In addition to executing the ETF instead of the short leg of the pair, we need to consider whether the position size of the short leg should change. The position size was intended to equalize the risk of both legs and was calculated based on the average true range of each stock price over the recent past. If we subst.i.tute an ETF with significantly different volatility, then the long and short positions will not have offsetting risk. Then it seems logical that we would recalculate the position size using the volatility of the ETF.

In many cases, only the closing prices are available for the ETFs. If the high, low, and close are available, then the volatility calculation would be performed in exactly the same way as the underlying stocks, using the average true range. However, if only the closing price is available, as it was for our data, then the average true range becomes the average of the close-to-close differences, which would be a much lower volatility value. With only the closing prices, we went ahead and tested the use of both ETFs as a subst.i.tute for short sales. The results, using the standard values of a four-day momentum and 40 entry threshold, are shown in Table 3.24.

TABLE 3.24 Comparison of ETF shorts using ETF volatility to determine position size.

Although the chart shows that ITB and XHB are very similar, and the correlation of their returns is .957, the effect of subst.i.tuting them for the short sales yields very different results. The ITB ETF performs better than the benchmark case, and the XHB results are much worse. Profits per share increase from 1.7 cents to 4.0 cents, and the ratio gains from 0.131 to 0.149. At the same time, using XHB turns those profits into net losses. When we look at the individual pairs, we see that the ratio for KBH-TOL jumped from 0.589 to +0.919 using ITB but was little changed with XHB, and both ETFs performed very badly when used for LEN-HOV. In fact, ITB has a very different performance overall.

We can conclude that using an ETF as a subst.i.tute for short sales can work, but each ETF needs to be tested because just looking at the chart doesn't tell us enough.

PORTFOLIO OF HOME BUILDER PAIRS.

The final step in creating performance expectations is to build a portfolio from the results of the 10 pairs. This will show our antic.i.p.ated returns and the risk a.s.sociated with those returns. We expect that the risk will be reduced due to diversification, but we also know that these stocks are highly correlated, and the reduction might be small.

Because the construction of a portfolio is important, we will go through the process in six steps. The values for the pair LEN-TOL will be used as an example. The first steps can also be followed in Table 3.25.

1. Create a series of daily profits or losses for each pair of stocks. The net profit for the pairs is the sum of the gain or loss in one stock (the number of shares held on day t times the change in price from day t through day t + 1) plus the gain or loss in the other stock. If you are using capped results, then the daily profits and losses reflect the net return after capping the position size. The capped results are shown in the second column (B) of Table 3.25.

TABLE 3.25 Construction of NAVs from daily profits and losses for LEN-TOL.

2. Find the annualized volatility of the daily profits and losses in (1). For this pair, first create a column with the daily differences in the PL (column 3). Then find the standard deviation of those differences, 18.03. To annualize, multiply that value by to get 286.21. Note that this annualized change seems lower than expected because there were many days with no trading, giving zero returns on those days.

3. Find the investment size needed to trade this series at your target volatility. If the target is 12%, divide the annualized standard deviation in (2) by 0.12. The capped series yields an investment of $2,385. The capped series actually has lower volatility and a lower investment than the noncapped returns. This lower investment will translate into an increase in portfolio returns. When using futures in the next chapter, we will be able to adjust the leverage freely while keeping a close eye on risk.

4. Create the volatility-adjusted returns in column 4. Divide each PL difference (column 3) by the investment size calculated in step 3.

5. At this point, we have volatility-adjusted each pair to the target volatility of 12% using the capped returns, so that each PL series has an equal risk. We now create a NAV series (column 5) from the capped PL for each of the pairs. Starting with the value 100, we multiply the previous day's NAV by the value 1 + Returns for the current day. For example, on January 13, the strategy had a positive return of 0.00461, or about 46 basis points. Up to this point, there were no trades. We get the new NAV value as follows: On the next day there was another gain of 0.00587. The new NAV would be: 6. Repeat steps 1 through 5 for each pair of stocks.

When all the changes are processed, the final NAV for the pair LEN-TOL is 343.56. Because this is a compounded rate of return, results will sometimes increase faster or slower than the simple profit and loss returns, as seen in Figure 3.20, which shows the NAVs of both the noncapped and capped results.

FIGURE 3.20 LEN-TOL comparison of capped PL and capped NAV.

FIGURE 3.21 All 10 capped NAV streams from home builder pairs.

Putting the Portfolio Together When the individual pairs results have been converted from PL to NAVs, we get the NAV streams shown in Figure 3.21. All show good returns at the same target volatility of 12%. The final step in creating a portfolio is to begin with the daily returns for each pair, shown in column 4 of Table 3.25. Our portfolio will equally weight the returns for each pair because we believe that all have the same chance of being profitable in the future. You may know that modern portfolio theory states that a portfolio should be maximized using the information ratio, the annualized returns divided by the annualized risk. Then those pairs that have a better payoff (higher return for the same risk) should be given more of the investment, which is the same as giving them a larger weight or larger allocation.

Although modern portfolio theory was religiously accepted when it was first proposed by Markowitz, the years have tempered enthusiasm for it. No one has actually proved that it has predictive ability, only that the optimized returns are better than any other combination. We shouldn't be surprised that optimized returns are better.

We prefer to a.s.sume that we don't know which of the 10 pairs will give the best returns next year. They all seem good, and the economy, as well as individual corporate management, always seems to surprise us. The best company this year can be out of business next year. Had we heavily favored Enron in an energy portfolio, we would have been both disappointed and broke.

This portfolio will equally weight all pairs. This approach is called removing returns from the picture. It's simpler than portfolio optimization, we can do it on a spreadsheet, and it a.s.sumes less. Because of that, we also believe that the results, or expectations, are more realistic. The steps are simple: If you don't already have the daily returns for each series, you can start with the NAVs and calculate the daily returns, r, as For each day, average the returns of all 10 pairs. This is the same as equally weighting the results. We'll call the average daily return of the portfolio of 10 pairs R. Note that you will not equally weight the pairs if the liquidity of one or more stocks is restrictive.

Create the portfolio NAVs from the average returns using the same formula that was given in step 5 when creating the individual NAVs. The final portfolio NAV using capped returns is 327.77 and using noncapped is 173.36, both shown in Figure 3.22.FIGURE 3.22 Final portfolio NAVs for home builders using noncapped and capped returns.

Find the annualized rate of return The ending capped NAV was 327.77, and the total number of days was 2,530, then The number of years is a decimal number, the result of dividing the total number of trading days by the number of trading days in a year (typically 252). The annualized return for this portfolio is 12.55%, without commission costs or other fees.

The information ratio, the final measurement of return for risk, is A ratio of 2.34 is comfortably high and likely to remain above 1.5 even after costs are deducted. Any value over 1.0 means that you are getting more return for each unit of risk. The information ratio for a pa.s.sive investment in the S&P index may be as low as 0.4 over a long period of time, and even lower in recent years that include the 2008 decline.

The final NAV stream for the portfolio of capped pairs is smoother than the individual streams because of the unexpected diversification gained from the different pairs and the final annualized volatility of only 5.35%, down from 12%. Unless you choose one of the available leverage options, such as financing part of the position with borrowed funds, you can't boost the returns because the capped result is already using all of the investment.

EXECUTION AND THE PART-TIME TRADER.

Success of a short-term trading program depends on the timely execution of orders. Even though this program uses only the closing prices to generate signals, it's not likely to be successful if calculations were done after the market closed, then entered the next morning on the open. An extreme price on the close is very likely to have corrected by the next opening. It might work if executions were done in the aftermarket on the same day, provided trading was done in small numbers. However, we could be pleasantly surprised.

The most likely way of executing this program is to enter prices shortly before the close, calculate the new positions, and enter those orders for execution on the close or as soon as possible. As long as the pair satisfied the entry threshold, you should have a good trade. If there are differences between the price entered and the final closing price that would have affected the position size, adjustments can be made after the close; however, that may be unnecessary.

You may also want to choose a different time of day to trade, preferably a few minutes after a key economic report, such as a Federal Reserve Open Market Committee (FOMC) meeting. Statements of interest rate changes and policy are released at 2:15 P.M. on the second day of the meeting, usually a Tuesday. Capturing prices and trading somewhere between 2:20 and 2:30 P.M. is likely to take advantage of price distortions during a short but volatile reaction. Similarly, you might want to trade after the official close on a day when the first chip manufacturer announces quarterly earnings. There is a 15 minute window where you might find greater price distortions between similar companies.

Pairs trading does not require that you enter a trade on the close. The method holds up, or might even be better, if you can take advantage of obvious shifts in prices. Exiting from a trade, however, is still best on the close.

STOP-LOSSES.

Many traders are concerned about unexpected price moves causing large losses, and they often try to solve this problem using a stop-loss. Normally, you would use a stop-loss for a trend trade or fundamental position, one where you are net long or short. For a pairs trade, you have equal, balanced positions, long and short, in related stocks. Any price shock will affect both stocks and should cause offsetting profits and losses unless that shock was related specifically to only one of the stocks in the pair. Whether one stock of the pair moves more or less than the other is arbitrary.

But what if the pair continues to post a loss and that loss gets larger? How do you deal with controlling the maximum loss? It's not possible to handle this with a stop-loss because both stock prices are moving, and the loss is relative to the difference between the two. You could monitor the profits or losses and close out the trade if the losses persisted. But the nature of the method is that the stochastic indicator adjusts to higher or lower prices and establishes a new norm. Then a smaller relative price change will trigger an exit.

Using stops changes the performance profile of a system. Pairs trading has a high probability of a profit, and there are many smaller profits and a few larger losses. If you use a stop, then there will be more losing trades, and in some cases you will have captured a loss when the trade would have eventually produced a profit. The balance of the system will be altered, and there is no a.s.surance that the final result will be profitable.

TRADING INTRADAY.

Our application has used closing price data; however, intraday price changes can generate many more trades, and taken to an extreme, it is similar to the high-frequency trading done by the big investment banks. Those trades are entered and exited in milliseconds and costs are negligible, but the principles are the same. In the previous section, "Execution and the Part-Time Trader," we discussed that trading after economic or corporate announcements could be an advantage.

A compromise between daily and milliseconds is hourly data. The strategy posts prices each hour and looks for the stochastic difference to generate a trade. Of course, profits per trade would be smaller because the holding period would also be shorter, so your costs would be the limiting factor rather than the opportunities.

KEY POINTS TO REMEMBER.

This chapter was as much about the process as about the strategy. It was intended to be a step-by-step explanation of the process needed to take an idea and create a trading strategy.

We began with what we believe to be a sound premise, that pairs trading is based on the fundamental concept that two stocks in the same sector, affected by the same macrofundamentals, will perform similarly. Because of that, we skipped the process of using in-sample and out-of-sample data, which would be a requirement if we were exploring for a new solution. Instead, we tested our method on one market and one sector, then applied it to other markets and other sectors. It is a weaker out-of-sample approach but we felt that it was sufficient.

We selected our set of markets by what seemed reasonable. We picked those that were most liquid. We did not reject any because their charts looked bad. We calculated the cross-correlations but did not find those values useful. Had one of the stocks been negatively correlated, we would have rejected it.

We created the trading rules based on simple calculations that showed the relative, not absolute, differences between the pairs.

We volatility-adjusted the position sizes to equalize the risk of both legs. In doing that, we avoided a price move in one leg overwhelming the results.

We ran tests on multiple pairs but looked at the average results of all pairs, not at the individual profits and losses. When we selected a parameter value, it was because it improved the net performance of all pairs.

We confirmed our belief in the robustness of this strategy because nearly all combinations of parameters were profitable for the average results. We observed that the pattern in performance was continuous when parameter values were increased or decreased.

You don't need perfect performance to have a profitable trading program. When we selected our final parameters, some of the pairs showed losses. We did not remove those from the set because we don't know what will happen in the future. If we could justify removing the losing pairs, we could have justified removing all but the most profitable pair.

We volatility-adjusted each of the pairs to the same target volatility.

We tested a low-volatility filter, under the premise that small price movement is the worst-case scenario for this strategy.

By creating a profit and loss series, we found the investment size that was needed to trade each pair at our target volatility. We then checked to see if the cost of trading the volatility-adjusted positions on any day exceeded our investment size. It did, in about a third of the days.

We needed to cap the size of the positions on days when they would have exceeded our investment size. We scaled down the position size to satisfy the cap and adjusted the daily returns by that ratio. The result was that the capped performance exceeded the original returns before capping.

Using the capped profits and losses, we found the new investment size and created return series for each pair, each with its own investment size. We averaged the daily returns and found that the risk had dropped by about 40%. We created the new portfolio NAVs from the aggregate investment size and the average daily returns. Because of the nature of stock investments, we were unable to leverage our returns to the target volatility and needed to settle for about half of that, or 6%.

Chapter 4.

Pairs Trading Using Futures.

Although pairs trading clearly works, the returns for many stocks are small, and the demands on good execution are high. In the previous chapter, we looked at airlines and home builders and found that airline stocks did not have enough volatility, and home builders are more active but give only marginally good results. Using volatility and distortion filters will improve results but at the cost of fewer trades. Capping provided an unexpected gain while reducing the risk. The concept of trading pairs, however, is fundamentally sound and can be considered robust because it shows profits across a wide range of parameter values. What is needed is more volatility or leverage.

It is possible to leverage returns in stocks by borrowing part of the capital needed to trade. If interest rates were very low and trading returns were high, then borrowing would be a profitable alternative. It is also possible to use stock options rather than stocks. The companies that we are trading are mostly large and liquid, so that options could be a viable choice. In addition, options have no restrictions on going long or short. In fact, by selling (writing) one leg, you receive the premium, offsetting the cost of buying the other leg. Given the bullish or bearish bias in the market, costs may be kept very small. It would be necessary to evaluate the slippage, or bid-asked spread, combined with the net premium of the two legs, to know the viability of using options. That is not our focus here.

FUTURES.

Futures contracts offer a completely different approach, satisfying all liquidity and leverage problems but providing far fewer choices in the markets that can be traded. While there may be 30 to 50 different companies within the energy complex, there are only six viable energy futures markets: crude oil, natural gas, heating oil, and unleaded gasoline trading at the New York Mercantile Exchange (NYMEX) in New York and Brent and gas oil traded on the International Petroleum Exchange (IPE) in London.

Another complication is that the trading hours are different in New York and London. Even though crude oil has become a 24-hour market, the bulk of liquidity occurs during the normal business hours in the country where the commodity is being traded. Crude oil in New York is officially open from 10:30 A.M. to 2:30 P.M.; in London, the pit session (open outcry if there are still humans on the floor) is open from 10:02 A.M. to 7:30 P.M. On the ICE (the all-electronic Intercontinental Exchange), trading is from 1:00 A.M. to 10:00 P.M. Given the five-hour time difference betwee n New York and London during all but one week in the spring, we see that both the New York and London markets are open during the entire time that New York trades its pit session, from 3:30 P.M. to 7:30 P.M., and they both close at the same time.

Our approach has been to trade on the close, which also a.s.sumes that prices posted on the close are realistic approximations of the price of two markets at the same time. Even then, no one ever gets a fill at the settlement price. If you're a buyer on the close, expect to get a price above the settlement, and if you're a seller, below the settlement. A bid-asked spread exists no matter what time of day you're trading and no matter how liquid the market.

In futures, the settlement price is a volume-weighted average price of all trading during the last 30 seconds of the session. This is similar to the close of the stock markets. Because of that, getting the closing price for your execution is not realistic, and on volatile days, the range of the last 30 seconds can be wide. It may be safer to trade a few minutes before the close to have better control of your fill price. Unfortunately, because this trade may have one leg in New York and one leg in London, it's not possible to enter the order as a spread, which guarantees a minimum differential between the two prices. For pairs on the same exchange, spread orders are the best way to go.

Technically, the popularity of 24-hour markets and extended pit sessions makes it possible to trade both legs of a spread at the same time, at almost any time, albeit with different amounts of liquidity. But it is more important to know that the prices used to decide the trade actually occurred at the same time. If the two markets that make up the pair are not actively trading, the screen price will not be the price you would get when you execute the trade. For example, you think there is an opportunity to arbitrage gold because a COMEX (Commodity Exchange) deferred contract is trading at $1,150 and London forward of a comparable delivery is showing a print of $1,154. The cost of delivery is $2. But London gold hasn't traded for two minutes. If you were to look at the bid-asked, you might see $1,149.5 and $1,150.5, showing that the next trade would bring prices back together, eliminating any arbitrage possibility.

Even with these complications, the closing price is the most realistic a.s.sumption that both markets have traded at the same time. The close tends to have very high liquidity. Still, when you buy one leg, you often get a price in the upper half of the closing range, and as a seller, you get the lower half. It's necessary to be sure you wait for a spread difference large enough to absorb a poor fill and still net a profit.

For the equity index markets, which we will apply in these examples, only the U.S. and European markets will be used. In April 2005, EUREX changed its trading hours to remain open until the close of the U.S. sessions. The German DAX, EuroStoxx, French CAC, and other European index markets (although not the London FTSE) all close at 10 P.M. in Germany, or 4 P.M. in New York, except for one week in the spring when the United States adjusts to daylight savings time earlier than Europe or Great Britain. This allows us to a.s.sume that the closing prices are close enough and base our trading signals on the close.

Different Holidays.

There are many days in which either the U.S. or European markets are not open due to holidays. Over the years, many countries have aligned their holidays so that the markets are open and closed at the same time; however, London does not close when the U.S. celebrates Independence Day, July 4, even if there is little trading. It's the principle.

To account for these differences, and the errors it would cause if the strategy didn't recognize that one market was closed and the other open, we have adopted the rule: If either market posts no price change, we cannot enter or exit a trade.

There will be days when the market actually closes unchanged, and the system will not change its position. These differences seem small compared with entering or exiting with a large profit (or loss) because one market traded and the other did not. Those traders who are more ambitious can keep an accurate calendar of market holidays, which are available on each of the exchange web sites.

Trading Habits in Different Countries.

Although markets now stay open at all hours to accommodate traders, most activity is still concentrated during local business hours. That is to our advantage. The results will show that the best performance is when we trade a U.S. against a European market. There are two primary reasons.

While the fundamentals of the economy are different, globalization causes traders to push similar markets in the same direction. When the release of a U.S. economic report causes the U.S. stock market to jump higher, European index markets tend to follow before deciding to what extent that report actually affects them.

The second reason is that traders in Germany just don't want to stay up until 10 P.M. to trade the U.S. close. Naturally, professional hedge funds hire traders to do just that-to keep the same hours as U.S. markets. But that's not enough to make the volume high.

Check the Data Carefully.

Many futures markets have day sessions (once called the pit session, before most of the pits became extinct), night sessions, and overnight sessions. For convenience, these will be called open outcry and electronic sessions.

For a pit session, or open outcry session, the trading day starts at the beginning of business hours for the country in which the futures market is trading. In the United States, stocks and stock index markets open their pit sessions at 9:30 A.M. New York time. The stock market closes at 4 P.M., but futures close at 4:15. The extra 15 minutes allows reaction to some earnings reports released just after the close of the NYSE. The close of the pit session is also the price used for marked-to-market accounting, the settling of all profits and losses for the day.

After a pause of from 15 minutes to one hour, the electronic session starts. For some markets, there is side-by-side trading. For example, there is an S&P pit session trading a big contract worth $250 per big point change, and an electronic e-mini contract with a $50 per big point change. The e-mini contract trades 24 hours and has most of the volume.

"Big point" or "handle" is the insiders' way of saying that the price to the left of the decimal point has changed. In the days when the pit session was active, traders used hand signals to indicate the price at which they wanted to buy or sell. Those signals showed only the decimal place, not the handle. Even today in crude oil, trading is so active that prices could stay between, for example, $71.00 and $72.00 for hours, so that only the cents are needed to show your bid or asked price.

When you look at combined session data, which are very common now, you are seeing the trading from both the pit and electronic sessions. For the U.S. markets, the new trading day starts after the close of the pit session when the electronic session opens (for the S&P) at 6 P.M. the same evening. Both the pit and electronic sessions close at 4:15 P.M. the next day.

For Europe, the electronic session becomes an extension of the current day. But even in Europe, the official settlement price occurs between 2:30 P.M. and 4:15 P.M. local time, essentially tied to when the banks traditionally closed for the day and posted all debits and credits.

It is important to know that, in Europe, the DAX day session closes at 5 P.M., electronic at 10 P.M., and some data series have the high and low until 10 P.M. but the official settlement at 5 P.M. Whoops! That doesn't work for your trading system because the high or low could have occurred after the 5 P.M. close, and you would be using data that did not exist at 5 P.M. You'll need to be sure that you have a data series that has the last price instead of the settlement.