Rank, the order of stocks when sorted by slope, strongest (1) to weakest (5).

Shares, the number of shares traded.

Chg pos, the change in the number of shares (used for calculating costs).

Returns, the daily returns in dollars.

c.u.m ret, the c.u.mulative returns in dollars.

Home builders suffered the brunt of the economic downturn; therefore, it will not be surprising that the period we are using, from 2001 to May 2009, shows wide, volatile price swings. Figure 8.1, repeated from Chapter 3, should be a good reminder.

FIGURE 8.1 Prices of five home builder stocks. All five react in a similar manner to the economic changes.

Ranking on Day 1 Table 8.2 shows the prices of the five stocks in row 2 (price) and the corresponding index value in the row 3 (xprice). The index prices started at 100 on January 2, 2000, but Pulte had already dropped more than 12% by January 13 and Lennar had gained more than 13%. Stocks trading at low prices often move in surprisingly large percentages. We'll see the same phenomenon at the end of the test period, when prices drop back to these levels with increased volatility. While volatility is generally good for mean-reverting programs, volatility also means risk and must be treated carefully.

We calculate the slope using an 8-day period in order to have enough data to smooth the direction but not too much as to cause a sluggish reaction to price changes. We also keep in mind that a mean-reverting strategy should only hold a position for a short time, looking for a fast correction in relative price distortions. The longer the holding period, the greater the risk. We have the advantage of knowing that a 7-day period performed well in tests that will be shown later.

Row 4, slope, clearly reflects the direction of the past few days and is consistent with the change in the index value from its start at 100. Pulte, which lost 12%, has the steepest downward slope, while Lennar, which gained 13.5%, is the strongest. Toll Brothers and Hovnanian, with their index value still near 100, fall in the middle.

Selecting the Specific Stocks to Trade Having ranked the five stocks, the next decision is which to buy and which to sell short. With only five stocks and a mean-reverting strategy we have two choices: 1. Sell the two highest-ranking stocks, and buy the two lowest-ranking stocks.

2. Sell the single highest-ranking stock, and buy the single lowest-ranking stock.

We choose the first option, buying and selling two stocks rather than one stock. Although we have a very small group, four stocks (two pairs) offer some diversification over a single pair.

The two stocks with the most negative slopes, Pulte and Hovnanian, get ranks of 5 and 4, respectively, seen in row 5. The two most positive slopes, although not as positive as the others are negative, are Lennar and KB Homes, with ranks 1 and 2.

Basing the decision entirely on the rank, we buy Pulte and Hovnanian and sell short Lennar and KB Homes. We do not take a position in rank 3, Toll Brothers.

Number of Shares to Trade Keep in mind that the fundamental purpose of a market-neutral program is to be risk neutral; that is, the risk of the long positions must equal the risk of the short sales. Otherwise, when the market makes a uniform move up or down, you are not protected.

There are two accepted ways to balance the risk: 1. Trading equal dollar amounts for each stock.

2. Volatility adjusting the position size.

We start with the first option because it is commonly accepted on Wall Street-and far easier to calculate. Allocating equal dollar amounts to each stock relies on the loose but generally valid premise that volatility increases as price increases. Although this is generally true, the specific volatilities of two stocks trading at the same price could vary by as much as 50%. If one stock is in the news and the other is under the radar, then the visible stock normally experiences a short-term surge in volatility.

Stocks at very low prices also have erratic volatility, often much larger than those stocks trading at higher prices. It was not surprising to see Bank of America (BAC) or Well Fargo gain 25% near the bottom of the financial crisis in early 2009. A jump from $3.00 to $3.75 is a large percentage for one day, but still a small gain after a drop from above $50. Figure 8.2 shows the path of BAC during the recent financial crisis, and Figure 8.3 shows the corresponding annualized volatility based on a 20-day calculation period. Volatility reaching 250% is unheard of and can be sustained only over short time periods; however, this surge of volatility, in excess of 100%, is now approaching one year. This unusual scenario makes any measurement of volatility subject to problems but is an opportunity to prove that a strategy can survive under stress.

FIGURE 8.2 Bank of America showing increased volatility at lower prices.

FIGURE 8.3 Bank of America annualized volatility increases as prices drop.

The initial investment for this strategy is $10,000. We will always trade four of the five stocks; therefore, each stock will get an allocation of $2,500. On day 1, we calculate the number of shares to buy for Pulte as $2,500/$3.87 (the closing price) or 646 shares, rounded down. For KB Homes we get $2,500/$7.31 = 342, Hovnanian $2,500/$3.09 = 809, and Lennar $2,500/$2.44 = 1,025.

We also apply the transaction cost of $0.005 per share. That gives the cost of selling short 1,025 shares of Lennar as $5.12, which is shown only in dollars in Table 8.3; however, the cents are acc.u.mulated in the returns. As you will see, transaction costs are very important because the returns per share are typically small for market-neutral strategies in the stock market and the changing of positions may make it difficult to apply costs afterward.

Day 2 At the end of day 1 we have 2 longs and 2 shorts, each committed with $2,500. At the end of day 2 we will do the following: Update the prices.

Calculate the next index value (see Table 8.4). TABLE 8.4 Day 2 of calculations.

Find the 8-day slope of each index series.

Rank the stocks according to their slope.

Choose the two strongest markets to sell short and the two weakest to buy.

Determine the number of shares by dividing $2,500 by the current price.

Enter orders based on the difference between yesterday's positions and today's.

Calculate your profit or loss for each stock as the shares held yesterday ("+" for long and "" for short) times today's close minus yesterday's close.

Subtract the cost of trading applied to the difference of today's position size and the previous day's size.

On day 2, the stocks kept the same relative strength; that is, Pulte and Hovnanian remained the strongest, Lennar and KB Homes the weakest. Because the prices changed, the size of the positions changed. Lennar's price increased; therefore, the number of shares dropped from 1,025 to 1,011. Because we were short Lennar, 14 shares are bought to reduce the short position. Only three of the four stocks had small adjustments, but the cost of trading is reflected in the PL. The net position lost $24 on day 2, for a c.u.mulative net loss of $38.

The Last Trading Day Moving forward, the results of the last trading day are shown in Table 8.5. Prices have increased significantly for all but Hovnanian, although the path between the start and end dates of this test was extreme. Hovnanian and KB Homes are the strongest, and therefore they are short, while Pulte and Lennar are the weakest and are long. The c.u.mulative returns on the last line 9 show that three stocks were net profitable and two losing, for a final gain of $10,207. Based on an initial investment of $10,000 that's slightly more than double over a period of about nine years and three months, a simple return of 10.9% per year.

TABLE 8.5 Results of the last trading day, May 15, 2009.

When all the statistics are reviewed, we find that this strategy returned only $0.013 per share, less than 2 cents, certainly not enough to be profitable, even after costs of $0.005 per share. But then, making money isn't easy.

Choosing the Critical Parameter Up to now, we've referred to the 8-day slope calculation, used to rank the indexed values. But the number of days used to find the slope is the critical parameter for this market-neutral strategy. When the number of days, n, is small, the slope jumps around but is responsive to relative changes in price. As n gets bigger, the linear regression line is smoothed and the slope becomes more stable, changing less often. The size of the returns for each stock is directly related to the calculation period and the holding time.

In Chapter 2, the concept of price noise was discussed. The conclusion was that certain markets were noisier than others, stock markets being the noisiest, and that shorter observation periods emphasized the noise. As the calculation periods get longer, the prices are smoothed, and the trend begins to show. Over the short term, say, 2 to 8 days, there is no trend, only traders reacting to news and investors entering and exiting through large funds not particularly concerned with market timing.

Because we've chosen a mean-reverting strategy, we'll test this strategy for a range of calculation periods, from 3 to 10 days, to be consistent with the concept of noise. The results, shown in Table 8.6, are based on a $10,000 investment, positions in 4 stocks (2 long and 2 short), and an equal dollar allocation to each stock. The four columns show the number of days in the slope calculation, the annualized rate of return (AROR), the profits per share, and the return ratio (AROR divided by annualized risk). This test covered 10 years ending May 15, 2009.

TABLE 8.6 Market-neutral basic test of the slope calculation period using four stocks.

The results of this test have both good news and bad news. The good news is that all tests are profitable, showing that the concept is sound. One of the best measures of robustness is the percentage of profitable tests. In this case, the test spanned the full range of calculation periods that seem reasonable for a mean-reverting strategy; therefore, when all tests show profits, we can conclude that the strategy is sound. As long as the percentage of profitable tests is above about 70%, it would be considered a success.

The bad news is that the profits per share, after taking out costs of $0.005, peak at only $0.013, between 1 and 2 cents per share. That doesn't leave much room for error, and the returns of about 6% don't make this worth the risk. We'll need to explore some alternatives, remembering that we were able to get more than $0.13 per share using pairs trading.

Filtering Low Volatility The most obvious way to boost profits per share is to remove trades taken when a stock is doing nothing. "Doing nothing" usually means that prices are exhibiting very low volatility. We found this method successful for various other pairs trading. The filter is based on a 10-day standard deviation of the returns, expressed as a percent. The 10-day standard deviation is annualized by multiplying by , which we've done before. Unless the current volatility is above our threshold, no trades are entered. Without a filter, the strategy returned 6.82% annualized and $0.013 per share. The results of using the volatility filter are shown in Table 8.7.

TABLE 8.7 Using a volatility filter on the 8-day slope strategy.

The low-volatility filter is shown in whole percent in column 1. To keep it simple, these tests all use the 8-day slope. When the 10-day annualized volatility is below 1%, performance increases slightly to a return of 7.43% and $0.015 per share. Notice that the volatility filter works as expected: The AROR generally increases as the filter increases, and the profits per share move from our no-filter case of $0.013 to a maximum of $0.063 when the filter is at 14%. The ratio also increases, showing that the low-volatility filter is a legitimate way to approach the problem. To confirm our belief that the filter is working correctly, we plot the resulting NAV with and without a 6% filter, shown in Figure 8.4. Having not seen the original NAVs, we find it surprising that the strategy had a long declining period; however, the volatility filter shows that the decline was the result of low volatility-exactly the result we are looking for. The filtered results show a much stronger performance at the end because both NAV series are adjusted to 12% volatility. If we leave them unchanged, then the original NAV series would simply have no trades in the middle years.

FIGURE 8.4 Home builders original NAV and results using a low-volatility filter of 6%, both adjusted to a target volatility of 12%.

Applying a volatility filter turned out to be very useful, but a profit per share of slightly over $0.06 is still marginal. The $0.13 using pairs remains more attractive. However, the filter has all the right characteristics and will be used for other strategies to boost results. A filter has the added advantage of reducing costs.

Quantizing to Reduce Costs Another technique for reducing costs is to hold the same position size unless that size changes by more than a threshold percentage. In engineering, this method is call quantizing. For example, without the filter, if we are long 1,000 shares of Lennar and the next day the volatility declines by 5%, we would add 50 shares and be long 1,050. With a 10% filter threshold, we would not add the 50 shares but wait until volatility had dropped by 10%. There is a good chance that volatility will increase tomorrow and we would just be resetting all or part of today's trade. Instead, the volatility must change by at least 10% to trigger an additional purchase or sale of 100 shares or more. This method won't be tested here, but it is a common and successful way of minimizing trading costs.

Price Filters Another filter that might increase returns per share is a simple price threshold. A stock that falls under $2, perhaps even $3 or $4, might be considered unstable. At low levels, prices make much larger percentage moves than stocks trading at, say, $30 per share. This is particularly true if it was once trading at a much higher price, such as Bank of America. Removing the low-priced stocks from the mix might save both aggravation and risk, along with some costs.

Trading Only the Extremes In an effort to increase the profits per share, we must return to an earlier decision to trade the two strongest and two weakest of a total of five stocks. Instead, we'll trade only the extremes, the two stocks that show the steepest positive and negative slopes. Naturally, trading a set of only five stocks has its limitations. One of those limitations can be that none of the stocks show enough volatility to generate sufficient profits per share. With a larger set of stocks, we might get both diversification and volatility.

The first step is to repeat the basic test of various linear regression calculation periods, this time only going long the one weakest stock and short the one strongest stock. We are sure that the profit potential is greatest for this pair, but we don't know by how much. By limiting the number of stocks trading, we also expect the risk to increase. In this case, we have removed all chance of diversification. The results are shown in Table 8.8.

TABLE 8.8 Home builders mean-reversion strategy using only the two stocks with the strongest and weakest slopes.

Compared with the results of using two longs and two shorts, these are clearly better. The average AROR, profits per share, and ratio for the earlier test, using four stocks (Table 8.6) were 6.70, $0.0107, and 0.558. The averages for these tests are 7.99, $0.0152, and 0.666. The ratios are much higher and very stable, showing that we have improved the risk profile. Unfortunately, the best per share return is only $0.022, a 69% increase, but still too low.

When applying the low-volatility filter to the 2-stock case, we again see that the numbers are better, but not as much of an increase as we saw when we used two longs and two shorts, a total of four stocks. The results are shown in Table 8.9. The best returns per share are not as large as when we used two longs and two shorts, and the averages are all marginally lower.

TABLE 8.9 Applying the low volatility filter to the 2-stock case.

VOLATILITY-ADJUSTING THE POSITION SIZE.

At the beginning of this chapter, we discussed two ways of calculating position size. The first was allocating equal dollar amounts to each stock and then dividing by the current price to get the number of shares. This relied on a general relationship that volatility increases as price increases.

This general risk relationship is not very accurate because some stocks trading at about the same price can be much more volatile than others. In this section, we'll find the volatility of each stock and adjust them so they have the same risk. In addition, we'll be sure that the net long positions have the same risk as the net short positions, in the event there are an uneven number of longs and shorts.

We feel strongly that this is the correct way to size positions. At the same time, we hope that this increases the profits per trade. Even if it does not improve returns, there is a clear element of chance introduced when you trade equal dollar amounts of each stock. If you can fix the problem, you are obligated to do so.

Beginning again on day 1, this time the annualized volatility is calculated over the same 8-day period that is being used for the slope. The annualized volatility is the standard deviation of price changes over the calculation period multiplied by the square root of 252 for annualization. Row 5 of Table 8.10 shows PHM at 0.624, an annualized volatility (AVOL) of 62.4%. When we use a very short calculation period, the annualized volatility can exceed 100%, but over a long period, even these numbers will average out to the same value as those using a longer-term calculation period.

TABLE 8.10 Calculating the number of shares from the volatility.

The steps for finding the number of volatility-adjusted shares follow, and the values a.s.sociated with each step are shown in Table 8.10.

Calculate the 8-day standard deviation of price changes for each stock, i.

Calculate the annualized volatility for each stock as Create a volatility adjustment factor, VAF, for each stock equal to your target volatility divided by the annualized volatility: where the default target volatility is 12%. Note that this inverts the volatility so that markets with higher volatility will get smaller allocations. Actually, the target volatility can be any number such as 1, which will invert the annualized volatility. Later, we will scale this to the investment size.

Set VAFi to negative if the position is to be short (highest ranks).

Normalize the volatility factors by finding the sum of the long factors, LFV, and the sum of the short factors, SVF. Then divide each of the long factors in (4) by LFV and the short factors by SFV. TABLE 8.11 Step-by-step process for finding volatility-adjusted share size.

6. Calculate the number of shares for each stock as In row 6 of Table 8.11, the investment of $10,000 is divided by 2 in order to get the amount allocated to only longs or shorts; then the number of shares for PHM is In Table 8.11 we show the number of shares rounded up, but in the actual trading strategy, we truncate the number of shares to avoid exceeding the investment size.

TABLE 8.12 Day 2 calculation and performance detail for home builders.

If we compare the number of shares from this volatility-adjusted method with the equal dollar allocations (from Tables 8.3 and 8.10), the numbers are sufficiently different. PHM is only half the equal dollar method, and HOV is more than 20% different. This difference should be enough to change the final outcome.

On the second day, the stocks that are long and short remain the same, but the position size changes slightly. This can be compared to day 2 using the equal dollar method, shown in Table 8.12.

In the end, using the volatility-adjusted position sizes seems to be the right method, but the annualized returns were 4.45%, and the per share profit was only $0.008, less than the approach using equal dollar allocations. We'll need to look further.

ARBING THE DOW: A LARGE-SCALE PROGRAM.

A large-scale market-neutral program trades a basket of long positions against a basket of short sales. Our example of only five stocks in one group is not a fair indication of its success. When you have many markets, there should be bigger divergences and more volatility among the stocks. This will lead to larger profits. The previous small example was just an exercise to show how a market-neutral program is constructed and traded.

One important point to remember is that pairs trading uses a timing device, some form of momentum indicator to find relative differences. We've used both a simple stochastic difference and the stress indicator. Trades are not entered until that indicator reaches an extreme, which provides entry timing. Of course, that extreme could occur under conditions of high or low volatility. If it's low volatility, then the per trade returns would be small, but over the entire period, entry prices will be at relative extremes and average higher returns than methods that choose entry points more arbitrarily. Pairs trading will also be improved by filtering out low-volatility situations.

Market neutral has no timing. Each day, the slopes are calculated, and the steepest slope is sold. If you use some other form of ranking, then the highest-ranking stocks are sold. Although a mean-reversion entry benefits from timing, at no point did the market-neutral method try to decide if this was a good place to sell or if the relative price of one stock was at an extreme compared with its own history or the price of another stock. As soon as one stock moved into the top zone, it was sold. If it continued to strengthen relative to the other stocks, it would produce a loss rather than a better opportunity for entry.

Success in market-neutral trading requires two key attributes: high volatility if the method is mean reversion and a reversal in the strength or weakness of one stock compared to the others. If trend following, then we would want continued strength or weakness of each stock, keeping them in the buy or sell zones.

Are the Dow Components Trending or Mean Reverting?

We need to decide if the large-cap stocks in the Dow tend to move away from each other, exhibiting trending, or whether they keep switching from being in the strongest zones to being in the weakest, a mean-reverting trait. We have two pieces of information to help us. If the Dogs of the Dow work, then we should expect any large company that has underperformed its peers to rotate back up to the top. To have stayed on the top, companies such as Microsoft and General Electric seem to figure out how to evolve.

More important, our study of price noise in Chapter 2 showed that the stock markets in all developed countries exhibit a large degree of erratic price movement. The equity index markets were the noisiest of all markets and resisted profits based on trending methods. Most individual stocks also show the same characteristics. Then mean reversion is the likely scenario, and we'll take that approach.

Specifying the Rules of the Market-Neutral Method To make this as transparent as possible, we will begin with the following rules: An investment of $100,000.

Equal investment in each of the 29 stocks, $3,448 (we found a data error in one of the Dow components and removed it).

Buy and sell zones each with 13 stocks, a neutral zone with 3 stocks.

Cost per transaction of $0.005 per share.

We know that if we use a very short calculation period, then the trades will all be held for a shorter time, and the returns per share will be small; therefore, we tested periods of 5, 10, 15, and 20 days. Trading began on January 11, 2000, and ended on June 18, 2010. Table 8.13 shows the detail of calculations and positions on the second day of trading.

TABLE 8.13 Second day of trading the DJIA.

Table 8.13 is simply a bigger version of Table 8.12, which showed the home builders. The symbols are along the top and the rows are: Stock price.

Stock index value (converted from price).

Linear regression slope using the index price.

Annualized volatility of price using the index.

Sorted rank based on the linear regression slope, where 1 is the strongest stock.

If long, the size is the volfactor divided by the sum of the volfactors of all long positions; if short, it is the volfactor divided by the sum of the volfactors of all short positions.

Number of shares entered based on equal investment sizes.

Number of shares that changed from the previous day.