Once we have those values, shown in Table 4.15, we take the target volatility of 12% and divide that by the annualized volatility of each pair to get the investment size that would make the profit and loss changes equal to 12%. The total portfolio investment is the sum of the individual investments, $1,280,915. We could also have started with an investment size and converted the daily profits and losses to returns using that value, but we will cover that method later.

TABLE 4.15 Annualized standard deviation and corresponding investment needed to have an annualized volatility of 12%.

Step 3. Calculate the Returns for Each Pair The second panel of Table 4.14, Daily Returns from Total Investment, is simply the daily profits and losses in the first panel, divided by the total investment size, the sum of the individual investments.

Step 4: Create the NAVs The final NAVs must reflect our target volatility, the amount of risk we are willing to take. When we calculate the annualized standard deviation of the portfolio return column, we get 0.2236, more than 22%, an unacceptably high number. To adjust to our 12% target volatility, each return must be multiplied by 0.537. Then the NAVs begin at 100 and each subsequent NAV is The final NAV stream at 12% volatility is shown in Figure 4.15. Although the performance flattens out in 2009, a portfolio of more pairs would add diversification and consistency. All pairs using the S&P are likely to have similar performance.

FIGURE 4.15 Portfolio of four S&P pairs adjusted to 12% volatility.

An alternative approach to constructing this portfolio, and one that will be useful when using diverse sectors, is to a.s.sign an arbitrary or an actual portfolio investment size. In this case, we can still use the sum of the investment sizes calculated using our current method, $1,280,915. We then Calculate the returns of each pair by dividing the daily profit or loss by the investment size, the same as the returns shown in panel 2.

Next, find the annualized volatility of the returns of each pair. Only the one with the highest volatility, the FTSE, will show 12%. The others are The factor in the table is the multiplier that changes the volatility to 12% for each return's stream.

We calculate the portfolio returns by using Excel's sumproduct function, which multiplies each daily return by the factor and adds them together to get a portfolio return for that day.

Again, we find the annualized standard deviation of the new portfolio returns and get 0.025. Dividing the target volatility of 0.12 by 0.025 gives a new return factor of 4.863.

The final NAVs are created using the last formula given, but the return factor is 4.863 instead of 0.537.

You might think that this second method is unnecessary because it gives the same results as the simpler method shown first. However, when dealing with portfolios composed of diverse sectors, such as interest rates, equity index, and foreign exchange (FX), you must have each sector adjusted to the same volatility before you combine them into the final mix. For example, if you are allocating to these three sectors equally, and the interest rates have an annualized volatility of 5%, the equity index markets 20%, and FX 25%, then your risk exposure is 10% for rates, but 40% for index and 50% for FX. The profits and losses from interest rates will have only a small impact on the total portfolio. To maximize diversification, you must have all sectors adjusted to the same risk level before you apply your portfolio allocation percentage.

LEVERAGING WITH FUTURES.

During the construction of the futures pairs, a target volatility of 12% was used. That means we have the ability to vary the number of futures contracts that we trade without increasing the portfolio investment size. There is a limit to the amount of leverage you can get using futures, but most investors should expect that their exposure (the face value of the futures contracts being traded) can be 4 to 6 times their investment. That's not possible with pairs using only stocks, where the cost of trading is the number of shares times the price per share. We don't consider stock margining, in which you borrow part of the funds.

We have discussed that margin in futures is very different from equities. Margin is a good faith deposit, where you are obligated to invest on average about 10% of the face value of the purchases. For example, if you buy one contract of crude oil at $80, you own 1,000 barrels, a value of $80,000. Typically, you need only a deposit of $8,000 unless you are a qualified commercial trader (in the oil business, a hedger), in which case your margin might be only 5%, or $4,000. Commercials are often able to provide a bank letter that guarantees any losses; therefore, it is not clear how much they put up as margin or what leverage they get.

Spreads are also given preferential margin. The exchange recognizes that buying one oil product and selling another related product has less risk than an outright position; therefore, the margin is also less, perhaps 5%. For exact amounts and which combinations of spreads qualify, you will need to refer to the exchange web sites, as well as contact your broker. Brokerage firms must conform to the exchange minimum requirements for margin, but they may choose to ask for more if they feel that the risk is higher.

Cross-market spreads, such as the S&P-DAX may not get lower margins, although some exchanges have cross-margin agreements. But let's use energy pairs as an example of leveraging.

Leverage Example An example will make this clearer. Going back to the crudenatural gas pair, we calculate the annualized volatility of the daily profits and losses using the standard method: the standard deviation times the square root of 252. We get $132,671. An investment of $1,105,593 is needed to give that performance a volatility of 12% of the portfolio. However, with stocks we found that the amount needed as an investment was fixed, based on the number of shares times the share value. That's not the case for futures. Table 4.16 gives the necessary calculation.

TABLE 4.16 Calculations needed to understand leverage in futures markets.

For the pairs trade in crudenatural gas, we often took positions of 10 contracts in each leg; therefore, we will use that quant.i.ty here. The first column of Table 4.16 is the current price of crude and natural gas, followed by the size of the contract and the total value per contract. If we buy or sell 10 contracts, we are trading $800,000 worth of crude oil and $500,000 in natural gas, at total exposure of $1,300,000. The exchange considers a spread in two related markets as having less risk and will give lower margin requirements, for example, 5%. The last column shows that the amount needed to buy 10 crude and sell 10 natural gas is only $65,000.

If we could invest only the margin, our leverage would be 20:1. The volatility of $132,671 would be more than 200% rather than our target of 12%. However, brokerage firms require more than margin, although the exact amount varies from firm to firm and is also based on the net worth of the client and the amount of trading activity expected. Ironically, a good client, such as a hedge fund that trades in a large number of contracts, will have lower requirements, and those deposits might be a small amount of cash plus a credit note or some other form of collateral. As with Long-Term Capital Management, the bigger you get, the more you can negotiate, regardless of the exposure to risk.

But let's say you are an ordinary investor and the brokerage firm requires 4 times the margin as your deposit. That would change the $65,000 margin to a minimum investment of $260,000 and reduce the 200% volatility to a fourth, or 50%. Therefore, the amount required in your investment account for trading futures determines the amount of leverage you can achieve. In this case, $260,000 to buy $1.3 million in energy futures gives you a leverage of 5:1.

Varying the Leverage In trading the crudenatural gas pairs, we varied the number of contracts but averaged no more than 10 per leg. Most investment managers target 12% volatility, or less, the same level we have been using in our examples.

Starting with the volatility of 50% based on a margin of 5% and brokerage requirements of 4 times the margin, we can reduce the annualized volatility of the portfolio to 12% by dividing the current volatility of 0.50 by the target volatility, giving a factor of 4.16. Then the investment needed to trade an average of 10 contracts of each leg is 4.16 $260,000 = $1,083,333. The less you invest, the greater the leverage and the greater the risk. Remember that an annualized volatility of 12% means that there is still a 16% chance of seeing a loss greater than 12% in one year, and a 2.5% chance of having a loss greater than 24%. And if you don't see that loss in your first year, the chances are greater that you'll see it in the next year. By anyone's standards, that can be a lot of risk.

LONDON METALS EXCHANGE PAIRS.

The six nonferrous metals traded on the London Metals Exchange (LME) are another group of futures markets that seems natural for arbitrage. This group of markets consists of aluminum, copper, lead, nickel, tin, and zinc. In different ways, each of these markets is related to each of the others through commercial and individual home construction, copper for plumbing, others for stainless steel. It is said that the floor traders will arbitrage any combination of these markets because, during an expanding economy or real estate boom, they all move in the same direction. Figure 4.16 shows that the six markets have similar movements, although nickel seems to be out of phase with the others.

FIGURE 4.16 LME nonferrous metals, from January 2000 through April 2009, quoted in USD per tonne. There is similar movement in all markets, but nickel seems to be out of phase with the others.

Trading on the LME is different from trading on other exchange-traded futures markets. First, trading occurs during four sessions, when each metal is traded for a relatively short period in turn. The traders aren't standing in a pit yelling and using hand signals; most often, they are directors of large metals firms sitting in a circle, called a ring. Although there could be a lot of active trading when prices are moving quickly, it could also look more like a poker game during quiet times. Contract sizes are large, and delivery dates are designated during trading; they do not need to be on specific dates set by the exchange but can correspond to the commercial needs of the trader. For convenience, we can choose a delivery date for copper, for example, that corresponds to the same delivery date as the NYMEX copper contract, but any delivery date is acceptable when trading. For these tests, we use the rolling three-month contract, so that any position entered will deliver three months from the entry date. That gives enough time to exit the trade without the need to roll forward or be forced out because of contract expiration and delivery.

Using the same parameters and rules we have for other markets, we tested the six LME metals from January 2000 through April 2009. The results, shown in Table 4.17, are dismal. Not one combination was profitable. As we noticed in the price chart, Figure 4.16, nickel has the lowest overall correlation to the other metals, but the pairs aluminum-copper, aluminum-zinc, and copper-zinc all have reasonably good correlations. We would have expected them to be profitable. There is always something to be learned by looking at more detail. Sometimes you find a relationship that can make the entire process, including the good results we've already seen, even better-often by reducing the risk. With that end in mind, we will look at the pair of the two most liquid markets, aluminum and copper, at the top of the table.

TABLE 4.17 LME pairs results, January 2000 to April 2009, using parameters 50 x 10.

To understand the price relationship between the aluminum and copper, we need a more accurate view of prices, found in Figure 4.17. The two metals have very similar moves, although copper is significantly more volatile. The bull market that started in 2003 first peaks in 2006 but doesn't collapse until the subprime crisis in 2008. While the copper price drop in 2007 is not seen in aluminum, many of the other moves are similar. Given volatility adjustments for position size, these two look as though they are a good candidate for pairs trading.

FIGURE 4.17 Price moves in aluminum and copper have many similarities that would make them a good candidate for pairs trading.

The returns, however, don't reflect what we think should be true. Profits were generated steadily through late 2006, followed by a large drawdown, and finally another profitable period beginning in late 2007, shown in Figure 4.18. What happened to cause the large losing period? Our first thought is to look at volatility.

FIGURE 4.18 c.u.mulative profits/losses for the aluminum-copper pair. Profits drop sharply beginning in October 2006 and then rally in late 2007.

The standard calculation for volatility is the standard deviation of returns times the square root of 252. Although 20 days is standard for option volatility, we use 14 days to correspond to the same period as the stochastic indicator used for momentum. Using 14 days rather than 20 will be a little less stable; that is, fewer days will tend to show more volatility. In Figure 4.19, we see that copper has been much more volatile than aluminum, spiking to about 80 in 2006, when aluminum reached only 40, and peaking at almost 110 in 2008, when aluminum reached about 45. However, neither of these spikes corresponds to the problem dates seen in the c.u.mulative profits and losses. In fact, when the subprime crisis drove the price of copper sharply lower, the pairs trade was adding steady gains to the total returns. But when a pairs trade has a large imbalance in the position size, then there is potential risk and potential loss. For example, if we had 20 contracts of aluminum and 5 of copper, then we can say that copper is four times more volatile. Because volatility is measured over only 14 days, that value can change quickly. If aluminum became volatile and copper quiet, we would be exposed to large swings in returns. We can say that it would be prudent to limit the trades entered when there is a large position imbalance.

FIGURE 4.19 Volatility of aluminum and copper as measured by the annualized standard deviation.

Dangers of High Leverage As a side note, the large discrepancy between the position sizes means that we have a.s.sumed much greater leverage in one market based on relatively low volatility. This situation is similar to what happened in the hedge fund industry in 2010 for those trading interest rates.

As yields fell to unprecedented low levels, they also showed exceptionally low volatility. If rates are only one of a number of sectors in the hedge fund portfolio, then the returns of that sector must be leveraged up to have the same impact as other sectors and provide both comparable returns and diversification. Then, as interest rate volatility fell, positions got bigger, and at some point they were disproportionally large compared with other sectors. This poses a real threat of disaster due to event risk. A price shock that moves the market against you can wipe out all your gains-and perhaps your entire investment.

The only way to avoid this is to cap your leverage, that is, put a limit on the size of the position you can take based on low volatility. That would reduce the risk, but capping one sector or a.s.set puts the portfolio out of balance. Interest rates would not contribute the right amount to the diversification of the portfolio, and the hedge fund has reduced its diversification.

If maintaining the diversification is most important (and it is an important way to control risk), then all positions in the portfolio must be reduced in the same ratio when the interest rates are capped. That way, the shape of the portfolio remains the same. Unfortunately, potential returns are reduced along with the risk, but you can't have both. Keeping the integrity of the shape of the portfolio at the cost of lower returns and lower risk is the best choice for the investor.

FIGURE 4.20 Volatility of aluminum and copper as measured by the average true range.

Alternate Way of Calculating Pairs Risk We are now concerned that large differences in the annualized volatility, combined with equally different position sizes in the legs of the pairs trade, can result in unacceptably large risk. But the position size is calculated using the average true range (ATR), so we need to see if that tells us anything. Figure 4.20 shows that a structural change took place in early 2006 that may correspond to the drop in performance. Copper volatility spiked above $600/tonne while aluminum reached only $150/tonne, a ratio of 4:1. Although the change in volatility looks clear, we might also find an imbalance before 2006, possibly at the beginning of 2004, when copper volatility touched $100/tonne and aluminum was well down, near $30/tonne. Using the ATR instead of the traditional annualized volatility measure gives us much more information.

To distinguish between high volatility and extreme volatility, we will create a rule that combines both the annualized standard deviation and the average true range.

Do not enter a new position, and exit any existing trade, if both of the following two factors hold: 1. The ratio of the position sizes is less than the position ratio threshold (this can be written as min(size1/size2,size2/size1) < position="" ratio="">

2. The annualized volatility of leg 1 > annualized volatility threshold or the annualized volatility of leg 2 > annualized volatility threshold (called the "high" filter).

To see if our premise works, and to find the value of the thresholds, we run a series of tests on values of both thresholds. For the annualized standard deviation, the choices are 15% through 30% in varying small steps of 5% or less (the high filter), and for the average true range, we test from 0.20 through 0.90, in steps of 0.10. The results are given in Table 4.18. The pattern of improvement is consistent for the ATR, but much narrower for the annualized volatility (STD). We also looked at trading only when the volatility was high-that is, removing trades below the threshold-as shown in the part of the table marked STD Low. Results were actually more consistent than STD High, but the number of trades was very small. If we look at the c.u.mulative profits and losses of the best filter tests, compared with no filters and with both filters, we can see the results in Figure 4.21.

Can we use these results? The ATR and the ATR plus the STD low filter look good, and the result is just what we wanted. The statistics for the combined filters, STD 30 (low) and ATR 35, had 31 trades, $47,071 in profits, 6.5% annualized return, and a ratio of 0.539. There was no trading during the period of loss in 2006 and 2007, and the c.u.mulative profits move steadily higher. That's all good.

On the negative side, we have clearly fitted the results. It's good that the ATR had progressively better performance and consistent ratios, but the positive results from the annualized standard deviation were in a very narrow range, and the best results used the most volatile prices when we TABLE 4.18 Test results showing the comparison of filter thresholds for aluminum-copper.

FIGURE 4.21 Aluminum-copper c.u.mulative PL with volatility filters.

were trying to filter those periods. For that reason, we would reject the annualized volatility results (both STDs) and use only the ATR filter. In Figure 4.21, the combined filters results are identical to the ATR filter results except at the very end, where the use of the STD adds some value.

Distortion Filter In Chapter 3, we introduced the high-distortion filter, or distortion ratio. This is a simple way of recognizing when two markets are acting very differently based entirely on the relative position sizes needed to equalize the volatility of the two legs. The implication is that when the two legs have very different volatilities, then: The leg with the lower volatility and the larger position may have an unusually big move, causing an unusually large profit or loss relative to the other leg.

Or the leg with the lower volatility and the larger position may not move at all, thus not providing the market-neutral effect that we are trying to achieve.

The distortion ratio is simply the minimum of the leg 1 position size divided by the leg 2 position size or the leg 2 size divided by the leg 1 size, so that the distortion increases as the ratio gets smaller. For example, if the leg 1 size is 10 contracts and the leg 2 size is 7 contracts, then the distortion ratio is 0.7; or if leg 1 is 7 and leg 2 is 10, the ratio is also 0.7. In Table 4.19, the distortion ratio is tested against both the 14- (left) and 4-day (right) momentum results of the aluminum-copper pair. In both cases, the results are significantly improved by choosing trades that have similar volatility (higher ratios); however, these levels are different for each of the calculation periods. The position sizes are based on the same calculation periods as the momentum, so it should not be surprising that these levels are different. By fixing the period for determining position size, rather than using the same period as the momentum calculation, these results will be more uniform and interpreted as robust.

TABLE 4.19 The distortion ratio applied to both the 14- and 4-day momentum results for the aluminum-copper pair.

It is equally important to view the results of using the distortion filter. Figure 4.22 makes this clear. The filter does exactly what it is intended to do: remove trades that could present higher risk, and keep those trades with more balanced positions. The 4-day momentum produces twice as many trades as the 14-day, so there are more remaining after the filter is applied. As we can see from the horizontal lines in the figure, there are long periods of time with no trades, although fewer with the 4-day momentum. Our conclusion is that the filter works, but we're not interested in trading it because there is not enough activity. We have learned two important points: 1. The distortion filter has merit.

FIGURE 4.22 Net performance of the aluminum-copper pairs after applying the distortion filter.

2. We can find a way of turning a loss into a profit.

But have we just completed an exercise in overfitting? We could go on to test if the filter works on other LME pairs, but the overall results of pairs trading are poor. In the next chapter, we will change our approach and look at trends in these pairs, but before moving on, there is another example of volatility distortion that makes this distortion index more relevant.

Lead and Zinc If we think of this as an in-sample test, we now want to know if using the ATR as a filter works for the other pairs. We've rejected the use of the annualized standard deviation. We choose to look at lead-zinc because it has an average number of trades (72), average correlation (0.549) compared to the high correlation for aluminum-copper (0.723), and poor performance, shown in Figure 4.23. Returns follow the same pattern as aluminum-copper, gaining small, steady profits until 2006, then dropping precipitously before a recovery begins. Given the same pattern, we would like the exact same solution to work. Unfortunately, that's not the case. While the original test with no filters had 72 trades, a loss of $9,304, and a ratio of 0.100, the ATR filter of 0.35 had 61 trades, a loss of $26,354, and a ratio of 0.178. Why are the results so different?

FIGURE 4.23 c.u.mulative PL for lead-zinc shows the same general pattern as the aluminum-copper pair.

Because we are filtering volatility, a look at the relative volatility of lead and zinc, compared with aluminum and copper, should show why our filter didn't work. In Figure 4.24, the annualized standard deviation is much the same for the two markets, unlike the extreme volatility of copper. We see that lead was more volatile in late 2008, but zinc had been more volatile in mid-2006. Overall, there is not enough of a difference to distinguish one from the other. But then, we are no longer using the annualized volatility.

FIGURE 4.24 The annualized volatilities of lead and zinc show that they are very similar and would not be useful to distinguish risk.

On the other hand, the ATR in Figure 4.25 again shows that it's a better measure of volatility. In 2006, zinc spikes well above lead, much more than copper above aluminum. With its peak at about 270 and the corresponding peak in lead at about 60, zinc is more volatile by a factor of 4.5. Unlike copper, which remained more volatile than aluminum, zinc fell back into its previous pattern and even trades marginally below the volatility of lead.

FIGURE 4.25 ATR volatility comparison of lead and zinc shows a different pattern from aluminum and copper.

TABLE 4.20 Test of the ATR filter for lead-zinc shows profits once past the 0.50 threshold (twice the position size).

Is there a threshold value that turns lead-zinc losses into profits? To find that answer, it is necessary to run tests on various values of the ATR threshold, actually the ratio of the position sizes calculated using the ATR. Those results can be seen in Table 4.20. If you look back at the copper tests in Table 4.18 (the bottom panel), the returns jumped from a large loss to a large gain when the threshold increased from 0.25 to 0.35. Here we started from 0.35 because that value showed a loss, then moved up. As the threshold becomes larger, the relative difference between the position sizes that will cause the filter to be activated will become smaller. For this pair, the trigger value is 0.50, where one position is twice the size of the other. Any trade with one leg twice that of the other will not be entered, and if at any time during the trade, the volatility of the two legs is such that one is more than twice the other, that trade will be exited.

Using the ATR threshold of 0.50, the c.u.mulative profits are shown in Figure 4.26. Again, it is similar to the filtered results of aluminum-copper. It avoids trades during the highly volatile period in 2006, then begins trading again when the relative volatility of the two legs comes back to normal. It seems reasonable. Should you be comfortable trading this method? No.

FIGURE 4.26 c.u.mulative PL for lead-zinc using the ATR filter of 0.50.

VOLATILITY FILTERS.

This chapter gives a simple solution to a basic pairs trading concept. We enter when the two prices are relatively far apart. We exit when they come back together. We adjust the two legs so that we have equal risk and equal opportunity. That worked well for the housing sector and the equity index markets, where all pairs used exactly the same parameters.

We then found that the metals markets do not work well under extreme volatility. It is clear that the metals pairs we observed had very similar patterns and could be solved by creating a volatility filter-that is, a relative volatility filter-to identify when one of the legs was much more volatile than the other. We reason that the more volatile leg could overwhelm the profits and losses, or that the less volatile leg has the potential of becoming much more volatile (possibly due to event risk), causing our risk balancing to fail. Although the same situation occurs with each pair, the volatility threshold would also be different for each one. Is that an acceptable solution?

You may rationalize the results and say that we should expect those differences and that if the threshold is only shifted up or down by some small amount, then the answer is still valid. It may not be possible for anyone to say whether that is right or wrong. The market will tell you. For now, let us say that the better solution is when the parameters and rules are identical for all pairs.

INTEREST RATE FUTURES.

There are many similarities between equity index and interest rate futures. There are at least four exceptionally liquid markets in the U.S. (Eurodollars, 5-year notes, 10-year notes, and 30-year bonds), and there are comparable futures markets in Europe (Euribor, Eurobobl, Eurobund, British long gilt, and the short sterling). As you would expect, there are stronger correlations between interest rate markets than between equity index markets. Our experience says that with high correlations, profits would be very small and survival would depend on extremely low transaction costs, the venue of the professional traders. That does not turn out to be entirely true. There is opportunity trading interest rate pairs when one leg is a U.S. interest rate and the other is European.

Rather than include those markets in this chapter, we will look at them in Chapter 7, "Revisiting Pairs Using the Stress Indicator." Readers are encouraged to try these pairs using the momentum difference method covered in Chapters 3 and 4.

SUMMARY.

Although the rules were clearly specified in Chapter 3 and the same parameters were used for all examples in this chapter, the purpose was to show how a simple method can be used in different markets to create trading opportunities. No rules or calculations have been omitted, but some of the test statistics were summarized and not shown in detail.

To implement this, you will need to put this into a spreadsheet or computer program and verify all the results. You cannot rely on anyone else's numbers when it's your money that is at risk. You will need to understand the process, do the calculations, and place the orders with precision. To help you, there are sample spreadsheets available at www.wiley.com/go/alphatrading.

Chapter 5.