The Expectation of Having a Portfolio Drawdown. I do also find that Kelly helps greatly at preventing total portfolio loss going to zero.
What is a 'Maximum Drawdown (MDD)'
However, as bad as that sounds its just business as usual. We must accept, there will always be some global crisis or untimely misstep destined to drag the markets one way or another. Now, lets take a few minutes to identify what can be done when drawdown comes knocking.
Below are the only 3 steps you need to know to avoid, sidestep, or overcome account drawdown. This cannot be overstated.
All to often traders panic and start closing positions while consequently accepting a larger loss than would have occurred if they had done nothing at all. I know because I still struggle with this myself.
Unfortunately, that soothing comes generally in the form of escape. My suggestion, for whatever its worth is to stop and think. Refer to your trading plan for the reason why you initiated the trade in the first place.
If you believe it is, great, there is nothing else to do but ride the wave. If even possible given the emotional turmoil experienced from watching your account drown a slow agonizing death. 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. 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. Hi Andris, there are two return distributions that are important to a trading system developer: 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.
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. 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: 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: Andris, Thanks for the link to your paper. The Thorpe article I quoted is here: 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? Hi k1, For discussions on Ralph Vince's work, please see my earlier post: 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 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. Ernie- note that I asked a question about how the subaccount approach you describe differs from "fixed fractional" MM approaches, not "fractional Kelly".
This means I have 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 there are better ways to solve this constrained optimization problem other than the subaccount approach, I would be interested to hear about them.
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: 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: Hope this clarifies things. 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. 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".
Hence now the overall leverage is 0. Hence the difference is more than just psychological! 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: This concept is the same as what's known as "portfolio insurance" i. 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. Hi aagold, The subaccount method is applied to a strategy, not a 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, The term "portfolio" is very general. It applies to any set of liquid securities, either long, short, or market neutral.
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. But from what I have read, it seems to have a superficial similarity to the risk management scheme based on Kelly. 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. Kelly would recommend reducing the number of contracts to What action would portfolio insurance recommend the trader take? 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". Option Based Portfolio Insurance. I will look into CPPI to see if it can suggest any improvements on my method. Have you seen any publications besides your book, that deal with the sub-account idea you described?
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.
Hi Ernie, I'm a big fan of your book and your blog 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. 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: 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. 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. 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. 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.
Have you thought about whether there could be some kind of optimal trade-off between growth and volatility as a function of Sharpe ratio? 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.
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: This is a very interesting result: So it turns out humans are not irrational after all! Thank you for your insight. I welcome further thoughts from you.