Saturday, April 17, 2010

How do you limit drawdown using Kelly formula?

As many of you know, I am a fan of Kelly formula because it allows us to maximize long-term growth of equity while minimizing the probability of ruin. However, what Kelly formula wont' prevent is a deep drawdown, though we are assured that the drawdown won't be as much as 100%! This is unsatisfactory to many traders and especially fund managers, since a deep drawdown is psychologically painful and may cause you to panic and shut down a strategy prematurely.

There is an easy way, though, that you can use Kelly formula to limit your drawdown to be much less than 100%. Suppose the optimal Kelly leverage of your strategy is determined to be K. And suppose you only allow a maximum drawdown (measured from the high watermark, as usual) to be D%. Then you can simply set aside D% of your initial total account equity for trading, and apply a leverage of K to this sub-account to determine your portfolio market value. The other 1-D% of the account will be sitting in cash. You can then be assured that you won't lose all of the equity of this sub-account, or equivalently, you won't suffer a drawdown of more than D% in your total account. If your trading strategy is profitable and the total account equity reaches a new high watermark, then you can reset your sub-account equity so that it is again D% of the total equity, moving some cash back to the "cash" account. Otherwise, you continue to keep the equity in the cash account separate from the equity of the trading sub-account.

Notice that because of this separation of accounts, this scheme is not equivalent to just using a leverage of L=K*D% on your total account equity. Indeed, some of you may be too nervous to use the full K as leverage, and prefer to use a leverage L smaller than K. (In fact, the common wisdom is that, due to estimation errors, it is never advisable to set L to be more than K/2, i.e. half-Kelly.) The problem with using a L that is too small is that, besides not achieving maximum growth, the portfolio market value will be unresponsive to gains or losses and will remain relatively constant. Using the scheme I suggested above will cure this problem as well, because you can apply a higher leverage L_sub to the sub-account (e.g. use L_sub = L/D%) as long as L_sub < K, so that the portfolio market value is much more sensitive to your P&L while still ensuring the drawdown will not exceed D%.

Has anyone tried this scheme in their actual trading? If so, I would be interested in hearing your experience and see if practice is as good as theory.

59 comments:

Joshua said...

Will your broker recognize this cash as seperate if you experience a large drawdown? You'd have to have it in a sub-account or seperate account to be absolutely sure it would be safe.

But that's besides the point. It doesn't make any sense to begin with. If, for example, you set aside 50% of your account in cash and use 2-to-1 leverage for your system, because the kelly formula finds this to be the optimal amount, isn't the end result roughly the same as a 100% unlevered allocation to your system?

Matthew said...

Hi Ernie, I do something like what you suggest. I have a "Permanent Portfolio" which is usually stocks, gold, long bonds, and cash. One change is that I use a swing trading system to trade the "stock" allocation, instead of a S&P 500 index fund.

I use leverage to seek high returns with the swing trading system. I am more comfortable being 4x as aggresive with 25% of my account. I wouldn't be as confident trading 100% of the account with a less aggresive system, because what if something goes wrong with my system design or the markets!

Ernie Chan said...

Joshua,
The sub-account concept is purely for your own accounting purposes. In reality, you can trade in just one account.

The answer to the second part of your question is no. Initially, the market value will indeed be the same in both cases. But as your strategy incur P&L=P, the market value of your portfolio should increase or decrease by P*K=2P, if we assume K=2. On the other hand, if you simply use the unlevered allocation, the market value should only increase or decrease by P. In fact, this difference is the main point of using the subaccount method.

Ernie

Ernie Chan said...

Matthew,
What you use seems to be the classical capital allocation method using Kelly formula (see my book page 96). However, it is different from the subaccount method I suggest here.

As my answer to Joshua above shows, the subaccount method is purely conceptual: it involves no real division of assets, but it results in real difference in portfolio market value adjustments in response to P&L changes.

Ernie

Matthew said...

Hi Ernie, if I understand your post correctly that is roughly what my portfolio policy is. I don't allow the trading portfolio to draw down the other assets; I only re-balance out of the trading portfolio when the equity increases up to a re-balance band. I think that classic asset allocation would involve replenishing the trading system when it draws down.

Anonymous said...

Isn't that a similar approach to the Options and T Bills method outlined by McMillan (2002)?

Anonymous said...

Anyone saw the interview with Ernie Chan on Quantnet.com yet?
I haven't seen it posted here
http://www.quantnet.com/interview-with-ernie-chan

Ernie Chan said...

Hi Matthew,
Thanks for explaining your method. Yes, it seems to very similar to my proposal. One question: do you apply a constant leverage to your trading account? I.e. do you increase the order size if you earn a profit, and decrease it if you lose?
Ernie

Ernie Chan said...

Hi Anon,
I am not familiar with McMillan's method. Could you please describe it in brief?
Thanks,
Ernie

Anonymous said...

McMillan's "Options as a Strategic Investement" describes what he considers to be mathematically the best strategy as one where 90% of the "risk" capital is actually in T-Notes, and 10% exposed to the market (long calls or long puts based on highest prob of return). He looks at ways to annualize and weight what the 10% actually is, and how to re-adjust after a period of gain/loss.

It's a pretty well-known book for options people (I'm not one but I need to know about options), and that chapter is worth a revisit if you have it in your library.

I need to look at the math again but it looks pseudo-Kelly, and is designed to specifically limit annual(ized?) drawdown.

Ernie Chan said...

Anon,
Thanks for the recommendation for McMillan's book. I will take a look too. (I will also add that to my Recommended Books list on the sidebar.)
Ernie

Matthew said...

Yes, for better or worse, I allow the position sizes to increase as the system wins.

Anonymous said...

Hi Ernie

Thanks for writing about Kelly again. I am still a little unclear about how to apply Kelly when the returns are not continuous.

Maybe you or a kind reader can assist.

Say I have 3 systems, over one week system A returns 1% every day, system 2, 2% every day, but system 3 only trades Monday making 2% and say Thursday making 3%.

Its Monday of the next week and I get a signal for all three systems - what's the procedure for applying Kelly?

If I understand example 6.3 from your book, I would only calculate the return and cov matrix from the returns of Monday and Thursday as they are the only days when all three systems were in action.

Is this correct or have I missed something?

Thanks

John

Ernie Chan said...

Hi John,
On days that one strategy does not trade, then you should assign a zero return. Everything else follow.

Does that make sense to you?

Ernie

Anonymous said...

I'm not sure I get the point, I've reread the post a couple of times and I can't understand the difference between it and simply saying "don't trade your whole account and you won't blow your whole account...", or maybe that's the point? but if it is, it really has nothing to do with the way position sizing is made, or am I completely missing the point?

Ernie Chan said...

Hi John,
I am afraid you have indeed completely missed the point.

Calculating the covariance matrix and thus the Kelly leverage is based on historical data, and so you can put in zero returns for days that a strategy did not have positions.

Once you have determined the Kelly leverage, you will adjust your position size on a daily basis based on your current account equity. Even if a strategy does not trade that day, your hypothetical order size is still determined ahead of time.

Ernie

Anonymous said...

Hi Ernie

Thanks for the clarification regarding zeros.

John

talll.com said...

Are you saying half-kelly is already too low?

Ernie Chan said...

talll,
No, half-Kelly is usually the maximum leverage you should apply. But some people use a leverage much smaller than even half-Kelly, and for them, the sub-account method would be useful to ensure that the order size is more sensitive to P&L.
Ernie

Matthew said...

I was reading an interview with Jim Leitner where he talks about excluding tail risk from your portfolio. That caused me to think that this post is kind of like that: use Kelly to manage known risk based historical volatility for the trading account, but use the sub-account partition to limit un-forseen and possibly un-knowable risks that your system may encounter in the future. The bottom line is that if certain risk isn't included in your historical return distribution then Kelly can't help you avoid it.

Ernie Chan said...

Matthew,
Yes, it is very true that Kelly formula does not take into account tail risk.

But the subaccount method is more than about preventing tail risk. If it were, then we just need to lower the leverage to way below Kelly's. The subaccount method will prevent tail risk but at the same time enable our order size to be much more responsive to daily P&L.

Ernie

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Vipin Kumar said...

I dont know whether my Question fit in to the current discussion or not, but i will ask anyway,

How do you distribute your capital among your pairs?

For example: Lets say i have $ 100,000 with me and depending upon some criteria i get 2 pairs each day with holding period of 6-8 days. Now if i put $1000 on each side, in the first 6 days i have $24000 of my money blocked ( assuming none of them converged). So my account is showing $66000 as resting capital.

On the 7th day my first pair converged and i got my 2000 back with some P/L. Now on the account shows $68000.

My question is, how do i put the optimal money per pair so that i am utilizing my resting capital.

What i mean on the 7th day i have $68000 in my account and i want to use it fully in the future trades, something similar to optimal f ...but then i am not sure.

Ernie Chan said...

Vipin,
Kelly formula is not concerned with your "resting capital". To apply it to different pairs, calculate f for each pair, and allocate market value to each according to f and your total equity. If the total gross market value exceeds your buying power, decrease each pair's MV by the same factor. This own procedure is described on page 98 of my book.
Ernie

Andris said...

Ernie,
I should argue with you, because "that Kelly formula does not take into account tail risk" is not general truth. Since the Kelly formula is based on the maximization of expected log return (expected rate of growth) it implicitly takes the tail risk into account. This can be shown if we write the Taylor expansion of the log function, for example the fourth order one. In this case the expected log return decomposes into the expected return, expected second moment, expected third moment and expected fourth moment, where these moments have different weights and signs in the series. This decomposition shows that the first moment (expected return) has the largest positive impact on the growth rate, while the fourth moment has the smallest. Since the sign of the fourth moment is negative in the decomposition the strategy (Kelly rule) implicitly try to avoid investments which has large fourth moments. Because the fourth moment corresponds to the kurtosis, and large fourth moment means fat tail, the Kelly rule inversely incorporates with fat tails.

I am happy to read about the Kelly rule somewhere, great blog.

András

Ernie Chan said...

Andris,
Thanks for your insightful comment. I do believe that the general formulation of Kelly's formula has taken into account higher moments of the returns distribution, as you have explained. However, as a trader I am mostly concerned with an analytically simple formula utilizing the mean and standard deviation of returns, which is provided by Ed Thorp's article which I quoted in my book. I think Thorp's derivation has dropped the higher moments in order to arrive at f=m/s^2. Do you disagree?
Ernie

Matthew said...

Hi Andris, there are two return distributions that are important to a trading system developer:

1) The expected distribution based on historical data before the system starts trading.

2) The actual distribution achieved after X years of trading the system live.

Distribution #2 is always going to have tails that are equal to or greater than #1 in "fatness". That equates to "tail risk you didn't know you were taking". Since Kelly is based on historical data, it can't take the future into account even if it considers higher moments ;-)

Ernie's sub-account idea does allow a means to future-proof a trading portfolio against un-known events lurking in future tails. For example if "ruin" is defined as loosing 40% of the portfolio, then buy T-bills with 60% of the portfolio and trade the remaining 40% with full Kelly. This reduces tail events that can ruin you to things like fraud or civil unrest, rather than just an un-expectedly bad sequence of trades.

Andris said...

Ernie,
I agree with you. If the first two moments (expected return and variance) are taken into account the problem of maximizing expected growth rate is simplified into a quadratic programing problem instead of a complex maximization of logs. So, there is a point in this approach, however, unless you trade in every second the original optimization problem can be also resolvable. Although if I would be a trader I think I also would have a safety account beside the "Kelly" account to avoid unexpected risks which have not arisen in the training dataset. I have a paper which deals with some computational issues of the Kelly rule: http://www.szit.bme.hu/~oti/portfolio/articles/semi.pdf

Which paper of Thorpe have you referred?

Andris

Andris said...

Hi Matthew,
I also think that a trading system should run lower level of risk than the risk level which is dictated by historical data to be ensured against unexpected bad outcomes. But, I am afraid I couldn't catch your first argument: Why should Distribution #2 have fatter tails than Distribution #1?
The previous link correctly: http://www.szit.bme.hu/~oti/portfolio/articles/semi.pdf

Andris

Ernie Chan said...

Andris,
Thanks for the link to your paper.

The Thorpe article I quoted is here: http://www.edwardothorp.com/sitebuildercontent/sitebuilderfiles/KellyCriterion2007.pdf

Ernie

k1 said...

Hi Ernie, great post and some deep comments here. I'm curious to know if any of you have read Ralph Vince's work on Kelly ("the Kelly formulas are applicable only to outcomes that have a Bernoulli distribution"). Vince's work on optimal f takes off from there, although having recently heard Vince speak, I get the impression he is a bit scared of the implications of his math.

More to the point though, I don't understand why one would talk about trading Kelly formula money management with a portion of one's stake. It seems like the point of Kelly (or optimal f) is to maximize the geometric growth rate of your trading system(s). If you're not seeking to maximize geometric growth rate (ie your goals include Sharpe ratio or avoiding drawdowns or other constraints) then how does "partial Kelly" differ from "aggressive fixed fractional" or other, simpler MM?

Ernie Chan said...

Hi k1,
For discussions on Ralph Vince's work, please see my earlier post: http://epchan.blogspot.com/2009/02/kelly-formula-revisited.html

It is incorrect to state that Kelly formula applies only to Bernoulli distribution. Commenter Andris above has made a good point that Kelly formula works for any distribution, but a simple mathematical closed form can only be obtained for specific distributions such as Bernoulli (in Ralph's case) or Gaussian (in Ed Thorp's case).

As to why using my subaccount method differs from fractional Kelly, please see the paragraph in my original post that began with "Notice that because of this separation of accounts, this scheme is not equivalent ..." If you are still unclear about the difference, please let us know.

Ernie

Matthew said...

Hi Andris, I think the tails of a trading system in the future will either be the same as the present distribution or larger. The tails will be the same width if all events fit within what has been previously recorded for that system. However there is a non-zero chance that an observation will occur that will not fit in the previously assumed distribution - that will result in drawing a fatter tail to include the new observation. There is zero chance that future trading performance will un-do the past and create a smaller tail distribution for the system.

k1 said...

Ernie- note that I asked a question about how the subaccount approach you describe differs from "fixed fractional" MM approaches, not "fractional Kelly".

To more clearly describe my possible misunderstanding, I picture a strategy in which the Kelly formula suggests that 25% of my stake be at-risk, and that I must be prepared to survive 40% drawdowns.

This is too much for me, so I put 60% of my stake into T-bills, then apply half-Kelly to the remainder. This means I have 12.5% of 40% of my stake at-risk in my strategy, popularly known as 5%. :-) Which is why I ask my fixed-fractional question.

In the forex forums, one can often read how fixed fractional is a good MM approach for newbies and that 1-2% per position for a total of 5-10% at-risk at any one time are good rule-of-thumb limits. Thus, an MM approach that dictated 5% per position (or strategy) might be considered "aggressive fixed fractional".

Ernie Chan said...

Hi K1,
I don't believe that the fixed fractional approach that you described has a recommendation on what leverage to use on the risk capital that you are trading, am I right?

If that's right, then my subaccount method is different because it uses full or half Kelly on that fraction, resulting in a large daily change in order size or market value in response to P&L changes.

You can view the subaccount method as maximizing the growth rate subject to the constraint that the maximum drawdown not to exceed X%. If there are better ways to solve this constrained optimization problem other than the subaccount approach, I would be interested to hear about them.

Ernie

k1 said...

Ernie- I think the fixed-fractional MM I am describing implies the leverage, in that it specifies the amount of your stake that is at-risk, based around where the system says your stop-out point must be.

I don't want to turn this into a discussion about FF, so perhaps a better question is in order:

When your Kelly calcs tell you to adjust position size, what goal(s) are you pursuing?

I can think of a couple options:
- attempting to keep your at-risk capital the same as the probabilities move more into your favor (or against you)
- pressing your advantage (ala Thorpe's discussion of applying new buying power against convertibles) or hedging your exposure
- some other goal I've totally missed.

Ernie Chan said...

kl,
I don't think the implied leverage in the fixed-fractional MM that you described is the same Kelly leverage that I used in my sub-account method.

Kelly formula would not tell you how big a drawdown you will suffer, except that it is unlikely to be 100%.

As you mentioned, the goal of Kelly formula is to maximize compounded growth. To achieve this, one has to both cut back on risk exposure when a loss occurred, and increase risk exposure after a gain. My subaccount modification merely add an additional constraint: to limit drawdown.

Hope this clarifies things.
Ernie

Joshua said...

Ernie,

If we assume K=2 and we decide to segregate half of the account into cash, how does it ever increase in value at the "levered rate"? The only way I can see this is possible (and we're talking simple math here) is if you consider the part held in cash to either not be rebalanced or to be rebalanced less frequently than trades occur.

If that is the case, I still don't see what the point is beyond being a psychological trick.

Ernie Chan said...

Joshua,
Yes, in your example, we will not rebalance the 2 subaccounts until a new high watermark is reached in the full account.

I do not understand what you mean by "psychological trick". If K=2, D=0.5, the overall leverage is L=K*D=1, when we began trading. Suppose also we started with $1 overall account equity, and therefore a market value of $1. If we lose x=$0.1, then we decrease the market value by K*x=$0.2. Hence now the overall leverage is 0.8/0.9=0.8889, clearly different from the original leverage of 1. Hence the difference is more than just psychological!

Ernie

Anonymous said...

Ernie,

Sorry to post a comment so long after your post, but I only became aware of it a couple of days ago.

I haven't gone through all the mathematical details, but I suspect the sub-account method you propose to limit drawdowns is suboptimal. I believe the optimal method is the following: buy an out-of-the-money put option on your portfolio, where the strike price is set at the drawdown limit. This concept is the same as what's known as "portfolio insurance" (i.e., the system that was blamed for the 1987 stock market crash).

Of course there's no way to literally buy a put option on your portfolio, but you can "replicate" the option using a dynamic hedging strategy based on the Black-Scholes formula. Any of the Quantitative Finance books (e.g., "Options, Futures, and Other Derivatives" by Hull) explain how to do this.

-aagold

Ernie Chan said...

Hi aagold,
The subaccount method is applied to a strategy, not a portfolio. For e.g., it may be applied to a short-term trading strategy that does not hold positions overnight, or it may be applied to a strategy that trades a market neutral portfolio. And it is the strategy that may lose money day after day and results in a drawdown in the account. So unfortunately, there is no option that one can buy to insure against this type of losses, as opposed to losses incurred by a long-only portfolio.
Ernie

Anonymous said...

Ernie,

The term "portfolio" is very general. It applies to any set of liquid securities, either long, short, or market neutral. The problem you're trying to solve is exactly the same as the "portfolio insurance" problem, and as I said the solution does not require any buying/selling of actual options. The concept of buying a put option on your portfolio is a mathematical concept used to derive the dynamic trading algorithm.

Have you ever studied the concept of how to replicate an option payoff using a combination of cash and the underlying security? I think you'd find it useful and interesting.

-aagold

Ernie Chan said...

aagold,
I only have a pedestrian familiarity with portfolio insurance and dynamic hedging. But from what I have read, it seems to have a superficial similarity to the risk management scheme based on Kelly. I.e. it advises you to reduce the portfolio positions after a loss, and vice versa after a gain.

Let's consider a concrete example. If we are trading a high frequency strategy in E-mini, which takes long and short positions at different times. The order size is 100 contracts, and the account equity is exactly equal to 100*market value of a contract. At the end of the trading day, we have no position in E-mini, but the return on the day is -10% of equity. Kelly would recommend reducing the number of contracts to 90. What action would portfolio insurance recommend the trader take?

Ernie

Anonymous said...

Ernie,

Well, I did some more research on this topic and concluded that your method is probably better than what I was proposing.

What you described is what's referred to as "CPPI": Constant Proportion Portfolio Insurance. What you call a "drawdown" is what they refer to as the "cushion", and the leverage ratio (fractional kelly) you're using is what they call the "multiplier". What I was talking about is referred to as "OBPI" : Option Based Portfolio Insurance. It's not clear OBPI really buys you anything over CPPI and it's more complex.

-aagold

Ernie Chan said...

aagold,
Thanks for the research. I will look into CPPI to see if it can suggest any improvements on my method.
Ernie

Mark said...

Hi Ernie,

I'm a colleague of Andris, doing PhD about growth optimal investment strategies. Have you seen any publications besides your book, that deal with the sub-account idea you described?

Thanks,
Márk

Mark said...

Hi Ernie,

I'm a colleague of Andris, doing PhD about growth optimal investment strategies. Have you seen any publications besides your book, that deal with the sub-account idea you described?

Thanks,
Márk

Ernie Chan said...

Hi Mark,
You can just google "Constant Proportion Portfolio Insurance" as suggested by aagold above.
Ernie

Mark said...

Hi Ernie,

I've looked up CPPI before, but it only seems to leverage the risky part of our investment, so all together total exposure is never more than initial investment. (Of course we can borrow to buy CPPI, which solves this problem.)

What I'd like to investigate is creating growth optimal portfolios by borrowing, then using cushion to protect against large losses, while keeping large gains. This wouldn't make an insurance on the final value of the investment (only maximal loss per period), but would optimize log-utility.

Best,
Mark

Ernie Chan said...

Hi Mark,
You can read Ed Thorp's original research on edwardothorp.com
Ernie

Anonymous said...

Hi Ernie,

I'm a big fan of your book and your blog ... really thankful that you post the information that you do.

I've read much of your book and I have a question about Kellys formula. I was thinking of applying it to size the bets
made on an individual strategy. My question has to do with the meaning when Kellys allocation gives a different
directional bet than the base strategy. For example, say you are trading a moving average crossovers and on a given
day you get 10 buy signals over all the stocks you are watching. You then might use the correlated Kellys criterion to
determine the amount of your portfolio to allocate on each entry signal. In general, however, the inverse of the
covariance matrix can tell you to {\em short} one (or more) of the 10 stocks you originally thought were good candidates
to go long on. This is even if the average return on each instrument is positive (the M vector is all positive). What do
you do in this case? I can think of a couple of things:

1) Ignore the sign from Kelly and follow your strategies long direction but with the Kelly size
2) Ignore the direction from you strategy and short that stock since Kelly told you to.
3) Don't trade the given instrument at all (only trade with both strategy signals and Kelly signals are in the same direction)

Can you give me what would be the preferred approach to do in this case (it might not be one of the anwsers above)?

I do also find that Kelly helps greatly at preventing total portfolio loss (going to zero). Strategies that loose more
money than initially seeded with but if you keep trading are profitable can be saved from going to zero by using Kelly
(you mentioned this in your book).

Thanks so much for any help you can provide,

Wax

Ernie Chan said...

Hi Wax,
If your strategy already determines the allocation among the stocks, you can just use Kelly to determine the overall leverage and keep the allocation fixed to your strategy's decision.

I find Kelly allocation most beneficial when applied to allocations of capitals between portfolios or strategies, not among stocks within a portfolio.

Ernie

ezbentley said...

Hi Ernie,

I recently realized an interesting fact regarding Kelly Criterion that I overlooked before.

If you set your optimal leverage to fully Kelly, your standard deviation is exactly equal to your Sharpe ratio. See Thorp's paper. In other words, a Sharpe ratio of 2 will lead to an annualized volatility of 200% if operating at full Kelly. I know in practice you would be more conservative by using half or quarter Kelly due to estimation error etc. This result seems to suggest that even in an ideal Gaussian world, you would want to use low fractional Kelly to reduce drawdown(assuming volatility is roughly proportional to drawdown). And the higher the Sharpe ratio, the lower the fraction of Kelly if you want to keep volatility/drawdown roughly the same.

Have you thought about whether there could be some kind of optimal trade-off between growth and volatility as a function of Sharpe ratio?

ezbentley said...

Hi Ernie,

I recently realized an interesting fact regarding Kelly Criterion that I overlooked before.

If you set your optimal leverage to fully Kelly, your standard deviation is exactly equal to your Sharpe ratio. See Thorp's paper. In other words, a Sharpe ratio of 2 will lead to an annualized volatility of 200% if operating at full Kelly. I know in practice you would be more conservative by using half or quarter Kelly due to estimation error etc. This result seems to suggest that even in an ideal Gaussian world, you would want to use low fractional Kelly to reduce drawdown(assuming volatility is roughly proportional to drawdown). And the higher the Sharpe ratio, the lower the fraction of Kelly if you want to keep volatility/drawdown roughly the same.

Have you thought about whether there could be some kind of optimal trade-off between growth and volatility as a function of Sharpe ratio?

Ernie Chan said...

Hi ezbentley,
One thing to note is that the "standard deviation" you (and Thorp) referred to in this context is the standard deviation of the annualized compound growth rate, not the usual volatility of uncompounded returns per period that goes into the calculation of Sharpe ratio. In Thorp's paper, it is Sdev in Equation 7.6.

At optimal Kelly leverage, the compounded growth rate g=S^2/2, where S is Sharpe ratio. So if S=2, g=2, and indeed Sdev=2 as well. So you are right that in this case we have a good chance (within 1 stddev) of realizing 0% return in
that year. But if your S is 3, then g=4.5, but Sdev is just 3. So you can see that as S goes above 2, a 1 stddev fluctuation of g below the mean will still get you a positive number: profitable for the year!

This is a very interesting result: this means that S=2 is really an important threshold in more ways that I realized. From behavioral finance experiments, we already know that humans demands $2 profits for $1 risk. So it turns out humans are not irrational after all!

Thank you for your insight. I welcome further thoughts from you.

Ernie

P.S. To answer your question directly: No, I don't think that in the ideal Gaussian returns case, assuming perfect estimation of parameters, we need to worry about decreasing leverage below Kelly, given my reasoning above.

Patrick White said...

Ernie,
Great book and thank you for this post! I have been searching for a method that combines geometric portfolio growth with limited drawdown since about when you wrote the article, and was very happy to stumble upon it yesterday. Since that time I've been steadily running simulations and can confirm that drawdown is limited to a fixed percentage using this method with all the benefits of geometric growth on the sub-account. Amazing! You have effectively perfected Shannon's Demon (book: Fortune's Formula) not for theoretical market timing, but for practical use in determining trade size based on equity. Thanks again! You should publish this and all your academic colleagues would be amazed!

In your book you discuss "An Improved Method for Calculating Optimal Leverage." (p316) Can the method of dynamically altering position size be applied to the method discussed in this blog post to maximize median wealth? If so, how?

Ernie Chan said...

Hi Patrick,
Good to hear you have verified this numerically! I have also performed some Monte Carlo simulations and written about this in my new book.

I am not sure which book you are referring to when you said p. 316? It is probably not mine since it is shorter than 300 pages.

Ernie

Patrick White said...

Ernie,

You're right, my mistake. I was quoting from "Quantitative Trading Strategies" (Lars Kestner). I hope your new book's name is a bit more distinctive. ;) By the way, what is the title of your new book?

Back to the point, I think I saw on another blog post of yours talking about Kelly sizing for continuous position management on long term positions. Do you have any opinion on this technique. It would seem to be a high churn method. The idea is that if stock A (or portfolio A, strategy A) has appreciated you should buy more, and if stock B has depreciated you should sell more. On a short term basis I don't see how that could work given the mean reverting nature of shorter timeframes. On a longer term basis, by adding into the more you worsen your cost basis but make more if a trend continues. Also doing so says nothing about an exit strategy with the newly added positions if the market mean reverts.

Ernie Chan said...

Hi Patrick,
My new book will be called "Algorithmic Trading: Winning Strategies and Their Rationale".

Continuous position adjustment based on Kelly is mathematically proven to result in the maximum compounded growth rate, irrespective of whether the underlying returns series is mean-reverting (anti-serial-correlated) or trending (serial-correlated). The proof is simple if we assume gaussian distributions of returns (see edwardothorp.com). Otherwise, you may have to look up Kelly's original information theoretic paper to find the proof.

It has also been applied to practical portfolio management by numerous people, not least by Ed Thorp himself.

Ernie