Risk of Drawdown

A couple of volumes on my bookshelf discuss drawdowns and how to calculate their probability.

Balsara, in Money Management Strategies for Futures Traders. publishes tables of calculated risk of ruins based on different parameters.

However risk of ruin is different from risk of drawdown. Ruin is usually defined as a fixed capital level, representing a large percentage loss on initial capital. For example, a risk of ruin at 60% is the probability that your initial capital falls to 40% of what you started with. As the equity grows, the risk of hitting that ruin threshold decreases.

Risk of drawdown, on the other hand, stays constant regardless of how high the equity grows, because the drawdown capital barrier keeps moving up in line with the equity.

Both risk of drawdown and risk of ruin increase as the track period or backtest length increases. However, the risk of drawdown tends to 100% as track period length increases, whereas the risk of ruin is bounded at a value determined by the characteristic of the trading system results (probability of win, payoff, trade risk, etc.).

Vince expands the concept of risk of ruin by modifying the calculation to derive the risk of drawdown (in The Handbook of Portfolio Mathematics )

Monte-Carlo simulation allows for estimating the risks of drawdown and ruin by iterating a random process governed by characteristics such as probability of win, payoff ratio, percentage of capital risked on each trade.

Risk of Ruin formula

Perry Kaufman, in New Trading Systems and Methods. presents a formula to calculate the risk of ruin. This is more convenient than having to run a Monte-Carlo simulation but it does not allow for calculating a risk of drawdown.

The formula is as follows:

21 Comments so far

Jez // May 28, 2010 at 8:02 am

Suhendro My bad, I had chopped some of the code on that page (as well as the Kaufman formula).

Stefan // Nov 16, 2010 at 7:31 am

Could there be 2 errors in the above formula?:

Jez Liberty // Nov 16, 2010 at 8:31 am

@Stefan,

The formula was taken from Kaufmans earlier book (New Trading Systems and Methods ) so Ill have to check this (I do not own Smarter Trading though).

When I tested this, I also compared it with the MC simulation result and the results seemed to converge for large values of iteration (but I realise there is a problem in the code above as the number of iterations is ignored and defaults to 1,000).

Ill try and revisit this and make sure the formula is correct.

I think that Kaufman formula cant be correct for an easy reason:

If I put this parameters:

Prob. Win = 0.5

Win/Loss Ratio = 1

Risk Amount = 50%

Number of periods = 1

Loss Level = 50%

The result should be logically 50% and the risk of ruin given by Kaufman formula is 100% when MC formula gives a 50% (each time I press calculate it changes, but moving around 50%).

Also if you try to put higher and higher win/loss ratio you wont see that the Risk of Ruin goes each time lower and lower, you will see it going down and up.

Each time you trade the risk of ruin is:

Risk of Ruin = (Loss trades/ Total Trades) ^ ( Total money / Risked money)

If you Risk the 25% of your money and you system have loss trades the 50% of the time your risk of ruin is logically:

1/16 times or 6.25%,

If you riks the 100% of your money and your system loss trades the 25% of the time your risk of ruin is:

1/4 times or the 25%

It doesnt really matter if your ratio proffit/loss is 10 or 100000, if you trade and loose 4 times consecutively you will loose everything, and loosing 4 times consecutively happens 1/16 times. The thing is that each time you win money your (Totalmoney /Risked money) will get lower and that will reduce your risk of ruin. But is a formula that should be actualized each trade.

CR // Sep 4, 2011 at 12:30 am

Super Article, Jez Thanks a million.

I had no idea anyone had a way to calculate Risk of Ruin without simulations! I so glad I chanced on your blog.

I had a tremendous time playing with your applet above, and discovered how close the results between Monte Carlo (MC) and Perry Kaufmans formula were if you run enough MC simulations.

A couple questions: Does Iterations 1000/1000 means that there were 1000 trades per simulation? Im not sure what the divisor / is for

Periods stands for how many actual simulations were done, is that correct?

With respect to Sickness1s comments, I believe his result is simply because he ran only one simulation (Period =1). If this is changed to 10,000 the results from the MC and the Formula are identical.

Also, I was able to duplicate his anomaly where sometimes the formula result would go up slightly if you increased your win to loss ratio. This is counter intuitive. However, I was only able to get this to happen with very impractical parameters like 50% bet size, etc. I noticed that with more normal parameters, the results seemed very consistent with logic, and the MC simulations. I havent thought about the remainder of his comment yet, and I do wonder why this anomaly exists. Ideally it shouldnt.

I have an email into Mr. Kaufman about the discrepancy between the formulae that appears to exist based on Stephans comment. I think it would be very good to clear this part up!

Kind Regards CR

CR // Sep 4, 2011 at 7:11 pm

Jez,

I received a reply from Mr. Kaufman, and have included a partial quote from him concerning the Risk of Ruin calculation in his book Trading Systems and Methods. He modestly writes:

Thanks for your note, but you overestimate my role in risk of ruin. I simply researched methods that had been done before and presented the best one. The one in Trading Systems should be more recent than Smarter Trading and Im sure that I worked through some numbers on a spreadsheet to be sure they looked reasonable. I would use that one over the older one. Im not really

sure at this point how the results differed. There is probably a reference in Trading Systems as to where it came from. Many other things in the book are original, but not that. Regards

Perhaps if you and Stephan could list any footnote to the formulas listed it may shed some light on the purported discrepancy.

Kind Regards CR

Jez Liberty // Sep 5, 2011 at 5:37 am

Hi CR Thanks for following up with Mr Kaufman. I do not have the book at hand at the moment so Ill have to check next time I get back to my office in a couple of weeks (this is where the Kindle comes in handy if books are available on it!).

Iterations 1000/1000 this is a poor man attempt at a progress bar. When I first developed this, it was taking a bit of time and the code updates from 0/1000 to 1000/1000 1 by 1 after each MC (ie it supposed to mean 1 of 1000, 2 of 1000, 1000 of 1000). Now that browsers are so fast, the execution is immediate and displays the result straight away.

Periods stand for the number of trades (number of simulations is the number iterations)

CR // Sep 6, 2011 at 2:51 pm

Jez,

Im currently reading Larry C Sanders paper on Monte Carlo simulations. It can be found at tradelabstrategies.com I highly recommend it! His explanations of expectancy and the more important role winning % plays compared to win/loss ratio in your systems performance are EYE OPENING. This finally explains why high % scalpers can do so well (often better than swing traders who hit home runs, but do so less than 50% of the time). HIGHLY RECOMMENDED. His paper is actually better than Van Tharps Definitive Guide to Position Sizing, which actually raises more questions than it answers.

One problem I kept having was using too many trades on each of your simulations. I thought period was the number of simulations, and that iteration was the number of trades per simulation. Now that I know its reversed, that helps to explain the results I was getting

As I made period higher (toward 100,000), the chances of a specific drawdown occurring (e.g. 25%) would go to 100%. This was frustrating, as it required very small money management to avoid (sometimes .25% position size). I began asking How many simulations are needed to get a good estimate of your chances of drawdown?

But, my question was incorrect. The question is how many TRADES do you need to include in each simulation, to get a meaningful outcome. The problem is, the more trades you add to each simulation, the more the likelihood of a long losing streak. Its probability at work.

As Sanders explains, your chances of getting all 10 flips of a coin tails is 1/2 to the 10th or 1 in 1024 an unlikely event. Now, if you flip the coin 10,000 times the chances of getting 11 tails in a row becomes very likely. In fact, a maximum streak of 10 tails is only at the 5 percentile, meaning 95% of the time the streak will be longer than 10.

Take your simulator, above for example. If you enter 60% wins, with 1:1 Win/Loss ratio, 2% position size and drawdown 25%, and you run 100 trades, your chances of a 25% drawdown are just 1%. If you make 10,000 trades it jumps to 88%.

Sanders states the way to resolve this is to make sure the number trades in each MC simulation is realistic.

This raises a new question: Over what period do I wish to analyze the risk of drawdown? In a one month period, I may have only 50 trades. Over the year, that number becomes 600 and during a 20 year career, it becomes 12,000 This is a question I suppose each trader must answer for his or herself.

Jez Liberty // Sep 8, 2011 at 6:11 am

Hi CR Thanks for the recommendation on this paper. Ill check it out when I have the chance.

www.automated-trading-system.com/vince-leverage-space-model/ ) in which he explains that the risk/probability of drawdown always tends towards 1 as the probability of reaching a drawdown of X% is a (increasing) function of the number of periods. Actually, in his LSPM approach, he uses drawdown as a risk constraint for the optimization, but it is expressed such that probability of a 20% drawdown should not exceed 0.5 on a time horizon of 12 months for example.

So you still need to choose the numbers that you like over a specific time horizon. Which is different for each trader.

Jez

CR // Sep 8, 2011 at 11:33 am

Hi Jez,

I finished the Larry C. Sanders paper on money management. Its the best, most sensible thing I have read on the subject. He even devotes a section to Vinces optimal f.

There was so much buzz on the net about optimal f, I really felt I needed to use it, or at least understand it. At the same time, I kept wondering Do I really?, because NO ONE was using it, as it stakes too much and people kept saying youd blow out

Well, Sanders very succinctly describes what it is: optimal f maximizing 50 percentile returns without blowing out assuming future returns are identical to historic returns. He described it as getting as close to a cliff as possible, without falling off.

Further, he said its premise is entirely inaccurate that a trader would want to optimize to reward (disregarding risk) and that your future trades will be drawn from your historic trades only ONCE. As can easily be imagined those are faulty and dangerous assumptions.

Sanders then goes on to describe how to position size using Monte Carlo for what is important maximizing your profit AND chances of staying alive.

It appears that Sanders actually traded, or at least understands whats important in staying in the game. It annoys me when experts make a living talking about things they have never done before espousing methods that real traders would never use. Im so glad I read Sanders it saved me a lot of wasted time and money on optimal f. In my book, F is for failure!

I read your blog about LSPM, and I LOVE the graphs, especially the truncated one its very cool. Maybe in his new method, Vince has learned the importance of risk management.

Alas, I am day trading one position at a time. I am glad I dont need to think about adding additional complexity to my position sizing model! Ill read Vinces paper per your suggestion when I run a portfolio.

Im curious if you even feel that one could get stuck learning about money management for portfolios. It deals with so many variables, and unknowable correlations. Sometimes I wonder if it may just be best to find a simple, sensible model one feels comfortable with and go back to trading. My 2 cents

Kind Regards,

Michael Harris // Nov 2, 2011 at 9:35 am

Hi Jez,

I found this older but very interesting post.

It is my understanding that the Kaufman formula applies only to biased experiments, i.e. when P(win) >o.5.

You will see that if you plug in Prob. of win 0.4 with win/loss ratio equal to 1 then you get R> > 1, which is impossible of course.

In addition, it appears to me that this formula cannot apply to trend-following systems because there you may have Prob. of win < 0.5 although a large win/loss ratio. In this case the results of the calculator are wrong IMO.

What do you think about the risk of ruin probability of trend following systems? Is there an easy way to calculate it with a formula?

Virtualmoney // Nov 9, 2011 at 5:43 am

DowBoy // Aug 22, 2013 at 5:26 pm

Hi Jez,

Maybe a calculator you might find useful.

I created this sometime ago.

Ill rephrase that;

I completed most of the formatting, but excel guru SHG got the initial sheet going, then probability guru BruceZ worked his magic.

Heres the link;

Hope it helps, or at least you find some value in it as it is a similar subject to your post here.