September 12, 2009

This article is a follow-up to the position sizing article. It is available HERE.

What preceded position sizing was the general mathematical framework where an optimal trading strategy was simply summarized as a share-holding matrix,holding matrix, where the initial quantity held in inventory could grow at the portfolio's delayed average exponential raterateThis "optimal" trading strategy is not unique; it depends on each stock's appreciation rate and as such will be different for each subset of stocks selected as a portfolio.

Since the inventory growth rate also depends on other controlling factors, it will create as many "optimal" payoff matrices as there are, not only arrays of factors but also mixes of stocks with varying holding times. This is saying that everyone will get a different answer to the portfolio "optimality" problem simply because most of the factors under consideration will depend on personal preferences such as portfolio size, quantity, and quality of the stock mix, risk aversion preferences, determination (guts, convictions or strength of belief system) and trading skills.

And since all these considerations are dynamic in nature and vary in time, the term "optimal" could be quite vague. Therefore, the holding matrix H+ might be more appropriately termed an enhanced position sizer, which would provide more latitude and variety of responses to the ongoing inventory control problem than the "optimal" trading strategy. But, most of all, as will be emphasized, the decision process itself will be at the center of portfolio optimization. It is the responsibility of the decision process to trigger trade entries and exits, and therefore it will be this process that ultimately will control portfolio optimization and performance.

To recap, the simplified "optimal" payoff matrix was presented as:


which came from a more complex equation as it tried to summarize equation (16) from my first paper2. It is restated here using part of Schachermayer's matrix notation:

payoff matrix equation

which formula, for simplicity, was reduced to the Schachermayer equation representing the payoff matrix:

 payoff matrix

The improvement over the Buy & Hold will come from the factors introduced to control the inventory accumulation process. Therefore, H+as an enhanced strategy, will be greater than H by the factorfactor, which tries to resume the effects of all the controlling factors. My second paper, the "Jensen Modified Sharpe Ratio"2, is a direct consequence of this trading strategy.


What I wanted to do was elaborate a trading strategy that would outperform the Buy & Hold. It is from the results obtained from tests, where I needed a logical explanation for the observed data, that these formulas were elaborated. They served not only to understand what was going on in the trading procedures but also to verify that the obtained results were mathematically plausible and could fit within a rigorous theoretical framework.

Removing the inventory growth rate from the equation would naturally return the "enhanced" payoff matrix to its Buy & Hold origin.


ΔP may be the most unpredictable of time series. Long-term, on average, day-to-day price variations should approach a zero mean with a slight upward regression line that will account for the average upside bias. Any short-term consistent predictive ability you may have could be transformed into the most lucrative of pastimes. However, short-term predictive accuracy is hard to come by. Most find that their long-term hit rate, trading short-term, is not that far away from 50/50. Even professional money managers have a hard time beating the averages. In fact, over 75% fail to do so.

However the price variation matrix ΔP may be generated, you have to design a trading strategy that will adapt to these unpredictable price variations in such a way as to be the most profitable possible. You can always know what you could have done in the past to outperform anyone, or any trading system, hindsight is indeed a very powerful tool. But what really counts is what you will do now and in the future to shape your portfolio performance. Don't expect that backtesting "unrealistically" your favorite dip-buying script (like ignoring bankrupt companies) that it will outperform going forward. You will find out, soon enough, as they say in all those disclaimers, that past results are no guarantee of future performance.

So, let's say you are faced with an unknown future regarding price series evolution. Short-term price predictions will approach 50/50, making it hard to generate an enhanced trading strategy H+, since the payoff matrix, long-term, will tend to be average: payoff As a matter of fact, the more the market is considered as a random walk, the more the "enhanced" payoff matrix will tend to add zero to performance. And the more the enhanced payoff matrix tends to produce nothing additional to your portfolio, the more you should consider doing something else.

Long-term, one can hold a highly diversified portfolio with the expectation that the probability of a positive payoff matrix is positive and, furthermore, with a probability approaching 1 asymptotically. This is like holding, for the long term, the market average. Notwithstanding, a single stock's long-term forecast can be found to be far from probabilistically accurate. What the price of any 100 stocks might be 20 years from now is anybody's guess. It is just as a group, over the long term and only over the long term, that you can describe, using averages and statistics, the general stock price behavior.

Whatever you do, you must at least beat the most trivial of all trading methodologies, which is the Buy & Hold. Underperforming the Buy & Hold simply translates into unnecessary time and effort deployed to achieve said underperformance. You might as well have done absolutely nothing but your initial purchases and put your time into more creative and enjoyable endeavors. Therefore, should you, managing your own portfolio, underperform the Buy & Hold using whatever mix of trading techniques and/or strategies you could devise, you might as well quit the game. I would suggest that you either pass the torch to someone else who might or might not be able to do a better job, or you could simply adopt an enhanced Buy & Hold strategy.

The Decision Surrogates:

All trades are not created equal. The quantity to be traded in any one stock at time t should be determined by a decision surrogate and other controlling factors as to the suitability of trade; adjusting the trading basis to the degree of desirability, executability, and sustainability. This needs to be calibrated in accordance with the relative importance and conviction level in a particular trade as well as its relative size in the portfolio, all the while maintaining relatively high diversification. This may sound confusing, but the point is that one should look at all parameters that may affect a trade in the context of a whole solution to the portfolio optimization problem over the whole investment period. The solution is not one size fits all. Each stock should be treated on its own merits and its prospects to generate a profit; otherwise, it might not be the best trade for your portfolio. The market is not lacking in opportunities, nor is it lacking in treachery.

The holding function H(), needs to be controlled by a decision surrogate D+(t) which has for unique function to determine all buying and selling decisions. But, there is more to it since the decision surrogate has to determine and control the value of all parameters to finally output a decision to buy, hold, and/or sell. And how many shares should be traded at any given time for a particular stock within the context of seeking a total portfolio solution? It is not a trivial problem since all stocks do not have the same price, the same opportunities, the same intrinsic value, and certainly do not behave the same.

The quantity traded over time can depend on an initial bet and the decision process to acquire additional shares as time progresses. This can be expressed as:

quantity bought 

The above equation states that the number of shares held QB(), at time t for a particular stock j, will depend on the initial bet IB0 placed on the stock to which will be added the dot product, or the sum of the ongoing purchases. It is the decision of the surrogate Dj(t) to determine if there is to be a trade and the trade-modifier TMj(t) to make any adjustments, up or down, to the trading basis. Therefore, the decision process becomes a sequence of logical decisions based on trading conditionals that will determine when a particular trade will be triggered and what quantity should be traded, how long the "hold" rating should be maintained, and finally, what will trigger the exit from the trade, if at all.

A trade simply becomes a quantity held over a time interval of a price series, all controlled by the decision surrogate. The stock price is considered as just another criterion in the decision process as trading decisions could be taken regardless of the current price. The price is what it is when your decision surrogate triggers a trade and nothing more. It is the decision surrogate that determines the trade and not necessarily the price. For example, in a moving average crossover system, the decision to enter a trade at the current price is made on old price data due to the lag of the moving averages themselves.

So, it all boils down to what dictates your market behavior. What process or concept will trigger a trade for you, what will make you hold for a time, and then what will persuade you to exit? It is, consequently, all under your control; it is your own decision process that you delegate to a decision surrogate that will mold all your trades and trading strategies. It is, finally, your convictions in your own belief system that will implement your trading strategy and shape your portfolio.

For someone not implementing these trading procedures, the above equation (1) simply reverts to the old standby: the Buy & Hold, since having Dj(t) = 0 over the investment period will result in no additional trades.

Some of the factors that could enter into the enhanced decision process D+() and also alter the trade modifier TMj(∙) are:


 (click to enlarge)

The decision surrogate D() can have only three possible outcomes: {1, 0, -1}, which stand for {buy/cover, hold, sell/short}, and this should be made clear by the context and the inventory on hand. In trading, you can only add, subtract to/from inventory, or leave it unchanged. The trade modifier TM(∙), on the other hand, can have a range of values that could be dependent on the conviction level, leverage, weighing of the stock in the portfolio, as well as other factors.

trade modifier

The current value of shares held would correspond to the accumulated shares to date at today's price:

current value

and where the decision surrogate and trade multiplier would be controlled by all or some of the listed factors and possibly others not mentioned. It should be noted that to know the current inventory value, it is not required to know at what price each share was purchased, only the evaluation price is required. However, to determine the total costs of all acquired shares, it is required to sum all trades at their respective acquisition price.

total cost 

Based on equation (1), each stock will have a unique trading signature due to the array of factors considered and will be performance-related as well as path-dependent, all in accordance with the pre-set trading objectives. The purpose and use of some of these controlling factors are provided below:

Initial Bet (IB): the purpose is to set the amount invested in the first position for a particular stock. It need not be the same as the trade basis; it only serves as a starting point. However, the initial bet can be optimized in a way as to minimize the needed capital requirements in accordance with pre-set objectives as prices evolve (see Alpha Power2). One can also elect not to place the initial bet  and just start with the trade basis when it will be triggered; this will have as secondary effect to modify the cash requirement equations. The size of the initial bet will depend on portfolio size and the number of stocks one which to trade. Some of the constraints being capital limitations, risk aversion and trade desirability.


Trade Basis (TB): the main purpose is to set a size as a trading unit (say 100, 200, 1000 shares or more). The trade basis can be different for each stock in the portfolio. A penny stock and a $200 dollars stock should not be treated the same. The trade basis can be considered as the ongoing bet size. It should also be related to the actual portfolio size and be within the limits of the capital constraints. The size of the bet should not be so high as to put 100% of equity in a single position and certainly not on a dumper. Remember that the one trade which suffers a drop of over 90% can put you out of business at any time. The trade basis can also be used to time slice a trade as a method of scaling in or out of position.


Stop Loss (SP): the stop loss is a requirement in this stock market game. It might be considered a zero sum game implying that it is a fair game. But nothing is further from the truth. It may be a zero sum game, but fair it is not. There is too much cheating and down right frauds to make this a fair game. A stop loss function is a necessity. It acts as a feeble attempt to preserve your capital. It should be considered as part of the unavoidable cost of doing business. The worst trading strategies you could ever design are those that take multiple positions on downers to the extent of putting 100% of the portfolio on the line in a single trade. Sure, most often you can catch a nice rebound but this can hardly compensate for the one time where your portfolio is obliterated following a 90% haircut on the downer that fails to rebound. So, I will repeat, a stop loss function is a necessity just to stay in the game.


Portfolio Size (PS): the available trading capital sets trading limits of its own. All shares purchased must be within available portfolio equity. The portfolio size will dictate the number of stocks that can be traded as well as the initial bet and trading basis in a manner as not to exceed available trading capital.


Number of Stocks (NS): this sets the limit of tradable stocks within the capital constraints. It is a function of portfolio size. It should be noted that it might be preferable to increase the number of stocks to be traded in order to approach an over-diversification status. The advantage being that long term the average price of the stocks in your portfolio will tend to mimic the market's average price.


Price Level (PL): sets the price at which a trade can occur. It can also serve as targets to be reached or in trading conditions to be met before a trade can take place. You are looking for price appreciation starting from your entry point and your trade should be managed on that basis.


Weighing Limits (WL): the trade basis enhanced by the conviction factor, overweighing and leveraging should not exceed the maximum bet size as a percentage of portfolio value nor should it represent a high percentage of the traded daily volume in a particular stock. Putting 100% of portfolio equity in a single position may lead to a debilitating drawdown and has no part of the trading philosophy presented here. The trade size should limited to such a level as to represent only a small fraction of portfolio value. The bet size could be maintained at less than the limiting value; however, this value can increase in time as the portfolio size or as the stock's relative performance increases.


Volume Objective (VO): this is used to preset the desired volume level to be reached in time. It is a function of price performance and therefore is not a guarantee of achievement. It only sets an objective; and since it is a time function, it can become a criterion for relative performance.


Conviction Factor (KF): this is to pre-set the higher profit expectancy for a particular stock; a conviction factor greater than one will increase position size on the more desirable trades. In this sense, the more you are certain that a particular stock might rise in price the greater the number of shares should be traded. And the higher your conviction factor relative to the other stocks in your portfolio, the higher the weight of this stock should be in the portfolio. This should be interpreted as trying to make bigger bets on what you expect most to rise the most.


Sector Overweight (SO): this is another factor used to increase position size. If the sector the stock is in is increasing relative to other sectors, it might be appropriate to increase the bet size in relation to its sector performance. This can be related to the conviction factor as well as the price's relative strength.


Position Overweight (PO): the purpose is to increase position size again. If the stock is increasing relative to other stocks in the portfolio, it sounds appropriate to increase the bet in relation to its over-performance. This also relates to the price's relative strength.


Leverage (LF): enables to increase bets beyond available capital. It increases risk but coupled with a high conviction factor may prove more rewarding. Leveraging has the same effect as adding more trading capital. Maintaining a portfolio leverage of 1.5 has the same effect, almost the same effect, as providing 50% more capital and multiplying the final outcome by 1.50.


Scaling Function (SF): this is the method used to scale in and out of a position. Setting up big positions requires a discrete hand. Volume and/or time slicing may be more appropriate to establish a sizable position rather than issuing a huge market order. But this is very much dependent on portfolio size. If your portfolio has at most a 100 shares trading basis, you will find scaling in and out relatively simple, fast and easy. The scaling function is therefore highly dependent on portfolio size.


Sunset Rules (SR): depending on the style of play, sunset rules should be designed to put a time or price limit on your exit in accordance with the reason and time frame selected to set up the trade. Gradually reducing holdings as time progresses or as targets are reached; and this may be for the short, mid to long term views. It all depends on the type of game you want to play. It is a way of saying: let's convert those paper profits into cash as the price of the stock is still rising. And if they want it all on a rising scale, then let them have it. For instance, some consider that real estate operates on a seven year cycle, therefore reverting to cash at the height of the cycle may seem appropriate. This is quite different from a trailing stop which tries to scale out after a price decline.


Alpha Multiplier (AM): is used to reinforce trading decisions that are out-performing the average. The main purpose is to increase the quantity of shares being held as the price goes higher. It can also be used to reduce the quantity held due to under-performance. In this sense, the a lpha multiplier seeks, in the selected group, "atlas stocks" and increases their weight relative to other stocks in the portfolio while the under-performers might see their share weight dwindle down, even to zero, for not providing any tangible profits. The Alpha Multiplier is used as a positive feedback loop, a reward for achieving better performance and since it is proportional to the price level reached it tends to amplify returns in the best performers. This was the subject of my first paper: "Alpha Power"2.


Control Function (CF): this function is responsible for regulating the stock accumulation process. It sets the range for the quantity of shares to be held as time evolves.


Desirability Factor (DF): this pertains to the trade desirability. Should this particular stock be a good trading candidate for the time period under consideration? A stock price that stays flat has practically no trading value and even if it is part of your selection, it might not be the best choice as a trading vehicle. If your watchlist is composed of 100 stocks and you only intend to trade 50, then the Desirability Factor could help select the best candidates among the selected group based on some sort of overall rating system.


Cash Inflow/Outflow: there is a need to implement strategies to use additional cash inflows from other sources as time progresses. Procedures should also be considered when in need to extract cash from the portfolio as this could affect not only the capital constraints but also force the liquidation of some stocks at an inappropriate time. Depending on the amount added or withdrawn; it could affect other factors such as the trade basis, leverage and weighing method.

There you have it, some of the factors to be considered when setting up multiple positions in multiple stocks. Each of these factors should be adapted to each of the stocks being traded and other controlling matrices should be set up in order to better frame the procedures into following the desired objectives. The decision surrogate matrix, even if it has a simple outcome (buy, hold, or sell), is more complex since it has to adapt to each stock and to all others in the group. This is without necessarily knowing which stocks will eventually outperform or underperform.

Trading Window Selection:

The second symbol in the payoff matrix stands for the stock's price variations. It too, could have optimality searches in order to find ways to improve performance.

stock selection

where S+, is a subset of all price variation series available under S that outperform the average; while S*, is simply the "Atlas stock", the best-performing stock in the selectable stock universe. Whereas, S-, is the set of underperforming stocks. This is saying that there is this one stock that will outperform all others, then there is a group of stocks that will outperform the average, and the last group that will perform below average. To this last group, we should add Szero: the list of all the stocks that went down to zero (bankrupt). It should be noted that that list is far from empty. Nonetheless, there is no way beforehand of knowing which stock will outperform or not 20 years from now. Lehman Brothers had been in business for over 100 years, yet it took less than 18 months to go from its historical high straight down to zero; a probability which, for some, only 5 years ago would have been unthinkable.

Then, the real question is: long term, could you outperform the Buy & Hold strategy? And the answer there is definitely yes. This is a rather bold statement, as much of the financial literature expands on the notion of no arbitrage. The statement is, therefore, in contradiction with the no-arbitrage or the No Free Lunch with Vanishing Risk (NFLVR) arguments of the Stochastic Portfolio Theory (SPT). If there is no arbitrage possible, no one can devise a trading strategy that will tend to escape from the gravitational pull of the market average. And yet, the enhanced holding matrix H+, is making such a statement.

It should be clear that the whole trade selection process - the part of any price series where shares are being held - is the payoff matrix. From what appeared to be a simple expression (only two symbols, after all), the payoff matrix will start to be much more complicated; it has to include many other variables in order to lead to a more effective, controlled, and regulated portfolio.

One point that should be emphasized is that the payoff matrix is all there is. The whole portfolio optimization problem is, therefore, a big position-sizing problem. It is not solely a matter of finding better stocks or better holding intervals; it is more of finding the proper size of each trade's holding time within a portfolio and where the future may still be an unknown.

Holding all shares in S over any time interval is tantamount to holding the whole market, and therefore, your performance should be somewhat the same as the market average. There is nothing you can do to enhance any of the price series that will be dealt in the future. All you can do there is maybe pick better-performing stocks or better time slices to hold shares. But, whatever you do, whichever stock you may pick, for any period of time, there is nothing you can do that will change or alter the price series itself. The job remains to outperform the Buy & Hold. The job remains to find arbitrage where there is none to be had. And I believe this can be achieved by changing some very basic notions concerning the holding matrix, which, after all, is under our control to do with as we deem feasible. What is sought is a position sizing function that can deliver more than the Buy & Hold:


even if under many modern portfolio theories, any long-term path taken leads to:

an alpha scenario

This might be the case for many trading strategies that try to outguess quasi-random price series. As if one could develop a winning strategy by playing heads or tails. The more price series can be considered random, the more the above equation tends to hold. And in that case, SPT makes its point; there can be no enhanced payoff matrix. Consequently, the alternative seems to be to increase the holding matrix itself as time evolves, meaning that the inventory on hand increases based on a predetermined equation. Explicitly, then, about any time, increasing the holding function should do the job, providing the function is related to price performance.

We could easily write:

doubling performance

where the holding matrix, at terminal time T, has doubled as a result of time-slicing additional purchases over the trading horizon. With a time horizon of 20 years or more, the additional shares to be purchased could amount to per period, per year or so. And if we can multiply the holding matrix by a constant, then why not by a time function? Reinvesting 3.5% of excess equity buildup will accomplish the portfolio doubling scenario. Whereas, reinvesting 8.5% of excess equity buildup will multiply portfolio performance by 5.

enhanced payoff matrix

This is the cornerstone of what is proposed in my two previous papers. It is our ability to control the inventory and modify our position sizing over time that will make a difference. When looking at the above equation, it is not a payoff matrix that is pulled towards the gravitational forces of the market average that we see, but one that escapes it. And, at an exponential rate, at that. It jumps over the efficient frontier with an increasing Sharpe ratio, as was presented in my first paper.

An enhanced trading strategy was stated earlier as a combination of an improved holding function and possibly an above-average stock selection:

enhanced trading strategy

What was presented up to now on the decision surrogate pertained to trade size modifiers thereby adjusting the bet size based on predefined conditional settings. There is another set of conditionals that needs to be addressed and that is straight conditionals: the do it or not kind. This set is based on fundamental and/or technical indicators that can help determine the yes or no of trade execution. It was already present in the first series of factors by the use of the confidence or conviction level as well as the desirability of trade factors, which, in a way, could curtail or enhance the execution of a particular trade. But this set of conditionals pertains only to the yes or no of a trade, the criteria that open and close trade windows by time-slicing price series.

To all of this, we should add a gaming factor that could alter the whole set of parameters because of your experience of the game. It does not exclude, for example, that when a stock behaves in a pattern that you have seen so many times in the past, you simply throw out all the above considerations to only play your current hand based on your knowledge of the game. You simply make a bet, of which size is determined by your experience, guts, or conviction in your analysis of the present situation.

All the controlling factors presented are time function matrices that can be pre-set to meet specific objectives. First and foremost, all your trading procedures will depend on your portfolio size, which in turn will determine the trading basis in accordance with the number of stocks you wish to trade in your portfolio.

Ultimately, what you want are small bets on losers and big bets on the winners, with the biggest bets for those that have really outperformed all others in your selection. You have to do this in an environment where the future is unknown and where cheating is prevalent. It's like playing poker with some of the opponents having aces up their sleeves. You have to protect yourself first. The only way to achieve this is to start with small bets relative to your portfolio size, use stop loss, and up the ante only when your hand is the strongest.

So, it is not a simple game where I buy this, and I sell that. It is more complicated as all these considerations should be addressed in all trading procedures, and your trading script should be accountable for implementing trading decisions within these constraints at all times. Each stock has its unique price signature, and it is your decision surrogate to set the trading signature in such a way as to outperform the simple Buy & Hold strategy. In the end, it is your decisional process that matters, your conviction in yourself as well as your belief system. Let the game teach you how to play the game! The market is your playground, learn how it behaves, and then learn how you should play so that at endgame you are left standing and the winner...

Again, this installment took a direction of its own. It was not what I intended to cover in the first place; the subject was supposed to be the implementation. However, I think that, as this thing evolves, it should lead there.

I am currently being sidetracked again by my research, which is on a price-modulated version of my last paper. It is complicated, but I think it should lead to something quite interesting. It will take time to develop... so for those interested, what can I say, stay tuned...


1 Introduction to the Mathematics of Financial Markets.

    S. Albeverio, W. Schachermayer, M. Talagrand: Lecture Notes in Mathematics 1816 - Lectures on Probability

    Theory and Statistics, Saint-Flour summer school 2000 (Pierre Bernard, editor), Springer Verlag,

    Heidelberg (2003), pp. 111-177.

    Available as a pre-print here.

2 Alpha Power: Adding more Alpha to Portfolio Return.

Jensen Modified Sharpe Ratio to Improve Portfolio Performance. 

Created on ... September 12, 2009,   © Guy R. Fleury. All rights reserved.