Part 8 (1/2)

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 relations.h.i.+p 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 relations.h.i.+p 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.