Optimal f

Discussions about Money Management and Risk Control.
lperepol
Senior Member
Senior Member
Posts: 30
Joined: Thu Feb 26, 2004 12:50 pm
Location: Castlegar, BC, Canada
Contact:

Post by lperepol » Tue Mar 09, 2004 1:28 pm

Kevin
What leads me to say this is that markets (i.e. unique symbols) trade differently.
...
assigning a fixed stop amount to different markets

Do the original turtle rules distinguish stops and entries for different markets?

I am also puzzling over this idea. A stock like MU trades at lower entry and exit values than most other stocks. I thinking over a method of categorizing stocks. Hmm -- beta might work. Currently I do not make such distinctions. When I crunch the bet fraction using multiple stocks I look for a bet fraction that has a low variance -- the result comes in much lower then what optimal-f gives.

Lawrence

ksberg
Roundtable Knight
Roundtable Knight
Posts: 208
Joined: Fri Jan 23, 2004 1:39 am
Location: San Diego

Adaptive Techniques

Post by ksberg » Tue Mar 09, 2004 2:01 pm

Do the original turtle rules distinguish stops and entries for different markets?
Yes. Stops are adaptive based on volatility (2*ATR). While the 1st unit entries are based on rules that don't differ between markets (e.g. 20 day high), the entry scaling is also based on volatility (0.5*ATR). There have certainly been other well-known systems that apply adaptive techniques to entry and exit as well.

Using adaptive techniques allow you to work consistently across markets and still account for market differences. Most often, when a system developer claims they have a single system that works across markets without optimization, it will use adaptive techniques of some kind. IMHO: that's the way to go.

Cheers,

Kevin

zidane
Contributor
Contributor
Posts: 3
Joined: Sat Aug 28, 2004 4:50 pm

comparing strategies

Post by zidane » Sat Aug 28, 2004 4:54 pm

Hello,

Vince say that you have to calculate optimal f's for different strategies (market systems) en then use the geometric mean (which follows from the f) to compare the strategies to see which strategie performs best. Why can't you just compare the returns of some stocks for the strategies? Why can't you say that the strategie which yields the highest return is the best, without calculating the optimal f?

thanks

ksberg
Roundtable Knight
Roundtable Knight
Posts: 208
Joined: Fri Jan 23, 2004 1:39 am
Location: San Diego

Re: comparing strategies

Post by ksberg » Sat Aug 28, 2004 6:15 pm

zidane wrote:Vince say that you have to calculate optimal f's for different strategies (market systems) en then use the geometric mean (which follows from the f) to compare the strategies to see which strategie performs best. Why can't you just compare the returns of some stocks for the strategies? Why can't you say that the strategie which yields the highest return is the best, without calculating the optimal f?
You certainly can, and many people do. However, when you compare returns you are only considering the end data point, and not how you got there. With equal returns, one system could have a smooth equity curve compared to another with a highly volatile equity curve. Each of these will exhibit radically different Optimal-f values. In this light, Optimal-f is related to other system measure discuss on this forum.

Cheers,

Kevin

zidane
Contributor
Contributor
Posts: 3
Joined: Sat Aug 28, 2004 4:50 pm

how to choose 1 contract (unit)

Post by zidane » Sun Aug 29, 2004 4:01 am

Hello,

By dividing de biggest loss by the negative f, one obtains the dollar amount to invest. This means that one should buy 1 unit for the dollar amount. This says Vince. But how to choose 1 unit? Isn't there any rule to choose it?

I don't also understand the word 'drawdown' ? I am a Dutch student and I find nowhere what it means. Can someone explain it to me?

thanks

ksberg
Roundtable Knight
Roundtable Knight
Posts: 208
Joined: Fri Jan 23, 2004 1:39 am
Location: San Diego

Post by ksberg » Sun Aug 29, 2004 11:00 am

Zidane, you will find formulas a few pages earlier on this forum thread. Study those formulas and you will see the answer right in front of you (hint: think "contracts" vs. units).

As for drawdown I suggest you start looking through books and articles on Technical Analysis to provide additional background. Drawdown is a core concept in trading: the amount of money lost from a new equity high.

My recommendation is to become familiar with core trading concepts before tackling Optimal-f.

Cheers,

Kevin

zidane
Contributor
Contributor
Posts: 3
Joined: Sat Aug 28, 2004 4:50 pm

how much is one contract?

Post by zidane » Sun Aug 29, 2004 12:22 pm

Hello,

My problem was what the meaning is of '1 unit'. I searched the internet and found that the capitalisation per contract equals
largest observed loss/(-f)

Then, the number of contracts equals
account balance/capitalisation per contract

For example, if the largest observed loss equals -1000, optimal f is 0.25 and account balace 10000, then the capitalisation per contract should be equal to 4000 and one should buy 2.5 contracts.
Can anyone confirm the correctness of this?

thanks

PS
Contributing Member
Contributing Member
Posts: 8
Joined: Mon Sep 13, 2004 9:21 am

About the optimal betting system

Post by PS » Mon Sep 13, 2004 9:27 am

Is the Kelly's System Optimal?

On the one hand there are many articles with some proofs of this, but on the other hand there are many fixed fraction bets calculators, which show different values than Kelly's values.

Can anyone to clarify this point?
Which betting system is the optimal one?

Thank you in advance,

PS

Forum Mgmnt
Roundtable Knight
Roundtable Knight
Posts: 1842
Joined: Tue Apr 15, 2003 11:02 am
Contact:

Post by Forum Mgmnt » Mon Sep 13, 2004 10:18 am

I'm not trying to be rude, but did you read the posts in this topic?

They answer your question pretty well and address the Kelly Formula specifically.

enigma
Roundtable Fellow
Roundtable Fellow
Posts: 99
Joined: Tue Apr 29, 2003 6:56 am
Location: UK

Post by enigma » Mon Sep 13, 2004 6:06 pm

Hi Forum Mgmnt,

I remember that you mentioned writing a paper on robust position sizing (correct me if I'm wrong) quite sometime back. Will you be posting it on this site? Cheers..

Louis

PS
Contributing Member
Contributing Member
Posts: 8
Joined: Mon Sep 13, 2004 9:21 am

Post by PS » Mon Sep 13, 2004 11:14 pm

Dear Admin,

>I'm not trying to be rude, but did you read the posts in this topic?

I very carefully read the posts in this topic. And this is the reason why I had placed my post. By the way, did you read my post CAREFULLY?

>They answer your question pretty well and address the Kelly Formula >specifically.

If you will run search on google using words like "fixed-fraction betting calculator", you will find links to different calculators. If you compare the calculations performed on these calculators with the Kelly's values you will discover that the results will be different. If you can write simple programs to test the results you will see that Kelly's values are not optimal according to such criteria as maximization of the probability to reach the goal and maximization of the expected value subject to previous restriction.

Now, let me repeat my question.
Which betting system is the optimal one?
I hope there are experts on this forum who will be able to clarify this point.

DrHendricks
Roundtable Fellow
Roundtable Fellow
Posts: 81
Joined: Tue May 13, 2003 12:21 am

Post by DrHendricks » Tue Sep 14, 2004 3:00 pm

Which betting system is the optimal one?
I think part of your problem in receiving an answer is that you haven't supplied enough information in the question. My initial reaction was "Optimum as defined how"?


Most people in the final analysis define optimum as a function of their goal. Are you trying to turn 10,000 into a gazillion in 1 year or are you trying to attract investors with low Sharpe, Sterling, Sortino MAR etc, ratios?

Another factor is that the optimum amount to bet (by whatever formula) is that amount which gives your equity a ride tame enough for you to stick to the system through drawdowns. This is a function of your own psychology.

Hope this helps.

PS
Contributing Member
Contributing Member
Posts: 8
Joined: Mon Sep 13, 2004 9:21 am

Post by PS » Tue Sep 14, 2004 3:20 pm

Thank you for your reply.

I agree with you that the answer depends on the criteria of optimality.

As far as the topic is fixed-fraction betting and we want to reach our goals, so there are at least two criteria.

The first criterion is to minimize the probability to lose the inital capital (bankroll).
The second criterion is to maximize the probability to reach the target profit or sum.

Any fixed-fraction betting strategy satisfy to the first criterion (at least theoreticaly). If we will try to maximize the probability to reach the target profit we will discover that there is a set of optimal strategies (which satisfy to both criteria).
For this reason there is an opportunity to maximize on the last set the expected value of our profit/return. (This is not possible in Kelly's approach.)

It seems these criteria are practical criteria. In the Kelly's approach the artificial function (log) is introduced to avoid trivial solution (at infinity).
There is no way to connect our goals (target profit) with Kelly's criterion.
Kelly's formula gives the same number for different targets.
For all these reasons it seems that Kelly's strategy is not an optimal strategy from the practical point of view.

DrHendricks
Roundtable Fellow
Roundtable Fellow
Posts: 81
Joined: Tue May 13, 2003 12:21 am

Post by DrHendricks » Tue Sep 14, 2004 3:30 pm

For all these reasons it seems that Kelly's strategy is not an optimal strategy from the practical point of view.
There you have it.

ksberg
Roundtable Knight
Roundtable Knight
Posts: 208
Joined: Fri Jan 23, 2004 1:39 am
Location: San Diego

Re: About the optimal betting system

Post by ksberg » Tue Sep 14, 2004 7:38 pm

PS wrote:Is the Kelly's System Optimal?

On the one hand there are many articles with some proofs of this, but on the other hand there are many fixed fraction bets calculators, which show different values than Kelly's values.

Can anyone to clarify this point?
Which betting system is the optimal one?
Please see Kelly vs. Optimal-f and the graph preceding it. The most optimally aggressive one can be is to optimize growth curve without constraints. Since Kelly bets more than optimal fixed fraction growth, Kelly can never be optimal.

However, optimization is always defined within a context of constraints: change the context and you get different optimal results. In optimization, this is often called the fitness function. The number and extent of fitness functions is only limited by the imagination. Hence, what is optimal to one person may not be otpimal to another (very literally in the mathematical sense). That is why asking which is the optimal betting strategy is a bit nebulous. If you're asking about unconstrained optimal growth, the answer is Optimal-f.

Hope this helps,

Kevin

PS
Contributing Member
Contributing Member
Posts: 8
Joined: Mon Sep 13, 2004 9:21 am

Optimality???

Post by PS » Tue Sep 14, 2004 10:24 pm

Thank you for some clarification.

I am trying to consider the optimality from a practical point of view.

Any gambler and/or trader want to maximize her/his chances to reach a target on a time interval and minimize her/his chances to loose her/his capital at risk (bankroll). The preferences (risk tolerance, risk/reward ratio, etc.) are reflected in the levels of the target and capital at risk.

On this stage we have multi-criteria optimization problem. The solution of this problem is a set of fixed-fraction betting strategies. On this set it is possible to consider the next optimization problem: maximize expected value of the total sum (bankroll+profit/loss).
This gives us an optimal solution wich will be different from Kelly's solution and optimal-f solution.

This solution is free from artificial assuptions about the criteria.

I agree with you that if we choose other criteria, the optimal solution will be different. But why we should use other criteria, which to some extent artificial?

ksberg
Roundtable Knight
Roundtable Knight
Posts: 208
Joined: Fri Jan 23, 2004 1:39 am
Location: San Diego

Re: Optimality???

Post by ksberg » Tue Sep 14, 2004 11:54 pm

PS wrote:I am trying to consider the optimality from a practical point of view.

[...]

On this stage we have multi-criteria optimization problem. The solution of this problem is a set of fixed-fraction betting strategies. On this set it is possible to consider the next optimization problem: maximize expected value of the total sum (bankroll+profit/loss). This gives us an optimal solution wich will be different from Kelly's solution and optimal-f solution.

This solution is free from artificial assuptions about the criteria.
What you ask, an optimal solution without assumptions about critieria, maximizing the bankroll + profit/loss is EXACTLY what Optimal-f does. There are some deep investigations into what you're asking, but it requires some serious focus and study.

Also, consider that constraints are what give us "practicality". For example, I might impose a constraint that I never see more than a 75% drawdown, or that I lose no more than 60% of my initial capital, or expose no more than 25% of total capital. IMO, I really don't want to consider optimal results without some constraints ... I prefer something useful over theory. Does that sound artificial?

Since the optimal growth solution has already been addressed for fixed fraction, I think it may be more useful to ask the same question about non or mixed fixed fraction models. This could include damped scaling such as Mark Johnson's aggressiveness model, tiered models similar to Tharp's Timid/Bold strategy, or equity model modulations such as Notional Equity or even equity reserves. These open up whole new dimensions that impact an "optimal" sizing solution. In my experience, these factors can play a significant role.

Cheers,

Kevin

PS
Contributing Member
Contributing Member
Posts: 8
Joined: Mon Sep 13, 2004 9:21 am

Post by PS » Wed Sep 15, 2004 10:03 am

Please note, there are some difference in the both problems.

I am trying to consider the realistic model in which we have not only probabilities of Win/Lose and corresponding amounts W/L, but also fixed time horizon , our target (profit we want) on this horizon, and our bankroll (money we are willing to risk on this time horizon).

>What you ask, an optimal solution without assumptions about critieria, >maximizing the bankroll + profit/loss is EXACTLY what Optimal-f does.

Please note, that in my case maximization of the expected value is acomplished on the set of optimal fixed-fraction betting strategies, where optimality was defined as a solution of the two-criteria optimization problem. In your case the optimization is accomplished on the set of all possible strategies.

Let me use your own words from this forum.

"Optimal-f is fixed fraction position sizing that is optimal in a very precise, mathematical sense. The methodology returns the value which maximizes geometric growth."

It seems that maximization of geometric growth is not a practical criterion.
Optimal-f does not take into account time horizon and our preferences (target, risk-reward ratio, etc.)

Optimal-f is greater (or equal ??) than Kelly's value. But Kally's value is greater than the optimal values (received from multi-critera optimization problems). It is not possible to use Optimal-f and Kelly's methods in cases with negative expected values (in the long run). But as we know from probability theory there are deviations from the expected values, so we can be profitable (on the fixed time intervals) even if the expected value is negative.

There is no restriction in multi-criteria optimization problems. The restriction are introduced, when we use some methods to reduce the initial problem to a one-dimensional problem. This adds some subjectivity in definition of optimality.

In any way, thank you for all your comments. Now I have a better understanding of this topic.

With kind regards,

PS

MCT
Roundtable Knight
Roundtable Knight
Posts: 102
Joined: Fri May 16, 2003 7:27 pm

Post by MCT » Wed Sep 15, 2004 1:06 pm

The first criterion is to minimize the probability to lose the inital capital (bankroll).
The second criterion is to maximize the probability to reach the target profit or sum.

Any fixed-fraction betting strategy satisfy to the first criterion (at least theoreticaly). If we will try to maximize the probability to reach the target profit we will discover that there is a set of optimal strategies (which satisfy to both criteria).

Controlling risk is the absolute number one priority, not optimization. Although important, optimization is only secondary in importance.

What you are attempting to do is fix in place what’s by nature a moving target. Remember, unlike a game of black jack the rules that generate these statistical properties are continually changing with time. And time, my friend, makes it difficult to determine what’s really optimal. You can't determine what your profit will be, but you can determine what your loss will be.

When it comes to optimization, what someone ones said applies quite well:
“You navigate by the North Star, but you don’t expect to get there.â€

ksberg
Roundtable Knight
Roundtable Knight
Posts: 208
Joined: Fri Jan 23, 2004 1:39 am
Location: San Diego

Post by ksberg » Wed Sep 15, 2004 6:46 pm

PS wrote:I am trying to consider the realistic model in which we have not only probabilities of Win/Lose and corresponding amounts W/L, but also fixed time horizon , our target (profit we want) on this horizon, and our bankroll (money we are willing to risk on this time horizon).
It's clear you're using mechanics of Kelly, which don't express what you're after. The mechanics of Optimal-f speak about initial wealth (W), Terminal Wealth Relative (TWR) and a Geometric Mean (G). Using trade frequency, you can derive number of trades expected in a fixed horizon (N), and then the expected TWR by taking W * G**N. I would use something like Monte Carlo simulation to derive the chance of actually hitting TWR in the future.
PS wrote:Please note, that in my case maximization of the expected value is acomplished on the set of optimal fixed-fraction betting strategies, where optimality was defined as a solution of the two-criteria optimization problem. In your case the optimization is accomplished on the set of all possible strategies.
Maybe it is useful to start enumerating the bet size strategies you are interested in. Fixed fraction is simply (Equity*Fraction)/(Equity at Risk), and there are numerous interpretaions and methods which derive a fraction.
PS wrote:It seems that maximization of geometric growth is not a practical criterion. Optimal-f does not take into account [...] our preferences (..., risk-reward ratio, etc.)
I agree (given the edits).
PS wrote:Optimal-f is greater (or equal ??) than Kelly's value.
In my experience, Optimal-f is always less than Kelly value.
MCT wrote:What you are attempting to do is fix in place what’s by nature a moving target. Remember, unlike a game of black jack the rules that generate these statistical properties are continually changing with time. And time, my friend, makes it difficult to determine what’s really optimal. You can't determine what your profit will be, but you can determine what your loss will be.
Agreed. That is where Ralph delves into how the probabilities of blackjack are nothing like trading, and why Kelly misses the point. If I remember correctly, when you do the mathematical redux, Optimal-f becomes Kelly for binomial distributions.

The other difficulty MCT mentions is that optimal changes with the series of trades, so that Optimal-f varies. This makes it highly likely that what you thought was optimal is actual trading too hot (or too cold ... but too hot's the one that bites back).

Cheers,

Kevin

Post Reply