January 19th, 2017

Stock Trading Strategy Math I

In my last article, A Stock Trading Strategy Signature, I presented a 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.*ΔP). Making: Σ(H.*ΔP) = n∙u∙PT. As a matter of fact, if you break down the payoff matrix Σ(H.*ΔP) 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.*ΔP).

This generalization was 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 fix dollar amount you put on a trade. Its equation: 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 life span. 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 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 is easy to determine: Total sales = Σ(H.*ΔP) + n∙u. The total profit plus the cost of all trades. This, whatever the composition of the portfolio or its duration. Also, your objective, as a trader, is to have: Σ(H.*ΔP) / n∙u > 1.0. 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 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.*ΔP)

They all end up with the same number: the total profit or loss generated over the trading interval, the life span 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, better get ready to trade a lot.

The equation: A(t) = A(0) + n∙u∙PT also state 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 in 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.

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