## On Performance Metrics

Some of the usual performance metrics became irrelevant if not useless due to the nature of the trading strategy used. For instance, because you are holding on to a major portion of your trades, they tend to show high hit rates when in fact, you are just holding the bag on many still opened positions. So you get hit rates in the 80 and 90% with corresponding high profit per trade and comparatively small losses on losers; not because you accurately forecasted future prices but because you held on a long time on your highest winners and a much shorter time on your losers.

For instance, see the motrader simulation for some examples. Oups, rewind here, that sounded much like: let your profits run and cut your losses. Sorry, the cliché was not intended, it is just a side effect of the methodology.

Even if I had the actual Sharpe ratio, it would only be a point on an exponential curve that I could not forecast going forward and since I do use a lot of random entries, the number would change for every scenario tested. So, I can't use the Sharpe to regulate the portfolio in any manner; let's be kind and say I have not found a way to use it yet and most probably won't even look for one. Anyway, over its 50 years existence, the knowledge of the Sharpe ratio does not seem to have improved average portfolio performances since the secular trend hasn't changed over the period.

The Sharpe ratio would be on an exponential curve simply because when Jensen's alpha is not included in the formula; it then hides in the Sharpe with everything else that provides the premium.

## Sharpe Ratio

The Sharpe ratio increases exponentially because of the added Jensen alpha which becomes exponential in this case.  The plain Sharpe ratio formula stays what it was. In my papers, I usually refer to a Jensen Modified Sharpe ratio to imply that the Jensen alpha is also modified. The chart presented to Silviu has that exact formula. It is when adding Jensen's alpha that you can convert what was seen by Jensen as a measure of added performance due to skills to an exponential function simply by re-investing generated profits. The Jensen measure was intended to be a constant, not an exponential function. All it was designed to say was: of the performance achieved, this much could be interpreted as due to the portfolio manager; and this way we can compare portfolio manager results on their skill levels. A very interesting idea, I would say brilliant.

But here you are, you just transformed a constant into an exponential function simply because you are re-investing the accumulating profits generated by your Buy & Hold strategy. And to help Jensen's alpha to go even higher exponentially, you can trade over the accumulation process with the objective to accumulate even more shares within the limits of maintaining positive equity. In all my tests, all the trading procedures can only add shares to existing holdings if and only if there is cash in the account. There is no margin or leveraged used. The WL4 simulator does not allow it.

## Jensen Ratio

When in 1968 Jensen presented the notion of alpha, I think he was somewhat surprised to find it had a negative value (-1.1). This meant that the value of any type of skills brought to the game where not only worthless, they had a negative impact on one's portfolio. So some explained the negative value was due to fees and commissions. Even accepting this thesis amounted to saying that no amount of talent (how ever it is expressed or applied) could increased long term expectancy, let alone profits. We were doomed from the 60's to have as most likely long term performance: the market average. I understand why the financial industry has put the Jensen ratio under the rug. Imagine the ads: When we invest you listen. We re-balance on Jensen's ratio every week, and we can assure you we will lose your money faster than you can.

But then, you have someone like Mr. Buffett that has achieved over his career some 12 alpha points over market averages. His performance can not be explained using the Sharpe ratio, but it can using the Jensen ratio. Therefore, to me, Jensen was right, skill mattered and could be partially measured since I think Mr. Buffett did put a lot of talent in managing his portfolio. So my first question becomes: where or how did he get his alpha points? Because I want them too. And second, can I improve on his methods and get even more alpha?

All my trading philosophy is summarized in a single matrix equation: Σ(Q.*ITRADE.*ΔP), which when reduced to its simplest form is a Buy & Hold strategy: Σ(Q.*ΔP).

Any trading methodology H, and I do mean any, can be described using Schachermayer's pay-off matrix equation: Σ(H.*ΔS). My equation is just more elaborate but it says the very same thing with: H = Q.*ITRADE. As a matter of fact, your own or anyone's trading strategy could be expressed in this manner.

The Buy & Hold strategy, which I consider often hard to beat, is like saying Mr. Buffett's strategy is hard to beat, could also be expressed as: Σ(H.*P(1+r)t) since ΔP= P(1+r)t) and where the holdings in each stock appreciate at their respective rates of return.  However, if you start re-investing the accumulating profits by buying more shares, then the equation is more like: Σ(H(1+g)t-1.*P(1+r)t). And that is where you start accelerating alpha exponentially simply because you are re-investing part of the generated profits following in step with your price appreciation curve. This means that the matrix ITRADE = (1+g)t-1 and that the inventory is increasing at an exponential rate.

The profits generated by the Buy & Hold scenario start to produce some return instead of sitting idle in the portfolio. And I think that is where part of Mr. Buffett's alpha points come from. He is always reinvesting the accumulating profits in new businesses or increasing his current holdings and thereby is compounding his long term profits at the same time as his long term holdings. Basically, the ITRADE matrix does the same thing and more. The matrix is described in more detail in the on seeking alpha part III article.

## Increasing Performance

But you can compound even more. Instead of just accumulating new shares with the accumulating profits and stopping there (close to Buffett style), you could trade over the accumulative process itself; thereby generating cash that can be fed back to the account to increase buying power. This would result in an ITRADE matrix with an increased performance rate proportional to the skills applied in your trading procedures: ITRADE = (1 + g + T)t-1 . In this equation where g is limited by the delayed utilization of the generated profit rate r, you have added the equivalent of a new rate which is the direct effect of the deployed trading skills. This too will be compounded since as each price cycle you will be increasing the amount of cash pumped back in the account while at the same time netting an increased inventory. And depending on the amount of trading skill applied, you should see the holding matrix increase at an even higher exponential rate. Like kicking performance a notch higher since T can also be exponential.

Since each purchase is intended to be held long term from the start, I use a Buffett style stock selection process. Therefore, all short term price fluctuations are of little concern. But if along the way, I get a nice profit, I'll take it and start buying anew. And do this each time the price gyrations are sufficient to touch my profit points to cash in the profits and repeat the operation.

Look at it this way. You fix your end price point 20 years in the future (I know I can not forecast that price, but I can put it at the average secular market trend in probability) and then fix the entry point. Lets say my end point is \$100 per shares and now shares are available at \$10; this would represent an appreciation of 12% compounded annually. But not being very precise, I purchased at \$15; this would have reduced my performance to 10% instead of having 12%. So even if I am wrong on the entry price, the long term effect could be considered minimal. Sure you would prefer having the lowest purchase price possible but then it is no big deal even if you are off the mark by 50% or more. And since you are holding long term anyway, waiting for that \$100 mark, then that the stock dips 50% below your price won't change your performance results if you continue to hold. You should accept the loss only if your 20 year view has changed. But that would be on a stock basis, not on your entire portfolio.

All the latest charts presented in the LinkedIn forum (except the first 5 based on the original script) have seen their trading component increase level by level; each time increasing their respective alphas. Compounding their performance higher and higher as I designed new trading procedures not only to capture more profits but mostly to increase the number of such profitable trades which are at the core of the methodology as described in my piece on trade acceleration. I have shown with all these charts, that one can push performance way beyond the Buy & Hold, way beyond the limiting Sharpe ratio and even way beyond the Jensen ratio; in fact, my version of alpha has gone exponential. And I am still exploring the limits as explained in a previous post.

To achieve performances at these unprecedented levels I had to devise some peculiar trading methods. One, for instance, is a ratchet like function that will trigger to jettison some position. I know how the ratchet works but I don't know when it will be applied or at what price having set it to trigger at random. What a weird way to close a trade: I don't know when, I don't know at what price and I don't know if it is profitable or not. But it has its purpose. One thing is sure, if I take it off, overall profits drop.

I use 10 to sometimes 20 or more trading strategies each competing to get the next trade. A trade might be triggered by one strategy and inhibit another like saying its not your turn, get at the back of the line and wait for your next opportunity. The trade might be triggered by an explicit function or the result of a random process; it will still push back the competing decision surrogate.

I have a function that will buy shares at random like sprinkling its entries on the flip of a coin. I don't know when it will trigger, but then again, I don't know what is the best price and I can not forecast what it could be. Having selected stocks that I am ready to hold long term, that my entries be random-like and even not controlling the entry price is perfectly acceptable since long term it will not make that much of a difference. That is why I use a lot of random entries, in the end it will have been more important to be in the position than to have been right on the price or missing the opportunity.

## Trending Signal

On a \$50 stock, the impact of the average long term market average representing the drift is about \$0.02 per day. That's the mean's 200 years historical contribution to the stock price. That you compare that 2 cents to what ever number, I think it becomes meaningless. How could I guide my trading a \$50 stock on its 2 cents drift?

The rest is the error term, the random component of the price series and has an expected long term cumulative sum that tends to zero. In my Jensen Modified Sharpe ratio paper, I use stochastic differential equations which had as first part a linear regression representing the mean ( the rate of return) and a second part as the random component. Like separating the signal from the noise. In digital signal processing, there are methods, numerous methods, to filter out the noise and extract the signal; and those methods work well. They work well because the noise is a small component of the signal. But, in stock prices, the noise is so intense that it completely buries the signal. And if you wanted to extract the signal anyway; all you would get is rubbish. But there is still a signal.

As a side note, and an answer to Andrey's initial question on his thread: Automated Trading Strategies:”Anyone test trading strategies on Brownian process?”  Well yes, and a lot, and to top it all, we do it all the time; and on real data. That you trade real life data that is over 90% random-like, or that you simulate on data 100% randomly generated, the difference is not sufficient not to consider them both as highly, and I mean highly random.

And therefore, what ever simulation that is made, be it on simulated data, past data, or real-life data will not show much difference unless you start curve fitting and over-optimizing to the point that past data do finally show what you wanted to see: a correlation over some random data that most certainly will evaporate when going forward testing live. Its like analyzing the evolution of a 100,000 coin toss. We will find trends, cycles, patterns of all sorts, statistics, and even Fibonacci retracements... But it all has no value for your next bet or the 100,000 after that.

We are all entitled to lose our money any which way we like, especially in a democracy. We can even design programs that will do it for us automatically, now that is nice. The other day, I tested a script published in a trading magazine and on 43 stocks tested, only one showed a very small profit; the author provided his PayPal account number so that you could send him some money... for his extraordinary trading strategy.

## Alpha Power

In my first paper (2007): Alpha Power, all the testing was done on randomly generated data. And it is from that paper that the ITRADE strategy emerges. It is described as equation 16 and over the years, the equation has evolved to say the same thing but in a cuter way. I had to convert my formulas to adhere to the Schachermayer notation once I had read his course notes. I was very impressed, his formulation was the most concise I have found. I took, I think, some 20 minutes to make the conversions; everything was already almost following his formulas.

In Alpha Power, all the stock prices were randomly generated. It was like picking 50 stocks at random out of an infinite universe with no replacement; giving each stock a random trend (up and down) and not knowing which would trend in which direction. I did not use a random seed, therefore any test I did could not be repeated. Each test would be different and independent, each price series would go its own way, follow its randomly fluctuating path from start to finish.  A random uptrend was also defined to randomly average around \$0.02 per day; but that too was buried in all the noise.

I was not suppose, under the above test conditions, to be able to extract a cumulative profit. The test data structure was such that what ever alpha might be hidden in the data it would tend long term to zero. Up to 28% of stocks could fail during a single test; never knowing which one it could be or avoided as price progressed. Not only there would be no alpha, but you would be unable to escape from the efficient frontier, Sharpe would prevail. I even had “fat tails” designed in the random price generation. Those are severe test conditions. You can not test anything on the same stock twice.

I knew that if I cheated in any way, shape or form, I would be only cheating myself. So there was no peeking at answers, no trying to find the trend, no trying to find which stock would fail. Each test was independent and would run its course in just a few seconds. And there was nothing I could do about it.

When you accept the high degree of randomness in price series, then you also have to accept that the long term signal is buried deep in the surrounding noise. And all that noise has for long term cumulative sum zero. This implies that long term I can not make money out of it. Sooner or later, the cumulative sum of the noise will again tend to zero. But this does not say that I can not win the game.

What ever analysis I may make of past data, can only represent statistical views of what was and not of what is to be. So how should you play future random-like data? Well, here is a little contradiction: by analyzing past data. Not only that but you will need to make price predictions which you know in advance might be far from right. But that is not the real point. The real point is: based on what ever decision surrogate you may have, it is better to be in the position than not being in it at all. You have to design “your” game with a vision of the future. In my design, I WANT my 2 cents.

This is not a critic of anybody's trading technique, I approve any method you may be using since we are all playing on what is mostly random-like data. And at times, what ever the basis for our decision surrogates, our decision with be in the same direction as the price movement. Not necessarily based on our foresight or great predictive powers but simply by coincidence.

## What is really being said here.

1. The Buy & Hold is not dead and should serve as a foundation to build upon. Solution: a Buffett style stock selection methodology. Equation: Schachermayer's pay-off matrix: Σ(H.*ΔS).
2. Reinvest the accumulating profits. But that is implied in the Buffett style of investing. Equation to be used: Σ(H(1+g)t-1.*ΔS).  Result: an exponential Jensen alpha.
3. Trade over the accumulation process. Equation: ITRADE = (1 + g + T)t-1 resulting in: Σ(Q.*ITRADE.*ΔP). And that is where you get the Jensen ratio on steroids.

You do #1 to beat the Buy & Hold. You beat by a better stock selection with higher prospect of surviving your investment horizon. Technically, you are saying ΔS+; an enhanced selection process, better than random. Result: CAGR 10-20%.

You realize that Mr. Buffett does not only hold, he accumulates and reinvests part of his accumulating profits; so you do #2. In reality, the equation should be more like: Σ(H(1+g)t-1.*ΔS+). Mr. Buffett did not need any formula to do his style of investing, but the pay-off matrix does explain his over-performance. Result: CAGR: 20-45% because you are more efficient , more systematic at using the excess equity build-up.

You design a trading process, not to replace your accumulation program but to complement it. You stop seeing the world as either you Buy & Hold or you trade stocks. You do both. You accept profits from shorter price swings to feed your accumulation appetite. By implementing your own ITRADE matrix, you recognized that even achieving over-performance as in #2 is not enough, you want more.

But at the same time, as you progress in designing over-lapping trading strategies you realize that you can start controlling performance levels. You can add trading functions that extract more and more profits out of all the price swings and you use that profit to feed the other monster: your accumulation program. Result: CAGR: 60-200% and up.

The equation in step #3, to be consistent should read: Σ(Q.*ITRADE.*ΔP+) since it is performed on a better stock selection.

When you review all the presented simulations I have performed since last April, you can see a progression in the numbers and the one number that really counts is the CAGR. All the rest that describe what happened are just statistics of what was; and even if I knew these numbers, none could guide going forward since I don't know how the market will behave in the future.

I know that as I increased the number of trades, I could increase overall performance. I would add new procedures and see performance increase, not for just one stock, but across the board. I would add boosters and push performance even higher. I have shown on the Automated Trading Strategies forum that you could even add thrusters and push performance to such levels that they become incredible, and still at least for me, not reach the limit, the boundary within which I must stay. You should have seen this thing pass in warp drive, that is one chart I will not show.

Everyone has his/her ITRADE matrix. There are millions of them. I think anyone could design their own ITRADEZ matrix to out-perform even mine. What you have here are guidelines that could help.

By following the three steps above, I can assure you that you are bound to see your long term portfolio return go ballistic.

Regards

P.S.:

The impact of the ITRADE equation are far reaching. With a simple equation I challenge the very foundation of acceptable Modern Portfolio Management Theory of the last 50 years. It is no longer a universe controlled by a Sharpe ratio which has very little meaning or value. It is not even about showing that the Jensen ratio was a better measuring stick. It is about you, as an individual, who can follow in the footsteps of a giant like Mr Buffett and even improve a bit on his methodology by going exponential not as a result of your brilliance, but as a consequence to having adopted a singular trading methodology.