August 4, 2009

Before proceeding with this next section, I would like to make a few comments. This is not intended for commercial publication; for me, it is just a way of keeping a public record of what I think has an intrinsic value (the formula for the Jensen modified Sharpe ratio - equation (16) in my first paper). I write first for myself and sequentially, meaning from the top down, and when I miss something or other, I will go back to include what I think is missing to make the current passage more understandable.

Usually, I don't know exactly where my text is going. I only try to make sense as I go along. For instance, currently, the section after this one is blank. I do know what I want to say, but I often hit a snag when trying to express a concept or other in a mathematical format. I think that all the mathematical formulas I use are of the 2+2 = 4 type. So, there is not much to argue on those except maybe if they do or do not express clearly and correctly the concept under study. After reading Schachermayer's lecture notes, I was practically forced to re-interpret my writings in that light. It sidetracked what I was doing then, as this was quite a change in perspective. The representational simplicity used in his matrix notation for the portfolio optimization problem fitted with what I had done like a glove. I was working with time vector iterations, and there he was dealing with matrices and showing how simple a payoff matrix expressed the problem to be solved. I had nothing to alter; all my equations fitted without change. All I had to do was plug them in at the right place, and voilà. The whole problem is represented in a single matrix, talk simplicity. I wanted to cover the position sizing in a certain way, but as the text evolved, it took its own direction. It became too big a problem to fit in a small package. Therefore, an upcoming section will deal with strategy implementation, the constraints, and the betting system. So let's get on with it.

First, let's restate the basics from the previous sections. (HΔP) is the payoff matrix resulting from all the trading operations. It has size t x j, (with ) and expresses the selected trading strategy H available from the set of all admissible trading strategies with the constraint that at all times, the wealth process Wt(H) using this strategy H is strictly positive; otherwise, you are out of the game, having lost it all.

This t x j payoff matrix can be used to summarize any trading strategy, methodology, or philosophy you wish to pursue over any time interval and for any number of stocks you select for your portfolio. When looking at the portfolio optimization problem using the payoff matrix, it becomes clear that one should look for a total solution covering the whole of the investment interval. It becomes a matter of filling the total H matrix in such a way as to maximize performance.

The objective, over the long term, is to achieve the highest possible "terminal" wealth HΔP for this payoff matrix:

as before. The notation (∙.) stands for the matrix inner-product, a term-wise multiplication.

This is the same as saying that the objective is to maximize an expected wealth utility function. Since all the interest is on the terminal value of the payoff matrix, then the maximum outcome should be considered path-independent in the sense that what is of interest here is the final value, not how it got there. However, how it got there must be within the limiting constraints, and therefore, even how it got there actually matters.

If the holding matrix H is made a constant,  all the column entries in the matrix will have for value the initial quantity of shares bought in each stock h0j. The payoff matrix will then have the same behavior as a Buy & Hold scenario and would represent the profits generated using this particular trading strategy H. The behavior won't change even if you start with a higher or lower initial capital,  the matrix is scalable on the inventory side. The weight of each stock in the portfolio is:

So the H matrix could be view as a weight shifting mechanism even when it is constant.

When looking at the portfolio optimization problem in this (HΔP) matrix form, it does not leave much leeway as only two variables are at play: the quantity or inventory on one side and the price variations on the other. There is not much one can do to change the price behavior side of this matrix. All one can do is select d stocks out of an m-stock universe, knowing that as d increases, the portfolio's average price movement will tend to mimic the market average. There are, however, two other variables at play in the (HΔP) matrix, which are not directly seen. They are the time interval over which the trading is to occur and the number of stocks that will be traded. Both can affect considerably the outcome of any trading strategy.

Playing the market should not be considered a short-term endeavor. It is designed so that an investor will need many years of trading to achieve meaningful results. When looking long-term, total market exposure should be taken into account just as trading frequency. The Buy & Hold scenario has total exposure, with technically just one trade. Trade time-slicing will have a tendency to reduce exposure and thereby reduce expected long-term performance. All these factors should be taken into consideration when designing trading strategies over long-term horizons. The intention here is not to deal with high-frequency trading but more with mid to long-term investment strategies.

A stock's final price can be expressed as the initial price to which is added the sum of all price variations over the holding interval:

This implies that wishing to increase performance will necessitate better price series, meaning, on average, a better stock selection. You can only play, in any one stock, the sum of price fluctuations over the holding interval. One should not consider this game as a one-period game; prices fluctuate all the time, and it is the sum of these variations, which can span over many years, that counts. What really matters is when you get in and when you get out of a trade, which in turn suggests that some form of market timing might somewhat improve performance. But this is not enough, there is the notion of bet size which must be addressed as it can also have a major impact on performance.

If the stock selection process is better than average, then you can expect better performance on the price differential side of the equation. However, picking n stocks out of an m-stock universe will tend to select a portfolio of stocks that approaches asymptotically the market average as n increases, as presented in the previous section. Actually, the n-stock portfolio will tend towards the center of gravity of the m-stock market universe.

Whatever your stock selection, you should expect some 66% falling within one standard deviation from the historical mean return, not of the market, but of your own selection even if, statistically, it should tend towards the market average. Some 95% of your selection will be at most 2 standard deviations away from your historical mean. If the market has a long-term historical sigma of about 16%, this should tell you much about your expected portfolio variance.

It is not solely a matter of better stock picking that governs the payoff matrix: the inventory on hand as time progresses can play a major role. With it, one can not only attempt to time the market but also pre-determine the size of each bet. There is, therefore, a need for a decision surrogate that will have the task of determining when and how much should be put into a specific trade. This decision process has the mission to control the stock inventory on hand with the specific goal of outperforming not only the Buy & Hold strategy, but also outperforming your own portfolio selection using any other method. All this has to be done within your trading constraints: limited capital, risk-aversion preferences, uncertainty relative to the quasi-random nature, and variance of price movements.

All stocks are not created equal and should not be treated as such. Each stock in the portfolio should be assigned a relative strength, which should affect its initial weight, a kind of likelihood measure of overperformance. You want the stocks with the highest measure to also possibly have the highest initial weights.

Of all the stocks in your selection, you will have some 2/3 performing relatively close to the historical mean, while the remaining 1/3 will be divided into those stocks that really outperformed on the upside and those that performed so poorly that many of them went right down to zero. The only real problem is that you do not know beforehand which one will be in which category. Therefore, your trading method will have to adapt and detect as it goes along which stocks should be over-weighted and which ones should be under-weighted. What you want to do is put the heaviest bets on the stocks that rise the most while overly neglecting underperformers. In other words, you want your smallest bets on the losers and your biggest bets on the winners - those outperformers - the ones that can propel your portfolio to outstanding and stellar performance.

On the price side of the matrix, the price can be viewed as a compounded rate of return which leads to an equation of the form:

where each stock has its initial price P0j appreciating at its own rate of return of r j percent per period.

The price might evolve in a quasi-random fashion and seem uncontrollable; however, the inventory process itself can be made to change under controlling functions. It is this ability that can make a difference. Inventory control, in this sense, can be made predictable as you always know how many shares will be held over the next trading interval. Whatever the price matrix, much can be done on the inventory side to improve performance. One of the simplest methods to improve on final results is to start with more capital:

As a side note, one should always consider the impact of providing more initial capital versus waiting for capital appreciation.

For example, increasing initial assets by a factor of 10 will give you ten times the final result, which is equivalent to increasing the compounded rate of return by 12.2% over a twenty-year period. And increasing the initial stake by 100 will have the same effect as adding 25.89% to the base rate of return. This is another way of saying that one would need to achieve a compounded rate of 25.89% over and above the base return to produce the same results as providing 100 times the initial capital appreciating at the base rate r. The question is, which is easier to do: raise more capital or produce a higher long-term return?

While on this subject, also as a side note, one should consider the opportunity cost of delaying strategy implementation. The initial delay might appear minimal when considering long-term results. However, the initial delay propagates throughout the investment interval, and its effect on terminal value can be considerable.

Depending on the initial capital and the achievable rate of return, this delayed opportunity cost should be considered when designing trading strategies. For instance, if you start with 1 million to operate at a 20% rate of return and delay for one year, the above equation shows an opportunity cost of 6.3 million for year 20 (and this is scalable).

Another way of looking at the price process is with the Stochastic Portfolio Theory (SPT) notation using the following partial differential equation (PDE):

where the price increment is composed of the drift to which is added a standard Weiner process. This was the notation used in my previous paper2 before applying the Jensen-modified Sharpe ratio. This last equation emphasizes the importance of the random component compared to the drift (the rate of change), which is the linear regression of the time series itself.

Removing the drift, one is left with a totally random data series whose expected mean value is zero. In trading terms, it says that the expected long-term profit generated from all trading operations on this random component will tend to zero.

where the expected value of the payoff matrix is reconsidered using only the standard Weiner process as a price surrogate. Any implementable trading strategy would have little value facing a totally random data series. The only way to outperform the zero expected mean would have to almost entirely reside in pure luck. This is not what a trader is looking for as encouragement to his addiction.

I would like to emphasize this last equation. It says that no matter what your trading style or strategy: be it dip-buying, price breakouts, moving average something cross-over-under something of something, any form of position sizing or trade time-volume slicing; all of them would have an expected value of zero. There would be no expected benefit from all the trading activity. And faced with such a prospect, might as well put your money in the bank, or an index fund for that matter, and save yourself all the time and emotional aggravation of being a "trader". On the other hand, you might like the game for what it is: a gambler's paradise, the casino of choice, where you get almost even odds.

The game we play, however, is not an entirely random game. It is a "quasi-random" game where there exists a soft underlying upward bias. This bias is slightly in favor of the long-term holder who diversifies as he knows that his chance to profit increases asymptotically towards one the longer he holds on to 30 stocks or more.

Another way of expressing the payoff matrix using SPT would be:

where the stochastic nature of price movement is reconsidered. This puts it all back to square one. You might have a simpler expression in the matrix, but it still does not resolve the optimality problem. However, removing the random component, which has an expected value of zero, one would be left with the drift's linear regression over the price series, which could be restated as:

this, again, is another equation for the Buy & Hold scenario expressed in capital appreciation form. What this equation says is that the profits generated by this payoff matrix will be close to the long-term market average, especially if the number of stocks (d) is large. This last equation stresses that the average drift might be the main reason for the price appreciation over time since the expected value of the sum of the random fluctuations is zero.

Then it does not really matter how a price series is represented. It will not change what the future price will be on any one security. It is as a group only that some form of prediction can be made and only for the long haul. The more you trade short-term, the more the explanation for the price movement can be found to have a quasi-random origin.

Over long-term horizons, US stocks, on average, have appreciated at about 10% per year, including dividend reinvestment. In fact, there have been only a couple of rolling twenty-year periods where the US market has shown a loss. Does this make markets predictable? Certainly not! However, in the long run, one should expect, on average, to have some capital appreciation (simply due to this secular trend).

We could also represent the payoff matrix as:

since it is assumed to be path independent which again would be another expression for the Buy & Hold should H be held constant. So what is a trader to do? All his/her performance depends on price movement, and there is not much that can be done to improve predictability. And, as recent history has shown, it is not because a corporation has existed for 100 years or more that it is immune from bankruptcy or that the market can not drop by 50% in a single year.

Then, to resume, the payoff matrix, in its simplicity, presents the entire portfolio problem. In the finance literature, much is done to find trading strategies that, in one way or another, try to predict future price movements even when the signal is almost completely drowned in market noise. There are many studies that, in turn, will conclude that it is best to imitate the market average using indexed funds. Maybe, one should look more closely at the inventory side of this matrix.

So, the objective is to find a trading strategy H+ among all admissible trading strategies such that:

where the wealth process Wt(H) has to remain strictly positive. One can try predicting better price series, but that might not be the only place where overperformance lies; maybe it might be in finding a more optimal trading strategy. The quest is to find the best inventory holding function H+, whatever the selected stocks may be, which leads to the best performance possible for that particular stock selection. This does not mean that one should not try to make the best stock selection possible, only that whatever this selection may be, there are ways to improve performance by optimizing the holding function.

A basic question could be: who can predict the price of a stock 20 years in the future with any form of accuracy? I, personally, have not found anyone with that ability. However, I have found many that will take the bet with a high degree of confidence that the market as a whole, in all probability, will be higher 20 years from now. And that their stock selection, on average, will have a better-than-average chance of surviving long-term. Over a 20-year investment span, a stock can be sold, replaced, or a new one added to a portfolio at any time for any reason.

Let's recall one of the payoff matrix equations presented earlier:

where the price matrix is composed of d stocks appreciating at their own rates of return.

Now, assuming that price variations could be considered quasi-random time functions, the emphasis will need to be placed on the chosen decision surrogate, which will have to translate long-term objectives into short-term decisions. Then, the real issue is inventory management within the constraints of capital limits, risk aversion, and uncertainty. By using over-diversification, we can reduce the price component of the equation to its most probable outcome in the sense that the portfolio's average stock price will tend to mimic closely the market's average price movement. Taking a position in the 30 DOW stocks (in the same proportions) will closely mimic the DOW 30 index.

Therefore, it is to the decision surrogate (D) to determine the best trading procedure to be undertaken at all times. It is not a matter of guessing what the next price variation will be. It is determining what will be the final result of the sum of all the trading decisions over the long-term trading horizon. Therefore, what really matters is (D+HΔP)T which represents all the elements that will be part of the trading activities. When looking at this optimized trading matrix, it should be sufficient to show that  meaning that there exists a trading strategy such that it will outperform not only the Buy & Hold but also any other within its probability context.

What kind of trading activities are permitted in such a scenario? From all possible and strictly positive trading strategies, it is sufficient to select the subset where H+ > H holds should such a subset exist, and if it exists, it will definitely outperform.

And there is the main quandary in financial literature; such an animal does not exist. The Stochastic Portfolio Theory will demonstrate and prove, without a shadow of a doubt, that the Growth Optimal Portfolio (GOP) is the optimal and, in probability, the most achievable portfolio. Whatever you do trading, in the long run, you should expect that your performance will tend to the GOP simply because you stayed in the game that long. What is proposed here is that there exist whole families of controlled trading functions that belong to H+ where H+ > H and, therefore, will outperform.

The trading strategy His not the ultimate trading strategy; it is only the best you can do based on your selected stocks. The ultimate trading strategy, call it H* is composed of very few stocks if not just one: the "Atlas stock". The one that will outperform every other stock in this m-stock universe. And the ultimate strategy is very simple: put all your capital in that one stock and wait. Should you want to even outperform the Atlas stock, applying the Jensen-modified Sharpe ratio to it will do the job more than nicely. The trading strategy His a compromise of sorts for not finding H*. You select a number of stocks you believe to have the potential to survive long-term; since you might have estimated that your chances of selecting the Atlas stock were minimal. And remember, no stock comes with a guarantee of long-term survival.

## Partial Equity Buildup Reinvesting:

This is not a new concept; partial equity buildup reinvesting can be done in a number of ways. For instance, a fixed percentage of equity on each trade will do just that: the size of the bet increases as the portfolio increases. However, increasing the percentage of equity on each trade also increases the risk of ruin, especially if a high percentage of equity is placed on downers. Some rare events will see a stock suspended to reopen with an 80% haircut, which could almost annihilate one's portfolio. The difference presented here will be in the method used to determine the size of this partial equity buildup reinvestment, as it will be made conditional on relative performance.

All the preceding descriptions have led to plugging directly the main equation of my first paper into the payoff matrix, which will show that by simply reinvesting part of the equity buildup, one could transform the last cited equation:

in to

which some might recognize as equation (16) from Alpha Power2. Instead of putting all the emphasis on the price series, much effort is deployed in controlling the inventory on hand. All that was presented in both my papers applies to the above equation. It does resume all trading activities within the constraints presented. And it was that simple to put equation (16) in the context of the payoff matrix.

As a matter of fact, any time increasing inventory function will be sufficient to improve portfolio performance.

This time-increasing function will have to maintain portfolio integrity in the sense that at all times, the wealth process has to remain strictly positive and also survive within the available capital constraints. Therefore, it is not all time-increasing functions but the subset of such functions that adhere to the trading limitations just as the framework of equation (16) tries to do. Equation (16) is certainly not the only equation that will fit the bill; whole families of such subsets can be devised to survive and thrive within all the trading constraints.

My favorite form of payoff matrix has an exponential inventory function:

in fact, the above equation summarizes equation (16). You have the holding inventory increasing in time at a delayed growth rate of each stock's appreciation rate. In the beginning, the payoff matrix will be of the same performance as the Buy & Hold scenario. It is with time that the separation becomes more evident as g and t increases. The last equation could also have been written as:

which might make it more evident that  and there lies the power of the Jensen-modified Sharpe ratio, as presented in my previous paper2. Any increasing controlling inventory function would improve overall performance as long as, , or that  Reinvesting part of the equity buildup forms a feedback loop for the wealth process and transforms the inventory on hand into a controllable exponential function.

So, there you have it. You have the mathematical framework on which to build your own trading system. You can reverse engineer the procedures from all the clues left in both preceding papers. I encourage you to study carefully all the implications of such a methodology.

What is implied in this growth-oriented inventory is that as prices rise, one accumulates more shares based on the individual stock's performance. It is not a one-size-fits-all; it is an adapting reinforcement process where the best performers are rewarded while the worst have their portfolio weights continuously reduced as punishment for non-performance. See the analogy of the horse race in the first paper: Alpha Power2.

Also, from the equation presented, there can be subsets of strategies that will adapt to your stock selection to make your optimal trading strategy H+ > H perform better than not having implemented these procedures. Not only can you design a better-performing strategy, but you can also implement controlling procedures to limit risk.

The next installment will deal with: the strategy implementation, the constraints, and the betting system that are part of equation (16) as presented above.

_____
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, 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.