November 27, 2008

My latest trading formula. Over the past few weeks, I have been working on ways to mathematically express some of the performance behavior of my underlying trading philosophy in an attempt to better understand the whole process. My trading methodology uses partial excess equity build-up to acquire more shares on the way up and consequently slightly leverages the portfolio. See my **Jensen Modified Sharpe **paper for a more elaborate exposé.

This is kind of a follow-up to the previous page and an attempt to provide more clues as to what to look for, even if it is out of the beaten path.

Here is my latest equation:

It says: the current value of stock holdings minus the total cost of said holdings equals the sum of current net profits. The middle section of this equation is part of a class of equations that provides a link between my trading philosophy and performance. It lacks some of the refinements proposed in my last paper, such as protective stops, equivalent switching alternatives, and return enhancers. However, even in this simple form, it should be sufficient to make my point crystal clear. It is rare that we can write a trading methodology that can span years into a simple mathematical function except maybe for the most obvious of cases like the Buy & Hold.

The middle equation, being a power function, stipulates that net profits are on an exponential curve and of a simple form: Ax^{2} +Bx + C. Thereby, you can predetermine the amount of trading profits that the trading philosophy can generate based on how much capital will be put at risk (meaning how much money you want to put on the table) and how you intend to trade. The sum of net profits on this exponential curve will be determined by the time it takes to reach (P_{t}) from (P_{o}) and the size of your initial (i_{q}) and ongoing bets (a_{q}).

To have exponential profits, the above formula can use leverage to a certain degree. This is done by reinvesting part of the accumulated excess equity similar to dividend reinvesting. Again, see my **Jensen Modified Sharpe **paper for more on this subject.

Without the reinvestment of excess equity build-up (meaning a_{q} = 0), the equation would be reduced to:

where only the initial quantity bought and the price differential would matter, resulting in the simple Buy & Hold scenario.

The Buy & Hold strategy can be represented as:

where the end value depends only on the price appreciation over time. What is wished for here is to transform this equation into a more dynamic form:

where the quantity is put on a compounded delayed fractional rate of return (r'): resulting from reinvesting part of the ongoing profits in the pursuit of still higher profits.

There is nothing unprecedented about the previous equation. However, very few traders seem to put it in motion for what it really states. And that is: if P_{t} can be represented as a growth rate over P_{o}, so can Q_{t}. And there lies the beauty of the new equation at the beginning of this note. You can characterize a trading philosophy in terms of pre-determined betting procedures with quantifiable results. You can pre-determine how much you want to put up and calculate your net performance according to the price level reached the same as setting long-term targets.

It gets interesting when one compares methodologies trying to find the strengths and weaknesses of each.

Let's say you have a goal of reaching one million dollars, and let's make it simple: let the stock appreciate tenfold. Under the Buy & Hold, you need to buy 10,000 shares at $10 for $100,000 and wait for the stock to reach $110 in order to make your million. Over a ten-year investment period, this would result in a 25.9% rate of return and 12.2% should it take 20 years to get there. Something in between, should it take less than 20 years time.

Using the re-investment method (top equation), you can reach the same objective by buying 700 shares to start with and buying 235 shares on an ongoing basis as the price rises to achieve the same one million dollar goal. The required capital to accomplish this would be about $12,400, with an initial investment of only $7,000. The return on your $12,400 investment would be 55.1% over the ten-year interval and 24.5% for the 20-year case.

Using the same initial capital of $100,000 for both methods, the Buy & Hold would reach its one million objective while the re-investment method would be able to initiate 8 such series to generate 8 million for the same advancement in price. This would translate into 55.0% over ten years and 24.5% over 20 years on its $100,000 investment. The re-investment method can produce 8 times more than the Buy & Hold (for the case where the advancement in price is 100 points).

Now if the spread in price increases to $150, meaning that the price differential is $150 over the investment period. The Buy and Hold with the same initial investment as before (10,000 shares at $10) would generate one million and a half. On the other hand, the re-investment method using the $12,400, as in the first example, would see the net profit value rise to $2,384,555. Should 8 series of this type be used to simulate the same $100,000 initial investment as for the Buy & Hold, the total return would go up to $19,076,444. This would amount to a 69.1% compounded return over a ten-year period compared to 31.1% for the Buy & Hold strategy. Over a 20-year investment period, the numbers would be 14.5% for the Buy & Hold and 30.0% for the re-investment method.

I am following some stocks that I think will appreciate over 300 points over the next seven to twenty years. Using the Buy & Hold formula, this might translate into a cool 3 million dollars profit. The uncertainty would be not knowing when, over the 20-year timespan, this would occur (stock prices fluctuate). With the same uncertainty, using the partial excess equity re-investment plan, one could start with 6 900 shares as initial shares bought (i_{q}) with 1,900 shares added (a_{q}) on an ongoing process which in total would require the same initial capital of $100,000 as the Buy & Hold method. The result of the same price advancement would be $81,620 500. Simply plug the numbers into the equation at the top to obtain the same results. An alternative would have been to start 8 series as in the previous example, which would have resulted in $100,506,750 for the same price appreciation as if more diversification was to bring higher performance.

A 300-point advance over a ten to twenty-year time horizon might seem high until you start doing some stats on the market. Berkshire Hathaway started at 10 to go as high as 151,050 to last trade at about 100,000. But Berkshire is not alone; hundreds of other stocks far exceeded the 300 points in appreciation.

If you combine this with the trading methodology presented in **Jensen Modified Sharpe**, where best performers are rewarded while laggers are neglected, you are improving the odds in your favor to achieve outstanding returns. And if you add protective stops, alternative switching, parabolic exits, and return enhancers, you are starting to have more than a trading methodology: it starts to develop into a trading philosophy. And since the trading is done on the way up, even if a stop loss is executed, it should be when your net profits are still positive.

Created on ... November 27, 2008, © Guy R. Fleury. All rights reserved.