August 27, 2011

The most concise mathematical formulation for profits generated from trading stocks that I have encountered is provided by Schachermayer's payoff matrix:

Profits = ∑ (**H** . * ∆**S**) Schachermayer's Payoff Matrix

where **H** is the holding matrix (the number of shares held in each stock over time) and ∆**S** the matrix of price differentials from period to period.

Multiplying element-wise (• .* •) and summing will result in the payoff or sum of profits generated by the holding function **H**. These calculations can easily be done using any spreadsheet where columns are stock price variations and rows are the inventory held in each stock by date.

This article is a work in progress and should be viewed as such. I write sequentially and presently I do not know where this article is leading to. At times I fear its coming conclusions as I think it might end with a contradiction; a kind of catch 22. Nonetheless, even at mid-point, I thought it might be of interest as what is presented is the premise to the second part. And it is all based on common sense or acceptable knowledge. |

Profits generated by any trading method can be represented using Schachermayer's notation. For instance, the Buy & Hold strategy would be:

Profits = ∑ (h_{o}**I** . * ∆**S**) Buy & Hold Pay-Off Matrix

where h_{o} is the initial quantity in each stock and **I** a matrix composed entirely of ones with again ∆**S** the matrix of price differentials.

Only three variables are involved in Schachermayer's matrix equation: the quantities held in inventory, the stock price differentials and time. Both matrices **H** and ∆**S** are of the same size and time ordered. You have no control over time or the future price differentials for that matter; they will be the same for everyone.

Therefore, based on the same stock selection, if you want to outperform, produce more profits, it will most likely have to be by improving the holding function **H** itself, other things being equal. And this means implementing a trading strategy in such a way that it improves, enhances and/or controls the inventory level as a time process. And as such, the trading strategy (the way the inventory level is controlled) becomes a critical part of your trading plan. The position sizing algorithm will take center stage as it alone can exercise control over the holding function **H**.

It should be noted that one can also improve performance by selecting better long-term performing stocks. As the number of stocks in the portfolio increases, the average stock price differentials will tend more and more to approach ∆**M** (the average market price differentials). It is considered sufficient to have some 30 stocks or more in a portfolio to be diversified at the 95% level. And as my view of the game uses over-diversification (50 stocks and more), the average portfolio price for the selected stocks will also tend long term to move in sync with ∆**M**. Nonetheless, a better than average stock selection will lead to higher performance when compared to the average Buy & Hold strategy.

Generating alpha becomes almost synonymous with enhancing the holding function **H** and will need to result in generating more profits. The more you improve the holding function, the more you will outperform. One seeks a holding function such that **H**^{+} > **H**. To the extent that **H**^{+} is greater than **H**, it will generate alpha as a byproduct (**H**^{+} - **H** > 0). Having a higher inventory level than the Buy & Hold strategy appears sufficient to outperform the averages and whatever position sizing algorithm is used to accomplish this task will be solely responsible for the alpha generation, other things being equal.

In my first paper: **Alpha Power: Adding More Alpha to Portfolio Return**, I use a simple linear regression to represent a price series:

**P**(t) = **P**_{o} + **ax** + ∑**ε** where the sum of residuals: ∑**ε** → 0

And therefore, after detrending, the most expected value for the price will be **P**_{o}.

**P**(t) = **P**_{o} + (**a** - **a**)**x** + ∑**ε** since again ∑**ε** → 0

This raises the whole question of the value of predicting prices. There is no need to try predicting the error term since its most expected value is zero. There is no need to make estimates of **P**_{o}; its value is already given. So, one is left with trying to predict the value of the regression line with slope “**a**”; and this is very much time dependent. The shorter the time interval, the more noise is predominant; and the longer the time interval the more the rate of change will tend to “**a**”, the slope of the regression line. And the longer the time horizon, the less accurate these estimates will be. Making 20-year predictions on single stocks should immediately appear as just a wild guess at best.

But still we need to predict prices; we need a reason to enter a trade. We need to have a minimum expectation of a profit. Otherwise, why take the trade?

The Schachermayer payoff equation could be rewritten as follow:

∑ (**H** . * ∆ (**P**_{o} + **ax** + ∑**ε**))

where ∆**S** is being replaced by its regression line without loss of generality. Thereby, profits would accrue based on the slope of the regression line.

∑ (h_{o}I. * **P**_{o} ) + ∑ (h_{o}**I** . * ∆**ax**) since ∑**ε** → 0

The above equation accepts an initial position at **P**_{o} (the invested capital) to which is added the sum of incremental profits generated by the regression line differentials. Therefore, this equation is just another representation of the Buy & Hold strategy.

## Price Differentials

I will not argue the quasi-random nature of stock price movements. I will simply accept them as is: quasi-random like. Taking a linear regression on a long-term Buy & Hold portfolio, we should expect a slope in the neighborhood of 10% (the prevailing (over ≈ 200 years) secular US market average). Performance, over time, would come from price appreciation and reinvested dividends. The 10% average long-term return would imply that the most expected outcome for a $50 stock is to appreciate by about $0.02 per trading day (assuming 250 trading days a year) in order to reach $55 at year's end. For a $25 stock, the underlying trend would be a penny a day. Detrending the price would leave only the error term, the random-like component of the price movement with the usual bell-shaped distribution around the mean.

A $50 stock moves a lot more than 2 cents per day! Indeed, but whatever is left after detrending the price series is only noise: an error term considered randomly distributed around zero and where its cumulative sum over the time interval will also tend on average to zero. On this premise, one should not expect to extract (long-term) a profit from the error term since it is mostly randomly distributed data with an expected mean value of zero.

This means that whatever trading strategy one may devise to enhance performance, because of diversification and the very nature of stock price movements; the long-term performance will tend most likely to the average market return. In other words, long-term, **H**^{+} → **H** and therefore, **H**^{+} - **H** → 0. And this translates into alpha → 0. No alpha generation, no over-performance.

This has for consequence, for me at least, that with no alpha generation all my writings would be worth absolutely nothing. Don’t worry, this is not the case; I will make my point that alpha generation is relatively easy to come by.

Much of the portfolio management literature over the past 50 years has adopted this stance that if there is some alpha, it will tend long-term to zero. It is a byproduct of Modern Portfolio Theory as well as more recently Stochastic Portfolio Theory. And, the Growth Optimal Portfolio (GOP) turns out to be the most coveted outcome in portfolio management which leads directly to trading indexes. However, this is saying the same thing as all you can hope for, on average, long-term, is to achieve something close to the market average. Even with such a low objective; more than 75% of portfolio managers have a hard time beating the average and come in short of their goals. Indirectly, this justifies the pursuit of the GOP.

Are there trends after detrending? No, since that was the whole purpose: finding the best linear fit, the error term will tend to zero.

The amount of noise in stock prices is so considerable that it literally drowns the signal to such an extent that there is practically no need to extract it on a daily basis (except maybe theoretically). When viewed from a long-term perspective, a 10% move on the $50 stock in a single day would show noise to account for over 99% of the price movement. Even a 1% move in price, which is not unusual, would drown the signal in 96% noise.

## Predicting Prices

The job of predicting prices becomes at most an extrapolation of the linear regression since that will be the most expected outcome should past trend prevail.

Therefore, predicting the trend becomes the ability to detect within all the noise the 2 cents of expected daily appreciation on the $50 stock. Again, this trend is buried deep in surrounding noise. On a daily basis, the long-term trend has such minimal value that it is untradable.

But what about the rest (the noise) in the price data series?

Detecting the signal over all the ambient noise requires changing one’s perspective and allowing that even random-like price series can have at times detectable trends; which may or may not last. There lies the real problem; our inability to extract from the price movements what constitutes a trend or predicting to some extent the magnitude of the next price move. We might as well concede that prices are totally random after all. Over the long-term trend, I would be off by about only 2 cents for the day on a $50 stock (or by 0.04%)!

Disregarding the long-term trend in calculations will have little impact on estimates of short-term price movements. Short-term price movements will tend to show more of their random-like nature. But then again, discarding the long-term trend will leave only the random-like component of price movements. However, over the short term, there is no obligation for the random-like component to staying within boundaries or have its sum of variations tend to zero for that matter.

By the very nature of stock price movements, anyone can declare what might constitute a trend based on whatever principle and see that, at times, their trend definition holds. As a matter of fact, all types of patterns: triangles, wedges, flags, pennants, trend-following indicators, oscillators and much more can be detected by those seeking them. You will even find them in totally randomly generated price series. It is not our own personal definition of a trend that the market needs or has to follow; it will follow its own course whatever our trend definition.

Is not a sub-set of a random-like price series, a random-like price series in its own right? Isn’t a series of a hundred toss of a fair coin taken from anywhere within a series of 10,000 tosses still a random series?

## Playing Prices

A short-term trader is not that much interested in the 2 cents per day thing. There is too much price movement in the error term to be ignored. Put a lot of lines on a price chart and you are bound to see the price cross one or another at some point. You will start seeing all kinds of relationships and interconnections. Sometimes price will go through a particular line and at other times bounce right off. As to the meaning or interpretation of the move, that is a totally different question. What interpretation can be given when most likely the cross of a line has for origin a random-like phenomenon? You can find all kinds of explanations for past price movements, but very little that will help you anticipate future price moves except on a larger scale and even there you will not know how long it could or would last.

Nonetheless, even under the random-like nature definition for price movements, one can, just by looking at a chart, see trends develop, cycles, support and resistance levels and a lot more. Most of which will operate about half the time; as if at whatever price, one can only go up or down even when the most expected change is no change at all.

But there are short, mid and long-term trends! They are easy to see after the fact. A short-term trader is interested in the continuation of the trend; that is where he can make money. But then again, from the same price, contrarians will bet that the continuation will fail. Only time will show who gets the reward. Both players make their bets, but only one will win, and next time may produce the same result or a complete reversal.

## The Long-Term Investor

For the long-term investors, daily price fluctuations are not necessarily a concern; what they are looking for is the 2 cents a day of upward trend. They know over the long term, their expected portfolio performance will tend to approach the long-term market average. They appear ready to accept what seems like all the market has to offer. The best can make better stock selections and push their returns to 4 or 5 cents per day on average which will translate to 20 – 25% per year.

Technically, the long-term investor is playing the regression line while the short-term trader is mostly playing the error term (the noise). No wonder, why at times, the short-term trader has for nickname: noise trader.

The long-term investor/trader plays the Buy & Hold strategy which was expressed earlier as:

∑ (h_{o}I. * **P**_{o} ) + ∑ (h_{o}**I** . * ∆**ax**) since ∑**ε** → 0

His portfolio appreciation is based on the slope of the regression line; he plays time (hold). He knows that if he waits long enough, he is bound or at least on his way to reaching the market average or better at some distant point in the future.

The short-term trader is faced with a totally different problem.

## Trading on What?

… to be continued…

Created on ... August 27, 2011, © Guy R. Fleury. All rights reserved