FIGURE 7.2 Relationship between correlation and information ratio for index pairs using the momentum difference method in Chapter 4.

Figure 7.3 shows the same relationship 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 relationship. 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 relationship 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.

TABLE 7.11 Annualized volatility of daily profits and total investment of interest-rate pairs.

THE PORTFOLIO SPREADSHEET.

We began with aligning the daily profit streams, a small part of the data shown in Table 7.11. These values have all been converted to USD at the time of the trade so that no currency conversion is needed during the portfolio construction process. At the bottom of each daily profit stream, columns BH, is the annualized volatility, calculated as for column B, the 30-year bond and the Eurobund. The result is $80,465. Doing the same for the other pairs and adding all seven gives a total investment of $396,332. This may not be the final investment size because there are other steps that increase or decrease leverage, but it is necessary to have an investment size to calculate percentage returns. Alternatives would be to arbitrarily pick the investment size or to use the investment amount that you have in mind, and then the final numbers will adjust to that investment and show the returns at your target volatility. That will be seen as we move forward.

Using the investment of $396,332, we simply divide the daily profits and losses by the investment size and post them in columns IO, shown in Table 7.12. Note that each return is divided by the total investment, not by the investment needed for that one pair.

TABLE 7.12 Annualized volatility of daily profits and total investment of interest-rate pairs.

The portfolio returns in column P are now the sum of the seven returns in columns IO. We again calculate the annualized volatility at the bottom of column P and find that it is 63.9%, far above our target volatility of 12%. To reduce that volatility, we need to deleverage by decreasing the position size while keeping the investment the same, or by increasing the investment size while trading the same number of positions. The factor at the bottom of column P tells us that we would need to reduce our returns, and our position size, to 18.8% of the current size to bring volatility down to 12%. That means, if we were originally trading 10 contracts, we need to trade only 2. That's not good because we won't be able to balance the risk of both legs with only 2 contracts, so the alternative of increasing the investment size is better. Another alternative is to compromise and reduce the position size by half, to 5 contracts, while increasing the investment size by a factor of 2.5 instead of about 5. You must reduce position size, increase the investment, or some combination in order to target the right volatility.

The final step is to create the portfolio NAVs based on the returns in column P and the volatility adjustment factor (VAF) of 0.188. Starting with a NAV of 100 in column Q, the next entry is The final NAV is 197.28 (there was a small amount of trading done after March 3, 2010, resulting in a profit). That makes the annualized rate of return 16.6% and, based on a target volatility of 12%, the information ratio is 1.37, a good number. If you remember, this includes the inactive period from November 2005 through mid-2007. If you lower the target volatility, the ratio remains the same, but the rate of return drops proportionally.

SUMMARY OF PAIRS TRADING.

Pairs trading has been covered for stocks, futures, and a combination of both that we called cross-market trading. Overall, it has lived up to expectations; that is, it found the right entry and exit points and often returned a profit. It failed only when the volatility of the markets was too low to overcome costs. That could be avoided by waiting patiently for volatility to increase, but traders are not known for their patience. They want to trade all the time.

Futures markets are more interesting than stocks because of both higher volatility and the ability to vary the leverage. Futures trading inherently allows high leverage, and you can tailor it to your personal level of risk. Globalization has been a benefit to the spread trader because U.S. and European markets are now open at the same time, although volume is always highest during the business day in the location of the market being traded, so European volume drops off after 10 A.M. in New York (4 P.M. in Frankfurt). Still, there is enough volume to trade ahead of the U.S. close in both zones. Even now, some Asian markets can be traded 24 hours on an electronic platform, and arbitrage should be expected to increase. Volume is understandably light in Singapore at noon in New York because it is 12 hours earlier. Most people trading in Singapore at that hour are U.S. customers.

Opportunities for trading pairs surface and disappear, but some are always there at any one time. It's not as profitable as it was 10 or 20 years ago, but then we couldn't enter orders electronically, and fills were often disputed. As we've made it easier to trade, the compet.i.tion has also increased. But that is just a challenge to stay aware and become more technical about how you handle the opportunities and the decisions. Pairs trading is fundamentally sound; it is the basis for the extraordinarily profitable high-frequency trade. Unless all the stocks and futures markets merge into one company, these opportunities are not likely to go away.

Chapter 8.

Traditional Market-Neutral Trading Any strategy that balances longs against shorts can be called market neutral. It really means directional neutral, where you are not exposed to the risk of outright price direction. You will be able to avoid being long while the stock price is plummeting. It's well known that during uncertain economic times, investors shift to larger, more secure companies, and during boom times, they throw money at small-cap stocks. If that relationship holds true and the economy is robust, you should be able to generate a profit by buying the Russell 2000 and selling the Dow or S&P. That profit would be the result of the relative gain of the Russell over the Dow, regardless of whether the Russell and the Dow went up or down.

One of the greatest advantages of trading a market-neutral strategy is that it is, for the most part, immune to price shocks. If you were long the Russell and short the S&P on September 11, 2001 (volatility adjusted), the losses in the long Russell position would have been offset by the short profits in the S&P. You wouldn't have worried about how much you would lose over the days when the New York Stock Exchange (and most of the other exchanges) was closed. For example, when the markets reopened on September 17, 2001, the S&P had dropped 4.98%, the Nasdaq 5.80%, and the Russell 6.51%. At the lows on September 21, the S&P was off 15.8%, Nasdaq 14.0%, and the Russell 12.3%. Without considering adjustments needed to correct differences in volatility, the exposure would be small. Even leveraged up in futures, typically four times, the exposure would have been smaller than the declines in outright long positions.

In this book, we have focused primarily on pairs trading. The advantages are that you can choose a small set of markets and a modest investment. In addition, the calculations involve only two stocks or futures markets at a time and can be done on a spreadsheet. However, traditional market-neutral programs are much broader, serve a somewhat different purpose, and are not correlated to pairs trading, even when both methods are active in the same sector.

This chapter will use similarly related stocks but apply a more general hedging technique. We can then take this method and apply it to stocks that are not closely related, and then to futures markets. The technique used for both stocks and futures will be identical.

HOME BUILDERS.

To keep the example of a market-neutral trading program as simple as possible, and still learn as much as we can, the first choice will be the five stocks in the home builders group that were used for the pairs trading example: Hovnanian (HOV), KB Homes (KBH), Lennar (LEN), Pulte (PHM), and Toll Brothers (TOL). The correlations between these stocks can be found in Table 8.1. With pairs trading, we were able to return better than $0.14 per trade; therefore, we have a benchmark to judge the success of the market-neutral approach.

TABLE 8.1 Cross-correlations for home builders, 10 years ending March 2009.

TREND OR MEAN REVERSION?.

If you have a set of stocks and plan to go long some of them and short others, it is necessary to decide whether you take a trending position or a mean-reverting one. Do you believe that those stocks that are stronger will stay stronger and those that are weaker will stay weaker than others in the same sector? If we think back at the major trends, it is easy to remember Enron. Once it started down it kept accelerating. More recently, the financial stocks suffered the fastest, steadiest decline on record. We would expect to have produced remarkable profits by selling the financials short and buying everything else (well, maybe not housing) during the second half of 2008.

If you don't believe that trends persist, then you may believe in the Dogs of the Dow, a theory advocating that, with qualifications, the weaker companies in the Dow Jones Industrial Average will cycle to become the stronger ones in the near future. After all, they are all solid inst.i.tutions and just need to get past some short-term issues. At least that was true before the summer of 2008.

How can we decide whether trend or mean reversion is the best approach? For pairs trading, we automatically chose mean reversion, buying when the spread was oversold and selling when it was overbought. But there is always a choice. In Chapter 2, we discussed the level of noise in most markets and how we might use that information. Of all markets studied, the index markets were cl.u.s.tered together and represented the noisiest sector. We also know that applying mean reversion to pairs trading was profitable. The conclusion that was drawn from this was that mean reversion was preferred for stocks and that shorter calculation periods would emphasize the noise. One caveat, however, is that the spread between two noisy stocks might not be as noisy. We'll need to keep that in mind. In Chapter 5, we had a chance to apply a trend to some pairs, but we tilted the game to our advantage by using longer calculation periods, which emphasized the trending nature of some stocks and futures markets.

Another way of deciding whether a trend or mean-reversion method is best would be to find the profitability of a simple trend-following system on these stocks. Although the market-neutral strategy profits from the spread between two stocks, knowing whether the individual stocks trend might make the decision simpler. We'll save that for the next example, applying this method to all Dow components.

BASIC MARKET-NEUTRAL CONCEPT.

The first step is to find a measurement that allows us to compare each of the five stocks in the home-building group to decide which are stronger or weaker. There are many ways to do this, and they may all prove to generate similar results. One approach would be to use the same moving average on all stocks, say 20 days, and then compare the 1-day change in the moving average differences. That is done by looking at the change in the moving average from the previous day, MAt MAt 1, or over n days, MAt MAt n. Those stocks with the largest positive change are the strongest; those with the largest negative change are the weakest.

Instead of a moving average, we'll use the slope of a linear regression line. The regression line, the tool of choice for the econometrician, is simply a straight line drawn through the past n daily prices. The slope is the angle of increase or decrease of that line, and it's the same at any point on the line; therefore, if St is a point on the line corresponding to day t, and St 1 is the previous point on day t 1, then St St 1 is the slope of that line.

Putting All Markets into the Same Units Comparing the slopes of the lines through price points won't work unless all of the stocks are expressed in the same units, in this case percent. For example, if a stock priced at $10 went up $1 each day, the slope would be $1. If another stock priced at $5 went up $0.50 each day, the slope would be $0.50. If we compared those two stocks without converting to percent, then the higher-priced stock would have the biggest slope and appear to be stronger, when they are actually both rising at the same percentage rate.

Prices are normalized by indexing, which is the same as converting the raw price changes to percentages. Starting each series at 100, the next index value Xt is: where Pt is the current price and Pt 1 is the previous price. When the linear regression slope is calculated on each of the five converted home builders, they can be readily compared. Remember that the indexed series are needed only for finding the relative strength of the sector components. When we calculate profits and losses, the original prices are used.

Relative Strength and Weakness Having chosen to use the linear regression slope to compare the relative strength of all stocks in a sector, we first converted each of the stock series to percentage returns using indexing. The slope is now easily found using the spreadsheet function slope(y, x), where y is the column of dependent variables and the x is the column of independent variables.

Finding the slope of a price series requires only the prices. The independent variable, x, is actually the time. We would use the date for the xs except that stocks don't trade on weekends, and there would be gaps every five days. Instead, we just use the integer sequence 1, 2, 3, ... , n for as many days as needed in the calculation. For this strategy, we'll use n = 10 and apply it to the cash S&P 500 index, or SPX. Table 8.2 shows an example of the slope calculated using Excel. The first valid calculation is in cell D10 and is =slope(B1:B10,C1:C10), giving the result 4.60. That means the SPX is declining at an average rate of 4.60 points per day over the past 10 days. Over the next four days, the index drops even faster, as seen in column D.

TABLE 8.2 Example of a 10-day linear regression slope calculation applied to SPX.

The slope can be programmed directly using the formula: where x is an integer sequence (as in column C) and y is the price (column B). The term x-bar () is the average of the x.

Ranking and Choosing Which Markets to Trade It's time to do some of the calculations and be more specific about the trading rules. Table 8.3 shows the first day of calculations beginning January 10, 2000. The date and the stock symbols are shown in row 1. The next rows are: Price, the current price of the stock. TABLE 8.3 The five home builders showing the calculations needed on day 1 of the first two days.

Xprice, the indexed price.

Slope, the value found by using a linear regression.