Hello Guy, if you don't mind can you disclose what has been your actual CAGR for past 10 years on your portfolio? And on what kind of size. like above 1 million or above 10 million or 100 million. And are you sharing your strategies that helped you get those kind of returns in this book ?

Thanks,

Rajiv,

Building your Stock Portfolio is all about the mathematics of the game. It states that a trader will have to live by the following equation: A(t) = A(0) + n∙u∙PT. That he likes it or not, that is it. And as such, it also says that the vast majority of trading strategies, over the long term, will do about the same as market averages, or less. The book states that if you want more, you will have to do more, literally.

If you don't, well, that is a matter of choice. But at least, I made my part, and that is to show how it can be done. What are the tools to do it. And why over the long term an individual, or an organization, will achieve more than just average.

You don't need to be a millionaire to design trading strategies. You only need to understand what you want to do, and then program. Just as I don't need to be rich or an engineer to drive a car, I only need a key.

You want more, or not. It is always a matter of choice.

@Guy..You state, 1."The strategy was expected to do about the same number of trades, had a constant trade unit, and PT tended to a limit." 2. "Doubling profits over a 20-year period is a 3.5% CAGR." First, equity is doubled, not profits. Second, A constant constant contract basis does not allow compounding of returns, as assumed in CAGR. I highly recommend Ralph Vince's, "The Mathematics of Money Management" series. What I see in your articles so far is the addressing the stability of the decision making strategy.

John, maybe some misunderstanding.

No notion of contracts, it only dealt with stocks. n∙u∙PT, is the strategy's total generated profit.

For the case presented, over a 20-year test period, the strategy's signature did: A(t) = A(0) + n∙u∙PT. If you let it run another 20 years, it would do about the same: A(t) = A(0) + n∙u∙PT + n∙u∙PT, due to its large n. Doubling profits over a 20-year period is a 3.5% CAGR.

Profits doubled, not equity since that would require: A(0) = n∙u∙PT. That too is a 3.5% CAGR. We should program for at least 10 times more over a 20-year period: n∙u∙PT > A(0)∙10.

n = number of trades is a counter, u = trading unit (u=q*p) a constant, and PT = net average profit percent per trade (tending to a limit).

As for decision making stability, you still would not know when, which stock, at what price, in what quantity a future trade would be taken.

You state, "n = number of trades is a counter, u = trading unit (u=q*p) a constant, and PT = net average profit percent per trade (tending to a limit)." The limit is the asymptotic argument with respect to the number of times that the game is played . "PT" is the expected value or average returnof the distribution of the decision making strategy. "u = trading unit (u=q*p) a constant" is the constant contract basis that I referred to earlier. Your statment that "n*PT" is the number of times the game is played times the expected value of the decision making strategy. A binary game of a fair coin toss has 2 outcomes, betting that the event is H is the strategy, gain .5 for a H and lose .4 for a T with P(H) = .5 and P(T) = .5. PT is .5(.5) - (.4)(.5) = .25 - .20 = .05 = expected value. Thus playing the game COMPLETELY 2 times yields an EXPECTED value of (n =2)(PT = .05) = 2 * (.05) = (2 *.05) = .1 by linear property. Note an expected value is a mode of the distribution or the average

Now, we come to the errors in your logic. Let's address the binary game. A Head is a winning trade and a Tail is a losing trade. Suppose the game is played 2 times. A P(T) = .5 implies that the strategy loses 1 time. A P(H) = .5 implies that the strategy wins 1 time. The expected value is calculated from the ENTIRE distribution of returns of the trading strategy. This means that the expected value, PT, is calculated from playing the game n times. ( .5 + (-.4))/2 = (.1)/2 = .05 . If the distribution is stable for 4 plays, (.5 + .5 + (-.4 + .4))/4 = (1 - .8)/4 = .05. PT = (losses + gains)/n = average of decision making strategy = Sigma(decision making strategy)/ number of trades. This means that you implement the decision making strategy over n trades, and PT is the mode of the distribution. Clearly, you do not know the future trade outcome or particulars of trade. The assumption is that the distribution of the decision making strategy is stable or consistent.

The PT is a sample mean or sample average return. You want to show that a stable decision making strategy's sample mean converges in distibution to the population average trade. This is the limit of n * PT.Here is how you show this from the kind folks at Penn State. http://sites.stat.psu.edu/~dhunter/asymp/fall2002/lectures/ln02.pdf

Now, we have cleared up the difference between the sample average return and the population average return. continued....

Let's address the differences between reinvesting and a constant contract basis. In https://www.linkedin.com/pulse/stock-trading-strategy-math-i-guy-r-fleury , you state, "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." You state "n∙u∙PT, is the strategy's total generated profit." This is true ONLY with zero reinvestment. u*number of trades*((sum of returns)/number of trades) = u*(sum of returns). PT is the arithmetic average return. CAGR uses the geometric average. The geometric average is used only in reinvestment of returns. Succinctly, your analysis applies to only zero reinvestment. Thus, compounding does not apply. The guys over at NYU do a much better job than I, http://people.stern.nyu.edu/wsilber/Geometric%20Average%20Versus%20Arithmetic%20Average.pdf . continued...

You state, "For the case presented, over a 20-year test period, the strategy's signature did: A(t) = A(0) + n∙u∙PT. If you let it run another 20 years, it would do about the same: A(t) = A(0) + n∙u∙PT + n∙u∙PT, due to its large n. Doubling profits over a 20-year period is a 3.5% CAGR." Proof by contradiction. Let A(0) = initial investment = 1. Let 3.5% be the arithmetic average growth rate. Suppose we use your zero reinvestment framework. Suppose that n*u*PT is 20% of A(0). Thus, A(20) = 1 + .2 = 1.2. This that the account is up by 20% in 20 years, which implies a 1% arithmetic average growth rate over the next 20 years. Thus, 1% is less than 3.5%. QED. A few side notes, the Arithmetic average return is always greater than the geometric average return. Differences exist between trading strategies with reinvestment and zero reinvestment.

I recommend Stanley Pliska's book, "Introduction to Mathematical Finance" and Ryan Jones's book, "The Trading Game".

Good Luck

John, I don't know how to answer your 6 posts. Some of it holds, some not. Like, you start with: <...Your statment that "n*PT" is the number of times the game is played > No, n is the number of trades, the number of bets made over the portfolio's lifetime. The expression: “n*PT” does not make any sense... n is unlimited, while PT tends to a limit due to a large n. All “n*PT” says is: a fraction of the bets. Okay, so... what...

You say: <...A binary game of a fair coin toss has 2 outcomes > Yes. 50/50, expectancy: ZERO. Short term, long term, whatever: expectancy: ZERO.

Then, you design a biased game with no demonstration of where the edge comes from, as if out of the blue. To me, totally irrelevant to the subject. It does not make the 0.4 valid. You are playing a binary game or not.

I'll pass on the others, too time consuming.

@Guy..(1) Do you have a revolutionary probability analysis to bring to fore? Given the current probability framework, your analysis is fraught with errors. I have provided references to the current framework to support my statements in my previous posts. I will go in reverse.

https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter6.pdf

First, I had a typographical error. I typed, "This is the limit of n * PT.Here is how you show this from the kind folks at Penn State." This was supposed to be, "This is the limit of PT. Here is how you show this from the kind folks at Penn State."

You state, "You say: <...A binary game of a fair coin toss has 2 outcomes > Yes. 50/50, expectancy: ZERO. Short term, long term, whatever: expectancy: ZERO" The expected value of the presented binary game differs from zero. Sigma represents the greek letter Sigma, which signifies addition. The current probability framework defines expected value as the Sigma(Probability(Outcome(a))*Value(Outcome(a))). This is how I computed the Expected Value = .05 in the binary game. continued....

You state,"you start with: <...Your statment that "n*PT" is the number of times the game is played > " No, you left out the rest that followed. I stated, " Your statment that "n*PT" is the number of times the game is played times the expected value of the decision making strategy." I was putting into words the symbolic representation of your definition.

You state, "The expression: “n*PT” does not make any sense... n is unlimited, while PT tends to a limit due to a large n. All “n*PT” says is: a fraction of the bets." continued...

No. First, you state, "n∙u∙PT, is the strategy's total generated profit." This is a true statement with zero reinvestment. You state, "u = trading unit (u=q*p) a constant. " Note, the expected value = PT is the sum of the probability of the outcome times the outcome. The probability is (1/n). By linearity, n*PT = Sigma(n * (1/n) * outcome) = Sigma( (n/n) * outcome) = Sigma(outcome). By linearity, This is what makes true your statement, "n∙u∙PT, is the strategy's total generated profit."

All of my other statements hold true, unless you have (1). I am ALL ears to learn something new. Good Luck

John, this is starting to be a futile debate. What I wrote is so clear and simple.

A stock trading strategy can have:

A(t) = A(0) + Σ(H.*ΔP), using payoff matrix notation, or

A(t) = A(0)∙(1+r+α)^t, using a compounded rate of return, or

A(t) = A(0) + n∙u∙PT, using a trading strategy's unique signature

They all give the same answer, the final result over the same time interval. They are simple bean counters. Nonetheless, you can extract information from all three.

For instance, using n∙u∙PT, you will have the number of trades taken for the entire trading interval. For a trader, this number might run in the thousands, if not the hundreds of thousands over the years. And as such will practically force the trading strategy used to have a relatively stable long term edge (PT>0). This edge can also be viewed as the net average profit per trade (u∙PT) given by any of the simulation software we might use.

… continued

The advantage of the expression: A(t) = A(0) + n∙u∙PT, is that it reduces the total outcome of a particular trading strategy to 3 of its portfolio metrics. And with it we can make some estimates which the other two did not provide.

If a particular trading strategy does 100,000 trades in its portfolio over a 20 year period, it might do about the same over the next 20 years. Because that is the trading strategy's signature. It is all it does. On average, n trades per trading interval. Here, 5,000 trades a year, with an average profit nearing (u∙PT). Extrapolating, one should expect about the same going from period to period, or going forward.

Design another strategy and the signature will be different. But the bean counting will be the same. This other strategy will have an average profit nearing its own (u∙PT), and do a different number of trades of its own. But the equation: A(t) = A(0) + n∙u∙PT will stand.

… continued

You run one of my trading programs. I say you will get the same answers I did, which would be: A(t) = A(0) + n∙u∙PT, since it is that trading strategy's signature. You would not get more, you would not get less, but exactly the same results.

What my book is all about is pushing a trading strategy further. Using an existing trading strategy with its own signature and making it do more, as in: A(t) = A(0) + (1+g)^t ∙n∙u∙PT. And this, will change your game. All that will be required is g>0, and since g is under your control, it is easy to make it positive. And therefore, improve on your own trading strategy by making it do more.

Anyone can transform their own trading strategies in this way, not by optimizing the code by overfitting it, but by designing what I consider simple administrative procedures that would work even if done by hand.

https://www.youtube.com/watch?v=0BYRyEB-tb4