Another way of considering the optimal portfolio problem is as in Schachermayer^{1} where he represents a trading strategy as a matrix of size
is the matrix of the quantities held in each stock (1,...,d) over the investment interval with final time horizon T. Each in this matrix is a time series. Therefore, the matrix holds the quantity of shares held in inventory in each stock for all given time periods over the whole investment interval having for terminal time horizon T. And as such, constitutes a complete trading strategy.
The stock price process, using the same matrix structure, can be represented as:
while the held asset values can take the form:
which is an valued stochastic process with being considered as the cash, bond or risk free account which in turn can serve as numéraire. The stock values process by itself can be expressed as:
and
Schachermayer^{1}, in his lecture notes, defines a portfolio as:
and where is a possible strategy belonging to the set of all available trading strategies with strictly positive outcomes over . denotes the inner product in whileis being used as the cash account. The beauty of this formulation is its simplicity which still states that one profits from the sum of price differentials on the inventory on hand.
As in Schachermayer^{1}, the payoff function at time T can be expressed as:
for some meaning for someone pursuing trading strategy H. This is a simple termwise matrix multiplication and as such represents "the" solution to any portfolio over any holding interval. You can't get any simpler than this; two symbols representing your entire trading history whatever the methodology used.
In trying to optimize the above equation, there is not much one can do to change S in the future, or in the past for that matter, therefore one is left with only one controllable variable, and that is H: the trading strategy itself which can be made predictable in the sense that one can always know at period t  1 the quantity that will be held over the next upcoming time interval. One can always make changes in the inventory at time t (buying or selling) for the next period t + 1. And with such a simple formulation, it becomes understandable that one should look at the total picture when designing a profitable trading strategy. It is not the matter of finding a strategy that works on one stock in particular or on a small group of stocks; it is finding a strategy that will work on all your selected stocks for the whole investment interval with the objective of having a terminal wealth as high as possible and all this without knowing what the future will bring.
Again as in Schachermayer, the whole problem is to solve the following equation:
where is the initial available capital and is the payoff, or profit matrix, from all the trading operations.
It is easy to see that the result of the inner product represents the profit generated from the trading strategy over the entire investment period. Therefore the wealth process can be made to be selffinancing by making sure that remains positive for all .
with the restriction that at all times.
Therefore, let H be such an applicable trading strategy which maintains this portfolio integrity. Meaning that there exists a trading strategy H where you don't lose it all.
All this implies that the most important aspect in these equations is related to the position sizing over time as price evolves. It is in the how well one manages his stock inventory that seems to really matter.
The Buy & Hold strategy can easily be expressed as the initial position taken in each stock at time t = 0. This can also be represented as where is a matrix composed entirely of ones and is filled with columns of : the initial position in each stock. You want more profits; the only recourse seems to be to put more cash on the table to start with, make a better stock selection or add more cash en route to T.
Here, k is simply a scaling factor to show that final wealth is directly proportional to the initial capital. The final wealth will be:
where this last expression states that the final wealth will be a direct result of the initial investment plus the sum of profit generated over the investment period. This does reduce to just another representation of the Buy & Hold equation.
Under the efficient market hypothesis, the more stocks one selects for his/her portfolio, the more the average performance of this portfolio will tend to the average performance of the market. It is said in my previous paper that the average portfolio performance with tend to the market average as the number of stocks in the portfolio tend to the number of stocks in the market.
This implies that on the price side, due to the increasing stock representation, the average portfolio performance with tend to the average market performance over the long term should a Buy & Hold strategy be undertaken:
where is the initial quantity of shares being held until terminal time horizon T, bought at time t = 0 and at price . Therefore, under the Buy & Hold scenario, all the emphasis is put on the initial weighting of shares in the portfolio and thereby on the stock selection process itself. By using some form of overdiversification, one can almost be assured that the portfolio's average performance will tend to the market's average performance. Therefore, the do nothing scenario in the sense of no inventory change over the investment period has for most probable outcome the market's long term average return.
Should you want more than the secular average market return, then you will have to play with the inventory; make changes up and down in the hope of achieving higher returns. It is not a matter of designing a trading strategy for one stock, it is a matter of extracting from your selected stocks as much as you can from the entire matrix and this is done at the portfolio level and over the entire investment interval.
It might be within the simplicity of these equations that the portfolio optimizing problem can be best expressed: it is not only a stock picking problem; it is also an inventory management problem only limited by one's knowledge, skills, risk factors, capital constraints and an unknown future.
I'll leave the problem of stock picking for later. For the moment, my interest is in the trading strategy itself.
can be any viable trading strategy in the sense that at all times must be strictly positive . Any strategy that fails this requirement is simply out of the game having lost it all. As is, could be any generating function which maintains the minimum requirement of positive wealth. We could also add the requirement of selffinancing due to capital constraints. It should be noted that nothing should stop someone from adding more capital, or borrowing some, en route to terminal time horizon T.
Following from my previous paper (A Jensen Modified Sharpe Ratio to Improve Portfolio Performance^{2}) where a trivial trading strategy is expressed in equation (9) and adapting it to the Schachermayer formulation would lead to the following:
where the first I is a Poisson process matrix and where the expression in square brackets is a deterministic trading strategy governed only by the initial bet in each of the stocks and the ongoing added shares as prices rises. Here, has an equivalent meaning to as described previously as it represents the initial position taken in a particular stock.
As such, this linear trading strategy is bound to remain a positively increasing wealth process as it allocates funds to best portfolio performers while comparatively starving underperformers. Also, the stock selection process itself, just by being overdiversified, can assure the maintenance of the primary requirement of positive wealth at all times. Picking 50 or 100 stocks out of the available stock universe almost guarantees that not all of them will go down in flames at the same time with your portfolio. I have not included other controlling functions in the above equation in order not to obscure the point being made, however, be assured that just like in any trading strategy a form of stop loss is required as a form of insurance policy against portfolio oblivion. Note that the other controlling functions are discussed in my previous papers^{2}.
As was expressed in Schachermayer^{1}, the whole optimization problem resides in the following equation:
A fix inventory, like in the Buy & Hold only leads to average performance unless you have outstanding stock picking abilities. Overselection, leads to overdiversification which in turn leads to average performance.
However, simply in designing increasing inventory functions over the investment period will lead to an increased payoff matrix what ever the original selection may be. Keep in mind that the wealth process must remain positive over the entire investment period.
Through overdiversification, the average price process will tend to behave as the average price of the selected stocks which in turn will tend to behave as the average market price. Therefore, it is not necessarily only in the price process that one should seek higher returns; but in the combination of price and volume with the greatest impact originating from the volume side of the equation: the trading strategy itself. Optimal position sizing over all the selected stocks in the portfolio over the whole investment period then becomes the cornerstone of finding an optimal portfolio all within the boundaries of the limiting constraints as cited above.
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Notes:
^{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, SaintFlour summer school 2000 (Pierre Bernard, editor), Springer Verlag, Heidelberg (2003), pp. 111177.
Available as a preprint: here. 

^{2} 
Alpha Power: Adding more Alpha to Portfolio Return.


A Jensen Modified Sharpe Ratio to Improve Portfolio Performance.

I only uncovered Schachermayer's paper a month ago, even if it is dated 2000. It was like opening a new door, a new interpretation of the portfolio process. Its formulation simplicity was more than fascinating. I immediately started interpreting my previous papers in this new light and found that I almost had nothing to change. Everything fitted like a glove. It even enhanced my interpretation of my own work. I do thank the author for having made his lectures notes available on the web.
Created on ... July 8, 2009 © Guy R. Fleury. All rights reserved.