January 19th, 2017

In my last article, **A Stock Trading Strategy Signature**, I presented as model for a trading strategy, an equation. It is derived from the payoff matrix, another expression used to resume a portfolio's entire trading activity over its lifetime. This model has interesting properties.

It, too, resumes in just three numbers, the total outcome of any stock trading strategy:

A(t) = A(0) + n∙u∙PT

A(t), the total portfolio payoff, is equal to the starting capital A(0), to which is added all the generated profits and losses from all the trades taken over the strategy's lifetime (n∙u∙PT). And where PT is the net average profit percent per trade.

It has the same output as the payoff matrix equation: A(t) = A(0) + Σ(**H**.*Δ**P**). Making: Σ(**H**.*Δ**P**) = n∙u∙PT. As a matter of fact, if you break down the payoff matrix Σ(**H**.*Δ**P**) into its basic components, with some shuffling, it is what you will get.

The payoff matrix is an interesting model in itself. You applied a trading strategy **H** to a portfolio of stocks **P**. Using the price differential Δp, you got the total generated profit or loss: Σ(**H**.*Δ**P**).

This generalization works fine. However, it did not give what was in it. It stayed kind of a black box. It gave the right answer. But, gave nothing on its composition. Only its final result.

This is where A(t) = A(0) + n∙u∙PT is able to shine. It states that the output of the payoff matrix is equivalent to the number of bets taken multiplied by the net average profit per trade. The trading unit u (the bet size) is a number you set. A fixed dollar amount you put on a trade. Its equation is u = q∙p.

As such, n∙u becomes the total cost or value of all the trades taken over the life of the portfolio. This includes all closed trades as well as still-opened positions.

The payoff matrix could not give the total cost of all trades, only the total profit. But here, two numbers: n∙u, and you have it.

It also shows how much money will be put in the market over the strategy's lifespan. It will far exceed the initial trading account A(0). The intention is to trade. To flip the whole inventory forward as many times as we can. The trading account reserves are continuously replenished by the proceeds from the closed trades.

Nonetheless, we are missing the origin of the profits. They are what is left after all the sales, meaning the output of the payoff matrix. We could write A(t) = A(0) + Total sales - Total costs. The total sales are easy to determine: Total sales = Σ(**H**.*Δ**P**) + n∙u. The total profit plus the cost of all trades. This is whatever the composition of the portfolio or its duration. Also, your objective, as a trader, is to have: Σ(**H**.*Δ**P**) / n∙u > 0.0. Just like in any business, you want sales to exceed costs.

The expression u∙PT has the same meaning as x_bar, the average net profit per trade. Therefore, we could also translate: A(t) = A(0) + Total sales - Total costs, to

A(t) = A(0) + n∙(x_bar + u) - n∙u.

We now have something that the payoff matrix could not reveal. A byproduct of using a trading unit as a measuring stick. Making all the following expressions equivalent:

n∙( x_bar + u ) - n∙u = n∙( u∙PT + u ) - n∙u = n∙u∙PT = Σ(**H**.*Δ**P**)

They all end up with the same number: the total profit or loss generated over the trading interval or the lifespan of your portfolio.

Note how this puts much more emphasis on n, the number of trades taken. The more you shorten the average trade interval, the more n will play a major role in the outcome. If you intend to trade very short term, you better get ready to trade a lot.

The equation A(t) = A(0) + n∙u∙PT also states that by using a trading unit, you made your trading strategy scalable.

Double the trade unit, and you double the profit. You can scale up or down at will. Evidently, there are limits. We should remain within the feasible and practical side of things. So, forget about a $100 dollars trading unit or one that is larger than the stock's daily traded volume. There, like everywhere else, we need to maintain some common sense. Trading needs to be worthwhile, feasible, and market-executable.

What is the use of having these new expressions?

You can now make reasonable estimates using the payoff matrix. Implying that your long-term backtests might have value, and could be representative of what is to come. It is assuming you did not cheat in your backtests, evidently.

... to be continued ...

Spoiler. The next part should have a tremendous impact on how you will look at the survival of your own automated trading strategies. So, a suggestion, do not miss it.

Created... January 19th, 2017, © Guy R. Fleury. All rights reserved.