Alpha Trading Part 7
When the data are aligned by date and one market is closed, it is necessary to enter a zero in the cell if that cell represents returns, or copy down the previous cell if the stream is the c.u.mulative profits/losses.
Step 3: Target Volatility and Investment Size To generate the daily returns, we need to decide on an investment size and a target volatility. For example, we will invest $100,000 but only want a 2.5% chance of losing 24% of the investment. We choose 2.5% because it represents one side of a 2 standard deviation distribution. Because it is 2 standard deviations and the annualized volatility is based on 1 standard deviation, we know that a target volatility of 12% will satisfy our needs.
If we know what our target volatility should be, then we can arrive at these results with a few steps. We use 12% annualized volatility as the industry standard for risk.
To find the investment size needed to have 12% volatility, Find the standard deviation of the profits and losses for the entire performance period.
Multiply that value by the square root of 252 to get the annualized volatility in dollars.
Divide the annualized volatility by the target volatility (0.12) to get the investment size for this market (the market investment).
Repeat the process for each market.
Add the market investments to get the portfolio investment.
Scale all values to your actual investment size.
Alternatively, if you know your investment size, you can adjust all the returns to your target volatility: Convert c.u.mulative profits to daily profits and losses for each series by subtracting the value on day t 1 from the value on day t.
Calculate the returns, rt, by dividing the daily profits and losses by the investment.
Find the standard deviation of the returns for the entire series.
Multiply that value by the square root of 252 to get the annualized volatility in dollars.
Divide the target volatility (12%) by the annualized volatility to get the volatility adjustment factor, VAF. Note that we must always lag the use of VAF by one day to replicate trading.
Calculate NAVs starting at 100, with each subsequent All NAV series are now adjusted to 12% volatility. You can now combine them into a portfolio using your allocation percentages.
Table 4.8 shows the steps using the first set of rules. The three daily profit streams are in the columns under the heading "Daily Profits/Losses." They begin on January 1, 2000. Below those columns are three additional values marked "Annualized volatility," calculated as shown in the previous steps. The market investments based on the target volatility of 0.12 are shown below that. The sum of the three market investments is $4,331,319, but remember that this is based on trading 10 contracts of crude all the time. The investment could be smaller if the number of crude contracts is reduced, but we lose the ability to accurately balance the risk of both legs. In the long run, that might not matter, but then it might.
TABLE 4.8 Calculation of market investment and portfolio investment (on left) and market returns, portfolio returns, and portfolio NAVs (on right).
Step 4: Calculate the Portfolio Returns The next panel shows the market returns, which are the daily profits or losses divided by the total portfolio investment. The daily returns for the three markets are added to get the portfolio returns, shown in the last panel.
Finally, the portfolio NAVs are calculated in the same way as before, starting with 100, and then each subsequent value for day t is Next, calculate the annualized returns through day t (which begin with 100) as The result is a return of 6.73%. We then calculate the annualized volatility of portfolio returns and get 10.19%, lower than the target of 12% due to a modest amount of diversification. The information ratio, which is the annualized return divided by the annualized risk, is then 0.66. Most traders would like this ratio to be greater than 1.0, but then the S&P pa.s.sive ratio is closer to 0.10 over the past 10 years. When we combine energy pairs with other sector pairs, these results will improve.
In addition to being directionally neutral on price moves, a benefit of pairs trading is that most pairs are in the market less than 50% of the time, often closer to 25%. That means you are less exposed to price shocks and general market risk. No matter how good the system, the only way to avoid price shocks is to be out of the market. It's an important benefit.
Figure 4.7 shows the final portfolio NAVs as calculated using the steps just explained. The method gains steadily over the 10 years, but as we have discussed before, choosing the right time to trade will make a difference. With higher oil prices, volatility is likely to remain high and opportunities will be better.
FIGURE 4.7 Returns for the three natural gas energy pairs and the equally weighted portfolio, at 10.2% volatility, 10 years from 2000.
Energy Summary Although we found only three pairs that we would trade, it is a good sign that we accomplished that using the exact same method and could have used the same parameters as the original examples using stocks. Not many trading methods hold up across different markets, especially moving from stocks to futures. However, we needed to explain how we could justify removing the three pairs that included only crude and its products.
Market experience must play a role. The relationship between crude oil and its products, heating oil and gasoline, is called the crack spread. It is intended to simulate the process of cracking crude oil (breaking its hydrocarbon chain) into its products. Based on the amount of product that can be extracted from crude oil, crack spreads are most often done in the ratio 3:2:1, buying 3 contracts of crude and selling 2 gasoline contracts and 1 heating oil contract. It is also done in the ratio 5:3:2. To be done properly, the products should be traded one or two months out from the crude contract to give the correct delivery relationship, for example, February crude oil produces products that are ready for delivery in March or April. Trying to arbitrage the crack spread using our pairs trading method is the same as an amateur competing with professionals. It's a difficult game to win. Figure 4.4 showed how closely heating oil prices tracked crude oil, and Table 4.3 showed the long-term correlations, with crude and heating oil at .90, crude and gasoline at .86, and gasoline and heating oil at .84. There is no percentage for us to trade those relationships. We need to look at markets in which there is a fundamental relationship, or a psychological one, and the correlations are less than 0.80.
Heating Oil and Natural Gas When we began looking at energy pairs, our choice would have been heating oil and natural gas because they both serve the home heating market. In the end, that pair was very profitable, but not nearly as good as the crudenatural gas pair, which had higher average profits per contract, as well as a higher information ratio, shown in Table 4.5.
We can conclude after the fact that a sloppy correlation in markets that are fundamentally related can result in more opportunity. Figure 4.6 also shows that returns are much better when prices and volatility are higher, which we've seen since mid-2007, when the oil crisis began. In fact, returns prior to that period were very small.
If we could only limit our trading to periods with exceptional moves, our returns would be outstanding. That leads us to our next main topic, trading inflation pairs, those markets that get the most attention from the financial news networks and newspapers under the headline of inflation.
THE INFLATION PAIRS: CRUDE, THE EURUSD, AND GOLD.
If only we could trade during the extreme market moves and avoid the other times. It's possible that volatility is the key to identifying price regimes, but the reality is that it takes time to recognize a change in the market structure. With a lag at the beginning and a lag at the end, we've usually given up more than we gain by regime switching. One good example is trend following. The key to profits in trend following is the fat tail, the occasional very large profit from an extremely long trend that offsets many small losses that came before. If you use a stop-loss with a long-term trend, you exit the trade with expectations of saving money, but the trend is not over; that is, it hasn't changed direction, and it may only have taken a mid-trend correction. If you're wrong and the trend stays intact and eventually becomes one of the few big winners, you've lost your chance at net profits. Many of these strategies win by diversification and persistence. The performance doesn't look perfect because it's not perfect, but it will make money if you play by the rules.
Another opportunity seems to be in those markets perceived as causing inflation, representing inflation, or being a hedge against inflation, namely, crude oil, EURUSD, and gold. Although the U.S. government measures inflation without food and energy, energy prices have an impact on everything we buy. Because the price of energy is embedded in many other costs, the government thinks it would be double-counting to include raw energy prices in inflation calculations.
We'll take a simpler approach. Everyone knows that doubling the price of crude oil would have a material effect on all commodity prices, as well as disposable income. It also seems clear that the relationships between these three markets become stronger when inflation is in the news. Perhaps someday we will measure the number of square inches devoted to inflation on the front page of the New York Times or Wall Street Journal to determine when the time has come to trade the inflation pairs.
FIGURE 4.8 Components of inflation. EURUSD, crude oil, and gold prices using back-adjusted futures.
The U.S. dollar started to weaken against the euro at the beginning of 2006 and so far has moved from 1.20 to 1.50, a loss of about 25%, before recovering back to 1.20. Futures prices in Figure 4.8 show the relationship between the euro, crude oil, and gold, although it is somewhat different from the cash prices that we see on the news each day because they represent both future expectation, the cost of carry, and backward adjustment. But we will trade futures, so it's best to look at that data. All three markets peak in mid-2008, drop quickly, and then rally through the end of 2009. We won't try to decide if the relationship is led by crude oil or by the expectation of large U.S. debt, the result of the financial market bailout, diluting the dollar. Is there enough of a relationship to profit from trading these as pairs?
To find out if we should trade these markets, we need to answer some basic questions: Would pairs created from these three markets have made money using our strategy during the past three years?
Could we have known when to start trading them?
What parameters would we have used?
Different Values, Different Volatility Remember that these three markets have very different contract sizes and, therefore, different risk. A futures contract in the EURUSD has a face value of $125,000, crude oil is 1,000 bbls $80 = $80,000, and 100 troy ounces of gold $1,200 = $120,000 at the current price in May 2010. The volatility, expressed in dollars per day (the easiest way to put them all into the same common terms), must be used to determine the position size in order to equalize the risk on both sides of the pairs trade. Table 4.9 shows the imbalance between crude oil and gold in September 2008 during the subprime crisis. Both markets were volatile, but crude was more so, as shown by the smaller position, 10 contracts, compared with 17 gold contracts. The Net PL column also points out the very large equity swings from day to day, even though this trade netted a profit. Be sure to remember that these prices are the result of continuous back-adjusting of data, so they will not be the same as either the cash prices or futures prices on those days.
TABLE 4.9 Trade in the crude-gold pair during high volatility.
Different Holidays When we discussed trading futures at the beginning of this chapter, we pointed out that these markets can have different hours and may not be open on the same days. Because this is a systematic program, we need a rule that tells us not to trade when one market is open and the other is closed. This can be done in two ways: 1. If the data are omitted, that is, one date does not appear in one of the markets.
2. If today's data are identical to yesterday's data, we a.s.sume it was forward filled because there was no trading. Some data services will repeat the data on a holiday.
When either of these situations occurs, no trades are entered or exited. If only one market is closed, then the profit or loss is calculated for the market that is open.
Results from 2007 through 2009 We've been consistently looking at results that use similar momentum periods and entry thresholds. If we continue to do the same, we get the results shown in Table 4.10.
TABLE 4.10 Test results in terms of the information ratio for three inflation pairs: (b) Crude-EURUSD, (c) crude-gold, and (d) EURUSD-gold, plus (a) the average of those pairs.
These tests include the range of momentum calculations from 14 to 5 days, but only one entry threshold, 50. Our previous tests included 50 as the middle of the 4060 range, but here we will look at varying the exit threshold. An exit threshold greater than zero means that we exit shorts sooner. For example, if the crude-EURUSD pair shows a stochastic difference of 55, we sell crude and buy EURUSD. If the exit threshold is 20, then the stochastic difference needs to drop only below 20, not to zero as in our previous tests, to exit the trade. The trade-off is that we exit sooner and avoid the risk a.s.sociated with holding a position longer, but we will also have smaller profits. However, we might get more trades because the stochastic value can then increase again to above 50, generating a new signal. We would be taking advantage of market noise.
To avoid too much complication, only the entry of 50 is used. If the results of using 50 are poor, then it's not likely that we would be trading these pairs. In addition, Table 4.10 includes a breakdown of the results of each pair, as well as the average of all three.
Normally, we use the average of all pairs to decide the success of a group of related markets, but with these inflation pairs, it's not clear that they are related in the same way as, for example, energy markets. We have chosen three very different markets that we believe are used as an inflation hedge. The results will tell us how closely they track each other.
If we base our decision on the average of all pairs, shown in Table 4.10a, we find there were 15 of 18 profitable tests, and all the losing combinations used the longest calculation period of 14 days. Longer calculation periods imply longer holding periods for the trade, which decreases the advantage of noise; therefore, it is consistent with our concept. Our initial thought is that these pairs are good. The best ratios are at the faster end of the test, a 5-day calculation period. The exit threshold of 10 is very good, with a ratio of 0.728, but the exit of 20 is marginally lower and would get us out sooner. Therefore, we expect to trade these pairs with a momentum of 5, an entry of 50, and an exit of 20.
Fortunately, we also summarized the results of the individual pairs in parts b, c, and d. All three pairs had very different performance, and the average does not represent the sum of the parts.
Crude-EURUSD is the most consistent, with profits in every combination and ratios varying from 0.301 to 0.909. The row showing the detail for the calculation period of 10 is the best, but it looks as though it's an outlier because, without that row, the information ratio increases smoothly from the larger periods to the smaller. We prefer to pick anything in the lower part of the table. A momentum of 5 and exit of 10 seem reasonable and somewhat typical.
Crude-gold is overall a poor performer, with only two profitable combinations out of 18. Even though we thought this could be a good pair, we need to eliminate it. By including these results in the average, we might have distorted those values so that any choice based on the average would also be poor.
EURUSD-gold has the widest range of performance, from negative to very positive, and also has the highest average ratios. It also seems inconsistent because the rows for calculation periods 10 and 7 are much lower (but still profitable) than most of the others. Choosing the momentum period 5 and exit 10 from the lower section seems the safest.
That wasn't too difficult, but it was all hindsight. We picked the same parameters for the two pairs that seem to work. The c.u.mulative profits for the period from 2007 through 2009 are plotted in Figure 4.9. The EURUSD-gold pair is consistently profitable; the crude-EURUSD pair has a fast, steady run-up, followed by a period sideways, before volatility and profitability returns.
FIGURE 4.9 c.u.mulative PL for the three inflation pairs, crude-euro, crude-gold, and euro-gold, from 2007.
The answer to our first question is yes, we could have made money during the period when these markets were considered an inflation hedge by the public. The second question is more difficult.
Could We Have Known When to Start Trading These Pairs?
We know that the subprime crisis caused unprecedented volatility. We can measure that volatility using the formula most often used by the financial industry: In order to recognize a change in volatility, we calculate a rolling standard deviation using the past 10 days. We've discussed before that using only 10 days and annualizing the value will occasionally cause annualized volatility to be greater than 100%. Because our interest is the relative difference in volatility, that won't cause any problems.
In Figure 4.10, the annualized volatility of the three futures markets is seen to change in late 2007, synonymous with the spike in crude oil but a full nine months before the subprime crisis. Had we started trading these pairs at that time, we would have captured the big moves in all three markets. By September 2009, the volatility of the EURUSD and gold had dropped back to previous levels. Crude followed in December 2009. Had we stopped trading when the volatility returned to normal, we might have netted profits in all three pairs.
FIGURE 4.10 Annualized price volatility of the euro, crude oil, and gold shows spikes during the subprime crisis.
Using volatility seems to be a simple way to qualify trading in inflation pairs. The volatile periods generate higher correlation between the markets and also increase liquidity. More investors who would not normally trade these markets are attracted to them during periods of high volatility. They are also pushed along by the commentators on the financial news networks, who manage to convince the public of correlations (and so-called contagion) before they occur.
How Do We Decide, in Advance, Which Parameters to Use?
The most difficult question of the three is how to determine, in advance, which parameters to trade. The easiest answer is to find another period in the past when inflation and volatility were in the news and test that data. Certainly, the early 1980s would qualify, when gold rallied to $800/ounce and interest rates topped at 21%. But we would have to trade the Deutsche mark instead of the euro, which didn't exist at that time, and oil wasn't a factor because the price was so low between the oil shortage in the mid-1970s and the Iraq-Iran War in the mid-1980s that cost wasn't significant. Gold has also been an unreliable gauge of inflation. After the famous run to $800 in 1980, it came steadily down until it reached its low in September 2000. Buying gold at any price during those 20 years would have produced only steady losses.
We could not have found a similar situation using the same three markets, but there may have been other pairs that would show improved returns, and higher correlations, during volatile periods. If we look back at Chapter 3, the use of a low-volatility filter identifies similar situations, although much shorter time periods. It shows that pairs trading does best during periods of higher volatility.
For now, we'll be satisfied with using the past three years to identify the parameters, but if we can show that some other period had the same results, our confidence would increase tremendously. On the other hand, if we had no previous periods with the same profile, then we must base our decision of what parameters to use on the belief that higher volatility will generate the profits we need. There is often a point in developing a trading system when you must make a leap of faith after you've done as much work as possible. Tying performance to volatility seems to be a very small leap.
A Last Word about Inflation Pairs Trading during periods of high volatility can be very risky, but we expect pairs trading to profit during high volatility because prices move apart quickly and correct just as fast. During periods of low volatility, prices may not move enough to generate profits in excess of costs. Markets that have caught the interest of the general public can offer great opportunity, triggered by a clear increase in volatility.
Having decided which of these markets reflects the most public interest, experience seems to be the key. When we looked at the changing correlations in the energy markets, we found that the dramatic rise in crude prices corresponded to increased correlations between all energy futures. We could attach a fundamental reason for that change; however, the inflation pairs are much more deceptive. Figure 4.11 shows the rolling 20-day correlations during the same period as the annualized volatility in Figure 4.10. With volatility, we saw a clear increase at the end of 2007 a.s.sociated with profitable pairs trading. The correlations show that the crude-gold pair is generally more correlated and has moved to its strongest relationship at the end of 2009. But the crude-gold pair was the worst performer. There is a modest increase in the correlation of the EUR-gold pair from the beginning of 2008 until mid-2009, but afterward that pair is mostly negatively correlated, although by only a small amount. Yet EUR-gold was the most consistently profitable pair. If the correlations don't tell us anything, then we can only conclude that it's the money that moves the market. The pairs are profitable because the public believes these markets should react to inflation news, but their movements with regard to one another are unpredictable, just as price noise is unpredictable but profitable for a mean-reverting trader.
FIGURE 4.11 Rolling 20-day correlations of three inflation pairs. Crude-gold, which shows the highest correlation, had the worst performance.
EQUITY INDEX PAIRS.
We now come to the equity index markets, which are probably the most interesting for traders. During the past 10 years, these futures markets have seen a tremendous increase in activity from traders around the world. Although most of the European markets have been electronic since their inception, the U.S. markets have been slower to change, but the liquidity is now in the electronic contracts. Electronic trading has facilitated the ability to execute on the CME, EUREX, SIMEX, and nearly any other exchange in the world.
Many of the equity index markets can be traded as ETFs, but they are not as liquid or as efficient as futures markets. They do allow you to sell short with no restriction, and they do not need to be rolled when the contracts expire. The results shown in this section all use the smallest futures contracts traded, normally called minis. The commission costs for trading minis are much higher for the noncommercial investor than trading the original, larger contracts, but that's where all the liquidity is, so the increased commission cost may be offset by less slippage. The data start on November 21, 2005, when EUREX moved the close of the trading day to 22:00 European time (10 P.M. in New York) and ends on May 1, 2010. Given the windup needed to perform the calculations, this gives us nearly 4 1/2 years of performance for all markets. That can still generate a lot of trades because we hold positions for only a few days.
The markets used to form pairs will be the S&P, Nasdaq, Russell, Dow, EuroStoxx, DAX, CAC, and FTSE, a total of 8 markets and 28 pairs. Of these pairs, 6 are combinations of U.S. markets, 6 are European, and the remaining 16 are combinations of U.S. and European markets.
If you use the combined sessions for European markets, which means the original pit session followed by the evening session, the data will all end at nearly the same time. In the U.S., the evening session starts the new day for the electronic markets, which then ends at the close of the next pit session. In 2005, the European markets extended their hours to trade alongside the U.S. markets, so that they now close at 10 P.M., equivalent to 4 P.M. in New York. This facilitates pairs trading, even though volume on the European exchanges at the 10 P.M. close is considerably lighter than during European business hours. If Asian markets trade electronically 24 hours, it may be possible to include the Topix, Hang Seng, Nikkei, and others, but those markets won't be included here.
Again, we point out that if either leg of the pair does not trade because of a holiday, then no entries or exits are allowed, but daily profits and losses are calculated for the leg that is open. Futures trading is always marked to market each day. When we combine the results of the different pairs into the final portfolio, we must pay attention to any missing days, where there was no trading in one region but trading in the other. The results must be aligned by date.
We already know from previous discussions that shorter calculation periods are most likely to work because they capture more price noise, an advantage for mean-reversion strategies. In this case, we just test a small range of values to confirm that the same parameters work for the index markets as they did for others. We are hopeful that the concept is robust and will produce orderly results. There are only six tests, calculation periods of 4 and 5 days, and entry thresholds of 40, 50, and 60. A commission of 25 currency units (USD, EUR, or GBP) per round-turn per leg was charged to all trades, which is very conservative. Table 4.11 shows the results in terms of the information ratio.
TABLE 4.11 Average information ratio for 28 index pairs, from November 2005.
The results do not distinguish between good and bad pairs and are simply the average of all 28. Because of that, we consider these quite good and confirm our expectations that faster trading is better. The final decision will be whether the profits per contract are large enough to offer some comfortable cushion to absorb unexpected slippage above the $25 commission cost. We know from experience that the fastest parameters, a period of 4 and short entry threshold of 40, will have the smallest unit returns but the best ratio. We'll look at those results in more detail.
Table 4.12 shows the basic statistics for the fastest parameters, 4 40. Only 4 of the 28 pairs show net losses; three of those pairs are combinations of U.S. markets, and the other is NasdaqEuroStoxx. If we try to cla.s.sify the pairs into U.S. versus Europe and relate that grouping to the performance, we find some consistency. For example, the higher the correlation between the two legs, the lower the ratio and performance. If we sort Table 4.12 by the correlation column and then create a scatter plot of the correlations against ratios, we get the results in Table 4.13 and Figure 4.12.
TABLE 4.12 Results of all index pairs using a calculation period of 4 and entry threshold of 40.
Figure 4.12 is most descriptive. At the top left, the information ratio is greatest and the correlation smallest. At the bottom right is the opposite combination, high correlation and low performance. There seems to be a break in the chart at the horizontal line representing an information ratio of 1.0. Below that line, the values are spread out and less orderly; above the line, they form a clear pattern up and to the left. We can now look back at Table 4.13 and understand which pairs are most likely to succeed.
TABLE 4.13 Index pairs in Table 4.12, sorted by correlation, highest to lowest.
FIGURE 4.12 Scatter plot of index pairs showing the relationship between the correlation of the two legs and the ratio of the performance.
Starting from the top of Table 4.13, we can see that most of the combinations involve one U.S. index market and one European index market. The correlation of 0.431 between the Russell 2000 and the FTSE shows that these markets have only a modest relationship; that is, they move apart due to very different fundamentals but revert to the mean because all index markets react to global economic factors. Even more important than the global factors is the way traders buy and sell these markets, forcing them into similar patterns. While the economy of the U.K. may be perceived as quite different from that of the U.S., when the U.S. economic reports show unexpected strength or weakness, the FTSE reacts to that information. Similarly, there is a smaller but noticeable reaction in other countries if the Bank of England is the first to lower or raise interest rates after a period of stable rates.
Within Europe, the most profitable pairs are between the British FTSE and the German DAX or the EuroStoxx. These represent the widest difference in fundamentals within Europe, with the U.K. not part of the EU and seen as a weaker economy at the moment. But with most of the total trade occurring between these two regions, their economies are clearly linked, and their equity index markets must reflect that relationship.
The only pair of U.S. equity index markets to make the cut is the Russell-Nasdaq, showing the highest correlation, 0.933. All of the U.S. equity index pairs show very high correlations, making the unit returns necessarily small. At the bottom of the performance list are three U.S. pairs. Although none of the same companies are part of the S&P 500 and Russell 2000 (large cap and small cap), they move in the same way and provide no opportunity for profit.
FIGURE 4.13 Momentum values for the S&P and Dow minicontracts during 2008 and 2009 using a calculation period of 8. None of the momentum differences reaches 40.
A clear example is the S&P-DJIA pair. Of course, all 30 stocks in the DJIA are also in the S&P, and because the DJIA stocks have the largest cap, they represent a disproportionate part of the S&P. The correlation is shown as 0.949 in Table 4.13. Figure 4.13 shows the momentum difference, the basis for the pairs signals, based on a calculation period of 8 days. While the two individual momentum values range from 0 to 100, the momentum difference reaches near 40 only once since 2000 and peaks over 20 only a few times during the entire period. Table 4.12, which has the detail for the 4-day momentum, shows that there were only five trades over the past five years, and those did not generate enough profit to overcome costs.
FIGURE 4.14 (a) Sample PL for S&P pairs (4 40) with European equity index futures, converted to USD, and (b) sample NAVs for S&P pairs with U.S. equity index futures.
Pairs using the S&P futures contract are a typical example of results. In Figure 4.14, we see the NAVs of the seven pairs. The calculation of the NAVs will be given in the next section. For now, Figure 4.14a shows the best results are for the FTSE and CAC, with correlations against the S&P of 0.513 and 0.577, compared with the EuroStoxx and DAX with correlations of 0.823 and 0.791. However, all four results could be combined into a profitable portfolio. Note that the intervals with horizontal lines indicate there was no trading.
Figure 4.14b shows the results of the U.S. equity index pairs. The DJIA, the bottom line, shows long periods of no trading, confirming the high correlation exhibited in Figure 4.13. In the middle, Nasdaq has trades but can't overcome the commission cost, and at the top, the Russell shows positive returns, but all based on one distortion during September 2008. We would not want to base our expectations on having to repeat a 50% drop in the stock market.
Portfolio Diversification Combining performance into a portfolio shows the value of diversification. To make this a manageable example, we will use only the four S&P pairs with the European index futures, the CAC, DAX, EuroStoxx, and FTSE, shown in Figure 4.14a. We start with the daily profits and losses and combine them, first adjusting each to a target volatility and then adjusting the final portfolio to a target volatility. Each step is explained in the following.
Step 1. Align the Daily Profits and Losses Table 4.14 starts on the first day of available data. The first pairs trade is initiated on December 8, 2005, and doesn't show a profit or loss until the close of the next day. In the left panel, the daily profits and losses are aligned by date. When using U.S. and European pairs, there will be many days when one or the other market doesn't trade, but the alignment process affects the signal generation rather than the profits and losses shown in the table.
TABLE 4.14 Constructing a portfolio with an annualized volatility of 12% from daily profits and losses.
Step 2. Normalize the Volatility and Find the Investment Size To give each pair an equal chance to contribute to the portfolio profits, we need to equalize the volatility of each pair. First calculate the standard deviation (volatility) of the return streams (not the c.u.mulative profits) of each of the four pairs, and multiply each by the square root of 252 (the nominal number of business days in a year) to get the annualized volatility. Note that the profits and losses have first been converted to U.S. dollars using the daily spot exchange rates for the euro and sterling.