July 8, 2009
Another way of considering the optimal portfolio problem is as in Schachermayer1 where he represents a trading strategy H(∙) as a matrix of size T x d:
H(∙) is the matrix of the quantities held in each stock (1,...,d) over the investment interval with a final time horizon T. Each Ht in this matrix is a time series.
Therefore, the matrix holds the number 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 H(∙) 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 a-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:
Schachermayer1, in his lecture notes, defines a portfolio as:
and whereis a possible strategy belonging to the set of all available trading strategies with strictly positive outcomes over ,denotes the inner product in while A0 is being used as the cash account. The beauty of this formulation is its simplicity which still states that one will profit from the sum of price differentials on the inventory on hand.
As in Schachermayer1, the payoff function at time T can be expressed as:
for some, meaning for someone pursuing trading strategy H. This is a simple term-wise 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 a 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 the 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 A0 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 W(∙) can be made to be self-financing by making sure that Wt(H) remains positive for all .
with the restriction that Wt(H) > 0 at all times.
Therefore, let H be such an applicable trading strategy that 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 Revisited:
The Buy & Hold strategy can easily be expressed as H0 the initial position taken in each stock at time t = 0. This can also be represented as h0I where I is a matrix composed entirely of ones, and H0 is filled with columns of h0j, 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 the 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 will tend to the market average as the number of stocks in the portfolio tends 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 h0 is the initial quantity of shares being held until terminal time T, bought at time t = 0, and at the P0 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 over-diversification, 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 the 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 payoff matrix (H∙S). This is done at the portfolio level and over the entire investment interval.
The Optimization Problem:
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.
H(t) can be any viable trading strategy in the sense that at all times Wt(H) must be strictly positive Wt(H) > 0. Any strategy that fails this requirement is simply out of the game having lost it all. As is, H(t) could be any generating function which maintains the minimum requirement of positive wealth. We could also add the requirement of self-financing 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 T.
Following from my previous paper (A Jensen Modified Sharpe Ratio to Improve Portfolio Performance2) 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 iqj, in each of the stocks and the ongoing added shares aqj, as prices rise. Here, iqj has an equivalent meaning to h0I 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 over-diversified, 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, taking your portfolio with it. 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 kind of insurance policy against portfolio oblivion. Note that the other controlling functions are discussed in my previous papers2.
Breaking the Mold:
As was expressed in Schachermayer1, the whole optimization problem resides in the following equation:
A fixed inventory, like in the Buy & Hold only leads to average performance unless you have outstanding stock picking abilities. Over-selection leads to over-diversification 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 whatever the original selection may be. Keep in mind that the wealth process must remain positive over the entire investment period.
Through over-diversification, the average price process will tend to behave as the average price of the selected stocks which in turn will tend to behave like 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.
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.
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 lecture notes available on the web.
Created on ... July 8, 2009, © Guy R. Fleury. All rights reserved.