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Thanks Harry. Yes, you were right to assume I was describing a system that trades the same period of time.

I appreciate your answer.

Mike

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The following user says Thank You to Big Mike for this post:

I am not yet satisfied with the result of this thread. We have shown, that it is possible to build a model for the most basic case of a Bernoulli distribution , that is when you enter a bet

-> you either win the amount A or lose the amount B
-> the winning probability does not change
-> the winning probability is known

Finally, these three assumptions have little to do with real live. The edge or winning probability are never known in trading and you can only make a best guess, based on some statistical data. Therfore the outcome of our model is of limited value. However, it gives us some information on position sizing and the parameters it depends on.

Position Sizing Depends on the Acceptable Risk Level

There were two input parameters

-> definition of ruin, or otherwise put, the maximum acceptable drawdown as a percentage of the account balance
-> the probability of ruin, which is the likelihood that such drawdown actually occurs

which were needed for determining the position size. There is no way around the point, that these two parameters need to be individually chosen. Nobody can tell you, whether 0.1% or 10% is an acceptable risk of ruin, and whether you stop trading after a 30% or 75% drawdown.

Experiment Versus Model

For the more complex outcomes that are the result of real trading activity, it is difficult to develop a mathematical formula, which allows us to calculate the position size, which is in line with our risk appetite.
Instead of running a complex model, I would rather like to repeat my trading experiment 100 times and then study the outcome.

Let us assume that it was possible to trade the same strategy a hundred times. If I accept a risk of ruin of 5 %, I would then look at the worst 5 outcomes, and adjust my position size in a way that the worst 5 outcomes would exceed the maximum allowable drawdown. But how do I get those 100 different series with trade data.

The Monte Carlo Method

A Monte Carlo Simulation allows me to obtain the 100 different series of trade data by taking a backtest

-> and create 100 series of trades by rearranging the order of the trades with a random generator
-> or picking a large number of trades allowing for double picks in a random fashion

The two types of Monte Carlo Simulation can be combined wiht trades of a constant trade size, but can also be combined with a fixed fractional betting approach.

A weakness of the Monte Carlo Simulation is that it assumes that there is no correlation between two consecutive trades, assumption which is not always true. The advantage is that the Monte Carlo Simulation can produce statistical information for any betting system - not limited to our Bernoulli distribution - which in turn allows to optimize the position sizing based on our risk appetite.

Interesting enough, no optimal F calculations are required for this approach. You only need to believe that rearranging the trades in a different order is a useful tool for evaluation the maximum drawdown of a system.

Replacing a Single Backtest With a Monte Carlo Simulation is Crucial

A single backtest will usually not give you any useful information on the likelyhood of a large drawdown. To get that information, several thousand trades would be required, an unless you are a HFT maniac trading for years, this will not be available. Without that drawdown information optimal position sizing is not possible, and two trading systems cannot be compared, if their risk of a drawdown is not known.

Monte Carlo lets you increase the statistical data available, which in turn allows you to make an estimate - even if this is not perfect - of the downside risk. There was a link to a Monte Carlo Simulation in the opening of this thread by @Big Mike, and I just want to refer to it. We should further explore that road.

The following 12 users say Thank You to Fat Tails for this post:

That monte carlo simulator linked in the first post is rather robust it seems, thought it does not bootstrap an actual list of trades. I need to look at the guts of it to see how some of the calculations are done, but initially it seems like a good starting point for discussion. It might be nice to have the variability of trades taken into account, instead of just the average trade. I will have to give it some thought.

Luger

The following 6 users say Thank You to Luger for this post:

This thread has some great stuff in it, but if I could just add a bit for the less technical.
A new trader here asked me about my trading style and my answer if anyone wants to hear it is, trade small, and diversify. New traders only want to hear trading systems and how much they can make, while if you talk to traders in the big leagues all they care about is money management and risk. This week JP Morgan relearned the same old lesson most rookie traders need to learn -- control your risk first: News Headlines

If a trader can master the two ideas below, they will make more money than they can with any indicator or clever strategy:

I am new on the forum so hello to everyone.
I am looking for the formula of the Risk of ruin for Kelly for a fixed number of events.
If I have a trade system and I apply to it the formula of Kelly (f) n-times how much is the probability to reach X fraction on the bankroll?
If I have a different trade system not correlated with first and I apply Kelly (g) m-times with the same bankroll how much is the probability to reach X fraction on the bankroll with the 2 system applied together?
Thanks for the help
Rosario from Italy

Reply With Quote

The following 2 users say Thank You to rosariod for this post:

@rosariod: Please read #44, #46, #53, #65 and all other posts of this thread first. You will find the Kelly formula and its limitation to Bernoulli distributions explained. You will also find a spreadsheet allowing you to compare two different systems and adjust the risk of ruin to your risk appetite.

Nobody can answer the question that you have asked, as it is not specific enough. The Kelly formula does only apply to limited cases and cannot be used in a general way. Please work your way through this thread and then come back.

The following 6 users say Thank You to Fat Tails for this post:

I’ve got a trading system with the following information:
p is the probability of winning a trade
q =1-p is the probability of losing a trade
b is the average amount of winning trade
a is the average amount of losing trade
of course m=p*b-q*a>0 and for Kelly f=(((b/a)+1)*p)/(b/a).
With this information I can calculate the risk of ruin absolute and the risk of reduce the capital to a fraction before reaching a target but I would like to calculate the probability of reaching a fraction of the bankroll after a fixed number of iterations.
Any idea?
Thanks

The following user says Thank You to rosariod for this post:

As far as I know, you cannot use the average amount of a winning and a losing trade and apply the Kelly formula. The Kelly formula only applies to Bernoulli distributions, which means to a set of trades that only have two possible outcomes, either a win of b or a loss of a. If the outcome of your winning and losing trades is distributed around averages b and a, the optimal f would depend on the variance of the winning and losing trades.

Let us assume that you have a sample of 200 trades and that you want to answer your question. Then a better approach to determine the probability of reaching a fraction/multiple of your initial bankroll is a Monte Carlo Analysis. Let your fixed number of trades be k. Then you could randomly select k trades from your 200 trade sample (if k > 200, you can use an urn model with replacement, allowing to select any trade several times) and repeat that process N times (for example 1000 times).

NinjaTrader would for example allow you to define a strategy, backtest it, and then use the backtested trades for a Monte Carlo Simulation. You can select the number of trades per simulation (that is your number k) and then directly read the percentage from the chart.

Attached is a sample chart, which does 1,000 simulations of 200 trades from a sample of 372 trades (SuperTrend Strategy on Gold futures). You can read the probability from the chart. The likelyhood that your capital exceeds 110% of the initial bankroll after 200 trades is about 45%.

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The following 7 users say Thank You to Fat Tails for this post:

First of all thanks for your help and for your time.

For sure a Monte Carlo Analysis is a good approach to determine the probability of reaching a fraction/multiple of your initial bankroll and I’ve already done it but I would like to find a formula and then, generalizing, apply it to different trading system on the same bankroll.
If I trade k different trading system on the same bankroll with k different f(i)I would like to have a function P(X,n) that give the probability that the bankroll is X fraction after n iterations.
Is it possible to have such function or I have to use Monte Carlo Analysis.

The following user says Thank You to rosariod for this post: