tag:blogger.com,1999:blog-35364652.post3750711124624920965..comments2023-01-31T01:51:34.473-05:00Comments on Quantitative Trading: How do you limit drawdown using Kelly formula?Ernie Chanhttp://www.blogger.com/profile/02747099358519893177noreply@blogger.comBlogger62125tag:blogger.com,1999:blog-35364652.post-55412616014877671712016-12-05T07:06:34.229-05:002016-12-05T07:06:34.229-05:00Hi Hesam,
The frequency of rebalancing depends on ...Hi Hesam,<br />The frequency of rebalancing depends on the volatility of your strategy. Naturally, the more volatile it is, the more frequent the rebalance need to be.<br /><br />It is OK to reach $0 equity in the virtual subaccount - CPPI is designed so that if it does that, you will stop trading. But all you have risked (lost) is the max drawdown, not the total NAV of your account in this extreme scenario.<br /><br />ErnieErnie Chanhttps://www.blogger.com/profile/02747099358519893177noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-50274697297477161832016-12-05T03:10:50.741-05:002016-12-05T03:10:50.741-05:00Hi Ernie,
I now realized that you dynamically allo...Hi Ernie,<br />I now realized that you dynamically allocate through kelly. but may I ask how often do you do rebalancing of bet size? and that still leaves the possibility of ruin or huge drawdown in between the rebalancings. how do you deal with that?<br /><br />Thanks,<br />HesamHesamnoreply@blogger.comtag:blogger.com,1999:blog-35364652.post-84303112076318144482016-12-04T23:07:08.835-05:002016-12-04T23:07:08.835-05:00Hi Ernie,
I wonder if you have any update on this ...Hi Ernie,<br />I wonder if you have any update on this topic. I just stumbled on this topic on your blog. I think there is a big flaw with your scheme and that is , with Kelly ratio, the probability of 100% drawdown goes to zero however if you leverage it up, you can easily get wiped out in your sub-account. (e.g. for a leverage of 4, a 25% drawdown will wipe it away). It could be that I haven't completely understood your suggestion here, I would appreciate your comment.<br /><br />Thanks,<br />HesamHesamnoreply@blogger.comtag:blogger.com,1999:blog-35364652.post-76955322031092862922013-02-01T07:42:55.640-05:002013-02-01T07:42:55.640-05:00Hi Patrick,
My new book will be called "Algor...Hi Patrick,<br />My new book will be called "Algorithmic Trading: Winning Strategies and Their Rationale".<br /><br />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.<br /><br />It has also been applied to practical portfolio management by numerous people, not least by Ed Thorp himself. <br /><br />ErnieErnie Chanhttps://www.blogger.com/profile/02747099358519893177noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-25977478386416514062013-01-31T21:07:33.608-05:002013-01-31T21:07:33.608-05:00Ernie,
You're right, my mistake. I was quotin...Ernie,<br /><br />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?<br /><br />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.Patrick Whitehttps://www.blogger.com/profile/16029754567005056050noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-4732005796465042662013-01-31T17:38:05.734-05:002013-01-31T17:38:05.734-05:00Hi Patrick,
Good to hear you have verified this nu...Hi Patrick,<br />Good to hear you have verified this numerically! I have also performed some Monte Carlo simulations and written about this in my new book.<br /><br />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.<br /><br />ErnieErnie Chanhttps://www.blogger.com/profile/02747099358519893177noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-73916769060460140842013-01-30T23:25:33.071-05:002013-01-30T23:25:33.071-05:00Ernie,
Great book and thank you for this post! I h...Ernie,<br />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!<br /><br />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?Patrick Whitehttps://www.blogger.com/profile/16029754567005056050noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-85928165609384350532012-11-09T08:54:55.428-05:002012-11-09T08:54:55.428-05:00Hi ezbentley,
One thing to note is that the "...Hi ezbentley,<br />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.<br /><br />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 <br />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!<br /><br />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!<br /><br />Thank you for your insight. I welcome further thoughts from you.<br /><br />Ernie<br /><br />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.Ernie Chanhttps://www.blogger.com/profile/02747099358519893177noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-22333750916312223692012-11-08T23:04:59.601-05:002012-11-08T23:04:59.601-05:00Hi Ernie,
I recently realized an interesting fact...Hi Ernie,<br /><br />I recently realized an interesting fact regarding Kelly Criterion that I overlooked before. <br /><br />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. <br /><br />Have you thought about whether there could be some kind of optimal trade-off between growth and volatility as a function of Sharpe ratio?<br />ezbentleyhttps://www.blogger.com/profile/17414981365550570912noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-14678721665446602712012-11-08T23:03:40.006-05:002012-11-08T23:03:40.006-05:00Hi Ernie,
I recently realized an interesting fact...Hi Ernie,<br /><br />I recently realized an interesting fact regarding Kelly Criterion that I overlooked before. <br /><br />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. <br /><br />Have you thought about whether there could be some kind of optimal trade-off between growth and volatility as a function of Sharpe ratio?<br />ezbentleyhttps://www.blogger.com/profile/17414981365550570912noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-67136131285110714422012-06-13T07:49:27.416-04:002012-06-13T07:49:27.416-04:00Hi Wax,
If your strategy already determines the al...Hi Wax,<br />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. <br /><br />I find Kelly allocation most beneficial when applied to allocations of capitals between portfolios or strategies, not among stocks within a portfolio.<br /><br />ErnieErnie Chanhttps://www.blogger.com/profile/02747099358519893177noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-81287893540738167092012-06-13T05:55:19.549-04:002012-06-13T05:55:19.549-04:00Hi Ernie,
I'm a big fan of your book and your...Hi Ernie,<br /><br />I'm a big fan of your book and your blog ... really thankful that you post the information that you do.<br /><br />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<br />made on an individual strategy. My question has to do with the meaning when Kellys allocation gives a different<br />directional bet than the base strategy. For example, say you are trading a moving average crossovers and on a given<br />day you get 10 buy signals over all the stocks you are watching. You then might use the correlated Kellys criterion to<br />determine the amount of your portfolio to allocate on each entry signal. In general, however, the inverse of the<br />covariance matrix can tell you to {\em short} one (or more) of the 10 stocks you originally thought were good candidates<br />to go long on. This is even if the average return on each instrument is positive (the M vector is all positive). What do<br />you do in this case? I can think of a couple of things:<br /><br />1) Ignore the sign from Kelly and follow your strategies long direction but with the Kelly size<br />2) Ignore the direction from you strategy and short that stock since Kelly told you to.<br />3) Don't trade the given instrument at all (only trade with both strategy signals and Kelly signals are in the same direction)<br /><br />Can you give me what would be the preferred approach to do in this case (it might not be one of the anwsers above)? <br /><br />I do also find that Kelly helps greatly at preventing total portfolio loss (going to zero). Strategies that loose more<br />money than initially seeded with but if you keep trading are profitable can be saved from going to zero by using Kelly<br />(you mentioned this in your book).<br /><br />Thanks so much for any help you can provide,<br /><br />WaxAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-35364652.post-11211230195464062392010-12-06T09:17:15.146-05:002010-12-06T09:17:15.146-05:00Hi Mark,
You can read Ed Thorp's original rese...Hi Mark,<br />You can read Ed Thorp's original research on edwardothorp.com<br />ErnieErnie Chanhttps://www.blogger.com/profile/02747099358519893177noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-69619290070187846832010-12-06T09:13:44.287-05:002010-12-06T09:13:44.287-05:00Hi Ernie,
I've looked up CPPI before, but it ...Hi Ernie,<br /><br />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.)<br /><br />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.<br /><br />Best,<br />MarkMarkhttps://www.blogger.com/profile/09015114465783630364noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-29353655791677973742010-11-22T10:38:33.837-05:002010-11-22T10:38:33.837-05:00Hi Mark,
You can just google "Constant Propor...Hi Mark,<br />You can just google "Constant Proportion Portfolio Insurance" as suggested by aagold above.<br />ErnieErnie Chanhttps://www.blogger.com/profile/02747099358519893177noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-85226892882050996212010-11-22T08:57:22.594-05:002010-11-22T08:57:22.594-05:00Hi Ernie,
I'm a colleague of Andris, doing Ph...Hi Ernie,<br /><br />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?<br /><br />Thanks,<br />MárkMarkhttps://www.blogger.com/profile/09015114465783630364noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-32554336800745435302010-11-22T08:54:57.746-05:002010-11-22T08:54:57.746-05:00Hi Ernie,
I'm a colleague of Andris, doing Ph...Hi Ernie,<br /><br />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?<br /><br />Thanks,<br />MárkMarkhttps://www.blogger.com/profile/09015114465783630364noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-19696622120315477492010-10-12T08:58:34.511-04:002010-10-12T08:58:34.511-04:00aagold,
Thanks for the research. I will look into ...aagold,<br />Thanks for the research. I will look into CPPI to see if it can suggest any improvements on my method.<br />ErnieErnie Chanhttps://www.blogger.com/profile/02747099358519893177noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-57168260664840523572010-10-12T08:05:14.216-04:002010-10-12T08:05:14.216-04:00Ernie,
Well, I did some more research on this top...Ernie,<br /><br />Well, I did some more research on this topic and concluded that your method is probably better than what I was proposing.<br /><br />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.<br /><br />-aagoldAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-35364652.post-36557162905735879532010-10-11T15:27:26.868-04:002010-10-11T15:27:26.868-04:00aagold,
I only have a pedestrian familiarity with ...aagold,<br />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.<br /><br />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?<br /><br />ErnieErnie Chanhttps://www.blogger.com/profile/02747099358519893177noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-47118693796951730022010-10-11T11:42:27.964-04:002010-10-11T11:42:27.964-04:00Ernie,
The term "portfolio" is very gen...Ernie,<br /><br />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.<br /><br />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.<br /><br />-aagoldAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-35364652.post-83625809486103838782010-10-11T10:14:20.156-04:002010-10-11T10:14:20.156-04:00Hi aagold,
The subaccount method is applied to a s...Hi aagold,<br />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.<br />ErnieErnie Chanhttps://www.blogger.com/profile/02747099358519893177noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-65537459536934923872010-10-11T09:59:55.145-04:002010-10-11T09:59:55.145-04:00Ernie,
Sorry to post a comment so long after your...Ernie,<br /><br />Sorry to post a comment so long after your post, but I only became aware of it a couple of days ago.<br /><br />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).<br /><br />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. <br /><br />-aagoldAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-35364652.post-58700949309217793392010-05-18T14:16:58.747-04:002010-05-18T14:16:58.747-04:00Joshua,
Yes, in your example, we will not rebalanc...Joshua,<br />Yes, in your example, we will not rebalance the 2 subaccounts until a new high watermark is reached in the full account. <br /><br />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!<br /><br />ErnieErnie Chanhttps://www.blogger.com/profile/02747099358519893177noreply@blogger.comtag:blogger.com,1999:blog-35364652.post-34278651623754477742010-05-14T12:22:02.845-04:002010-05-14T12:22:02.845-04:00Ernie,
If we assume K=2 and we decide to segregat...Ernie,<br /><br />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. <br /><br />If that is the case, I still don't see what the point is beyond being a psychological trick.Joshhttps://www.blogger.com/profile/10241276842886529712noreply@blogger.com