# Tortured data - buyer beware



## Gary (24 Aug 2012)

For investment marketers, communications is key. But there’s a lengthy continuum from honesty, through “towing the party line,” to gilding the lily, and ultimately deceit. 

Over the years, I have witnessed marketing/sales people from the various product providers operate on every point along this continuum. Some are very plausible. I should probably declare at this point that I am a former, or should I say reformed, investment marketer myself. I am certain that I towed the party line. But I’d like to think I exited with my integrity intact. 

I have recently been through the glossy brochure for the latest guaranteed fund from Dolmen (safe harbour) and a chart in it stoked my ire.

I will admit to being cynical about the value in structured/ guaranteed products generally. But rarely to the point of opening a thread like this to vent my spleen. 

The brochure refers to a BNP strategy and provides back testing as far back as 1995. It then goes on to say that the strategy being back tested was created in 2011 and the back testing data “_includes estimates made by BNP Paribas in respect of unavailable data”._

Dolmen is not alone in the use of back testing of simulated strategies, so I am not pointing the finger solely at Dolmen, but there needs to be some debate around the use of such charts. 

The financial markets have only one history. And when one knows what that history is, it is a trifling matter to find factors that ‘explain’ it (a statistical activity affectionately termed ‘torturing’ the data). After all, it is said that statistics is the science of producing unreliable facts from reliable data. Investors should view history with their eyes wide open.

In a famous tongue-in-cheek 'experiment' on data mining it was found that 75% of the variation in the S&P 500 could be explained by butter production in Bangladesh! By adding cheese production and sheep population in the US, correlation rose to 99%! 

The link between butter production in Bangladesh and the S&P500 is obviously spurious, but if cloaked in terms of a complex strategy involving GDP forecasting, interest rates, volatility etc, it begins to appear plausible. And so when presented with a chart that ‘proves’ the strategy works, our greed takes over. 

This product may turn out to be very successful (in terms of performance not in terms of sales – this dichotomy in how the provider and the investor would view success is important, but alas another rant all together). The point of this post however, is to highlight that if it does ‘work’, it will be as a result of sheer luck. 

The irony about the famous experiment I refer to above, is that following its publication, the author received enquiries from people wanting to know where they could access data on butter production in Bangladesh. Can’t recall who said it, but it’s very apt; Regulation will not stop a fool from losing his money, but it should at least stop the average person from being made to look like a fool.


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## Brendan Burgess (24 Aug 2012)

> It then goes on to say that the strategy being back tested was created in 2011 and the back testing data “_includes estimates made by BNP Paribas in respect of unavailable data”._



That is the most extraordinary thing.

I had heard of "data mining" but "data torturing" is a new one for me Gary.


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## Rory Gillen (24 Aug 2012)

Interesting and disappointing observations. I guess when the incentive system is wrong to start with the customer has no real chance. We rely on the regulator to limit such instances but confidence that they are at hand to do is rightly non existent.


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## Monksfield (29 Aug 2012)

*Dolmen product & ilk*

More and more these products have at their core makey-uppy indices devised by investment bankers and mathematicians who have beaten up mountains of data.

It will be interesting to see what proportion of the current and recent 'vintages' of these products deliver a cent more than the guaranteed amount. My own guess is less than 25% will deliver bonuses and because of the various caps and volatility control mechanisms, the payouts will be small.

The idea that these are selling in large quantities is awful.


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## Marc (1 Sep 2012)

How to test statistical significance

The t-test was introduced in 1908 by William Sealy Gosset while working for the Guinness brewery in Dublin to evaluate the quality of the brewery’s ingredients.

In investing, the t-test can be used to determine whether any market beating returns are due to luck or skill. When we apply a t-test to the historical performance we get a statistical feel for how significant the results are.

A result from a t-test in excess of 2 is widely accepted to be a measure of statistical significance.

When we apply a t-test to the results of any particular fund or strategy we should always start with a "null hypothesis" that is to say that we don't believe a fund manager has any skill and we should set out to statistically attempt to disprove this proposition.

Here is the difficulty with this process. At any given time we have a limited data set perhaps the existing track record of any given fund manager or a backtested strategy, we also have the average excess performance achieved and we can measure the volatility of that excess performance compared to a risk appropriate benchmark. But the limiting factor here is the size of the data set such as the career of a fund manager. Typically we don’t get that many observations.

In another post, I recently pointed out that if I flick a coin 5 times and it comes up heads 5 times, there is a 3% chance that will happen just by luck. So, by the same token, 5 years of performance for a fund manager isn’t sufficient time for us to say with a high enough degree of confidence that here is the “Tiger Woods of fund management”.

We need at least 30 or 40 years of data to show statistically that, yes this fund manager or strategy has real skill above the market, allowing for the risks taken, and the persistency of the results is consistent enough to believe that they are really skilled.


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## Brendan Burgess (4 Sep 2012)

Monksfield has an interesting post on the Wealth Options  Global Absolute Return Fund which he posted in reply to this. 

I have moved it to a new thread to facilitate specific discussion of this product:  
Wealth Options Global Absolute Return Fund


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## ecstatic (4 Sep 2012)

I agree the wording is incredulous on the package etc !

However i find some of the displayed comments here even more incredulous.

"So, by the same token, 5 years of performance for a fund manager isn’t  sufficient time for us to say with a high enough degree of confidence  that here is the “Tiger Woods of fund management”.

" We need at least 30 or 40 years of data to show statistically that, yes  this fund manager or strategy has real skill above the market, allowing  for the risks taken, and the persistency of the results is consistent  enough to believe that they are really skilled. 	"

This all depends on the time series used to perform the testing (if thats what i read into what you are saying (as most managers will be retired in 20-30 years)). Time series can be based on tick so you may have 1.5 millions ticks per second (so 1.5 million per second well you get the idea of how much data there is / could be to model the strategy). 

To deem whether or not these methods are / can be succesfull through data mining i urge you to readup on 'Bill Dunn' | 'James Simons' . (If they are successful over a period they will dump there investors money and just run there own!)

Indeed if the dataset is purely backtested and then conditioned to be 'curve fitted' this is of no-use. You want too see what is used in relation to in-sample and out of sample data to proclaim whether a strategy is valid or not. 

I guess this prospectus has been written badly / checked reviewed very badly.

If i was to tell you that if you bought wheat on 1st august and sold it at end of august every year since 1968 that prices ended up higher 83.33% of the time would you believe me ? Is this a tradeable concept ? In itself no its not as i havent stated the drawdown between 1st - end of august but it is a basis for further testing. 

Does the market behave similiar over long historical periods (most definetly yes it does). Can i prove to you that market trade entries at best can be outperformed by random entries ? (yes i can as the exits are only thing that matters).

Can i prove to you that 93% of the time equity prices revert to previous prices ? Yes i can. Is it a tradeable concept in itself no with other things added into the mix yes.

Anyone read the 'hedge fund mirage' book recently released ?
In it you find compelling evidence that the average hedge fund return was less than the t-bond return over past 30 years or so (when fee's are taken into account). 

Does that mean that hedge funds are bad not really in that some 
'superstar's' produce genuine alpha. Does it mean that the average joe bloggs should just buy t-bills; yes it probably does.

A hedge fund manager once mentioned something smart to me; he said if i wanted to run a fund that looked great id just buy argentinian bonds and 1 every 35 years id blow the fund up (as they may default). For the other 34 years id look like a superstar with between 8-15% per annum return. Argentina defaulted on its debts 5 times in the last 175 years.

Any random walkers out there ?


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## Marc (4 Sep 2012)

So we seem to be mainly in agreement. 

I am perfectly willing to accept that there will be some superstar hedge fund returns (no more than I would expect by pure chance before fee and considerably less after fees) but that I have no reliable way of deciding who the winners will be in advance and that therefore most people on average are better off with T bills. I think it was bill Sharpe who first made this observation.

So in conclusion since I don't have enough manager decisions in any reasonable period of manager tenure I can't determine with a high degree of statistical certainty if a manger is skilled or just lucky. I am therefore better off (on average) with a simple buy and hold strategy.


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## ecstatic (4 Sep 2012)

Buy and hold strategy for t-bills yes an average investor seems much better off with this (historically speaking).   Then again who wants average or a C- grade on there paper. Average probably means just below inflation returns.

Buy and hold in terms of equity markets no way! we agree in some fascets but not in other fascets (think i spelt that wrong!).

If the fund has done correct back testing forward testing and in sample and out of sample then it probably has 'some' merit. 

There is no 'superstar hedge fund' there by pure chance over a prolonged period of time. If they are betting every day on asset prices to move up or down they wont be in business long based on 'luck' if there making 10 trades a day or so. 

There is little 'luck' associated with good professional trader's. 

Take for example that markets are probably now run by 60% quantitive techniques this means in a way that mutual funds are at a disadvantage and indeed retail. Mutual funds not been able to short (apart from weighting a basket to mimic a short) (however why wouldnt i just buy an ETF in this case). Retail due to the actual quant machines and indeed due to them machines always watching the markets for a price level threshold that triggers a trade. 

Check this out: http://www.trendfollowing.com/perf.html as an example of dunn capital. Is he just lucky ? I doubt it id say he has a statistical edge. How did he gain this edge from 'backtesting' his strategy. 

On an aside note: personally i have created 5 strategies myself from an algorithmic perspective. 

Of these 5 strategies 2 i dumped as they were crap and didnt work and 3 i run live. The data i analyse is probably around 4165516 minutes of actual market data in sample and 1/3 of that out of sample for each strategy. 

Why did the 2 fail ? 1.) Illiquid market. 2.) Crap strategy. What did i learn 1.) test in demo live mode for 6 months after production too see which did and did not work. Do i need too see why 2.) was a crap strategy yes i do.

So do i believe a manager would do better if they backtested 'most definetly yes'!

Do i think backtest's can provide some evidence of future performance, to a degree i do (but it has to be well thought out etc etc and a lot of work done on it ).  Do i believe a lot of backtests are invalid (yes i do) as id really like too see the out of sample data as much as the in sample tested data. 

Do i think fundamentalists may be able to add value ? Maybe in some cases if the company is not cooking the books so too speak.

In sum if you want c- go for bills it as not everyone would go to the effort i have done to manage there own money . Its more of an interesting passionate hobby to me so dont go down this route!

You can never know the winners in advance but to be honest in investment terms 1 in every three (investments) may be a winner 1 in three breaks even and the other 1 loses money that could be enough depending on how big winners - losers are!

Losers and winners are part of life very successful corporations have plenty of losers added to there own balance sheet through Mergers but they have a few winners also which is what makes them winners!


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## ecstatic (4 Sep 2012)

A good stethescope when looking at a fund manager is what the sharpe / sortino and calmer ratios are.

Would anyone get funded based on not 2 consecutive years of live trading history, no they would'nt (even though in this case this fund has not been live for long).

If i was looking for a fund manager id actually look at fund managers based on poker / bridge tournament performances.

An interesting correlation maybe as to what table they get to at a major  tournament in line with there fund performance. I wonder would there be a correlation ? Id never know unless i backtested it!


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## Marc (4 Sep 2012)

Some interesting misconceptions here.

Firstly the amount by which a winner wins is equal to the amount by which a loser loses since there are no orphan securities everything is held by someone. 

Therefore in order for you to prosper from any trade you have to find someone willing to take the other side. For you to win they have to lose.

When you trade you pay costs and taxes. The more you trade the more you pay in costs compared to the poor average buy and hold investor. How do you know when you trade you are not going to be the loser on the wrong side of the trade? You don't and on average market participants are just as smart as each other so you need to find people less smart than you to exploit in order to consistently prosper. This in a market full of research depts, investment banks and of course private investors with their own systems.

It isn't credible that winning trade are down to anything other than taking on more risk or simply luck.


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## Marc (4 Sep 2012)

As for backtesting.

Fund managers use a process known as incubation. They design several strategies and run them privately for a period with their own money. They dump the strategies that don't work (sound familiar?) and launch the strategies that seem to make money. Naturally they are hardly going to launch a strategy which doesnt appear to work. The problem is that as the funds attract investors the persistency of performance tends to drop off and real investors tend to be left holding a dog.

Rinse and repeat until broke


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## ecstatic (4 Sep 2012)

If you did some research you would see that there are investors who make  outsized returns. In fact the decent ones make / take all the money. 

I think you dont understand what i meant when i termed winner and loser.  If i bet 1K USD at what i believe is 50/50 to win 2K then my winner is  twice my loser's size. So effectivly i can do this every day for the  next 220 days (say the trade lasts 1 day) and on average ill have 110  winners and 110 losers. 

110 losers = 110000 
110 winners = 220000
220000 - 110000 = 110000 profit.

In another example just because i shorted something and someone else  went long something doesnt mean theres 1 loser and 1 winner . The fact  is we both may have different timeline horizons i may be in and out in a  second and he may be in and out a few minutes later. At the end of the  day price in a commodity term is essentially just a number that dictates  in some facets supply and demand.

"When you trade you pay costs and taxes. The more you trade the more you  pay in costs compared to the poor average buy and hold investor. " - 

If i know that the typical standard deviation movement of a stock is X  and it doubles the standard deviation movement in a downward wouldnt i  be best selling out of that stock probably. Would the buy and hold be  better off holding it, no they wouldnt as if they can buy it back  cheaper than before its better than if they hold and hope. 

Buy and hold equities havent a hope in todays markets as the bot's will  pick them off. Anyone who thinks different is sadly mistaken the bots  run the market and make most of the money.

In terms of "When you trade you pay costs and taxes. The more you trade  the more you  pay in costs compared to the poor average buy and hold investor. " -  Well in fact this could be deemed wrong as big funds get paid too add  liquidity to a market so if i was a huge fund id be getting rebates for  adding liquidity so in fact i get paid to be in a market and defining  it.

Theres different ways too look at things, certainly the product mentioned at start of this thread i wouldnt endorse.

If you look up historical figures off the top of my head there are 12  days of outstanding movement (major stdev movement) in the stock market  every year if you miss them / are on the wrong side of them you  basically miss out on the real fruits of been in the stock market. How  can buy and hold pick those days by just buying on a particular day ?  Impossible. How can they short it not possible either.

Still having said that if i was to be putting money towards retirement  id be just holding high grade government bonds. (My reasons differ to  others though as i have enough volatility in my portfolio to pick up  outsized returns with my strategies ) or well at least i hope i do!

For others a see saw balancing portfolio may work a lot better.


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## ecstatic (4 Sep 2012)

they wont launch unless they have two years of live results (you can pick through these results by looking at what they have etc.)

much better a backtest than a buy and hope!


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## ecstatic (4 Sep 2012)

strategies sometimes just dont work thats why it was eddison who said he found a million ways not to make a lightbulb or something like that!

 			 			Look try this link for a list of funds that probably have outsized  returns versus buy and hold maybe they are the poster children or maybe  we should learn to follow them somewhat!

http://www.automated-trading-system....rds-july-2012/

http://abrahamtrading.com/performance

Id guess by looking at there career opportunites section in any of them  you will notice they are looking for quants so its obvious they are  willing to put money on non buy and hold strategies!


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## ecstatic (4 Sep 2012)

In fairness my 2 failing strategies were due to my own naievty there isnt enough volume on oats in 1 case and just not enough testing in the second scenario!  We will see after two years live on the other 3 strategies. 14 months only at present.

If your interested on a book that shows some strategies etc and is probably worth the investment have a read of this:

http://www.amazon.com/Applied-Quant...antitative+methods+for+trading+and+investment


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## Duke of Marmalade (4 Sep 2012)

_Gary_ where have you been all my life?  That is one of the best posts I have ever seen.  I agree 100%.  Let me add a few comments.

We are presented here with what looks like a very plausible and sophisticated strategy to beat the markets.  It is like a system for winning at roulette.  No such system exists and no such investment strategy exists.  Of course it backtests well.  That was the reason for its choice.  No basis whatever in financial theory.

So now lets talk about backtesting.  Backtesting is banned by the Consumer Protection Code for Tracker Bonds.  This is a Tracker Bond.  However, strictly speaking the backtesting is being applied to the underlying and not the Tracker Bond itself so technically not in breach.  The CPC should ban all backtesting.

Now to my hobby horse.  The CPC is a toothless sham.  One of its key requirements on Tracker Bonds is to show clear disclosure of where the investor's money is spent and in particular to show the charges.  The disclosure on this product says charges are *zero*.  This is typical of Ulster Bank (it is they who approve the brochure).  I have complained in the past to the CBI on this.  No response.   

They do make a reference to the commission costs but this is probably only half of the total costs on this product.  The CPC will remain a toothless fig leaf until practitioners like me can make meaningful complaints about non compliance.


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## ecstatic (5 Sep 2012)

I find that interesting the CPC ban backtesting nice information! Thats interesting...

" We are presented here with what looks like a very plausible and  sophisticated strategy to beat the markets.  It is like a system for  winning at roulette.  No such system exists and no such investment  strategy exists.  Of course it backtests well.  That was the reason for  its choice.  No basis whatever in financial theory." 

However a roulette system would technically be feasible to work if you could martingale til infinity so as your point is sound on the surface its not sound when its mathematically evaluated!

Like a system for blackjack and poker also (obviously roulette systems do not work)?  - There are systems for both blackjack and poker its only recently that poker has been recognised in america as a game of skill. Is it any wonder that over the last 30 years or so the same faces frequent the final table in poker tournaments. 

Would this be skill or luck ?

By the way do you have a link to the underlying document at start of this thread.

Lets not forget backtesting is hypothetical its not actual!


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## Duke of Marmalade (5 Sep 2012)

[broken link removed] is the very brochure.  _Ecstatic_ I will wait till Poker is an official Olympic sport before agreeing your thesis


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## ecstatic (5 Sep 2012)

Well i looked at the brochure and believe _some_ of it has merit. It has plenty of exposure for 1 thing!

However some major issue's in this that i see is that either they are using the VIX Volatility Index or the deviation of prices to control whether its risk on or risk off. 

As we all know and appreciate the last 5 years have been very volatile within the stock markets however between 1999 - 2001 is less volatile. How was the volatility measured in fact as what was the way to measure volatility in 2000 is not the way now to measure volatility.

It doesnt show the maximum drawdown or the sharpe ratio so its very difficult to measure the performance against anything buy and hold included. Without knowing what risk is been taken you cannot decide whether alpha could be present. 

From the prospectus those fees are unreal !
"the fee to Dolmen will be equivalent
to 4.18% of the Investment Amount in the case of Option A and 5.46% of the Investment Amount in the case of Option B."

Also it states the usage off bollinger bands to see the price etc which to be frank is a slightly dated notion to modern quantitive techniques. 

Also without knowing the timeframe etc this product doesnt mention anything regarding slippage etc.

If the fund size affects the price etc for instance it has silver as a market to be in but thats not exactly a highly liquid market. 

Also it doesnt state which treasury bills to invest in.

Also it doesnt state rollover for futures contracts.

Also it doesnt state any fx risk or is this been hedged out ?

More questions than answers but the underlying principle seems soundish, (once you expect germany not too default, us not too default and all those currencies to be still around in a few years well).

 					 					 Based on your comment i am also waiting for hurling to be an Olympic sport also!


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## Gary (5 Sep 2012)

*skill and luck*

This thread seems to have rambled off a little. But to some recent points:

Duke, that is very interesting that the CPC code bans back-testing of products, but does not extend this to the underlying indices/strategies. That’s a rather large loophole for the regulator to have left. One product providers would be silly not to exploit.


Ecstatic, to your point about skill and luck; Poker most definitely is a game of skill as you say. But, like fund management, it also involves luck. And the paradox of luck is that in highly skilled environments, luck plays an even larger role. So I can quite confidently say, that at the final table of the world series of poker, the ultimate winner will be the luckiest at the table (a statement which in no way detracts from the skill of any of the players at the table). 
 

I don’t understand your martingale point about roulette. If you are suggesting doubling your bet in the expectation of eventual pay out, the existence of white (and double white on some wheels) plus table maximum bets makes this a very dangerous ‘strategy’. Likewise for blackjack, table maximums prevent this strategy working.

As to the various posts regarding the efficacy of trend following investment strategies. I am not sure if we could reconcile our investment philosophies. You appear to concentrate on trading of individual commodity futures, something I would consider to be closer to speculation than investment (which involves ownership of some underlying asset for a period of time long enough for fundamental value to reveal itself). Like the croupier at the casino’s poker table, the only long term winner at this game will be the market maker, those taking a consistent slice of the overall action. That is not to suggest you will not make money, but as Marc points out, there is no way for me to tell in advance whether that successful person will be you or someone else at the table.

To go back to the original point of this thread, which was the spurious causation drawn between simulated investment strategies and wonderful investment outcomes. An investor must ask him/herself if the investment proposition makes sense. You will never convince me that frequent trading of commodity futures is something at which I can consistently make money. Equally I remain unconvinced about the prospects for an algorithm which dictates when to buy ‘risk on’ assets and sell ‘risk off’ assets (admittedly a very basic summation of the BNP strategy within the Dolmen safe harbour product). 


I am unconvinced. That is not to say it won't work. This product may be bailed out by a large dose of luck. But intelligent investing, recognising the role of luck in the eventual outcome, attempts to reduce its importance.


The idea that anyone can make money consistently trading markets is a fiction convenient only to stock broking and spread betting firms. Stock brokers and their ilk are in the main, profitable after all.


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## ecstatic (5 Sep 2012)

I do agree with a lot of the points from all posters but i digress on some!

 "I don’t understand your martingale  point about roulette. If you are suggesting doubling your bet in the  expectation of eventual pay out, the existence of white (and double  white on some wheels) plus table maximum bets makes this a very  dangerous ‘strategy’. Likewise for blackjack, table maximums prevent  this strategy working."

If you have an infinite amount of money to bet then you would eventually win. The problem arises as the roulette table usually has a fixed upper treshhold of betting (as in max bet 500 bucks).  If i bet a million times and continue to use a martingale i will win eventually! (again depends on casino rule's). Roulette by itself doesnt define a max bet the casino rules do.

Indeed i agree with all your points including the 2 greens on a roulette giving an unfair advantage. However mathematically, and very much depending on the casino rules blackjack can give a slight edge to the person if for instance you could card count etc. This edge may be ever so slight so have to be executed brilliantly aswell as looking carefully at casino rules.

There is some luck alright in the winner which is why i suggested the final table is a better metric than the actual winner.

However if i trade 100 times a day for 6 years and am still in profit then i could have an edge (13200) trades for instance as the number of trades is not insignificant.  

I also strongly agree brokers do sheckle there clients to bits and in fact there was an interesting case whereby a client of a large broker managed to figure out the brokers algorithim and tricked it into taking profits from the broker. This went to court and the client lost (which i find incredulous)!

Portfolio rebalancing is / can be a good thing as many of us know too heavy exposure to 1 asset class is/can be a dangerous thing!


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## Duke of Marmalade (5 Sep 2012)

_ecstatic_ that's not my understanding of what a martingale is. My textbook says that it is a game, or a model of the financial markets where there are no biased strategies. No matter what your rules of engagement your expected outcome is the same - it will of course be different in delivery between different strategies - but each strategy has the same risk adjusted expectation from the beginning. 

Martingale theory is very relevant here for what is being suggested with this product is that by very careful analysis of past performance a system has been devised to beat the market in the future. 

Ever go into a Paddy Power shop?, he runs a lotto every 10 minutes and helpfully informs the punters of suggestive patterns from the last 30 spins - the screen blares at you the "hot" ball i.e. the one which has appeared most often. It also tells us the "cold" ball and points out various other wondrous patterns. I am sure the poor mugs all form different views as to which of these past observations are of most relevance to the imminent future. 

In a somewhat more sophisticated manner this product is attempting to persuade gullible investors that the past holds the key to future free lunches on the markets.

I don't understand your casino point. For those unfamiliar it goes like this. Imagine a casino which offers even money on a single number in a roulette game. Shocking and criminal odds you say. But with unlimited resources you can follow a strategy which bets on a single number (can be different each time) and provided you bet enough to recoup past losses and win say a euro your number will eventually come up so that you will certainly finish winning a euro despite the dreadful odds. I don't think this adds much to the current discussion.


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## ecstatic (6 Sep 2012)

Martingale keep doubling until u win the only issue is money is a finite resource and they finitly cap your earnings. In this case dont complicate use red and black.

Funnily enough i have never been too a paddy power shop weird huh ?

The strategy is soundish for diversifcation component of a portfolio however as to whether us govvies and bundesbank govvies will be safe havens in 10 years is anyone's guess. Statistically speaking they will be but then again wasnt the GBP the powerhouse of the world until panama.

Will markets still be irrational and fear based yes they will as long as humans are involved.


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## Duke of Marmalade (6 Sep 2012)

_ecstatic_. two definitions of martingale in Wiki

[broken link removed]

and

Betting Strategy

A key quote from the first is:


			
				Wiki said:
			
		

> In probability theory, a *martingale* is a model of a fair game where knowledge of past events will never help to predict future winnings. Martingales exclude the possibility of winning strategies based on game history, and thus they are a model of fair games.


This seems very relevant to this debate. Not sure of the relevance of the second meaning, yours, is.


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## Duke of Marmalade (6 Sep 2012)

I'll try again

Martingale in game theory


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