Part 7 (2/2)

Fortunately, we also summarized the results of the individual pairs in parts b, c, and d. All three pairs had very different performance, and the average does not represent the sum of the parts.

Crude-EURUSD is the most consistent, with profits in every combination and ratios varying from 0.301 to 0.909. The row showing the detail for the calculation period of 10 is the best, but it looks as though it's an outlier because, without that row, the information ratio increases smoothly from the larger periods to the smaller. We prefer to pick anything in the lower part of the table. A momentum of 5 and exit of 10 seem reasonable and somewhat typical.

Crude-gold is overall a poor performer, with only two profitable combinations out of 18. Even though we thought this could be a good pair, we need to eliminate it. By including these results in the average, we might have distorted those values so that any choice based on the average would also be poor.

EURUSD-gold has the widest range of performance, from negative to very positive, and also has the highest average ratios. It also seems inconsistent because the rows for calculation periods 10 and 7 are much lower (but still profitable) than most of the others. Choosing the momentum period 5 and exit 10 from the lower section seems the safest.

That wasn't too difficult, but it was all hindsight. We picked the same parameters for the two pairs that seem to work. The c.u.mulative profits for the period from 2007 through 2009 are plotted in Figure 4.9. The EURUSD-gold pair is consistently profitable; the crude-EURUSD pair has a fast, steady run-up, followed by a period sideways, before volatility and profitability returns.

FIGURE 4.9 c.u.mulative PL for the three inflation pairs, crude-euro, crude-gold, and euro-gold, from 2007.

The answer to our first question is yes, we could have made money during the period when these markets were considered an inflation hedge by the public. The second question is more difficult.

Could We Have Known When to Start Trading These Pairs?

We know that the subprime crisis caused unprecedented volatility. We can measure that volatility using the formula most often used by the financial industry: In order to recognize a change in volatility, we calculate a rolling standard deviation using the past 10 days. We've discussed before that using only 10 days and annualizing the value will occasionally cause annualized volatility to be greater than 100%. Because our interest is the relative difference in volatility, that won't cause any problems.

In Figure 4.10, the annualized volatility of the three futures markets is seen to change in late 2007, synonymous with the spike in crude oil but a full nine months before the subprime crisis. Had we started trading these pairs at that time, we would have captured the big moves in all three markets. By September 2009, the volatility of the EURUSD and gold had dropped back to previous levels. Crude followed in December 2009. Had we stopped trading when the volatility returned to normal, we might have netted profits in all three pairs.

FIGURE 4.10 Annualized price volatility of the euro, crude oil, and gold shows spikes during the subprime crisis.

Using volatility seems to be a simple way to qualify trading in inflation pairs. The volatile periods generate higher correlation between the markets and also increase liquidity. More investors who would not normally trade these markets are attracted to them during periods of high volatility. They are also pushed along by the commentators on the financial news networks, who manage to convince the public of correlations (and so-called contagion) before they occur.

How Do We Decide, in Advance, Which Parameters to Use?

The most difficult question of the three is how to determine, in advance, which parameters to trade. The easiest answer is to find another period in the past when inflation and volatility were in the news and test that data. Certainly, the early 1980s would qualify, when gold rallied to $800/ounce and interest rates topped at 21%. But we would have to trade the Deutsche mark instead of the euro, which didn't exist at that time, and oil wasn't a factor because the price was so low between the oil shortage in the mid-1970s and the Iraq-Iran War in the mid-1980s that cost wasn't significant. Gold has also been an unreliable gauge of inflation. After the famous run to $800 in 1980, it came steadily down until it reached its low in September 2000. Buying gold at any price during those 20 years would have produced only steady losses.

We could not have found a similar situation using the same three markets, but there may have been other pairs that would show improved returns, and higher correlations, during volatile periods. If we look back at Chapter 3, the use of a low-volatility filter identifies similar situations, although much shorter time periods. It shows that pairs trading does best during periods of higher volatility.

For now, we'll be satisfied with using the past three years to identify the parameters, but if we can show that some other period had the same results, our confidence would increase tremendously. On the other hand, if we had no previous periods with the same profile, then we must base our decision of what parameters to use on the belief that higher volatility will generate the profits we need. There is often a point in developing a trading system when you must make a leap of faith after you've done as much work as possible. Tying performance to volatility seems to be a very small leap.

A Last Word about Inflation Pairs Trading during periods of high volatility can be very risky, but we expect pairs trading to profit during high volatility because prices move apart quickly and correct just as fast. During periods of low volatility, prices may not move enough to generate profits in excess of costs. Markets that have caught the interest of the general public can offer great opportunity, triggered by a clear increase in volatility.

Having decided which of these markets reflects the most public interest, experience seems to be the key. When we looked at the changing correlations in the energy markets, we found that the dramatic rise in crude prices corresponded to increased correlations between all energy futures. We could attach a fundamental reason for that change; however, the inflation pairs are much more deceptive. Figure 4.11 shows the rolling 20-day correlations during the same period as the annualized volatility in Figure 4.10. With volatility, we saw a clear increase at the end of 2007 a.s.sociated with profitable pairs trading. The correlations show that the crude-gold pair is generally more correlated and has moved to its strongest relations.h.i.+p at the end of 2009. But the crude-gold pair was the worst performer. There is a modest increase in the correlation of the EUR-gold pair from the beginning of 2008 until mid-2009, but afterward that pair is mostly negatively correlated, although by only a small amount. Yet EUR-gold was the most consistently profitable pair. If the correlations don't tell us anything, then we can only conclude that it's the money that moves the market. The pairs are profitable because the public believes these markets should react to inflation news, but their movements with regard to one another are unpredictable, just as price noise is unpredictable but profitable for a mean-reverting trader.

FIGURE 4.11 Rolling 20-day correlations of three inflation pairs. Crude-gold, which shows the highest correlation, had the worst performance.

EQUITY INDEX PAIRS.

We now come to the equity index markets, which are probably the most interesting for traders. During the past 10 years, these futures markets have seen a tremendous increase in activity from traders around the world. Although most of the European markets have been electronic since their inception, the U.S. markets have been slower to change, but the liquidity is now in the electronic contracts. Electronic trading has facilitated the ability to execute on the CME, EUREX, SIMEX, and nearly any other exchange in the world.

Many of the equity index markets can be traded as ETFs, but they are not as liquid or as efficient as futures markets. They do allow you to sell short with no restriction, and they do not need to be rolled when the contracts expire. The results shown in this section all use the smallest futures contracts traded, normally called minis. The commission costs for trading minis are much higher for the noncommercial investor than trading the original, larger contracts, but that's where all the liquidity is, so the increased commission cost may be offset by less slippage. The data start on November 21, 2005, when EUREX moved the close of the trading day to 22:00 European time (10 P.M. in New York) and ends on May 1, 2010. Given the windup needed to perform the calculations, this gives us nearly 4 1/2 years of performance for all markets. That can still generate a lot of trades because we hold positions for only a few days.

The markets used to form pairs will be the S&P, Nasdaq, Russell, Dow, EuroStoxx, DAX, CAC, and FTSE, a total of 8 markets and 28 pairs. Of these pairs, 6 are combinations of U.S. markets, 6 are European, and the remaining 16 are combinations of U.S. and European markets.

If you use the combined sessions for European markets, which means the original pit session followed by the evening session, the data will all end at nearly the same time. In the U.S., the evening session starts the new day for the electronic markets, which then ends at the close of the next pit session. In 2005, the European markets extended their hours to trade alongside the U.S. markets, so that they now close at 10 P.M., equivalent to 4 P.M. in New York. This facilitates pairs trading, even though volume on the European exchanges at the 10 P.M. close is considerably lighter than during European business hours. If Asian markets trade electronically 24 hours, it may be possible to include the Topix, Hang Seng, Nikkei, and others, but those markets won't be included here.

Again, we point out that if either leg of the pair does not trade because of a holiday, then no entries or exits are allowed, but daily profits and losses are calculated for the leg that is open. Futures trading is always marked to market each day. When we combine the results of the different pairs into the final portfolio, we must pay attention to any missing days, where there was no trading in one region but trading in the other. The results must be aligned by date.

We already know from previous discussions that shorter calculation periods are most likely to work because they capture more price noise, an advantage for mean-reversion strategies. In this case, we just test a small range of values to confirm that the same parameters work for the index markets as they did for others. We are hopeful that the concept is robust and will produce orderly results. There are only six tests, calculation periods of 4 and 5 days, and entry thresholds of 40, 50, and 60. A commission of 25 currency units (USD, EUR, or GBP) per round-turn per leg was charged to all trades, which is very conservative. Table 4.11 shows the results in terms of the information ratio.

TABLE 4.11 Average information ratio for 28 index pairs, from November 2005.

The results do not distinguish between good and bad pairs and are simply the average of all 28. Because of that, we consider these quite good and confirm our expectations that faster trading is better. The final decision will be whether the profits per contract are large enough to offer some comfortable cus.h.i.+on to absorb unexpected slippage above the $25 commission cost. We know from experience that the fastest parameters, a period of 4 and short entry threshold of 40, will have the smallest unit returns but the best ratio. We'll look at those results in more detail.

Table 4.12 shows the basic statistics for the fastest parameters, 4 40. Only 4 of the 28 pairs show net losses; three of those pairs are combinations of U.S. markets, and the other is NasdaqEuroStoxx. If we try to cla.s.sify the pairs into U.S. versus Europe and relate that grouping to the performance, we find some consistency. For example, the higher the correlation between the two legs, the lower the ratio and performance. If we sort Table 4.12 by the correlation column and then create a scatter plot of the correlations against ratios, we get the results in Table 4.13 and Figure 4.12.

TABLE 4.12 Results of all index pairs using a calculation period of 4 and entry threshold of 40.

Figure 4.12 is most descriptive. At the top left, the information ratio is greatest and the correlation smallest. At the bottom right is the opposite combination, high correlation and low performance. There seems to be a break in the chart at the horizontal line representing an information ratio of 1.0. Below that line, the values are spread out and less orderly; above the line, they form a clear pattern up and to the left. We can now look back at Table 4.13 and understand which pairs are most likely to succeed.

TABLE 4.13 Index pairs in Table 4.12, sorted by correlation, highest to lowest.

FIGURE 4.12 Scatter plot of index pairs showing the relations.h.i.+p between the correlation of the two legs and the ratio of the performance.

Starting from the top of Table 4.13, we can see that most of the combinations involve one U.S. index market and one European index market. The correlation of 0.431 between the Russell 2000 and the FTSE shows that these markets have only a modest relations.h.i.+p; that is, they move apart due to very different fundamentals but revert to the mean because all index markets react to global economic factors. Even more important than the global factors is the way traders buy and sell these markets, forcing them into similar patterns. While the economy of the U.K. may be perceived as quite different from that of the U.S., when the U.S. economic reports show unexpected strength or weakness, the FTSE reacts to that information. Similarly, there is a smaller but noticeable reaction in other countries if the Bank of England is the first to lower or raise interest rates after a period of stable rates.

Within Europe, the most profitable pairs are between the British FTSE and the German DAX or the EuroStoxx. These represent the widest difference in fundamentals within Europe, with the U.K. not part of the EU and seen as a weaker economy at the moment. But with most of the total trade occurring between these two regions, their economies are clearly linked, and their equity index markets must reflect that relations.h.i.+p.

The only pair of U.S. equity index markets to make the cut is the Russell-Nasdaq, showing the highest correlation, 0.933. All of the U.S. equity index pairs show very high correlations, making the unit returns necessarily small. At the bottom of the performance list are three U.S. pairs. Although none of the same companies are part of the S&P 500 and Russell 2000 (large cap and small cap), they move in the same way and provide no opportunity for profit.

FIGURE 4.13 Momentum values for the S&P and Dow minicontracts during 2008 and 2009 using a calculation period of 8. None of the momentum differences reaches 40.

A clear example is the S&P-DJIA pair. Of course, all 30 stocks in the DJIA are also in the S&P, and because the DJIA stocks have the largest cap, they represent a disproportionate part of the S&P. The correlation is shown as 0.949 in Table 4.13. Figure 4.13 shows the momentum difference, the basis for the pairs signals, based on a calculation period of 8 days. While the two individual momentum values range from 0 to 100, the momentum difference reaches near 40 only once since 2000 and peaks over 20 only a few times during the entire period. Table 4.12, which has the detail for the 4-day momentum, shows that there were only five trades over the past five years, and those did not generate enough profit to overcome costs.

FIGURE 4.14 (a) Sample PL for S&P pairs (4 40) with European equity index futures, converted to USD, and (b) sample NAVs for S&P pairs with U.S. equity index futures.

Pairs using the S&P futures contract are a typical example of results. In Figure 4.14, we see the NAVs of the seven pairs. The calculation of the NAVs will be given in the next section. For now, Figure 4.14a shows the best results are for the FTSE and CAC, with correlations against the S&P of 0.513 and 0.577, compared with the EuroStoxx and DAX with correlations of 0.823 and 0.791. However, all four results could be combined into a profitable portfolio. Note that the intervals with horizontal lines indicate there was no trading.

Figure 4.14b shows the results of the U.S. equity index pairs. The DJIA, the bottom line, shows long periods of no trading, confirming the high correlation exhibited in Figure 4.13. In the middle, Nasdaq has trades but can't overcome the commission cost, and at the top, the Russell shows positive returns, but all based on one distortion during September 2008. We would not want to base our expectations on having to repeat a 50% drop in the stock market.

Portfolio Diversification Combining performance into a portfolio shows the value of diversification. To make this a manageable example, we will use only the four S&P pairs with the European index futures, the CAC, DAX, EuroStoxx, and FTSE, shown in Figure 4.14a. We start with the daily profits and losses and combine them, first adjusting each to a target volatility and then adjusting the final portfolio to a target volatility. Each step is explained in the following.

Step 1. Align the Daily Profits and Losses Table 4.14 starts on the first day of available data. The first pairs trade is initiated on December 8, 2005, and doesn't show a profit or loss until the close of the next day. In the left panel, the daily profits and losses are aligned by date. When using U.S. and European pairs, there will be many days when one or the other market doesn't trade, but the alignment process affects the signal generation rather than the profits and losses shown in the table.

TABLE 4.14 Constructing a portfolio with an annualized volatility of 12% from daily profits and losses.

Step 2. Normalize the Volatility and Find the Investment Size To give each pair an equal chance to contribute to the portfolio profits, we need to equalize the volatility of each pair. First calculate the standard deviation (volatility) of the return streams (not the c.u.mulative profits) of each of the four pairs, and multiply each by the square root of 252 (the nominal number of business days in a year) to get the annualized volatility. Note that the profits and losses have first been converted to U.S. dollars using the daily spot exchange rates for the euro and sterling.

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