April 28, 2013

## On Randomness

When looking at stock price series, you often hear that such series are random or almost random in nature and if such was the case, then such series would have little predictability if any. However, some interesting observations could be made depending on the model used to mimic random stock prices.

The following chart shows a randomly generated price series. It's easy to build: price varies within a range of +/- \$ 5. It uses Excel's rand() function:

Model 1:    P(t) = P(t-1) + (RAND() - 0.50)*10

 Model 1: Random Price Move

Even if the chart spans only 1,000 data points; one would be hard pressed to say that there is no trend in that chart. Yet, from any one of those points, even if a trend was in formation and can easily be seen, there was no way of predicting what would be the next data point or, for that matter, if the trend could continue, what ever it was in what ever direction. It is sufficient to press F9 and have a totally new and unique price series. Each time you would press F9, a new price series would be generated with no memory of any previous iteration.

You can have trends and chart patterns; they are real easy to identify when looking at past data. But the real problem is to identify future trends, and there, if you are looking for a profit, you might find that trends are just part of randomness. Based on the above price function, no matter what you would want to use as a predictive function, nothing would help, nothing at all. That you tried trend following, mean reversal or technical indicators of all sorts, nothing would give you an edge, all you could count on would be luck.

I would add that what ever the trading methodology you would want to use, your expected outcome would be zero profits; and you would also have to make deductions for frictional costs. What ever portfolio composition rules you might want to use and what ever the correlation between all the stocks under consideration, none of it would have value. Try finding the efficient frontier for a 50 stocks portfolio where all price series were randomly generated as per Model 1. You might find it on past data, but be assured that what ever you found will not hold and will have absolutely no value going forward.

The equation of Model 1 says that adopting a symmetrical trading strategy based on point change will tend to produce no profits in the short, mid and long term as the expected value of this model is no change what ever the period under consideration.

## A Game Changer

Now the problem will change with a slight change in the formula:

Model 2:     P(t) = P(t-1) * (1 + (RAND() - 0.50)/10)

where prices now change within a range of up to +/- 5%.

 Model 2: Random % Move

The price series will tend to degrade slowly with time (1.05*0.95 = 0.9975), which makes it quite understandable. In the first case, the expected long term value had for probable outcome P(t) = P(0) while in the second case, the longer the time series, the more it will degrade and tend to zero. And also the overall sequence of percent changes will matter.

The above could be translated to a portfolio level consideration like using a trading strategy taking up to 5% gains and accepting up to 5% stop losses. From the above point, the long term outcome of such a trading strategy would be total portfolio destruction! Nothing even close to hoping to get even in the long run as in the first case. It won't be a consideration that the trend seems to degrade with time, it just will.

It is clear that using (RAND() - 0.50) will give a mean of zero, with zero drift and zero correlation. Thereby, no predictive value whatsoever, be it short, mid or long term scenario; otherwise the random function would not be random. There is no strategy that can beat heads or tails except pure luck.

The above chart has a negative long term drift (trend) built-in. In fact, long term, ΣΔP = - P(0), playing the percentage game is more difficult since it is really biased to the downside. The picture is totally different that you play for points or play for successive percentage returns. In the beginning, both price series might show little difference to the point where it will be hard to tell them apart, but gradually the spread will increase with time.

The first data series says that you might or could reach zero long term with no certainty (luck playing a major role here) but still having a 50% chance of ending positive. While the second says you will reach practically zero with an asymptotic probability approaching 1. The difference originating from considering the game as a casino type game (playing for points) or a compounded return game (playing percent changes). Note that the game itself is a compounded return game, and one can easily design trading strategies that play this type of scenario. It's amazing how often I see this last scenario in naive trading strategies where the author does not seem to understand why his strategy is breaking down going forward.

For those that believe that, long term, the game itself is totally random, they are out of luck. Playing the first data series will have for expected profit: zero, no matter how long you want to play. While playing the second series, ultimately, the expected loss is the total capital put in the game.

Both scenarios make the game not worth playing, except if you want to play the luck side, meaning that all you want to do is gamble and have fun. I would even add, that one should not be surprised to be part of the also ran after losing their stake, they will have just played for a known outcome which in this case could cost up to the total capital available for play.

## Not Totally Random

The above should make a good point for the case that prices are not totally random, that they have an underlying long term drift: P(t) = μdt + σdw where the long term drift component might be small but still there. For instance, the historical long term drift has been about \$0.04 per day on a \$100 stock while its variance might be in the order of \$2.00 per day and often higher.

It all comes down to how are you going to trade and on what basis will your trading strategy evolve over time. I find it easy to design trading strategies with positive expectancy: Σ(H.*ΔP) > 0, but to be consistent, I also think that price series are not totally random. And it is also why I look at the long term when designing and back testing my trading strategies. Should I not know how my trading strategy will do long term, then I could fall into the second scenario trap which is not a desirable scenario and which could have been easily fixed from the very beginning. The first data series can in probability reach zero even if its expected outcome is no change while the second can not, although in the long term, it will definitely tend to zero.

## The Minimal Drift

A case was made that there is a long term drift in stock price series that needs to be accounted for and should not be neglected or taken lightly; as in: P(t) = μdt + σdw where μ is the drift. Adding a drift component to the first expression would result in:

Model 3:    P(t) = P(t-1) + (RAND() - 0.50 + 0.02)*10

which would generate an upside bias of about 2%. Such an upside bias should be sufficient to make it a game changer.

 Model 3: Point Upside Bias

The price series is still unpredictable and its long term trend should be easily visible. And extending the price series over a longer time interval would only, in all probability, push prices even higher. It should be noted that having a downside bias would push the price series down over time, for instance, consider the following:

Model 4:    P(t) = P(t-1) + (RAND() - 0.50 - 0.02)*10

which would produce a 2% downside bias, something like:

 Model 4: Point Downside Bias

This would make the point that an upside long term expectation would make even a random point variation model a suitable long term trading model. And it would be by looking at the long term horizon that one could win the game almost by default. This type of strategy would only require to have an eye on future prospects of the stock you invest in; and the Buy & Hold would start to look like a reasonable first assumption for a long term trading strategy.

The last chart shows that there is no need to be long in a stock that has lower future expectations since its price will most certainly follow its diminishing prospects. If fundamentally a stock has declining sales and declining profits, it might not look like a long term positive scenario.

The more you look at it, the more a Buffett style stock selection is applicable to the problem at hand. Make the best long term stock selection you can and watch to see if the stock lives up to its expectation. There sure will be trading methods that can help improve on this minimalist design, I know I can design some, therefore why not you.

## The Compounded Drift

When putting the upside bias as a percentage price move, the picture changes again. This time, everything is accelerated, consider:

Model 5:   P(t) = P(t-1) * (1 + (RAND() - 0.50 + 0.02)/10)

where the up bias is again 2% which would result in a chart like the following:

 Model 5: Percent Upside Bias

Naturally a down side bias would destroy a stock more rapidly:

 Model 6: Percent Downside Bias

Again the long term prospect should be the main concern. Even with a small bias, either to the upside or downside, can make a major difference in the design of a trading strategy that tries to take advantage of a future that is still there to unfold.

Based on these various price modeling functions, it should appear that the most desirable model would be the percent upside bias model (Model 5) which makes long term prospects increase exponentially in time. In case some might have doubts, they might consider the following long term chart of the DOW:

 Dow Jones: Long Term Trend

which just by its log scaling show that prices have increased at an exponential rate over time. (Sorry, I don't remember where I got that chart).

It's a total game changer. I find all this just another point made for a Buffett style of playing the game. There is much to learn from his trading/investment methodology. One can build on this basic system and add some trading features which will tend to push performance levels higher.

Increasing the upside edge to 5% would result in something like this:

 Model 7: 5% Upside Bias            P(t) = P(t-1) * (1 + (RAND() - 0.50 + 0.05)/10)

which really makes it a worthwhile goal.

Then the object of the game becomes to design long term trading strategies that have this upside edge designed in their very structure. When looking at the above chart, it is clear that the most efficient strategy was to buy from the very beginning and hold for the duration. Nonetheless, I found that it would be even more profitable to trade over a stock accumulation process, and using the trading profits to accumulate even more shares. Note that more refined models would require adding fat tails and the use of non-Gaussian price distributions. But these would be refinements and would not change the basic structure presented in this article.

So the conclusion would be that one is bound to find more randomness designing short term trading strategies, even for a strategy using Model 5, while designing strategies basically for the long term (using Model 5) is almost an assurance of out-performance just by exercising patience.