This latter characteristic is closely related to nonlinear dynamical systems and the Fractal Market Hypothesis. However, the pure Hurst process, as discussed in Part Two, does not have an average cycle length. The "joker" is a random event that can happen at any time. Because the drawing of random numbers from the probability pack of cards occurs with replacement, the probability of the joker's occurring does not increase with time. The change in "bias" truly does occur at random.

In the currency market, we see exactly these characteristics. In Chapter 2, we saw that the term structure of volatility for the yen/dollar exchange rate was different than for U.S. stocks and bonds. In Chapter 4, we saw evidence of a persistent Hurst exponent for the yen/dollar exchange rate. In this chapter, we will examine this and other exchange rates in more detail. The study will still be limited.

Besides currencies, it is possible that other "trading markets" are also pure Hurst processes, particularly in commodity markets such as pork bellies, which are known to be dominated by speculators. Other researchers will, I hope, investigate these markets.


Currency markets have the potential for Type I undersampling problems. Like gold, currency fluctuations in the United States did not occur in a free market environment until a political event—in this case, another Nixon Administration event: the floating of the U.S. dollar and other currencies, as a result of the Bretton Woods Agreement of 1972. In the period following World War II, the U.S. dollar became the world currency. Foreign exchange rates were fixed relative to the U.S. dollar by their respective governments. However, in the late 1960s, the global economy had reached a different state, and the current structure of floating rates manipulated by central banks developed. We therefore have less than 20 years' data. In the U.S. stock market, 20 years' daily data are insufficient to achieve a statistically significant Hurst exponent. Unless daily currency exchange rates have a higher Hurst exponent than the U.S. stock market, we may not achieve significance. Luckily, this does turn out to be the case.


We have already examined some aspects of the yen/dollar exchange rate in Chapters 2 and 4. This exchange rate is, along with the mark/dollar exchange rate, an extremely interesting one. For one thing, it is very heavily traded, and has been since 1972. The postwar relationship between the United States and Japan, and the subsequent development of the United States as the largest consumer of Japanese exports, has caused the exchange rate between the two countries to be one long slide against the dollar. As the trade deficit between the two countries continues to widen, the value of the U.S. currency continues to decline. R/S analysis should give us insight into the structure of this actively traded and widely watched market.

Table 12.1 summarizes the results, and Figure 12.1 shows the V-statistic graph for this currency. The Hurst exponent is higher than the daily U.S. stock value, with H = 0.64. This period has 5,200 observations, so the estimate is over three standard deviations above its expected value. Therefore, it is highly persistent compared with the stock market. However, no long-range cycle is apparent. This is consistent with the term structure of volatility, which also has no apparent long-range reduction in risk. Therefore, we can conclude that the yen/dollar exchange rate is consistent with a fractional brownian motion, or Hurst process. However, unlike the stock and bond market, there is no crossover to longer-term "fundamental" valuation. Technical information continues to dominate all investment horizons. This would lead us to believe that this process is a true "infinite memory," or Hurst process, as opposed to the long, but finite memory process that characterizes the stock and bond markets.

Table 12.1 R/S Analysis

Regression output: Constant

Standard error of Y (estimated) R squared H

Observations Significance

0.642 0.553 4,400.000 5.848



Regression output: Constant Standard error of Y (estimated) R squared

Number of observations Degrees of freedom Hurst exponent Standard error of coefficient Significance

0.018 0.998 24.000 22.000


0.006 4.797


0.009 3.440

0.027 0.995 24.000 22.000


Regression output: Constant

Standard error of Y (estimated) R squared

Number of observations Degrees of freedom X coefficient(s) Standard error of coefficient Significance

0.624 0.004 4.650

FIGURE 12.1 V statistic, daily yen, January 1972-December 1990.


The mark/dollar exchange rate, like the yen/dollar, is tied to postwar expansion—in this case, Germany, as the United States helped its old adversary recover from the yoke of Nazism. Interestingly, R/S analysis of the mark/dollar exchange rate is virtually identical to the yen/dollar analysis. H = 0.62, slightly lower than the yen/dollar, but not significantly so. This gives us a significance of more than four standard deviations (see Figure 12.2). Again, there is no break in the log/log plot, implying that there is either no cycle or an extremely long cycle. The latter is always a possibility, but seems unlikely.


The pound/dollar exchange rate is so similar to the other two (see Figure 12.3) that there is very little to comment on, except that, unlike the stocks studied in my earlier book, all three currency exchange rates have values of H that are virtually identical. This could prove to be very useful when we examine the Hurst exponent of portfolios.

FIGURE 12.1 V statistic, daily yen, January 1972-December 1990.

FIGURE 12.2 V statistic, daily mark, January 1972-December 1990. 2.2-----------

FIGURE 12.3 V statistic, daily pound, January 1972-December 1990.


The yen/pound is slightly different from the other exchange rates. Japan and the U.K. are not major trading partners; the currency trading that occurs between them is far less active. In addition, the forward market, where the majority of currency hedging occurs, is quoted in U.S. dollar exchange rates. Thus, the yen/pound exchange rate is derived from the ratio of the yen/dollar exchange rate and the pound/dollar exchange rate, rather than being quoted directly. As a result, the yen/pound exchange rate looks essentially random at periods shorter than 100 days. The other exchange rates have similar characteristics, but the yen/pound exchange rate is virtually identical to a random walk at the higher frequencies. Figure 12.4 shows how tightly the V statistic follows its expected value for less than 100 days.

Even though the yen/pound is not an exchange rate that garners much attention, it too has no apparent cycle length. The long memory is either extremely long or infinite.

FIGURE 12.4 V statistic, daily yen/pound, January 1972-December 1990.

FIGURE 12.4 V statistic, daily yen/pound, January 1972-December 1990.


Currencies have interesting statistical and fundamental characteristics that differentiate them from other processes. Fundamentally, currencies are not securities, although they are actively traded. The largest participants, the central banks, are not return maximizers; their objectives are not necessarily those of rational investors. At the same time, there is little evidence of cycles in the currency markets, although they do have strong trends.

These characteristics, taken together, lead us to believe that currencies are true Hurst processes. That is, they are characterized by infinite memory processes. Long-term investors should be wary of approaching currencies as they do other traded entities. In particular, they should not assume that a buy-and-hold strategy will be profitable in the long term. Risk increases through time, and does not decline with time. A long-term investor who must have currency exposure should consider actively trading those holdings. They offer no advantage in the long term.



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