July 4, 2012

In the previous chapter: **Changing the Game**, it was presented that even trading randomly over randomly generated stock prices could not only generate a positive outcome to the portfolio payoff matrix but that this outcome could generate exponential growth.

**P**and

**H**matrices having any significance, all the efforts should be concentrated on better understanding their respective behavior and interactions.

**P**is simply the set of daily closing prices of the selected stocks. It could be of any size; one could select the 100 stocks of the S&P 100 for example, and easily build the

**P**and Δ

**P**matrices for 250 trading days (1 year) or 5,000 trading days (20 years).

**The Implications**

**P**and Δ

**P**matrices for backtesting the S&P 100. All the numbers are out there. You can analyze past data series from every angle using any simulation software available. However, all past data will remain just that: past data.

**Alpha Generation Excel File**

The conclusion of the Excel file can be showcased in a single graph:

**Generated Profits**

(click to enlarge)

**H**applied to a price difference matrix Δ

**P**resulted in the above payoff. The two main aspects for this payoff curve are: first, generated profits increase in time; and second, to a high degree of correlation, this increase in time is quadratic, it increases at an exponential rate.

**F9**and every formula is re-calculated. Having prices as well as all the 2,500 trading decisions randomly generated, no two portfolios could ever be the same from one test to the next. For instance, pressing F9 again produced the following:

**Generated Profits - 2**

**Generating Random Stock Prices**

_{o}+ ΣΔP

_{o}) at t = 0, to which is added the sum of all price variations thereafter.

**Stock Prices**

**F9**in Excel will generate an all-new price series for each of the stocks in the portfolio. From any point in time of the generated price series will there be any type of statistical or technical analysis to help determine what is coming next.

**The Price Difference Matrix**

**P**matrix is a simple subtraction and should present little interest. It is nonetheless required for the calculation of the final payoff matrix. Its equation is: ΔP = P(t) – P(t-1). The Δ

**P**matrix is the difference in price from row to row (day to day) of the

**P**matrix; another way of expressing the variation in price from close to close.

**Stock Prices Variations**

**The Trading Strategy**

**Jensen Modified Sharpe**from page 28 to 35.

**Stock Holdings**

**Exectured Trades**

**The Payoff Matrix**

**H**to the price difference matrix Δ

**P**. The payoff matrix shows the daily profits and losses generated as time advance. Since the

**P**matrix had the ability to generate outliers, these can be found in the large drawdowns that appear here and there over some of the price series. There are also up gaps as can be seen in the graph. The point is that none of those outliers (gaps up or down) could be predicted.

**The Portfolio Generated Profits**

**H**.*Δ

**P**).

**F9**will generate a new scenario. And, in final analysis, this graph is the only one of interest. It says how much was won or lost for that particular randomly generated trading strategy as applied to the randomly generated price series.

**H**and Δ

**P**matrices as someone else. The future is unknown and this payoff matrix is surely a representation of this.

**F9**key is pressed, a totally new trading strategy is applied to the newly generated price series. Yet, the output of the payoff matrix Σ(

**H**.*Δ

**P**) remains positive. All the academic literature over the past 60 years says that randomly trading over randomly generated prices leads to zero alpha. And here you have this Excel file, not only generating alpha but exponential alpha.

**Conclusions**

All the above not only shows that one can produce alpha, even on randomly generated data, but that even using randomly generated trading strategies; this alpha will be of the exponential type. It implies that the rate of return will increase in time. Starting at zero, this rate of return will increase at an exponential rate.

It is not about achieving 100% per year from year one, it is about gradually increasing the rate of return to even exceed 100% as time progresses. It is in the nature of compounding returns and in the formulations presented in my original papers and in all my webpage notes for that matter. All my simulations also make the same point: accumulate and trade over the accumulation process. And by adopting such trading procedures your own trading strategies will also get a boost.

In the next few days, I will be adding a link to the Excel file that generated all of the above. I think it will enable anyone to “play” with my working model. I don't advocate trading randomly, there are trading methods that can do much better than what this Excel file tries to demonstrate.

… to be continued …

Created... July 4, 2012, © Guy R. Fleury. All rights reserved