September 14, 2011
For the short term trader, trading on what basis becomes a major concern. Fundamental data is of little help analyzing short term market activity. Technical data is good at saying what was, but has little short term predictive powers. Statistics seems to become the only source of filtered data that can bring an edge, but there again, all you will find will describe past price action.
The shorter the trading interval, the more a trader is confronted with the quasi random nature of price movements and therefore the closer his trading should be considered a subset of a random data series. And as a consequence, the more a short term trader will become a momentum, volatility or noise trader. He does not have much choice.
Trading what does not move produces no profits; the same as not trading at all.
First as was stated in part 1, profits generated by any trading strategy can be described using Schachermayer's PayOff Matrix: Profits = ∑ (H . * ∆S). I will try to remain within this notation as much as possible and use this equation as the basis for the profits generated by the Buy & Hold investment strategy.
Many academic papers state that what ever the holding function used, long term, the most expected overperformance over the Buy & Hold would tend to zero, such that any enhanced holding function H^{+} → H, and therefore, H^{+}  H → 0. Accepting this conclusion leads to no expected alpha generation as it too would tend to zero. And thereby case closed: buy index funds and hold them for the long term.
Even if out there, there is an elaborate No Free Lunch (NFL) theory, martingale measures and an Efficient Market Hypothesis; I will try to show that maybe dessert can be on the house, with coffee, and some free coupons for the next few lunches.
For the sake of brevity, I will use the list of stocks in the S&P 100 Index as stock universe for the experimental portfolio. This list or another would do just fine for the elaboration to be made. This one has the advantage that we will not have to discuss the matrix of price variations (∆S) nor its composition: it will be the same for everyone. This exposé will not take into account the historical composition of the S&P 100 Index; all I’m interested in is that we accept a known list of 100 stocks. This way, what ever trading method used will start on the same footing.
First order of business is market exposure. The Buy & Hold is 100% invested at all times which mean it has a 100% market exposure. If the holding function His in the market 50% of the time compared to the fully invested Buy & Hold strategy; it should be no surprise to see this portfolio generate less overall performance.
1/2 • Profits = ∑ (1/2 • H . * ∆S) < ∑ (H . * ∆S)
Putting half the available capital at work would be sufficient to reduce profits by one half; while being invested only half the time would depend on the period or periods when invested in full but in probability would still reduce performance which would tend to have about the same effect as reducing invested capital by half. A mix of volume and time slicing could also be used with end result that performance should also be cut in half.
To compensate for the under achievement would require performing 2 times better just to get even with the Buy & Hold strategy:
1/2 • Profits • 2 = ∑ 1/2 • (H . * ∆S) • 2 = ∑ (H . * ∆S)
Using time slicing, a smaller trade size or a combination of both, will result in under exposure and cut performance accordingly. The problem gets harder as exposure is reduced further:
1/10 • Profits • 10 = ∑ 1/10 • (H . * ∆S) • 10 = ∑ (H . * ∆S)
An average 10% exposure would require a holding function that is 10 times more efficient at extracting profits from the same data series. And having found a holding function that is 10 times more profitable, it would make no sense to accept such a low market exposure. On the contrary, you would want full and even leveraged exposure to this more efficient trading method; unless it was all it could offer in opportunities; in which case why go to all the trouble for the same performance as the Buy & Hold? I often see trading strategies with even less market exposure hoping to outperform in a big way. These trading strategy designers should realize how difficult it can be to outperform the average by a factor of 10 or more.
But even if your strategy had 1/10 market exposure, this does not imply that you have 10 times the capital to invest in 10 times the inventory to be held. So the dilemma remains. A compromise of sort is required, a balance between what can be done within the portfolio constraints, available resources and what the market has to offer. And with the added constraint that you are not playing alone and that the market is not always fair.
Full exposure requires being in the market all the time; this is the same strategy as the Buy & Hold and can only lead to: H^{+} → H and therefore, H^{+}  H → 0: the no alpha generation scenario. To generate alpha, you need to do more than just hold. The enhanced holding function H^{+} needs to be greater than H; trading volume needs to be enhanced in order to overperform such that: H^{+}  H > 0.
For the Buy & Hold investor, the trading choices are simple: from all the available market data try to find stocks with the best long term prospects; which means trying to find stocks with high expected rates of return “a”. Take initial positions (h_{o}) in these best prospects at the prevailing price (P_{o}), and then wait, wait and wait. This would translate in the following equation (see part 1) where available capital is used to establish initial positions and profits would accrue over time:
∑ (h_{o} . * P_{o} ) + ∑ (h_{o}I . * ∆ax) since ∑ε → 0
That the price will fluctuate short term is of little concern, as these price variations will be looked at as short term noise. Even the ‘87 crash looks like a blip on a long term chart. The overall performance will be almost path independent since the main concern is “a”, the slope of the regression line which technically will be known only at the end of the trading interval. The path taken, in the Buy & Hold scenario, is of little concern. However, there is stock allocation at work: ho (the initial position in each stock) will exert much weight on the final outcome.
Placing small bets on big winners and big bets on slowly appreciating stocks will produce totally different performance results when compared to placing big bets on big winners and small bets on low performers. All this is dependent on the “worthiness” of the predictors of “a”. Overestimating “a” (the long term rate of return) will lead to overweighing its initial allocation while underestimating “a” will place smaller bets on higher performing stocks.
The conundrum remains. Because the Buy & Holder needs to invest for the long term, he/she is forced to forecast stock prices for the long term as well. But there, the longer the time horizon, the less accurate the forecast may be. There is a tradeoff to be made between the desire to invest long term and the unknown or almost unknowable long term future; and to select stocks that have at present the best long term prospects, the highest expected probability of survival. One might not know what the price of a specific stock may be 20 years from now, but on a group of 100 stocks, one has a high probability that the average performance will tend to the Index average which also in high probability will be positive over the 20 year investment period.
When looking at Warren Buffett’s investment style, you notice he does just that: select stocks he thinks will be there long term and that are at the time of investing undervalued based on his valuation methods. He does not know which of his holdings will outperform 20 years from now; but as a group, he hopes to maintain his past average portfolio return of 20%+ per year. A good part of his superior results is achieved by reinvesting generated profits in buying new businesses.
Can anything be done to improve on the basic design of the Buy & Hold strategy? There has to be a way.
Most of the academic literature on portfolio management point to low and vanishing alpha generation, that it be with the No Free Lunch, the Efficient Market Hypothesis, the Growth Optimal Portfolio (GOP), or any other equivalent method. They all conclude that what ever trading method used the most likely long term outcome will be to achieve close to the average market return. This again means that H^{+} → H and therefore, H^{+}  H → 0; and no alpha.
Notwithstanding academic papers, to outperform the average one will most likely need to develop a trading strategy where: H^{+} > H, and H^{+}  H > 0. There is no way to change ∆S: it was what it was and will be what it will be and a single investor is too small to influence the price movement in a big way.
To outperform will require designing a trading strategy H^{+} that can withstand time, market price gyrations and that can live within specific portfolio constraints of which the most important being not to go bankrupt. This enhanced trading strategy H^{+} should be scalable and preferably controllable under trade automation.
So the real question is: can the holding function H be modified or designed in such a way as to tend to H^{+}? And if so, how much greater should H^{+} be compared to H to make it worthwhile?
A simple H^{+} solution would be to reinvest part of the equity buildup in buying more shares which would result in the following equation where the inventory is being increased at a fraction of the delayed price appreciation rate “a”:
∑ (h_{o} . * P_{o} ) + ∑ (h_{o}I(1+g^{(t1)}) . * ∆ax)
Doing so would enhance the holding function by the fraction of the delayed price appreciation rate:
H^{+} = H(1+g^{(t1)}) and thereby H^{+} > H for g > 0
showing that: H^{+}  H > 0, and that alpha generation is relatively easy to come by.
One could, for instance, add a daytrading program to the mix; and close all these added positions every day. This would result in a leveraged portfolio:
∑ (h_{o} . * P_{o} ) + ∑ ((1+L) • H(1+g^{(t1)}) . * ∆ax)
where L is the leveraging factor. Using up to the available equity on a daytrading program would result in up to twice the performance. Or using a leveraging factor of 3 as in the case where full daytrader margin is made available would result in:
∑ (h_{o} . * P_{o} ) + ∑ (4 • H(1+g^{(t1)}) . * ∆ax)
Both methods; reinvesting excess equity and using margin for a daytrading program only, are simple procedures that will enhance portfolio performance; and show that alpha can easily be generated. If you time slice on a daily basis: (buy the open, sell anytime before close), you are in fact almost 100 % in the market which is close to the same as a Buy & Hold strategy. All you would need is a profitable day trading program with positive expectancy on a daily basis since all leveraged positions would have to be closed prior to the end of day. Still, H+, the enhanced holding function would be improved by both factors:
H^{+} = 4 • H(1+g^{(t1)}) and thereby H^{+} > H for g > 0
The above methods are not the only ones available to increase performance. From my latest paper: x, here is the simplified Alpha Power wealth appreciation formula:
∑ (h_{io} . * P_{io} ) + ∑ ((1 + L_{i})(1 + B_{i}^{(t1)})^{(t1)} • H_{i}(1+g_{i}^{(t1)}) + T_{i }+ C_{i})^{(t1)} . * ∆a_{i}x)
Added to the wealth appreciation matrix are: a bet sizing algorithm matrix, a covered call program and a swing (short to midterm) trading component. Each contributes to overall performance.
Nothing has changed on the price side of the equation; ∆S is still the price variation matrix. It is only the inventory on hand H that has changed over the investment period, nothing else. And as such, it clearly demonstrates that to gain alpha points over the long haul, it is sufficient to increase the holding function H. It all seems so simple.
The size of overperformance then becomes partially controllable by presetting the composing elements of the Alpha Power wealth formula. It is this control that should be the focus for the trading procedures to be implemented in an automated trading system.
The objective is to implement a trading strategy that evolves according to the Alpha Power wealth formula. Since in the beginning available capital is limited, some compromise will have to be made. As presented in previous papers, the intention is to do it all at the same time: over a basic Buy & Hold strategy, accumulate shares at an exponential rate while increasing bet size as portfolio increases and at the same time trade mid to short term over the accumulative functions. Excess equity buildup is used to finance the acquisition process. In order to do everything at once, risk as little as possible and trade over the accumulative process, it is required to make some compromises.
Because of the unknown short term future, an overdiversification approach is used. In this case, the stocks listed in the S&P 100 are used and represent by their number more than is necessary to show sufficient diversification.
To minimize risk and maintain some available capital for trading, only a small position is taken in every stock. In fact, only 5% of available capital goes to setting initial positions and 95% remains in cash bearing interest while waiting for trading opportunities.
Buy orders will form a sparse matrix similar to the one presented in my first paper: Alpha Power.
Sparse Buy Matrix 

... 

... 
The Buy sparse matrix raises the whole question of trading decisions. On what basis are trades being taken? This depends on the trading formula being used. For instance in my second paper: Jensen Modified Sharpe Ratio, the following formula dictated at what price and what quantity should be added as prices evolved:
ReqCap = (a_{q} / 2)x^{2} + (1.35a_{q} 2i_{q})x + 1.295(i_{q} – 1.0772a_{q})P_{o}
The above equation also determined the required capital to accomplish the task and has for only variables: i_{q} the initial quantity purchased, a_{q}: the added trade basis, P_{o}: the initial price and x: the price differential. The result is a binomial equation and its derivative will answer at which price the maximum capital required will be.
The required capital equation is not the only one of its kind. In fact, whole families of such equations could be designed. The trading game is not a single solution game; a lot of trading methods, strategies and systems can be successful.
What the required capital equation implies is that at first, capital will be put progressively in the market up to the derivative reaching zero and then the system will sustain itself as prices continue to rise; to the point where no additional capital will be needed for the additional purchases to be done. And as price rises, the required capital equation will show what has been extracted from the market in the form of profits.
It is an amazing equation. It presets all the trading action to be performed based on price and bet size. The time element is not even considered except in the sense that for prices to rise takes time. All that is seen in the equation is the accumulation procedure and on what it is based. Really remarkable, you preset your rules of engagement telling the market how you will behave depending on price movements.
Applying the required capital equation to a Berkshire Hathaway scenario could maybe show the unsuspected power under the hood so to speak. I choose this example to skip the dividend and split adjustments that might be required otherwise. I also needed a recognizable stock. Maybe as a side note, over 70 stocks have outperformed Berkshire, so the example is not so far fetched (I am just a little lazy on the side).
The rules of engagement:
1. Buy 2,000 shares (iq) of BRK/a when it came out at $10 (Po).
2. For every 50 point rise, buy another 1,000 shares (aq) at the prevailing price using part of the paper profits (the excess equity buildup).
With BRK/a at $120,000 this modified Buy & Hold strategy would value your holdings at 288 Billions of which $144 Billions would be your net profit. This far exceeds Berkshire’s capitalization of some $180 Billions. By the time BRK/a would have hit $82,410; you would have become the sole owner of all outstanding shares.


When compared to the traditional Buy & Holder, he would still have made plenty as his equity would have grown to about $240 millions. Therefore, just by changing procedures, the Alpha Power accumulating strategy outperformed the Buy & Hold by a factor of 1,200 to 1. Note that all the accumulated shares were purchased using only part of the accumulating profits; and that no protective stops or other safety procedures where included in this simple design.
Both strategies start from the same point with the same capital and in the beginning perform the same. It is with time and as the price increases that the two strategies start to diverge and the more the price increases, the more they diverge.



Yet the rules of engagement are very simple, so simple that you always know in advance what to do for the whole duration of the investment period. You don’t even have to be precise. You don’t even need a pencil. You might not even have to follow the markets by using good until cancel orders (one is executed, set up the next one and wait).
The point being made is that by using a mathematical function one could have bought Berkshire for as little as $20,000. Imagine what more evolved and refined mathematical functions could do! The above was based on one single trading rule (buy 1,000 shares for every $50 rise). It uses absolutely no market indicators and is totally selffinancing after the initial setup. It is independent of market and price gyrations. This does not mean that the portfolio value does not fluctuate, it does.
Another interesting aspect of this procedure would be in the risk valuation. How should one value the risk taken by starting so small? Commissions to be paid during the life of the portfolio are greater than the initial investment in BRK/a! The minimum selffinancing initial capital for the Berkshire Hathaway acquisition scenario is $12,000. In the above scenario, doubling the trade basis while still remaining selffinancing would double overall performance. One side benefit of this trading procedure is that you can quit at any new price level reached knowing that you are ahead of the game.
Not all shares are created equal and one should consider the Berkshire Hathaway example just as an indication of what can or could be done. Nonetheless, it does show the value of accumulating shares of a rising stock and letting the market pay for it. And when combined with other trading procedures, one could reduce the effects of market swings on the portfolio value.
For those interested in the Berkshire Hathaway example, I have made a spreadsheet with the calculations available here.
The Alpha Power trading methodology is intended to do more than just accumulate shares over time. One of the components is to trade over the accumulation process as exemplified in the Berkshire Hathaway example. The accumulation process is view as a way to increase portfolio equity while the trading process is there to pump more cash in the system in order to increase the bet size as the portfolio grows. This is an intriguette process where everything needs to be balanced in order to minimize portfolio risk, maximize growth within the portfolio constraints and thereby improve overall performance.
With random noise representing the major part of price data series, one wonders what could be used to effectively design and implement a short term trading procedure that would not rely on luck. If the signal is totally drowned in the noise then what is the basis to execute a trade since buying implies that someone is looking for more than just a gamble, he/she is expecting a profit from such an operation.
… to be continued…
Created on ... September 14, 2011 © Guy R. Fleury. All rights reserved.