Part 14 (1/2)
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.