-- Posted Thursday, 30 October 2008 | Digg This Article | Source: GoldSeek.com

John Peterson
jcp@eskimo.com

Abstract

At almost any time of day during the normal business week, gold bullion is being actively traded somewhere on the globe. Many observers of the international bullion trade have commented that when heavy selling pressure appears, more often than not it seems to happen during the hours when the New York markets are open for business.

This article addresses the validity of this perception from a quantitative perspective. The results will be presented in a format that is simple to interpret and does not require any specialized knowledge of Mathematical concepts. The conclusion is that the perceived pattern of a tendency towards heavier selling during the hours of the day when the New York markets are open for business is not only correct, but the forces behind this selling are more "determined" than even many "believers" might have imagined.

What would that be like?

While pondering this apparent tendency for the heavy selling of Gold to occur more frequently during the hours when the New York markets are open, I wondered what the Gold price might be today if somehow there wasn't any trading during those hours. Gee... what would that be like? After a little thought, I realized that there is indeed a calculation one could perform that represents something a little different, but very similar in a meaningful way.

I utilized three readily available quotes for cash Gold priced in US$; the AM and PM London Gold fixes, and the closing price of the StreetTracks Gold Trust ETF that trades on the NYSE (herein referred to by its NYSE trading symbol GLD for brevity). These price quotes divide each 24 hour trading day into three non-overlapping trading sessions. As I am writing from North America, I define my full 24 hour trading day to begin and end when the NYSE closes. When I refer to the daily closing price of Gold I mean the closing price of GLD times 10 (recall that the GLD ETF represents 1/10 of an ounce of Gold). One could have chosen one of the London fixes to end the trading day, the choice is really completely arbitrary.

I will present a Mathematical representation for the price series made up of these three daily quotes in temporal order. For each of the three sessions I will use the Mathematical equivalent of a chisel to "chip away" the contribution to the Gold price discovery from the other two sessions. The resulting three hypothetical price series will meaningfully reflect how the Gold price would have evolved if trading was isolated to only the trading session the given series is associated with. In a sense I will "slice out" out the forces behind the price discovery process of each trading session and let them evolve on their own in hypothetical isolation.

Calculating the Price Series

In this section I provide a more detailed description of how the three hypothetical price series are calculated. If you don't like Math, you can skip to the next section without missing the central message of this article. However, the results of my calculations were striking enough I felt that any presentation of them absolutely demanded an explanation as to how I arrived at them.

Consider a time series of real price observations. I denote them by p(0), p(1), p(2) and so on. For simplicity restrict ourselves for now to price series that are sampled exactly once per day (such as the daily closing price of GLD for example).

The "trick" that makes the calculation possible is to realize that any price series can be reconstructed from a knowledge of the starting price denoted as above by p(0) and the daily returns that I denote by r(1), r(2), r(3) and so on. The daily return r(i) on any given day is defined to be the ratio p(i) / p(i-1) or the price on that day divided by the price from the day before giving a number close to one. On days when the price declines relative to the previous price the raw return is slightly less than one, and slightly greater than one on days when the price advances. The price on the n-th day is the given by the expression;

It's not difficult to see why this must be so. Notice that the product of the first two numbers p(0)*r(1) can be rewritten as p(0)*(p(1)/p(0)), which after algebraic cancellation of p(0) is identically equal to p(1). Note that p(1)*r(2) reduces to p(2) and so on. Repeat this process until no more returns are remaining and one ends up with the value of p(n) as claimed. In plain English; the price is updated by multiplying the last price by the return.

Instead of a Gold price series that is sampled just once per day, now consider a Gold price series that is sampled three different times per day. The sampling times of these prices divide the 24 hour trading day into three trading sessions. The process used above can be used to reconstruct this series of observed prices in much the same way. Given the observed price at the end of a session, multiplying that price by the return of the next session gives the observed price at the end of the next session. The process is repeated as needed to reconstruct all of the observed prices. To clarify my choice of terminology, here are the three tradings sessions and recipes for computing the associated returns;

Session One

Session one is the beginning of my trading day and starts when the New York Stock Exchange closes for business (4:00 PM New York time), and ends when the next London AM Gold fix is reported. The return for this session, denoted by r_1(i) is the ratio of the London AM fix, divided by 10 times the closing price of the GLD ETF observed at the start of the session. I also refer to this as the "Asian" session.

Session Two

This session begins when the London AM Gold fix is reported and ends with the reporting of the London PM Gold fix. The return for this session, denoted by r_2(i) is the ratio of the London PM Gold fix divided by the London AM fix. I also refer to this as the "London" session.

Session Three

This session winds up my trading day and starts with the London PM Gold fix and ends when the New York Stock exchange closes (4:00 PM New York time). The return for this session, denoted by r_3(i) is given by the ratio of 10 times the closing price of the GLD ETF divided by the London PM fix from earlier the same day. Note that this session ends exactly one day after the first session begins. I also refer to this as the "New York" session.

It is not difficult to show that the daily closing prices of the GLD ETF can be reconstructed from a knowledge of the returns of the three trading sessions above. To be specific, the closing price of GLD on the n-th day is given by;

The easiest way to see why this must hold true is to notice that for any given day, the product of the returns (r_1(i)*r_2(i)*r_3(i)) is identically equal to the return of GLD for that day (substitute the definitions of the returns for the sessions and note that the AM and PM London fixes cancel out).

The reason for using such a method for reconstructing an observed prices series will now become apparent. To "chisel" away the effects of price discovery during unwanted periods of the trading day, we simply drop the returns from the sessions associated with them from the expression above. For example, consider the price series that is constructed by repeated multiplication by the returns associated with session one only;

At the risk of sounding redundant, it should be abundantly clear that the daily returns of this price series are identical to the returns for just session one. In this sense, the series is a meaningful representation of what the Gold price would be without the influence of trading sessions two and three. We can easily construct a price series for the hypothetical situation where trading only occurs during session two, and another series for the situation where trading only occurs during session three.

Note that it is possible to "glue" the hypothetical series back together in a manner of speaking. It was noted above that on any given day, the product of the returns (r_1(i)*r_2(i)*r_3(i)) is identically equal to the return of GLD for that day. This statement holds true for any arbitrary n-day time period. On the n-th day, calculate the n-day total return for each of the hypothetical series and for GLD by dividing the price on that day by the price at the start of the calculation, or p(0) to be specific. The product of the n-day total returns for the hypothetical series is identically equal to the n-day total return for GLD. I found this relationship to be of practical use as a way of providing a comforting sanity check on the accuracy of my calculations.

A Battle of Titans

In this section I present the results of calculating the hypothetical price series for the three trading sessions. I started the calculation of all of the series on Monday 29 November, 2004. The closing price of GLD on that day was quoted as 45.40 $US which corresponds to a Gold price of 454.00 $US. This is a few days after the StreetTracks Gold Trust ETF first began trading on the NYSE. Unfortunately, it is clearly impossible to choose starting dates that go back in time before the GLD data was available. The calculations used data up to and including the price quoted for GLD on Friday 24 October, 2008 which was 72.21 $US and corresponds to a Gold price of 722.10 $US.

Figure 1 below shows the three hypothetical series together with 10 times the closing price of GLD on a plot with logarithmic scaling of the vertical price axis. Logarithmic scaling of the price axis is demanded by the fact that the ending values of the series span more than 6 orders of magnitude.

Figure 1. Hypothetical price series for the three sessions and the StreetTracks Gold Trust

I think this is a good time to interject for a moment on a subtle but important issue. Earlier in this article I posed the question of what the Gold price would be should a major exchange close shop. Presumably, some of those who were conducting business through the exchange would move their business elsewhere presenting a new influence on price discovery during those times of the day. Hopefully it is clear to you that while that is a perfectly reasonable question to ask, the results of my calculations don't really give any real insight of any kind into how that scenario would play out, they are not a prediction of any sort. In fact, I know of no real world scenario that they would accurately mirror. I have presented hypothetical Gold prices that are the result of "slicing up" the observed data in a meaningful way (the daily returns of the series are identical to those of the session they are associated with). Enough said.

The hypothetical price series representing the Asian session (session one) and the series representing the New York session (session three) present a dichotomy of rather striking proportions. On Friday 24 October, 2008 the hypothetical closing price of the Asian session was 886,798.32 $US, and the New York session closed at 0.5546 $US. My comments in the paragraph above were intended to dissuade you from making any conclusions that relate to the absolute price levels given here.

On the other hand; a visual inspection of the prices series shows that they exhibit very smooth trends up and down respectively, with only modest corrections against their longer term trends. They leave little to the imagination as to the very "determined" nature of the buying and selling forces associated with these two sessions. It is also evident that the slope of the trends are becoming steeper (concave up and down respectively, and on a log scaled price axis where a straight line corresponds to exponential growth or decay). This suggests a battle of opposing forces that are both very determined and intensifying as well. A War of the (Gold) Worlds? Time will eventually tell.

A few other observations are worthy of note. The first 7 months of the calculation up to the start of July 2005 stand out; all of the series exhibit a (relatively) choppy and quiet sideways character. It is also curious to note that over the entire time period covered the London session exhibits a more or less sideways character (relative to sessions one and three). However, the London session has the shortest duration of the three. This is most certainly a factor that contributes to its relative calm. On balance to date it has contributed a net downward force on the Gold price.

If you are concerned about the lack of sharp corrections in the series for sessions one and three, rest assured that there are down days during the Asian session, and there are up days during the New York session. Part of the issue here is that the vertical axis in Figure 1 spans an enormous 6 orders of magnitude. The price of Gold (as given by 10 times GLD) as of Friday has corrected about 28 percent from its high of 1004.40 $US reported on 17 March, 2008. This vertical distance is visible in an inspection of the plot, but spans a small fraction of the full range of the vertical axis. This provides a ruler in the sense that it implies none of the corrections in the Asian or New York series have been anywhere near that violent in percentage terms. Table 1 below will help provide some insight into the over all character of the returns from the three sessions.

Most of the returns reported in Table 1 above are in units of percentage that most of you should be familiar with. To be specific; subtract one from the raw return, and then multiply that result by 100 to get percentages.

It should be noted that the mean values of returns reported in Table 1 above were calculated as geometric means of the raw returns. Because the raw returns represent a multiplicative process, the geometric mean better conveys what is "typical" as opposed to the more commonly used sample arithmetic mean that is best used for situations involving additive processes. See reference [1] for further details.

A quick scan of Table 1 above conveys a message similar to that conveyed by the price series plot in Figure 1. A typical trading day (as given by the mean return calculated over the time period spanned by the data) is a gain of 0.78 percent in the Asian session. The mood towards Gold in New York is generally much more sour with a typical day being a loss of 0.69 percent.

Summary

Some observers of the international Gold bullion trade have commented that when heavy selling pressure appears, it seems to happen more often than not during the hours when the New York markets are open for business. This article addressed the validity of that perception from a quantitative perspective.

A technique was presented for the calculation of three hypothetical price series, each one representing how the Gold price would have evolved if trading occurred only during a certain period of the 24 hour trading day. While the scenario that these series accurately reflect is purely hypothetical in nature, the series still convey information to those who correctly understand what they represent.

An inspection of the three hypothetical price series leaves little doubt as to the validity of the perceived trading pattern. The price series associated with the Asian session and the series associated with the New York session present a dichotomy of rather striking proportions. The opposing forces behind price discovery in these two sessions appear to be very determined and growing stronger (up and down respectively).

Acknowledgments

The calculations presented in this article were performed using R, an application specifically designed for statistical computation and graphics. It is almost 100 percent compatible with the S language [2] originally developed by John Chambers [3] of AT&T Bell Labs. R is open source licensed and is available at no cost. For more information visit the R project website [4].

Footnotes, References

1. Wikipedia: Geometric Mean; http://en.wikipedia.org/wiki/Geometric_Mean

2. S language home page at Bell Labs; http://cm.bell-labs.com/cm/ms/departments/sia/S/index.html

3. John Chamber's home page at Bell Labs; http://cm.bell-labs.com/cm/ms/departments/sia/jmc/

4. R project home page; http://www.r-project.org/

Appendix

In this appendix I have tabulated the first 20 values of each of the hypothetical price series for the benefit of those who might want to reproduce the calculations or just confirm their understanding of them.

A glance at the very end of Table 2 above brings forth a book keeping issue with the data. There are days when the London markets are closed for the day and the New York markets are open, and vice versa.

There is more than one meaningful way to address the issue of missing data. My convention was to discard all the data for any day when one or more of the three price quotes are not available creating a session that lasts longer than a single day (or weekend perhaps). In effect, my exchange honors all holidays that are relevant to the London fixes, or trading of GLD.

-- Posted Thursday, 30 October 2008 | Digg This Article | Source: GoldSeek.com

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