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Matched Pair Trades:

by Howard Simons, NQLX's Special Academic Advisor.

Covariance and the related measure of correlation are nothing more than the degree to which two variables, such as two stock prices, move together. Any stock has a correlation of 1.00 with itself, a correlation of -1.00 with its short position, and a correlation of 0.00 against something totally random.

The concepts are absolutely critical to both modern portfolio theory and to the emerging world of single stock futures (SSFs). For portfolio theory, the covariance is important because of its effect on total risk. Let's say you own both UAL and AMR. These two airlines are affected by the same macroeconomic and industry-specific factors, and we should expect them to have a high covariance. The formula for their combined variance would be:

Variance (UAL+AMR) = variance (UAL) + variance (AMR) + 2*covariance (UAL+AMR)

Over the past four years, the daily variance of returns for UAL and AMR has been .114% and .111%, respectively, and their covariance of returns has been .086% per day. Add these two together in their initial position sizes, and the daily portfolio variance jumps to a whopping .398%. If any of this sounds familiar to those of you who diversified by owning every tech stock available back in 1999, it should.

However, if we go long one and short the other, we now subtract the covariance term:

Variance (UAL-AMR) = variance (UAL) + variance (AMR) - 2*covariance (UAL+AMR)

The variance of this matched pair trade falls to .052% per day, less than half of each individual stock's variance. True, you won't make anywhere near as much as you would if you were right on both, but you won't lose as much if you're wrong, either.

Let's take the following group of nine pairs of stocks:

General Motors / Ford

Qualcomm / Nokia

Tyco / General Electric

Pfizer / Merck

Citigroup / JP Morgan Chase

Home Depot / Wal-Mart

Coca-Cola / Pepsico

Procter & Gamble / Johnson & Johnson

ChevronTexaco / ExxonMobil

Now let's construct a correlation matrix of each stock's returns, not only against those of their designated partner, but against the remaining sixteen stocks as well. The table below can be read like a mileage guide in a roadmap. Each stock has a 1.00 correlation with itself. The maximum correlation in each column is highlighted; in all cases, the maximum correlation is, unsurprisingly, with the designated partner.

Now let's take the pair with the highest correlation, Citigroup / JP Morgan Chase. We could trade this spread by buying the SSF of one and selling the SSF of the other. However, a simple one-to-one spread ignores the different statistical characteristics of these two stocks. A regression of Citigroup as a function of JP Morgan Chase over the past four years yields the following equation:

Citigroup = 25.57 + .30 * JP Morgan Chase

The highlighted coefficient, the .30 number, might be interpreted as saying Citigroup is roughly one-third as volatile in price as JP Morgan Chase, and therefore we may decide to trade three Citigroup futures for each JP Morgan Chase future. The two spread trades, one adjusted for this hedge ratio and the other left as one-for-one, are depicted below.

Which Should You Use?
SSFs can be used both for speculation and for hedging. If you think Citigroup is going to outperform JP Morgan Chase, the trade that reflects your judgment is to buy Citigroup futures and sell JP Morgan Chase futures. If you already own JP Morgan Chase and want to hedge your holdings, you might want to calculate a hedge ratio for the number of Citigroup futures to sell.

Either way, you'll be able to take advantage of SSFs and their ability to facilitate the short side of any stock transaction.

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