If you do Monte Carlo simulation, you basically reshuffle the trades and execute them in a different order. You can apply Monte Carlo simulations both to static bet sizes and to fixed fractional betting. Your simulation did not take into account the fixed-fractional approach. Here is the difference:
Static betsize: You reshuffle the trades and always bet the same amount. That is you always take the same absolute risk without looking at your equity. You would take the same bet if your equity is $ 100.000 compared to an equity of $ 500.000. Betting with a static betsize will lead to linear growth.
Fixed-fractional betting: Your risk as a percentage of equity remains constant. After a winning trade you increase your betsize accordingly, after losing trade you reduce your betsize. If you have a favourable bet, fixed-fractional betting will lead to exponential growth.
You can model both approaches with a Monte-Carlo simulation. The static approach is modeled with adding up returns. The dynamic approach is modeled by multiplying returns (or adding log returns).
Kelly betting is fixed-fractional betting using an optimal fraction of your account. This will lead to the fastest geometrical growth. However, fixed fractional betting is only possible, if the geometrical mean of (1 + r+) and (1 + r-) is greater than unity. In the case that I have described the geometrical mean is
and the system is not profitable. If this number exceeds the value of 1, fixed-fractional betting should lead to faster growth than betting the same amount irrespectively of returns. If the geometrical mean is smaller than 1, than you should not bet, as losing bets would significantly increase your risk of ruin, if you want to continue to play the game.
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I did not assume that I place 100% of my capital on each trade. I just assume that I rebalance the bet size to the downside after a loss and to the upside after a win. I know that I can compensate the adverse impact of rebalancing if I manage a portfolio and if the system is only part of that portfolio. In that case the rebalancing of the bet size depends on the return generated by the entire portfolio, the contribution of this bet as part of the portfolio might be positive.
But that was clearly not the question. If I run this system on a separate account and rebalance the risk after every trade I am going to lose money. And that is exactly the way leveraged ETFs work, which do daily rebalancing on a fixed-fractional basis.
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My simulations took into account fixed fractional betting as you had suggested.
If you are not convinced that this is free-lunch, we should play the game in real-life. Both you and me wire $100,000 to an escrow agent on the condition that we will numerically carry out your game for 10,000 rounds. To avoid bad luck, I think both you and me agree it is fair to repeat this game 1000 times and pick the median result.
For each time I lose, I'll pay you.
For each time I win, you'll pay me.
At the end of the game, we will exchange the net difference or the full $100,000, whichever is lower.
In fact, I've already written the code so we just need to decide on an escrow agent.
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This is not a fixed fractional betting approach. You have reduced the betsize to a fractional betsize, as you only want to bet 1% of your capital. In case that you win your account equity will move from $ 100,000 to $ 100,500.
Your bet size is calculated on the entire portfolio, but not on the separate equity of $ 1,500. Therefore you increase your betsize by 0.5% after a winning bet and not by 50% as the fixed fractional betting assumes. We are not playing the same game.
Also you have shown that the bet initially proposed can be profitable, if added as a component to an investment portfolio. A diversified portfolio reduces the volatility of returns, and it is diversified if the $ 99.000 are just sitting in your account as cash. However, if all the returns of the portfolio are positively correlated by 100%, the bet proposed may not yield a positive cash flow.
The outcome of your game would be close to
1.005 * 0.998 * 0.998 = 1.00098
during the first rounds, which is a figure > 1, whereas
1.5 * 0.8 * 0.8 = 0.96
is a figure < 1 describing the original bet. I am not playing your game.
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As this thread relates to "successful traders": They are at least risk neutral.
See definition of risk aversion according to Wikipedia:
Risk aversion is the reluctance of a person to accept a bargain with an uncertain payoff rather than another bargain with a more certain, but possibly lower, expected payoff. For example, a risk-averse investor might choose to put his or her money into a bank account with a low but guaranteed interest rate, rather than into a stock that may have high expected returns, but also involves a chance of losing value.
It can also be argued that successful traders have to be risk takers as there are no exact probabilities in trading. Hence, "calculating" the expected value of a trade is not possible (relating to the definition above). Even if you backtest a system for x years you just know the historical probabilities. These may or may not apply in the future.