March 22, 2012

Growth Optimal Portfolio VI

After having added a covered call and a naked put program to the trading component applied to the share accumulation routine, we were left with the following pay-off matrix:

  Σ [ hioI(1 + gi + Ti + CCi + NPi) t-1 .* Pio((1 + ri)t  - 1)]

where gi was the reinvestment policy rate, Ti the trading strategy contribution rate, CCi the covered call contribution rate for each of the individual stocks, and NPi the naked put program contribution. All these procedures combined to generate alpha at an exponential rate. (see On Growth Optimal Portfolio V for details)

From part V, gi, the reinvestment rate was capped. If fact, it was set to tend to ri; meaning that a fraction of the equity buildup would be reinvested in accumulating more shares. And this simple step gave access to positive exponential alpha generation. The last chart in part V show the reinvestment policy rate moving away from the Buy & Hold rate of return at an exponential rate which technically confirms the exponential alpha generation. A simple profit reinvestment policy was sufficient to gain exponential alpha.

Applying a profit reinvestment policy (PRIP) is relatively easy, it is about the same as adopting a dividend reinvestment plan (DRIP). As dividends are received, they are applied to purchasing more shares, or reinvested in other stocks having more potential, but reinvested nonetheless. And when looking long term, reinvesting accumulating profits is the same as averaging up since long term (like 20 years or more) average stock prices do tend to go up, on average. It does look like a scenario for a retirement plan.

The Game: Waiting It Out

Playing the stock market is not just getting in a trade, wait a few minutes, take your profit and run. It is a persistency game; you are exchanging risk and time for money. It is not like playing blackjack where all the odds are finite and you win or lose on the turn of a card. Playing the market is a variable pot game with variably quantifiable risk. The shorter your time span, the higher the uncertainty. The longer the time span, say 20 years or more, the more the probability of winning is approaching asymptotically to 1 if you had the foresight of spreading your bets over many candidates (meaning diversifying). But also, the longer the time span, the lower your expected return. As if everything you do or might do leads to a long term expected return that will approach about the same performance as the Buy & Hold. Short term, one can easy out-perform the Buy & Hold, but long term it gets harder and harder.

It is sort of a strange game. You want to outguess where the market is going short term, and you end up with a lesson or two, a reality check of sort, resulting in a reduced account. You want short term to go with the flow, but often find out that when you identified that the flow was moving in your direction and decided to enter a trade, the flow “changed it's mind” to force you to accept a stop loss. The shorter your trading interval, the more quasi-random price movements came into play. And the higher the randomness factor, the closer you get to simply gambling on the outcome of a specific trade.

You have to analyze your motives, the why you wish to take a trade on a certain stock at that particular time. And what ever you may decide, on what ever criterion, it might probably be just a coincidence that the market moved in the same direction as your bet.

The Hurdles of the Game

What are the hurdles in the game?

You know what you want, you know how much you got to play with, you know how much time you have, you know how much you will accept to lose on a single trade or a series of trades and you know how much you are willing to lose in all. You know there is a learning curve and that it is only with time that you will get better playing the game as if it were some kind of sporting event.

You also know that there is a limit to how many things you can do all at once, to how fast you can type, read the news or follow the tape on a 100+ stock portfolio. You find limitations left and right that will define your trading strategy not based on the ideal or optimal strategy but on your belief system of what you think you can do based on your current understanding of the game. It is your comprehension of what is happening before your very eyes that at times will determine for you your outcome of the game and not necessarily in accord with what was the real market outcome.

You had to setup and fund your brokerage account, select your trading platform, design and test your trading strategy. It wasn't that easy a task; a few brokers, a few software packages but thousands of strategies to choose from not counting the ones you intended to develop on your own after having studied the programming language of your preferred software package. You had to learn about what so many others had tried in the past, the bad, the too simple to the too complicated, and the few that showed some promise. You did what every other trader searching for a solution had to do, redo what all others had done before.

Even after all this, it was still not enough. You had to contend with Modern Portfolio Theory, the efficient market hypothesis and the capital market line. All of which try to curtail your enthusiasm for high long tern rates of return. You learned the limitations of the most expected long term outcome and ended up understanding that the efficient portfolio frontier is almost a brick wall that no one dares challenge even though all your research is oriented to breaking it. Otherwise, why take the challenge. Wouldn't buying an index fund be a lot less troublesome? At least, this Buy & Hold strategy would have for long term expected outcome the average secular return and as a result you would win the game.

The Basic Pay-Off Matrix

A lot can be said just based on the simple expression of the pay-off matrix:


It represents the total profits generated by all the stock holdings in the portfolio over the trading interval under consideration. Just expressing H as columns of constants volumes makes it a Buy & Hold benchmark, because that is what it becomes, a none changing inventory for each stock in the portfolio over the entire holding period.

In Excel, it is easy to implement, on one side you can have a matrix of say 100 stocks where all entries represent the quantity of shares held at any specific time over the trading interval. While the other matrix is simply the price variation from period to period (like the daily changes in price from close to close). Column wise multiplication for the matrices will result in the profit or loss for a given day and the sum of all columns will give the total profits generated for that particular strategy H.

Any function that modifies H, (the holding inventory matrix) will necessarily modify the final output: the total generated profits. Therefore, it is these functions that become the focal point for modifying or improving the output of the pay-off matrix. What ever the relationship with the price variation matrix, the holding function matrix will remain the central point even if it tries to outguess what ever the price matrix is doing or is going to do.

Using the same stock selection will result in the same price variation matrix; it becomes easy to compare two trading strategies over any trading interval:

 Σ(H3.*ΔP) >  Σ(H2.*ΔP)   >   Σ(H1.*ΔP)   >   Σ(H.*ΔP
In all cases, your trading strategy should perform better than the Buy & Hold, otherwise, why take the challenge? Also, your latest strategy H3 must outperform the best previous ones. Again, why adopt H3 if it does not outperform H2, or H2 if it does not out-perform H1? It is this series of iterations, finding ways to outperform previous strategies that enables one to search for new ways to implement trading procedures that can modify the holding matrix in order to achieve better results.

Probably the easiest way to improve performance of the pay-off matrix is to put more money at work:

 Σ(2H.*ΔP)   >   Σ(H.*ΔP)

Putting twice as much initial capital will produce twice as much profits. This is easy to understand, even on the Buy & Hold strategy starting with double the other guy's initial investment would most certainly produce double the other guy's profits, naturally assuming that both invested in the same stock selection using the same strategy. This is the same property as doubling the bet size by doubling the inventory.

What ever is done to modify the holding functions will have for effect of moving away from the Buy & Hold scenario and this from either side. If you put half as much capital as in the Buy & Hold, there should be no surprise to see half the Buy & Hold profits. Therefore, the relative degree of market exposure really matter.

The ideal holding function would be a mirror image of the price matrix, not the price difference matrix but the actual price matrix. This would mean having the lowest inventory levels at peak prices and the highest inventories at price lows. But this would require knowing what prices will do, and this is the same as forecasting prices which was admitted in the beginning as being most difficult to do with any accuracy.

In real life, what would happen having a price-inverse holding function would be: 1- selling on the way up until there is no inventory left thereby making some profits but also cutting overall profits short; 2- buying on the way down, accumulating shares on the losers and as a consequence pyramiding losses which is not a great way to win the game.

When examining the inventory matrix, you would find zero inventory in the stocks that when up the most and huge inventory levels on the stocks that declined the most. What sounded like a great idea in the beginning might not be that great after all. You would have made some profits on the rising stocks while accumulating losses on downers which might require a major rebound just to get even. And sometimes, it happens that stocks prices do not rebound at all forcing the acceptance of huge losses on downers.

Trading can be viewed as the product of the sequence of returns from all the closed trades, that they be profitable or not. You could view it as a long chain of rates of return:

  ..., (1+r)(1+r)(1+r), …, (1+r)(1+r)(1+r)(1+r), ...

Should any one of those returns (r) equal -1, then it is game over. It represents a major threat to a portfolio. In a single trade you could lose it all. As a consequence, you should never put your entire portfolio on the line on a single trade (unless the odds are really high in your favor). But increasing your bet size as the price goes down is just that, a way of pushing (r) to -1. Also, any single all-in trade with an (r) reaching -0.50, will cut your portfolio in half. Therefore, the entire portfolio on the line is not that good an idea no matter how one looks at it.

Then Back to the Game

The first question should be: is it a game, is the stock market a game? And I think the answer should be: if you want to. The shorter the time horizon, the more price variations are subject to randomness. And the more randomness comes into play, the more your next entry becomes a bet, a gamble as to where prices are going. That you play big or small, does not change the nature of the bet, in reality there is only up or down and that may lead to exiting your position with a profit or a loss based on whatever decision surrogate you may use.

The longer the time horizon, the more the method of play becomes viewed as investments. Long term indexes (20+ years) on average tend to have high probability for higher future prices. But they take a long time to get there.

There is a need to look for possible solutions.

Published ... March 22, 2012 © Guy R. Fleury. All rights reserved.

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