Part 14 (2/2)

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 relations.h.i.+p that volatility increases as price increases.

This general risk relations.h.i.+p 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.

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