Square Root Rule & Additional Investment

Hi all. Looking to expand my understanding...

The manual contains a description of the Square Root Rule for withdrawing and/or reinvesting: http://zorro-trader.com/manual/en/kelly.htm.

Once before in the Forum the question of adding additional funds to one's account came up, and the response was to simply treat this as a negative Withdrawal (W in the manual's formulas). (Unfortunately, can't now seem to find that thread/post... )

The question now in my mind is whether or not that's correct. Treating new investment funds as a negative Withdrawal has the effect of immediately subjecting them to the Square Root Rule, so that the mere action of depositing new funds makes a portion unavailable for withdrawal, which doesn't make sense to me.

For example, if I'd instead opened a new account with those new funds, wouldn't they have their own life and their own Square Root Rule progression?

So, bottom line: If new funds are added, whether to the same or a different account, doesn't each investment have its own independent timeline, Square Root Rule calculations, etc.?

Thanks.

Anyone can help clarify? jcl?

Thanks.

I think you're right. Adding funds is independent of trading time and thus they are not subject to the square root rule.

Thanks jcl.

Then would the correct process be:

1) From the add point forward, results (cumulative gain or loss) should be split pro-rata between the pre-existing investment(s) and the added funds.
2) Pro-rata results of the pre-existing funds would be subject to the same pre-existing square root rule.
3) Pro-rata results of the added funds would be subject to a new square root rule that starts at the time of the addition.

Is that correct?

Thanks.

Correct. The profit generated from your newly added funds are again subject to the square root rule.

Thanks!

Interesting, wonder how to adapt the original spreadsheet. It was useful. Another item in the to-do list...

Hi jcl, MatPed,

I thought more about this over the weekend, and my thoughts have shifted as follows:

a) While from the time point of view different capital fundings are different, in fact Balance is being used as a proxy for time & number of trades, and so time is absent from the formulas. If Balance is then pro-rata'd over the fundings, and Withdrawals as well, then the square root formula gives the same f factor for all capital fundings, regardless of when they were done!

b) One could drop down from the account level to the individual trades, since at the time of an additional capital funding it has obviously no trades and until a new one is opened all Profits apply to the old funding's trades. This would, however, be a lot of extra work for probably not really much benefit. Especially since over some likely relatively short time all the old trades would be closed, at which point everything is pro-rata again.

What should really be done when multiple fundings are involved? How should time be incorporated?

Put another way in terms of the manual's example:

Capital of $1000 has grown to $1300 account balance. The investment grew by factor 1.3; the square root of 1.3 is 1.14. Therefore $1140 must stay and $160 can be withdrawn.

But what if the $1000 came from $500 invested 9 years ago and $500 invested last year? How does this change things? Should 90% of the $300 Profit be allocated to the 1st $500? Or...?

Thanks.

Just an idea but adding a capital to a running system, looks like to start a new system from zero.
Following your example is like you start a TS 9 years ago with 500 and another one one year ago. So the square root, and the other relevant parameters, may be calculated just adding the values of the two systems.
I have not the capabilities to prove it mathematically but it make sense to me: if the square root rule is correct for 2 user starting 2 TS independently one 9 years ago and one last year, should be correct for the sum of them...

ciao

Thanks MatPed. I'm in the same boat - lacking the framework/understanding/statistics skills to answer this...

According to the manual, the real statistical issue under the Balance proxy covers is time/number of trades. Therefore it seems to me that the $500 that's been trading for 9 years will have a higher risk of larger drawdown than the $500 that's only been trading for 1 year. Therefore, it seems I would need to leave more $$$ in the account to protect the older $500 than the newer $500. From the withdrawal perspective, I can't withdraw as much for the older $500 as the newer.

Now what does this really mean? Some possibilities are:

- $300 is the right withdrawal amount, $150 to each $500, and the underlying statistics just works out fine, somehow.

- $300 is the right withdrawal amount, spread in some way between the 2 $500's based on time or ???

- Less than $300 (how much?) is the right withdrawal amount due to the increased drawdown risk of the older $500.

- Other???

Hopefully jcl can clarify.

Thanks!

MatPed is correct. Adding funds to an account is equivalent to freshly starting a new account with the added funds.

It is not so that the drawdowns in year 10 of a trading system are bigger than the drawdowns in year 1. The drawdowns grow not automatically any year, but their probability grows with the duration of trading. That is a subtle difference. You can freshly start a new account after 10 years trading the system and need not fear immediately a worst case drawdown, since it's more likely that it already happened in one of the years before.

Admittedly this is a bit tricky, but I hope it makes sense.

Thanks jcl. Yes, it is tricky!

What probability exactly increases over time? Probability of a drawdown happening? Probability of a greater drawdown? Probability of more frequent drawdowns accumulating to a larger total drawdown? It seems it must be one or a combination of some or all of these (& maybe more) to account for the need to keep square root rule capital on the account...

Similarly, let's take a closer look at the 2 $500 accounts that each now have a balance of $650 - the 1st after 9 years and the 2nd after 1 year. If there is some greater probability of larger drawdown that grows with the duration of trading, then I don't understand why I can take the same $80 out of each account. Since they've been trading for different lengths of time, their probabilities must be different, right? And so I can either take more out of the 1 year account or have to take less out of the 9 year account?

Thanks.

The probability of a bad drawdown DURING ITS LIFETIME is 3 times higher on account 1 than on account 2. But the probability of a bad drawdown in the same year is of course exactly the same.

I guess that the answer is purely mathematical. Somewhere in the discovery process of the square root rule the time variable have been eliminated. Too much time is passed from university, but it seams realistic to me. The objective of the research that brought out the square root was to find the optimal re-investing rule for a long term profitable investment, so the scientist was looking for a time independent rule.

my 2 cents

Thanks jcl, MatPed.

MatPed, you definitely win the prize for being a better student than me!

jcl, thanks for the last clarification! It helps, & I now remember it from years ago...

But now, as then, I struggle with how to transition from blue sky theory to what to do here in the real world. Specifically, how do these 2 reconcile:

1) If in every year the probability is the same for the 2 accounts, then in every year they should be treated the same in terms of withdrawal vs. funds left.

2) But, if in every year they are treated the same, since in each individual year the probability is identical, then because of the lifetime probability difference the older account is more likely to blow up, isn't it? Don't the accounts have to be treated differently somehow to prevent this?

Thanks.

Thank you DdlV,
I will forward you my details where to ship the prize I just won

I do not get you. The probability to blow out an account is not linked to the square root theory. The square root rules is linked on how much to re-invest a profitable TS correctly funded.
To be more clear if you start a TS under-capitalized with 500$ that has 1000$ of capital requirement, you may be profitable for a certain period, the square root will show the amount of money you can withdraw, but likely you will face a margin call. OptimalF can not save you.
At the TS will stop working because the inefficiency of the market is gone the square root can do little.

What I do not understand?
A fellow student.

Thanks MatPed. I could certainly be not understanding, but my understanding is that the probability to blow out an account IS linked to the square root theory. In fact, it's purpose is to prevent blowing up...

Specifically:

I started an account 9 years ago with $500 per the Test CR (i.e., not under-capitalized). Its balance today is $650. The square root theory says if I withdraw <=$80 I am not in danger of blowing up the account - it is still properly capitalized. If I withdraw >$80 then I have under-capitalized the account and it is in danger of blowing up. Is that not correct?

Then Part 2 is account #2 started 1 year ago also with $500 per the Test CR, so also not under-capitalized. It is also now at $650. And I'm still confused (not at all unusual ) by how to reconcile the questions above...

Thanks.

Ok, probably I am not able to write it in proper english, but the square root protect you from a Margin call maximizing profit, due to an increased probability of severe drawdown in a PROFITABLE TS. It does not protect you from a drawdown caused because your TS expire.

My perception is that you can not reconcile the 2 aspect because are 2 different things.
Just think at a TS that make 1000 trades in 5 years and no profit and no loss: even, flat . The probability of severe drawdown increases for sure, but you have the same capital, you can not apply the square root, because you have no capital to reinvest. The square root apply only if your TS is profitable and you want a proper money management.
Hope this helps. I do not think I can not tell you more on this matter
Ciao

Thanks MatPed! So far it seems your English & understanding are better than mine!

I think what you've said helps. But please let me clarify to be sure:

First, strategy expiration is not a part of this, unless I'm missing something like an unstated assumption. I'm assuming whatever strategy we're talking about is still good; otherwise the expiration more than likely overrides everything else.

Then, using your example of a flat account so that the square root rule is not a part of this: The account started at the beginning of year 1 with a certain risk of severe drawdown. To be explicit, x% chance of a drawdown triggering Margin Call.

There seems to be 2 parts to this: "risk" and "severe".

"risk": At the end of year 5 my risk is now (x+y)%. I suspect from what you've written that there's nothing I can do about this - this risk grows continuously as long as I keep trading - right? Is there some other formula/rule that describes this risk growth?

"severe": I also suspect that "severe" has in reality increased also, right? In other words, not only is there a larger % risk of an event; there's also larger risk that the event will be bigger & cause Margin Call even if I've added funds to the account?

To get to the bottom line of all this: What do I need to do (add how much more funds to the account?) at the beginning of year 6 so that I maintain the same x% risk of a drawdown big enough to trigger Margin Call?

Thanks.

Hi Ddlv the answer is in the manual:

"When modeling drawdown depth mathematically with a diffusion model, the maximum drawdown of a break-even system is proportional to the square root of the number of trades, and therefore also to the square root of the trading time. This also means that drawdowns have no limit. A trading system will suffer a drawdown of any depth when you wait long enough."

But, if you have a beak even system, why are you trading it? Stop it, and forget the square root rule. The point is what to do if you have a profitable system and want to keep it running and profitable (and remaining profitable), reinvesting the capital or withdraw some of the money. This is where the square rool plays its role.

Ciao

Thanks MatPed. Yes, I've read that - it's what generated the question rather than what answers it.

In any case, enough time on this. I'll take the concern off-line & try to resolve it other ways.

Thanks for your patience! Happy Trading!