### Intro

This is not your typical portfolio construction process. Presumably everyone understands the process wherein you weight the portfolio constituents based on expected returns relative to standard deviation, analyze the correlations between components, and develop a hypothetical 'optimal portfolio' that Markowitz or Sharpe would be proud of. That's not the FuseTrader portfolio.

Conversely, this website will be engineering an *optimal betting procedure* based on the Trade Model signals. The fundamental difference between methodologies is that we're not constructing a portfolio of investments, but rather a portfolio of trades, or bets, many of which may only last 1 day.

### Probabilities and Expectations

The following quote is central to the FuseTrader concept and the paper it's from is referenced here.

"The central problem for gamblers is to find positive expectation bets. But the gambler also needs to know how to manage his money, i.e., how much to bet."

-Edward Thorp

We believe the Trade Model provides us with the positive expectation bets that Ed Thorp mentioned. We're going to dig deeper into those expectations here, and then tackle how to manage our money later in the discussion. The model expectations are below:

The only flaw in the Trade Model is that these price arbitrage signals are so rare that often we will have less than 3 trades open at any given time, and that's while monitoring over 650 individual stock tickers in the market. It's hard to construct a "portfolio" with that small number of positions. We could adjust the signals so that we get more of them, but then the win rate would drop marginally and the average profit per trade would drop significantly. Therefore, for the time being, we are only modeling the highest quality trade signals for implementation of this portfolio, or betting model.

### Kelly Criterion

Now that we have positive expectation, we need to determine how to manage our trading bankroll and how much to place on any particular bet, or trade. The Kelly Criterion was created by John Kelly, a researcher at Bell Labs, who developed the formula in order to analyze telephone signal noise. A lot of the articles on the internet like to use a simplified version of the Kelly formula which is correct for gambling bets wherein a loss generates a loss of 100% of the capital which was allocated to the bet, think in terms of casino games like blackjack or roulette or sporting events.

**K% = W - (1 - W) / R**

Where:

- K% = The Kelly Percentage (portion of your bankroll to bet)
- W = Winning Probability
- R = Win/Loss Ratio

However, when betting in capital markets it's actually quite rare to lose 100% of your bet. Therefore, we are going to use the formula which takes into account the potential loss wherein you get some portion of your invested capital back, that formula is below:

**f* = p/a - q/b**

Where:

- p = win probability
- q = loss probability or (1 - p)
- a = non-win loss
- b = win profit

The simplest article I found on the subject which explains the math correctly is actually Wikipedia.

If you take the Trade Model stats from *All *the trades* *above and plug them into the formula as follows...

- p = 0.764
- q = (1 - 0.764) or 0.236
- a = 6.31%
- b = 7.55%
- f* = (0.764/6.31%) - (2.36%/7.55%) = 898%

You can see that the Kelly formula suggests that we should add almost 9 times leverage to the betting system for "optimal growth". On one hand that's a good sign the system expectations are favorable, but everyone knows you shouldn't go Full Kelly.

### Constraints

For the sake of conservatism, and to reduce volatility, I think we need to apply a few constraints to the betting model. It should be noted that not adding any leverage is already cutting the optimal Kelly percentage down drastically.

- Constraint #1 - No Leverage; Max Risk = 100% of available capital

Since we're now using a small fraction (100%/898% = 11%) of Full Kelly, I should therefore be comfortable betting 100% of the bankroll on the system at any given time. However, what if there's only one or two trades being held at that time, are we really comfortable betting 100% of our capital on only one or two positions? According to Kelly, maybe we should be! However, it's probably prudent to limit each single positions size.

- Constraint #2 - Max Position Size = 20%

This means if we only have two trades on at once, we will only be risking 40% of the bankroll. Some with traditional portfolio management backgrounds will not feel comfortable with position sizes that large however, I believe we will see that this is the primary factor in constraining the amount we put at risk. If each individual bet has approximately the same favorable probability of success vs. failure, then we should be comfortable with this amount of risk.

### Drawdown Risk

To minimize drawdowns, we are going to implement an anti-martingale betting approach. Martingale strategies are popular in casino betting, but it is generally well known that anti-martingale strategies have many benefits for stock trading. Here's how it works: if the previous bet was a win we're allowed the full risk of the model all else being equal; if our previous bet was a loss, we reduce the risk for the next bet. You can read more about Martingale and Anti-Martingale betting systems here.

To implement anti-martingale we're simply going to have the model run a query to determine if today's capital is larger than the previous day's, if the capital is >/= the previous day's capital then the full amount of risk is applied, if not then it's reduced by 25%.

- Constraint #3 - Reduce Max Risk by 25% on a losing streak.

### Correlation Risk

There is a lot to say about correlation risk with respects to betting and the Kelly Criterion however, we are going to simplify the computing requirements which would perfectly satisfy the optimal Kelly formula for our model. Estimating covariance between different stocks is going to be meaningless in our Trade Model application since we are attempting to arbitrage price extremes. For example, if we have one 'long' position and one 'short' position, their recent price action will most likely to have been 'down' a lot, and 'up' a lot, respectively. Therefore the covariance within the portfolio is really going to come down to whether the trades are 'long' or 'short', and more importantly the proportions thereof.

To implement this simply, we're going to assume that any ratio of long to short, or short to long >/= to 0.25 *is *diversified; if it's less than 0.25 then the portfolio *is not* diversified. For example, 8 longs a 2 shorts (ratio = 0.25) would be considered diversified, same with 1 long and 4 shorts. This is a pretty low figure, but we're approaching a level of being *too restrictive for *the portfolio* *overall.

- Constraint #4 - Reduce Max Risk by 25% if trades are Not Diversified.

### Summary

In general we are being very restrictive, often only allowing small portions of the available capital to be deployed. Since we just launched FuseTrader in January 2021, it will be interesting to see if our statistical expectations play out in real time. I plan on revisiting the portfolio trade weighting and risk constraining process on a routine basis in order to see if we're using the appropriate amount of risk. While we're not necessarily attempting to maximizing Sharpe ratios, everyone has their own risk profiles and we will review how to adjust the many variables up and/or down in order to assist the user in fine tuning their own risk while using the FuseTrader model for their own purposes.

Follow @FuseTrader and @Dyer440 on Twitter for any suggestions or inquiries.

*All *trading and

*hese trades*

*investing strategies come with the risk of loss, including this one. T**It should not be assumed that strategies which were profitable in the past will be profitable in the future or will equal the performance of the securities on this page.*

*may not be appropriate for your investment goals and requirements, and it is not investment advice.*