Oct. 10, 2020

The **previous notebook **put some emphasis on having an edge to overpower built-in long-term return degradation. There are many ways of doing this. The payoff matrix equations can have gazillions of solutions. They all depend on how you deal with the ongoing inventory matrix **H**. Trading implies doing a lot of trades, and doing so brings along with it the Law of large numbers.

It was shown in the notebook F(t) = F_{0} + n ∙ x_{avg} was an equivalent to the payoff matrix. This reduced huge payoff matrices to 2 numbers: the number of trades executed and the average net profit per trade. Hence, increasing either of those two numbers or both by whatever method would increase overall portfolio profits. Obviously, under your own risk constraints, available resources, and trade mechanics.

The first step is to make sure you have an edge: x_{avg} > 0. A negative xavg and you are losing at this game, no matter if it occurs early or late in the game. If ever, during this long-term trading interval, we have: n ∙ x_{avg} < -F(t), then game over; you lost it all. You can get there slowly or fast. I would say: *it is generally up to you*!

The following chart shows the impact this edge can have. It does put some emphasis on the strategy's hit rate since the normal distribution of returns is still a randomly generated series.

The above chart extends what was presented in the previous notebook. We start on the first line with the same 50/50 zero-edge scenario giving the same 0.10 long-term expectancy portfolio factor multiplier. Meaning that the ending portfolio value would be equal to: *factor ∙ init. capital*, in that case, a 90% portfolio loss after 5,040 trades.

The edge is progressively increased by 20 trades at a time, thereby slowly increasing the hit rate and the expected outcome. The same randomly generated 3% sigma was used throughout. The last column gives the percent of trades that were responsible for the outcome, the other trades mostly canceling each other out, even if not quite.

At the 52% hit rate, we have a factor of 41.80 times the initial capital. The advantage comes from the 100 positive trades that were added to the positive side and the 100 negative trades that were removed on the negative side. That is 200 trades total out of 5,040 representing only 3.97% of all taken trades. The other 96.03% of trades (4,840) were simply almost canceling each other out since an average up-down pair did not revert back to 1.0 but to something a little lower: (1+0.03)∙(1-0.03) = 0.9991 in fact, lower by 0.09%.

It is by increasing the hit rate that the final outcome increases considerably. Notice that the factor curve on the right goes ballistic near the end. Seeking this higher hit rate can make a strategy really fly.

This did not change the nature of the return series; it remained a 50/50 proposition. It is the trading strategy's procedures that have to provide the edge. And based on the above chart, it might not need to be that high.

According to the above chart, a 52.4% hit rate was sufficient to multiply one's initial stake by a factor of over 100 times. Less than 5% of trades account for the net profits. However, to get there, you still had to trade over those 20 years with the 3% standard deviation return. You could exceed 1,000 times your initial capital by reaching a hit rate of 53.2%. An edge is given by the 320 trades representing only 6.35% of the 5,040 trades. Notice the overall hit rate does not get that high, less than 60%.

**More Testing**

The following chart shows 100 portfolios based on the zero-alpha scenario over trading intervals of 1, 2, 5, 10, and 20 years. All portfolios were randomly generated using the code in the previous notebook. Therefore, you can duplicate that chart at any time. Evidently, your numbers will be different since all return series were randomly generated with no seeds. However, due to the number of tests in each column (20), you should get, on average, about the same results simply due to the randomness of a large set of random returns (100 ∙ 5,040).

In the chart above, the last column (20-year scenario) has only one of the 20 tests with a positive CAGR. You have some terrible results in those 100 tests, just as you have a few highly positive outliers. The problem is that you could not know in advance which would be which. The very nature of randomness.

We see the portfolio returns degrade with time, as should be expected. The reason is in the following expression: (1+0.03)^{n/2} ∙ (1-0.03)^{n/2} which assured this return degradation.

**Adding A Little Alpha**

If you add a little alpha to the mix, the long-term portfolio expectation goes up the more you increase it due to compounding, and this is as long as the added alpha costs less than what it generates.

F(t) = F_{0} ∙ (1 + r_{m} + α_{t} - Ʃ exp_{t} )^{t}

This added alpha can be generated in multiple ways.

In the 20-year scenario in the above chart, 18 of the 20 tests showed positive results. The averages at the bottom of the chart show that the average expectation was also positive. There were outliers, as should be expected when conducting these 100 randomly generated portfolios. No matter which trading interval you choose, there is nothing that would tell you which portfolio was the best.

**More Alpha**

Increasing the alpha further will tend to push overall results even higher, as illustrated in the following chart. Here, the alpha was raised to 0.0025 over the zero-expectation scenario. The alpha can only be generated by the strategy's trading procedures. Part of it can come from the luck of the draw (the return series were randomly generated, after all).

Another noteworthy observation from the above chart is that the general average return is now going up as the trading interval increases. And yet, the return series were randomly generated. All that was added was this positive alpha nudge.

What these charts show is that the added alpha can increase the bottom line over the long term. The more you add, the better the average performance level. You still do not know which scenario will prevail, but you can have a general idea of what might happen going forward based on your trading procedures. Note that even with the added alpha, you can still have outliers.

But there is more you can do. For one, compensate even more, not by changing the game, but by changing how you look at it. For instance, in my free paper: **Fixed Fraction**, you have a simple method, again with equations, to compensate for return degradation. It does not require much, but it will do the job. You should find the section on over-compensating even more compelling.

The stock market is not at some equilibrium, like half the stocks go up and the other half goes down. So, one thing to do would be to stop using the same percent move for your profit targets and stop losses. On a large number of trades, your profit target percent will tend to a constant, and so will the average stop loss. But notwithstanding, the profit target percent should be larger than the absolute value of your stop loss.

**Related Files: (files were available on the Quantopian website. It closed in October 2020.)**

**Playing a Long-Term Game - Part I**

**The Capital Asset Pricing Model Revisited**

**HTLM format: Playing a Long-Term Game - Part II**

Oct. 10, 2020, © Guy R. Fleury. All rights reserved.