Part 4 (1/2)

Filtered trades with low volatility to increase profits per share but also reduced the number of trades.

Removed trades in which the volatility of the two legs were significantly different, arguing that such a trade presented unexpected and unreasonable risks.

We believe that low volatility is a problem, but filtering the volatility removes too many trades. These setbacks are part of the reality of system development. At each step, there is another problem to solve, but continuing with specific solutions to each problem will eventually result in overfitting the data.

The market doesn't hand profits to you. There are some alternative ways of identifying entry and exit points, but none of them will create larger profits from low volatility. Unless we choose to believe that the two pairs, LCC-CAL and LCC-AMR, will perform in the future as they have in the past, and better than the other airline pairs, we need to refocus on sectors with greater volatility. At the moment, banking is very active, but share prices range from exceptionally low for those banks that were hurt in the subprime crisis (Bank of America) to unusually high (Goldman Sachs and J. P. Morgan & Company) if they were perceived as safe. That might produce good profits, but the shares needed to equalize the volatility of these companies will be highly skewed and could lead to unexpected, unpleasant consequences. Volatility adjusting might not be enough to control risk.

Instead, we can turn to the housing construction sector. Although it has been at the center of the financial crisis, companies have not been bailed out or nationalized, and stock prices continue to respond to market forces, but with greater volatility.

HOME BUILDERS.

Five of the biggest home builders are Lennar (LEN), Pulte Homes (PHM), KB Homes (KBH), Toll Brothers (TOL), and Hovnanian (HOV). The first three are part of the S&P 500. Their stock performance is an excellent view into the recent mortgage crisis and economic recession. Five companies give us 10 pairs, with the cross-correlations shown in Table 3.14. The average correlation is .58, which is nearly the same as the airlines, but there were fewer airline stocks and the correlations were as low as .39. Figure 3.11 shows the five home builders' stocks together. The run-up in prices parallels the real estate market, peaking in 2006. From the chart, these five stocks seem to be performing the same way, so there is no reason that we shouldn't use all combinations of them as pairs.

TABLE 3.14 Cross-correlations for home builders, 10 years ending March 2009.

FIGURE 3.11 Prices of five home builder stocks. All five react in a similar manner to the economic changes.

Testing the Home Builders.

We follow exactly the same process to test the home builders as we did for the airlines. From five related stocks, we get 10 pairs. Each of these pairs is tested by calculating the individual stochastic values, then taking the difference in the two stochastic values (SD, the stochastic difference). A new trade is entered when the difference exceeds our entry threshold, which, from our previous experience with airlines, should be between 60 and 40 for shorting the pair, and 40 to 60 for buying the pair. Shorting the pair means entering a short sale in the first leg of the pair and buying the second leg, while buying the pair means buying the first leg and selling short the second leg. The stochastic difference is the stochastic of the first minus the stochastic of the second. No costs were considered in the process, but results show the profits per share, net of both legs, so that it should be simple to reduce those results by your expected costs. Table 3.15 shows the results of tests covering a range of calculation periods and a range of entry criteria.

TABLE 3.15 Average results of 10 home builders varying the momentum calculation periods from 3 to 14 days and the entry criteria from stochastic values of 40 to 60.

As with the airlines, tests cover the most recent 10 years. In this case, all the stocks traded for the full period. Our first concern is the information ratio (annualized return divided by annualized volatility) resulting from trading each pair. We want an average ratio greater than 0.25; otherwise, we know that the final performance will be very erratic, with large drawdowns.

Table 3.15 shows the most important statistics for the 10 pairs: LEN-PHM.

LEN-KBH.

LEN-TOL.

LEN-HOV.

PHM-KBH.

PHM-TOL.

PHM-HOV.

KBH-TOL.

KBH-HOV.

TOL-HOV.

Columns 1 and 2 show the parameters. Column 1 is the momentum calculation period, ranging from 14 down to 3. The value 3 was included to show at what point the pattern fails. We must recognize that a stochastic calculation based on three days will jump from zero to 100 nearly every other day, making the results erratic. Column 2 shows the entry threshold of 60. Part a of Table 3.15 shows the results for an entry threshold of 60, and Tables 3.15b and 3.15c show the results of entry thresholds of 50 and 40, respectively.

The statistics shown in Table 3.15 are the number of trades, the annualized rate of return (AROR), the profits per trade in dollars, and the information ratio. We expected the relations.h.i.+p between the parameters to change in the following ways: As the calculation period decreases from 14 to 3, there would be more trades but the profits per share would get smaller. The less time you hold a trade, the less opportunity there is for gain. Similarly, we would expect the AROR to decline, but we cannot forecast the information ratio because both returns and risk will drop, but we don't know which will drop faster.

As the entry threshold decreases from 60 to 40, we will also get more trades, but we should see a smaller return per share and more risk. When a mean-reverting trade is entered sooner, we can expect prices to move against us both in magnitude and time. Both of these will affect the ratio. If there are many more opportunities at the entry thresholds of 40, those good results might offset the fewer trades in which the prices continued to diverge, but we will also not be able to tell the extent of that in advance.

If we also change the exit threshold, then, as it moves closer to the entry level (for example, entering a short at 60 and exiting at 0, 10, or 20), the size of the profits will decrease, the number of profitable trades should increase, and the total number of trades will increase. If we exit a short sale at 20, not having to wait for zero, and prices reverse to the upside, we would get another short sale that we would have missed had we needed to wait for zero to exit.

Our goal is to have very continuous results; that is, our statistics should move smoothly in one direction as the test parameter values change. We also need profits per trade that are large enough to net a profit after costs. Finally, we want enough trades to make it worthwhile to use this strategy, although that should be reflected in the rate of return. Our confidence in the results also deteriorates if there are too few trades.

It is easier to see the results as a chart. Starting with Figure 3.12, we can see that the number of trades increases as the momentum calculation period decreases. It does this uniformly for the three entry thresholds, but the number of trades also increases as the entry threshold gets smaller. The fastest trading combination, an entry threshold of 40 with calculation periods of 5 or less, generated an average of more than 100 trades in 10 years. Although that's only 10 per year for each of the 10 pairs, it is enough to give us confidence in the method.

FIGURE 3.12 Home builders, comparison of the number of trades.

The profits per share may be the most important statistic because, above all, it tells us whether we can net a profit after costs. In these tests, shown in Figure 3.13, we see that the pattern is good, but the highest average profit per share falls below 8 cents. That may be enough for a professional trader, but we would like it to be higher. When we consider that the entry threshold of 40 generated the most trades, we see that it cleared only 4 cents per share using the faster calculation periods. Because there were more trades, we can try to be selective by using a volatility filter.

FIGURE 3.13 Home builders, average profits per share.

The last statistic, the information ratio, is also important because it gives you an idea of how much risk you will take to get these returns. Figure 3.14 shows that, for the most part, the ratio continues to increase as the calculation period declines. We can explain this in hindsight by recognizing that pairs trading, like other mean-reversion strategies, flourishes in environments of market noise. Chapter 2 pointed out that a closer look at prices-that is, looking at hourly instead of daily data-accentuated the noise. Also, using shorter calculation periods focuses on more noise and less trend. The trend is emphasized by using longer calculation periods and less frequent data (weekly instead of daily). Figure 3.14 shows that ratios were above 0.25, our objective, for all calculation periods from 6 and lower.

FIGURE 3.14 Home builders, average information ratio.

The good news is that the three figures show consistency. As the calculation period declined, results changed in a very orderly fas.h.i.+on. Even better, they were nearly all profitable. If you remember, an important criterion of robustness is that a large number of combinations of parameters should produce profitable returns, given a reasonable test range. Our only problem, which could be insurmountable, is that we want larger profits per share.

Selecting the Threshold Levels.

The best choice will be some combination of the number of trades, profits per trade, and information ratio. We will use the average values of all pairs, even though looking at the detail of each pair would give us more information. We think that using the averages is an attempt to avoid unnecessary overfitting. However, the shape and consistency of the statistics have convinced us that this is a sound approach, so choosing any set of parameters, or more than one set, should be safe.

If we think back to the airline tests, we also saw that the performance ranges were very similar. For airlines, there were fewer pairs, so the results might be less consistent.

Because we plan to test a low-volatility filter that may remove up to 25% of the trades, we will choose the calculation period of 4 with an entry threshold of 40, one of the fastest trading combinations. We can now look at the detail of each pair, shown in Table 3.16. All but one pair, Pulte-KB Homes, was profitable, and all were reasonably consistent. Only one pair, Lennar-KB Homes, showed a return of greater than 10 cents per share, and all had at least 100 trades. The consistency, which is good, removes the temptation of discarding one or two pairs that performed badly or selecting a few that had large profits per share. An average ratio of 0.423 is very good if only we can increase the profits per share.

TABLE 3.16 Results of home builder pairs for momentum 4 and entry threshold 40.

Low-Volatility Filter for Home Builders.

As with the airlines, we can test the low-volatility filter. If results improve, it will confirm the method that we were unable confirm with fewer airline pairs. Using the same pairs and parameters shown in Table 3.17, we applied the low-volatility filter with values ranging from 0% to 120% and got the results in Table 3.17. Of course, 120% would be impossible except that we're using only four days to project an entire year of volatility, so an unusually volatile interval will produce a very large annualized volatility.

TABLE 3.17 Home builder pairs using momentum 4, entry 40, and a low-volatility filter.

It is easier to see the results of Table 3.17 in Figures 3.15 and 3.16. The first shows how the number of trades drops and the profits per trade increase as we filter out more low-volatility trades. However, the highest per share returns average only about 11 cents, while the number of trades drops to about 20 for each pair over 10 years, two per year. If we can accept 7 cents per trade, then we could double the number of trades. That's still not much.

FIGURE 3.15 The low-volatility filter for home builders shows a steady drop in trades and a corresponding increase in per share returns as more trades are filtered out.

FIGURE 3.16 The low-volatility filter for home builders shows a parallel decline in both returns and the information ratio as more trades are removed.

Figure 3.16 shows a parallel decline in both the annualized rate of return and the information ratio as trades are removed using the low-volatility filter. We can explain that because fewer trades spread over a longer time period will always reduce the rate of return. Similarly, if the risk of the individual trades remained the same but the annualized return was lower, then the information ratio would decline. Those results show that there is less to earn, but the most important statistics are the profits per share and the number of trades, both of which are too low to be very interesting.

At this point, we've gained confidence in the method but still need to find pairs with more volatility or those that allow some form of leveraging.