## Diversification Question

Discussions about Money Management and Risk Control.
C3PO
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### Diversification Question

Hello, I apologize if this is a beginner question, but I seem to be having trouble grasping how one should integrate fixed fraction position sizing with diversification. Below is an example.

Starting Capital: \$1,000,000
Optimal-f for market #1: .10
Optimal-f for market #2: .15
Optimal-f for market #3: .20
Optimal-f for market #4: .25
Optimal-f for market #5: .30

In this example, how does one position size for each trade in each market? How does the commonly used "2% risk per trade" fit into this example?

I am aware that people on this forum have warned against trading the optimal-f percentage. I am not debating that. I am asking about the mechanics of diversification with fixed fraction. I would very much appreciate it if someone can clarify this for me. Thank you.

ksberg
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### Optimal-f in a portfolio

Ok, I'll bite ... since you mentioned Optimal-f, take a look at page 159 in Portfolio Management Formulas. This section introduces optimization at the portfolio level, but for now let's just deal with f, the "optimial" fixed fraction. The book example shows 3 systems, where each considers a percentage of total equity, and the sum of the percentages always add to 100% (e.g. 40% + 30% + 30%). The simplest application of Optimal-f would apply 1/N of the capital to each system. In your example, you have 5 markets, so allocate 1/5 of total equity to each market system. Then use the reduced capital as the starting point to calculate the number of contracts for a specific market.

Optimal-f concept is independent of the 2% risk per trade rule; Optimal-f is a rule unto itself. To make a comparison, let's say you were trading 30 markets. The simple equity allocation is 1/N, or 1/30, which is 0.0333 of total equity per market. With this level of capital, Optimal-f could be as high as 0.6 to equal the 2% risk rule (0.0333 * 0.60 = 0.02).

Cheers,

Kevin

ps. Fixed fraction doesn't imply optimal, it's just a fixed fraction. Optimal-f is a fixed fraction, which is considered "optimal", in a specific sense of the word.

gbos
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I am afraid that the answer to that is very complicated. Also it is one of the most misunderstood concepts of money management. The solution of simply dividing your capital in proportions and trading each system independently is highly sub-optimal. For a â€˜correctâ€™ treatment, you have to know the joint-probability distribution for the returns of the five systems. This is close to impossible. An alternative is to estimate if your systems are independent or positive correlated or negative correlated. The systems will then trade using the SAME pool of money.

Example for two systems:

System_1 : f = 0.18
System_2 : f = 0.2

1 st case: The two systems are independent

When system_1 gives a signal allocate 0.16 of your capital to it
When system_2 gives a signal allocate 0.18 of your capital to it

2 nd case: The two systems are positive correlated

When system_1 gives a signal allocate 0.10 of your capital to it
When system_2 gives a signal allocate 0.12 of your capital to it

3 rd case: The two systems are negative correlated

When system_1 gives a signal allocate 0.20 of your capital to it
When system_2 gives a signal allocate 0.22 of your capital to it

If you had divided your capital 50-50 and traded each system separated then for the independent case you would had allocated 0.5*0.18=0.09 for the first system and 0.5*0.2=0.1 for the second system. Total allocation = 0.09 + 0.1 = 0.19. This is far from optimal because:

â€˜correctâ€™ Total allocation = 0.16 + 0.18 = 0.34 as in case 1.

Now, the final touch before start implementing your money management will be for you to decide what kind of drawdown are you willing to take for achieving your goal quicker?

Big: then use between 0.5 to 0.75 of above calculated fractions
Medium: then use between 0.25 to 0.5 of above calculated fractions
Small: then use < 0.25 of above calculated fractions

There are many shortcomings of using Kelly fractions or the equivalent optimalâ€“f fractions in your trading but this is not changing the fact that this is the BEST money management system out there if you are interested in maximizing the median (or in other words the most possible scenario) of your wealth. If you canâ€™t handle big drawdowns then the answer is simple, use a fraction of Kelly. The most risk averse, the smaller the fraction.

Regards

C3PO
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### Re: Optimal-f in a portfolio

ksberg wrote:Optimal-f concept is independent of the 2% risk per trade rule; Optimal-f is a rule unto itself. To make a comparison, let's say you were trading 30 markets. The simple equity allocation is 1/N, or 1/30, which is 0.0333 of total equity per market. With this level of capital, Optimal-f could be as high as 0.6 to equal the 2% risk rule (0.0333 * 0.60 = 0.02).
Thanks for the reply. I have been reading about the dangers of using optimal-f in other posts. However, I am not sure if those dangers apply in the situation above. Lets say you are in this situation. The optimal-f for this market is 0.6, but it does not violate your 2% risk rule. Should you be comfortable using optimal-f in this case? (I am not asking whether 2% is a good number to use. Say your risk rule is around 0.6% instead. So if optimal-f still allowed you to risk less than 0.6% of your portfolio equity, would you be comfortable using optimal-f in that case?)

C3PO
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gbos wrote: Example for two systems:

System_1 : f = 0.18
System_2 : f = 0.2

1 st case: The two systems are independent

When system_1 gives a signal allocate 0.16 of your capital to it
When system_2 gives a signal allocate 0.18 of your capital to it
Gbos, thanks for your reply. Lets stick with this example where the two systems are indepedent. Say you have \$1,000,000 starting capital. Then you would allocate the \$risk as follows:

System1: \$risk=\$160,000
System2: \$risk=\$200,000

Now say your System 1 stop was hit and System2 moved against you by \$100,000. Your account equity is now \$740,000. How would you size the next trade for System 1?

gbos
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C3PO wrote:
gbos wrote: Example for two systems:

System_1 : f = 0.18
System_2 : f = 0.2

1 st case: The two systems are independent

When system_1 gives a signal allocate 0.16 of your capital to it
When system_2 gives a signal allocate 0.18 of your capital to it
Gbos, thanks for your reply. Lets stick with this example where the two systems are indepedent. Say you have \$1,000,000 starting capital. Then you would allocate the \$risk as follows:

System1: \$risk=\$160,000
System2: \$risk=\$200,000

Now say your System 1 stop was hit and System2 moved against you by \$100,000. Your account equity is now \$740,000. How would you size the next trade for System 1?
This is the least of your worries. A conservative guy would assume that the trade which had gone already against him will turn out to be a loser and calculate the new risk as 0.18*640000. But you will have other more severe problems when trading many systems.

1st They donâ€™t produce the same number of trades per month so you have to calibrate for this just to make things comparable
2nd The triggering of the systems are not uniformly distributed in time (i.e. 4 trades per month doesnâ€™t necessarily means each month will produce 4 trades)
3rd You donâ€™t know which ones are going to trigger so you donâ€™t know what your optimal allocation will turn out to be.

You have to balance all this and many others in a way that makes sense and its also practical and manageable.

Regards

C3PO
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gbos wrote:A conservative guy would assume that the trade which had gone already against him will turn out to be a loser and calculate the new risk as 0.18*640000.
What is the rationale for using 640,000? System 1 is based on it's own rules that are independent of System 2. Without System 2, System 1 would have risked 0.18*840,000=\$151,200 on the next trade.

So System 1 was meant to risk

but System 2 changed System 1's base equity so that System 1 is now risking

Doesn't this throw off the "bet 0.18 of equity each trade" strategy of System1? I would be very interested in everyone's views on this. Thanks.

C3PO
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Kevin,

I have read P. 159 of Portfolio Management Formulas along with the rest of the chapter. Thanks for the info. However, it doesn't address my question about one system's equity affecting the equity of another system when using the same pool of equity. It just says to use the same pool of equity and to recapitalize each day.

Why isn't using a shared pool of equity a problem? Say for example I have \$10,000 bankroll to play 2 games. I'm supposed to bet 10% fixed fraction on the first game and 20% fixed fraction on the second game. I then start each game with \$5,000. Why would I want one game to affect what I'm betting on the other game?

I would very much appreciate it if you or someone can explain this to me. I suspect that it has something to do with optimizing the risk/reward of the total bankroll as opposed to the individual games, but I don't have a clear grasp of what's happening. Thanks.

ksberg
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C3PO wrote:I would very much appreciate it if you or someone can explain this to me. I suspect that it has something to do with optimizing the risk/reward of the total bankroll as opposed to the individual games, but I don't have a clear grasp of what's happening. Thanks.
I'll assume that you trade a portfolio of markets/systems to diversify, and that the reason to diversify is to reduce risk and possibly create a smoother equity curve. That only happens when you have losses in one system offset by the gains in another. If you don't recombine and recapitalize the shared equity, it is as if the systems are being traded in completely separate accounts and they will get no benefit according to modern portfolio theory (MPT). So, however it's done, you want to recapitalize: new trade sizing always comes from new equity calculations.

Look at your original #'s: does it really make sense to risk 0.10 + 0.15 + 0.20 + 0.25 + 0.30 = 1.00 = 100% of your capital on a single round? Keep in mind, it is possible to have 5 losers in 5 different markets. In your case it adds nicely to 100%, but many times a series of individual Optimal-f numbers will add to greater than 100% risk. This should be a sign that the application is not correct.

The topic can be as complicated as you choose to make it. I intentionally presented something simple, which is how one would approach using Optimal-f as a first step. That is, divide the account equity by the number of markets, then apply the Optimal-f fraction for the market/system to the slice for that market. You recapitalize (rebalance) the slices for every new trade. If you instead take the allocation out of 100% shared equity you will be trading FAR beyond Optimal-f. Why is this? The calculations for Optimal-f assume full use of trading capital, not shared use. Ralph adheres to this constraint when he shows the more complicated case where the equity slices are proportioned dynamically as well.
gbos wrote:For a â€˜correctâ€™ treatment, you have to know the joint-probability distribution for the returns of the five systems. This is close to impossible.
Not only is this not impossible, I do it on a regular basis for every portfolio run. What I find is that the joint probability distribution produces a combined optimal fixed fraction for which the total portfolio should not excede. This number is much less than the "correct" total allocation suggested in the post by gbos. I can only suggest that you get some software that allows you to experiment with this stuff yourself, and make your own conclusions. I recommend that the software include some form of distribution testing, like Monte Carlo, to see the possible outcomes beyond a simple portfolio run.

Cheers,

Kevin

BTW: Also keep in mind that the market doesn't respect your 0.10 risk number. In a large move (e.g. recent metals market price action) you could have incurred much more than 10% loss on a supposed 10% risk. If you have the capability of doing so, I'd recommend combining position sizing with worst-possible-fill slippage, then model the results.

gbos
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Without reinventing the wheel I will post the solution illustrated in Ed Thorps paper that can be found everywhere on the web.

Only one system available

We take a simple system: p (probability of winning) 60% , q (probability of losing) 40%

R-multiple probability
-1 40%
1 60%

f (Kelly) = [(-1)*0.4+1*0.6]/[abs(-1)*1]=0.20

Having a starting capital 100,000 and using a 1/4 (quarter) Kelly your first bet is 5,000.

2 independent systems available

We take the same situation as before but now we have available 2 similar systems and we trade them using the same pool of money. We now have to find the simultaneous f1 and f2.

Solution: maximize growth_rate(f1,f2) = p1*p2*ln(1+f1+f2) + p1*q2*ln(1+f1-f2) + q1*p2*ln(1-f1+f2) + q1*q2*ln(1-f1-f2)

Solving this we take f1=19.2% and f2=19.2%

Having a starting capital 100,000 and using a 1/4 (quarter) Kelly your first bet is 4,800 for the first system and 4,800 for the second system.

Suppose that your first system gives you failure and your second system gives you failure (an event with probability 0.4^2 =16% of occurring). Your ending capital is now 100,000-2*4800 = 90,400.

Your next simultaneous bet in the two systems from the common pool is f1=4520 and f2=4520.

I hope the example is clear enough.

Now, on the other matter of determining the joint probability distribution of a big number of systems. One can certainly calculate the numbers but the problem is how confident you are for that numbers. The sampling error is huge.
[/b]

ksberg
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### Not Kelly

gbos wrote:Without reinventing the wheel I will post the solution illustrated in Ed Thorps paper that can be found everywhere on the web.
I respectfully disagree with the positions put forth about using Kelly for trading. The statistical axioms under which Kelly may be correctly applied have been ignored. If you can predict the win and loss of each trading event, and expect your trading events to be normally distributed ... by all means, use Kelly. For some reason, my trading situations look nothing like those conditions.

Cheers,

Kevin

shakyamuni
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I agree with Kevin.
Last edited by shakyamuni on Sat Jan 08, 2005 4:59 pm, edited 1 time in total.

Forum Mgmnt
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At the start of this thread, C3PO wrote:I am aware that people on this forum have warned against trading the optimal-f percentage. I am not debating that. I am asking about the mechanics of diversification with fixed fraction. I would very much appreciate it if someone can clarify this for me. Thank you.
I think you are asking a question that might have an answer but that your approach to the problem is flawed; further, that this flaw is so large that it makes the entire question irrelevant.

First, in order to arrive at an "optimal" figure, there need to be some inputs, namely expectation (aka expectancy), win/loss ratios, etc. These inputs are in turn the output of some process, either actual trading, paper trading, or simulation. Unless you've been trading or paper trading for many, many years and have been religiously applying a rigid and well-specified set of rules, the results from trading or paper trading are suspect. This, in turn, makes any derived values like optimal-f or Kelly fractions suspect.

So the best source of the information is a simulation of the underlying trading system across historical data, since that is the only way you will be able to practically test the approach across a variety of markets and get a sufficiently large number of trades so as to have a statistically valid basis for drawing any conclusions.

Second, you really should be looking at the combined expectation anyway. Otherwise, you don't get to see the effects of correlation.

For example, if you trade only currencies, you will find that the benefits of diversification are lower than if you trade a basket of diverse futures, Crude Oil, Soybeans, Coffee, Japanese Yen, Eurodollars, etc.

Any formula that simply combines results from single-market tests will be in error. This means that your simulations should be portfolio-level simulations in order to get values for expectation and win/loss ratios that reflect the effects of diversification.

Since, A) You need to run simulations to get the inputs anyway, and B) those simulations will be portfolio-level simulations, why not just simulate the effects of various bet sizes across the entire portfolio and then decide which levels have the type of equity curve you personally prefer?

I think you'll find that's what the more experienced (read profitable) traders actually do.

The more sophisticated of them run hundreds or thousands of simulations at any given level using Monte-Carlo simulations, varying the start dates, etc. in an effort to understand the range of possible equity curves that a given bet size might demonstrate.

- Forum Mgmnt

P.S. I think that much of the lore of trading reflects the constraints of the tools in common use. I suspect your question comes more from the limits of products like TradeStation that don't allow you to do portfolio-level simulation than because the approach you are attempting to take makes sense in an objective sense.

That is to say, if you can't do portfolio-level simulation, then your question is a good one. If you can, then it becomes meaningless because you can get the answer to your question much more directly.

Since, there are tools that allow portfolio-level testing at price points in the range of everyone; as a practical matter, we can all do portfolio-level simulation; some of us have just chosen not to.

The correct solution to the problem of not being able to do portfolio-level simulation, is to get better tools; not to come up with some formula to compensate for the poor quality of the tools you have.

ksberg
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### Re: Optimal-f in a portfolio

C3PO wrote:
ksberg wrote:Optimal-f concept is independent of the 2% risk per trade rule; Optimal-f is a rule unto itself. To make a comparison, let's say you were trading 30 markets. The simple equity allocation is 1/N, or 1/30, which is 0.0333 of total equity per market. With this level of capital, Optimal-f could be as high as 0.6 to equal the 2% risk rule (0.0333 * 0.60 = 0.02).
Thanks for the reply. I have been reading about the dangers of using optimal-f in other posts. However, I am not sure if those dangers apply in the situation above. Lets say you are in this situation. The optimal-f for this market is 0.6, but it does not violate your 2% risk rule. Should you be comfortable using optimal-f in this case? (I am not asking whether 2% is a good number to use. Say your risk rule is around 0.6% instead. So if optimal-f still allowed you to risk less than 0.6% of your portfolio equity, would you be comfortable using optimal-f in that case?)
Sorry I missed your earlier question, so here goes. First, realize you're now asking my opinion rather than asking about mechanics. I do not use optimal-f directly for position sizing, although you can see from above that I have integrated it as a threshold boundary for total portfolio heat. This application is actually more related to something like the Turtle correlation rules. I have no attachment to the 2% rule, other than empirical and quantitative observation. The same goes for optimal-f.

Generally I find my personal tolerance is below 2%, and for one system my ideal actually is around 0.6% (that's not 0.6 as 60%, it's 0.006 as 0.6%). To me, your initial numbers like 10%+ are astronomically high. If I were trading optimal-f and the end result was less than 0.6% would I trade it? Maybe; it would depend on the rest of the analysis. The reason I don't have any attachment to the 2% rule is I've run the simulations I recomended above, and then I decided for myself. It makes a world of difference when you can inspect the information inside and out first hand. After that, you'll find the conversations about what you "should" do fall away like clatter.

Cheers,

Kevin

C3PO
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Forum Mgmnt wrote:I suspect your question comes more from the limits of products like TradeStation that don't allow you to do portfolio-level simulation than because the approach you are attempting to take makes sense in an objective sense.
I don't use Tradestation or products like it.

Kevin, thanks for your help. Those pages you suggested led me to find something else which gave a very clear and intuitive answer to my question. I wouldn't want to post something that's irrelevant to people, so if anyone else is interested just ask here and I'll pm you.

shakyamuni
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### Re: Optimal-f in a portfolio

ksberg wrote:The reason I don't have any attachment to the 2% rule is I've run the simulations I recomended above, and then I decided for myself. It makes a world of difference when you can inspect the information inside and out first hand. After that, you'll find the conversations about what you "should" do fall away like clatter.
Yes!

ksberg
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### Tools

c.f. wrote:P.S. I think that much of the lore of trading reflects the constraints of the tools in common use. I suspect your question comes more from the limits of products like TradeStation that don't allow you to do portfolio-level simulation than because the approach you are attempting to take makes sense in an objective sense.
I'm glad to see this situation start to change. Yes, it may cost a few bucks to go and purchase those programs, but at least traders have access to tools that can provide some answers.

Kevin