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Statistical approach to portfolio building

Introduction

In previous articles in this series, we became acquainted with the nature of stock pair trading (link and link). We have learned that some 60 - 65% of transactions are profitable and that thanks to the number of trading opportunities, a trading portfolio with a 95% or even 99% likelihood of profit can be put together. The objective of this article is to inform you in detail about the process of building a portfolio of stock pairs and about the statistical approach to trading.

Basic premises in portfolio building

Stock pairs manifest a stable 60 - 65% of profitable transactions, at an RRR of 1:1. The frequency of trading is relatively low – about 10 – 15 transactions per pair in one year. Stock pairs offer a great number of trading opportunities (see the previous part of this series). Thanks to that, it is easy to achieve broad portfolio diversification. By nature, a stock pair is hedged against marked fluctuations, it is a market-neutral strategy. This property is further enhanced in the case of trading a pair portfolio. Trading can be fully automated, making it undemanding in terms of time (approximately 10 minutes before the markets close). This makes trading effortless from the psychological point of view. The risk of error due to stress, human error, or loss of self-control is low.

Stock pair equity

Adopting the hypothesis that the profitability of a strategy is based on a statistical advantage based on the very nature of stock-pair trading, we can consider all stock-pairs equal. In practice, it is becoming evident that certain stocks or sectors are more suitable for pair trading, but we will get to that in one of the future articles in this series.
If all stock pairs are equal, their transactions must also be equal. The equity curve of a stock pair is then the sum of random transactions (of which XY% are profitable). Given that an average pair generates approximately 10 – 15 transactions per year, the annual equity of a pair may be rather variable (we are working with a small sample of data).

Examples of the equity curves of several stock pairs:

Stock pair equity portfolios

If we combine several stock pairs in a portfolio, the total number of transactions will increase and the equity becomes significantly smoother (we are working with a larger sample of data):
Sample of equity curve and performance of portfolio of stock pairs
Someone may think of improving a portfolio’s equity by eliminating pairs that stagnate or have recently noted a drop. That would, however, go against the initial premise that says that all pairs and all of their transactions are equal. We do not know whether the next transaction will be profitable or loss-making. We do not know whether a given pair will be profitable or loss-making in a given year. We only know that in the past, XY% of pairs (transactions) were profitable. The more pairs (transactions) we execute, the higher the probability that we will make a profit at the end of the year.
It is similar to playing the roulette at a casino. The casino operator does not know when and which player will have the number drawn. He does not need to know. It fully suffices that statistics are on his side. The more players and the more rounds played, the higher is the casino’s certainty of a final profit.

Likelihood of putting together a profitable portfolio

The above is the cornerstone of the building of a stock pair portfolio. Given that an analysis of an extensive set of historical data has shown in a back test that XY% pairs (transactions) are profitable, we can put together a portfolio with an expected profitability approaching 100%. How do we do that?
The graphs below show the likelihood of the selection of a P-element portfolio in which there are at least N profitable pairs. For example, the first graph shows the likelihood for a 50-element portfolio comprised of a set of pairs with a success rate of 50% (50% of pairs are profitable). We can see that in line with expectations, there is a 50% chance that we will choose 25 profitable and 25 loss-making pairs. The likelihood that there will be at least 30 profitable pairs is 10%. On the other hand, the likelihood that there will be at least 20 profitable pairs is over 90%.
Portfolio with 50 pairs, win percentage 50%, RRR 1:1
Portfolio with 50 pairs, win percentage 60%, RRR 1:1
Portfolio with 50 pairs, win percentage 65%, RRR 1:1
Portfolio with 100 pairs, win percentage 60%, RRR 1:1
Portfolio with 100 pairs, win percentage 65%, RRR 1:1
We can see that the likelihood that the portfolio will be profitable grows rapidly with the percentage of profitable pairs (transactions) but also with the number of pairs in the portfolio. With 60% of the pairs profitable, the likelihood of the profitability of a 50-element portfolio is 94.3%; for a 100-element portfolio, it is 98.4%.
Stock-pair trading is based on a statistical approach. We are essentially not interested in knowing whether the selected pair has just made or lost money. Just like a casino is not concerned about knowing when and which player wins, but tries to attract as many players possible. It knows that statistics are on its side and the more bets are placed on a repeated basis (transactions), the higher the probability of a profit. In other words: when trading in stock pairs, we do not bet on one pair or on a few pairs. We try to trade with as many pairs as possible.
In the next part of our series, we will introduce the Stock Pair Builder program designed for identifying stock pairs, for their detailed back testing, and for putting together a stock pair portfolio.