Part 13 (1/2)
FIGURE 7.2 Relations.h.i.+p between correlation and information ratio for index pairs using the momentum difference method in Chapter 4.
Figure 7.3 shows the same relations.h.i.+p using the stress indicator. While it is similar to the chart produced using the momentum difference (Figure 7.2), the correlations clearly cl.u.s.ter into two groups. We can see why this happens by looking at Table 7.3. All the pairs that have the CAC or FTSE as one leg post the strongest returns and have the lowest correlations. Knowing that, we can reason backward that the French and British economies are very different from the German economy, which dominates Europe and has greater impact on the EuroStoxx. This can be seen because the correlation between the EuroStoxx and DAX is .955, similar to the S&P and DJIA relations.h.i.+p. It is interesting that the top three performers are all European pairs, although in the top two groups, most of the pairs are combinations of U.S. and European markets.
FIGURE 7.3 Correlation versus information ratio for the index pairs using the stress indicator.
TABLE 7.3 Index pairs sorted by correlations, showing the information ratio.
A visual comparison of the c.u.mulative profits always adds to our information. In Figure 7.4, the results of the DAX-FTSE pair is shown for the original momentum difference method, the stress strategy, and the stress using the standard filter of 3.0. All three profit streams are similar. The filtered stress results, shown as the horizontal line from mid-2006 to mid-2007, also stops trading toward the end of 2009, where it is the middle horizontal line. The momentum difference method does not trade from mid-2009 until recently, when it starts again. These periods of low volatility show the main difference between the three methods. The unfiltered stress method will adjust to all volatility levels and continue to generate trades when it finds relative distortion, even if it cannot produce a profit after costs. By filtering those low-volatility periods, we get a more profitable strategy, but one that was out of the market recently as volatility fell to extreme lows.
FIGURE 7.4 Comparison of profits using the momentum difference, stress indicator, and the filtered stress indicator for the DAX-FTSE pair.
The momentum method is fine-tuned but requires that both legs move apart without the benefit of relative volatility. It filters trades much like the filtered stress method.
Of course, we all prefer that a trading system generate trades through all conditions and that those trades are profitable. However, that's unrealistic. Each trading method targets a particular type of trade and market condition. If it can do that consistently, even on a limited basis, it is a success. It is then up to us to find other markets, or other strategies, that work when these methods stand aside.
It is not necessary to go through the exercise of creating a portfolio of pairs from the best of the index group. The DAX-FTSE pair had a ratio of 3.0, about in the middle of the group of 13 best performers. If we created a portfolio from those pairs, it would look excellent, although the reliance on the CAC and FTSE would limit diversification, increase risk, and require that the portfolio is deleveraged to keep risk under control. It's a sacrifice we can all live with.
INTEREST RATE FUTURES.
Interest rate futures are among the most liquid markets, so any pairs trading that works will be an important a.s.set in a portfolio. As with the equity index pairs, we will compare the results of the momentum difference method in Chapter 4 with the stress strategy in Chapter 6.
Review of the Momentum Difference Method for Interest Rates For convenience, Table 7.4 is a recap of the momentum difference results for 14 combinations of the U.S. 30-year bond, 10- and 5-year notes, the Eurobund (maturity of about 10 years), Eurobobl (about 5 years), and the U.K. long gilt (about 8 years). Results are reasonably consistent across the selection of calculation periods and the one entry threshold of 50. However, for a period of 4.5 years, the number of trades, 22 to 31, is small and causes the annualized returns to be low.
TABLE 7.4 Original momentum difference average results for interest rate pairs, Chapter 4.
If we look at the results of the momentum difference method in more detail, we see that performance increases as the correlation between the two legs decreases. In Table 7.5, the most correlated markets-the U.S. interest rates-produced only 1 and 2 trades for the 10- and 5-year notes when matched with the 30-year bonds and 10-year notes. The 30 and 5 pair, which has the greatest difference in maturity, posted 10 trades and a good ratio.
TABLE 7.5 Results of the momentum difference for interest rate pairs are dependent on the correlation of the two legs.
As we look down the table at the declining correlations, we see that the biggest gains and the most trades come from pairs using the U.K. long gilt, a longer maturity that allows larger price fluctuations and a market that reflects an economy different from both the U.S. and European countries. The losing combinations all seem to include legs with the shortest maturity, the U.S. 5-year note and the Eurobobl. If we reason backward, one of our better abilities, those markets would have the lowest volatility. If the bobl is paired with the U.S. 30-year bond, the difference in maturity essentially puts them furthest apart on the yield curve and offers the most opportunity for profit, yet the information ratio was still low because the bobl leg would be generating only small returns per contract.
From these observations, we will select a smaller set of pairs for trading. We eliminate all pairs that use only U.S. markets and all pairs that use the Eurobobl-not because the methods pick the wrong entry points, but because, even with the best selection, the volatility of the pairs is not enough to generate profits. The final selection is 7 pairs out of the original 14 pairs. The averages shown in the following examples will include both all markets and the selected pairs. We want both those numbers to show improvement to be comfortable with the results and our conclusions. Table 7.6 shows the results of the selected pairs for the same momentum difference calculation criteria as Table 7.4. All the numbers are far better, although the number of trades is still fairly low, less than 10 per year for any pair. Trading less often should not affect the returns per contract but will lower both the absolute profits and the annualized rate of return. These returns of about 8.5% are not bad, and the few trades mean that the program is mostly out of the market and less exposed to price shocks and unexpected risk.
TABLE 7.6 Results of momentum difference method for seven selected pairs.
Results of the Stress Method Turning to the stress method, results show a very different profile from the momentum difference, even though the test criteria were comparable. Table 7.7 is compared with Table 7.4. The main differences are: The stress method had more than seven times the number of trades.TABLE 7.7 Results of the stress method for all 14 interest rate pairs.
The total stress profits were large in two of three cases.
The stress returns were lower.
The stress ratios were lower.
The stress returns per contract were too low for comfort.
In previous a.n.a.lysis, selecting trades using a volatility threshold has successfully raised these numbers to good levels. But before that step, Table 7.8 shows the selected set of seven pairs that did not include U.S. combinations and the Eurobobl. Clearly, the three criteria show greatly improved performance. If we compare these results with the same selection using the momentum difference, shown in Table 7.6, we get a very different opinion of the two methods. Now, the stress method still has many more trades and, at best, twice the profits, a higher annualized return, and a higher information ratio. The profits per contract are still smaller but better than before.
TABLE 7.8 Results of selected pairs for the stress method.
To allow readers to form their own judgment, the detailed summary for each pair, using the stress method with a 5-day calculation period and 95-5 entry thresholds, is shown in Table 7.9. These results include the U.S. interest rate pairs, as well as all the Eurobobl pairs.
TABLE 7.9 Summary of results using the stress method for a 5-day calculation period and entry thresholds of 95 and 5.
Volatility Filter In the past, we were able to use a volatility threshold filter to select trades that had, potentially, larger profits per contract. The threshold used for all markets was a factor of the average true range of price movement divided by the price, giving us a 3% threshold. The calculation period for the true range was always the same as the momentum calculation period.
When that same approach was applied to the interest rate futures, there were no trades! That is, the volatility of those markets is much less than 3%, so no trades qualified using that threshold. For example, a 3% move in 10-year Treasury notes, now trading at about 116 (this implies a yield of about 3.75%, where 100 is equivalent to 6%), would mean a move to either 119-16 or 112-16, a move so large that it is without precedent.
Instead of 3%, the threshold was lowered to 1%-still a large move, but more realistic. Using 1% generated from 3 to 4 times the number of trades in the momentum difference method and increased the returns per contract by up to 40% while lowering the ratios only slightly. A summary of results is shown in Table 7.10.
TABLE 7.10 Summary of stress method for interest rates using a volatility threshold of 1%.
Correlation of Momentum Difference and Stress Methods If we choose the 5-day calculation period for the momentum difference and the 5-day period (filtered) for the stress indicator, we can look at the c.u.mulative profits to see whether the pattern of returns is similar. Figure 7.5 shows that they are quite different, with the momentum difference returning infrequent but steady profits and the stress method showing a strong run of profits from mid-2008 through mid-2009, after a decline at the beginning of 2008. The correlation between the two profit series is .262, indicating that there is at best a weak relations.h.i.+p between the pattern of trading signals. It shows that a small change in technique, calculating the stochastic of the difference, materially changes the strategy. We could take advantage of that by trading signals from both methods. Figure 7.5 also shows the result of equally weighting the two methods. As we would expect, the result is better than either method, with a much smaller drawdown than the stress method and much greater profits than the momentum difference approach.
FIGURE 7.5 c.u.mulative profits for the U.S. 30-year bondEurobund pair using both the momentum difference and stress methods, plus the combined results of equal weighting.
A Portfolio of Interest Rate Pairs Although we created a portfolio of futures in Chapter 4, it seems instructive to go through that same process using the seven interest rate pairs. We first start with the daily profit and loss streams at the inception, November 28, 2005, for each of the seven pairs. We align the data by date because Europe and the U.S. do not always have the same holidays. Remember that the strategy does not trade either an entry or exit if one of the equity index markets is closed and just holds the positions until the next day on which both markets trade; however, there is a change in the returns based on the market that is open. If your data are forward filled, then a simple test of whether the open, high, low, and close today are identical to yesterday would be the same as recognizing a holiday.
Once the data have been prepared, you can acc.u.mulate the daily profits and losses into a net profit stream to get a visual understanding of the performance. The c.u.mulative profits, shown in Figure 7.6, will not be used in the calculations, but visual confirmation avoids simple errors. For example, we can see that the filtering of trades resulted in very little trading from the start of the data in 2005 through the middle of 2007. That was a period of very low volatility; therefore, we need to be aware that the 1% filter may keep us out of trading for long periods of time. Reducing the filter size would allow more trades but probably reduce the size of the profits per contract. Traders must decide what low threshold they can tolerate to increase activity.
FIGURE 7.6 c.u.mulative profits for seven interest-rate pairs filtered by 1% volatility.
Note also that the most recent data are also filtered, so that the beginning of 2010 may be inactive. This situation is not likely to continue once the central banks start raising rates, but continued concerns about economic recovery and the brewing debt crisis in some European countries might delay an increase in volatility. This is a normal trade-off in the decision process-more activity in exchange for lower unit returns.
It may be tempting to start the portfolio in mid-2007, when volatility and trading activity increased. That way, the annualized returns would be maximized and look better. But that is ex post selection, making the decision based on observing past data. In real trading, we will have periods when trades are filtered because of low volatility, and we can't erase those periods from our performance. It's not fair to do that now, so our risk and returns will include all data.