January 4, 2012

Growth Optimal Portfolio IV

Programming trading strategies is a succession of transforming an idea or ideas into procedures that when applied performs better than the previous iterations.

I am always looking for the best performer. I also know that I can always do better. But there are some basics. First, I know that what ever effort I deploy I must at least beat the long term Buy & Hold strategy. If I can not do that, then why do all the work?

I can not know now what the Buy & Hold strategy will bring 20 years from now. I don't even know which stocks will survive or prosper for that matter. However, I have to decide now on a particular trading strategy and live with it no matter what happens and should that strategy fail, then I better turn around fast, limit the losses, quit or go back to the drawing board.

As a consequence, I put my efforts in selecting trading strategies that have a high probability of generating profits. That is why I back test a lot. I require, as a bear minimum, that a potential trading strategy should at least outperform on past data. I now prefer to design strategies that use over-diversification as protective gear with long term trend following as underlying philosophy and use no price forecasting.

After spending years at evaluating and designing trading strategies of all sorts, I know pertinently well the dangers of over fitting and over optimization as well as the quasi random nature of price movements. I design long term trading strategies where the whole time spectrum, as it expands, is used to determine trades.

Lost Time

There are not that many choices; the future will happen only once. What ever the chosen trading strategy, if I win all is fine; and if I lose, I have wasted not only capital but most importantly valuable time. There is no re-run button on the future.

In a long term horizon compounding return environment, the most valuable year is not the first one, it is the last one. But to get there, patience is required. The 20th year of compounding returns can produce as much profits as the first 14 years of a portfolio's 20 year life. And if your trading system takes some 10 years to show its merits because of the long term holding objectives, then you might not want to wait that long for confirmation that your strategy works fine. It would be a very expensive decision: it is worth the discounted value of profits that could be generated between year 20 and year 30. Capital * [ (1+r)30 - (1+r)20 ]. It is much less expensive to over test your strategy over all kinds of market conditions.

On the other hand, searching for better strategies can be worthwhile. Here you are dealing with rates of return. Capital * [ (1+(r+x))20 - (1+r)20 ]. And depending on the value of the added performance x, the difference can be considerable. However, if the initial capital is small, one should consider not leaving his/her day job.

I often see people designing trading strategies that fail when they go live. But most often, after examining their strategies, all I find are defective trading procedures. And then they are surprised that a strategy fails or breaks down. It is not that a strategy fails or stops working; it was designed defective in the first place, it was designed to fail no matter what.

Strategies Exposure

There are thousands of trading methodologies that have been developed over the last half century. You would expect that the best of these strategies would be in the hands of professionals; the best portfolio managers out there. But when you look at long term  performance results, what you see is that about 3 out of 4 do not even beat the Buy & Hold. Either their trading strategies are not that great or the game of Buy & Hold is hard to beat.

So, what makes it so hard to beat? First of all, the Buy & Hold is a full market exposure strategy. It is 100% invested, 100% of the time. And second, long term (over 20 years), it has for most expected outcome the secular market average return which is around 10% dividends included. This means that just by sitting on your hands for some 5,000 trading days, you are bound, long term, to most likely win the game. And therefore, your real problem is simplified to a stock selection process. Take your best estimates on stocks you think have a long term positive expectancy.

Timing the Market

Any trading strategy that tries in some way to time the market by choosing which time interval is most appropriate to provide profitable trades will usually have less than a 100% market exposure. And, even if profitable, these trading strategies must  compensate with higher returns for their lack of market exposure. Taking one month or a year to produce a 20% return on a trade generates the same amount of profit.

If your trading strategy is less than 10% of the time in the market, it will have to produce its return over those 500 trading days or else it will under-perform. It is not that you can not do it, it is just that it gets harder as you reduce market exposure. Just missing the 50 biggest up days over the last 20 years (5,000 trading days) will almost wipe out all the profits the Buy & Hold would have made over that period.

Under exposure can also be achieved by using less than 100% of available capital. Again, you have to compensate for the lack of market exposure. Your capital has to work harder to achieve the same results as the fully invested scenario. If the Buy & Hold can give you 10% long term on 100% of your capital; how much return do you need using 40% of your capital invested 20% of the time to achieve the same result or better?

The Pay-Off Matrix

The Buy & Hold strategy using Schachermayer's pay-off matrix representation can be expressed as:


This pay-off matrix represent all the profit that can be extracted using what ever holding function I may develop. It is the composition of the holding matrix that matters.  When looking at the above mathematical expression, using the same price series, meaning selecting the same stocks over the same trading interval, there is only one option available left and that is to develop a better holding matrix, a better inventory management system that can take advantage of what ever price series is thrown at it.

There is then, in the case of using the same stock selection, only one way to outperform the Buy & Hold strategy and that is to provide by what ever means available an enhanced holding matrix: H+  >  H.

     Σ(H+.*ΔP)  >  Σ(H.*ΔP)

It is not necessarily a matter of predicting where prices are going. The more you diversify, the more your stock selection average price will tend to highly correlate to the market's average price.  You have to work on the holding function itself. And it must, at least, do better than the Buy & Hold strategy. If you succeed only marginally to exceed the Buy & Hold, then you have to evaluate if the added effort is worthwhile.

Viewing ΔP, the stock price variation matrix, as a 100 stocks by 5,000 trading days (about 20 years of daily data, some 500,000 price variations) can probably help in seeing the magnitude of the problem. You will have over the portfolio's life some 500,000 independent decision points; all part of a seemingly random data generation process where the future appears unpredictable. Yet, that is the universe in which you have to thrive.

Picking the same stock selection to trade as in a Buy & Hold scenario, it becomes almost evident that to out-perform, it is necessary to provide a better, an enhanced holding function: H+  >  H.

It is this enhanced holding function that needs work. To represent the Buy & Hold, the holding function could be written as:

 H = hioI  where I is a matrix composed entirely of ones (1).

and hio is the initial quantity of shares bought in each of the 100 stocks (i).  

A Profit Reinvestment Policy

The cumulative profit generated by the Buy & Hold strategy could also be expressed using compounded returns, as in:

 Σ [ hioI.*Pio((1+ri)t  -1)]

where the total profit generated is still the initial quantities of shares bought in each stock and where each stock price appreciates at its own averaged rate of return over time.

A profit reinvestment policy could be expressed using a delayed rate of return applied to the inventory as in:

 Σ [ hioI(1+ri)t-1 .*Pio((1+ri)t -1)]

The inventory of each stock would increase according to their respective delayed price appreciation functions. What this would mean is that by reinvesting the accumulating profits in rising stocks, one could achieve returns above the Buy & Hold by this delayed exponential factor.

It would be like having the original Schachermayer payoff matrix expressed as subject to a double exponential function. It would be like receiving interests on the accumulating profits (the excess equity buildup).

This has far reaching implications. It would mean that from an initial portfolio selection of 100 stocks (say the S&P 100) you could expect more, performance wise, than the Buy & Hold strategy (outperforming the index). In fact, you would be generating alpha points above the Buy & Hold by this inventory appreciation factor: the profit reinvestment policy rate. It would also imply that the generated alpha is not just a number as in the Jensen ratio, but a function, and an exponential one at that.

A simple profit reinvestment policy can be one source of exponential alpha. And would have for side effect to produce an exponential Jensen ratio residing above the capital market line (CML). Refer to my Jensen Modified Sharpe 2008 paper (Fig. 1 page 8) where the explanation for the following figure is provided.


A Jensen exponential alpha not only jumps over the CML, but it will also escape the gravitational pull of the efficient frontier at an exponential rate.

By adding a trading component to the above equation, one can push performance to an even higher level.

Published ... January 4, 2012 © Guy R. Fleury. All rights reserved.

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