On coming across the ‘Bismarx’ betting strategy (via the Bank Vault Report, published by, I decided to do some calculations on certain key factors (partly because it seemed too good to be true!). Chief among these was the question of liability control, which I could see would be an important factor for those traders (such as myself) with relatively limited capital.

The plan works on the basis of a ‘lay ladder’ of five horses, where successive selections are layed, with a fixed profit for the ladder as a whole, and stakes increasing to cover additional losses if the earlier selections win (i.e. the lay loses). The first question that may come to mind is: ‘what happens if all five horses win?’ The calculations and analysis below are essentially attempts to construct simple models to estimate a) how often this might happen, b) what the liability might be whenever this does happen. (It being a sound military strategy to have contingency plans to cope if things do not go according to plan!).

The first thing to note is that (from the point of view of simple profitability) it is not necessarily such a disaster if the bank is lost occasionally. Under the Bismarx betting plan, assuming 1/20th of the bank is bet on each occasion, then the bank renews itself every 20 days! So, mathematically, as long as the average rate of bank loss is significantly less than once every 20 days, then the Bismarx plan is highly likely to be profitable (this assumes fixed stakes, but also applies to ‘progressive’ staking strategies, where the stake is increased in line with increasing bank size, as long as profits are indeed begun to be taken out regularly at some point). In fact, I believe the current data is that there has only been ONE losing Bismarck (i.e. five successive winning horses on a day) over a period of well over a year (400+ bets?).

How often would one expect a losing Bismarck, statistically? The key factor here is the average rate of loss of the Bismarck selections. A statistic often quoted is that favourites will win ‘around one third of the time’. The Bismarck selections eliminate horses priced at either extreme (prices over 2.7 on the strict system, or prices under 1.7 - although in practise there is a little leeway given here), which mean that the ‘one third’ strike rate is perhaps a useful first approximation.

From simple probability theory, if each horse has a probability of 0.33 of winning, then the probability of five successive winners is: 0.335, or 0.0039. This represents an event which is only likely to occur once in every 255 days! Given that the bank renews itself every twenty days, then that seems to indicate a quite extraordinary advantage to those individuals trading via the Bismarx system. Although this calculation is just an estimate, empirical results seem to indicate it is of the right order of magnitude, with a single ‘losing Bismarck’ in over a year of trading.

It would be interesting to repeat the calculation using the actual percentage figure showing what proportion of the Bismarck ‘lays’ do in fact win (as this calculation is very sensitive to small variations in this). If the figure falls to 0.3, then the probability of five successive winners becomes 0.35, or 0.0024. This would only be expected to happen, on average, once every 411 days.

If this figure rises to a probability of 0.4, however, then the probability of five successive winners becomes: 0.45, or about 0.01. This represents an event which would be expected to happen roughly every hundred days. Given that there has only been one losing Bismarck in what (I believe) is well over a year’s trading, then statistically it would have been likely there would have been more than one losing Bismarck, if this was the case. So the empirical data would seem consistent with this probability falling below 0.4.

However, there does seem to be one slight anomaly, in that Bismarx report that (from memory) around 80% of lay ladders end by the second leg. This would indicate that around 20% of the time the first two horses win, implying: (probability of winning Bismarck selection horse)2 = 0.2. This figure relates to a probability of winning for a single horse of almost 0.45 (taking the square root of 0.2). If one then extrapolates this figure to the five legs of the lay ladder, probability in this case of a losing Bismarck lay ladder, i.e. all five horses winning = 0.455 = 0.0184. This would correspond to a losing Bismarck, on average, every 54 days. Despite statistical fluctuations, it is extremely unlikely that if this is the case there would only have been one losing Bismarck in well over a year of selections (one might have expected half a dozen or more).

Possible explanations for this apparent anomaly? Firstly, the “80%” estimate for Bismarck ladders which don’t go past the ‘second rung’ may not be precise; it may be rather higher than this. However, if one third of the Bismarck selections represent winning horses (in line with the earlier assumption), then one would in fact expect as many as 90% of Bismarck’s to be successful by the second rung (i.e. either the first or the second horses loses).

The second possibility (which I suspect may be more likely) is that the simplifying assumption of this model - that all horses are equal - may in fact be over-estimating the probability of a failing Bismarck. In practise, selections will be made with quite a variation in price (and hence relative likelihood of winning). There will be a major difference in strike rate between horses at decimal odds close to 3, than horses priced at 1.7. It may be the case that this unevenness in fact enhances the probability of the Bismarck being successful, over and above that from the estimates above based on a simple assumption of horses being equal. To investigate this more thoroughly, the precise statistics over the entire run of the 400+ Bismarck ladders to date would be very useful. The key statistics in relation to investigating this particular factor further are:

1) the overall strike rate of all Bismarck horses (i.e. how many do in fact win, irrespective of whether they are before or after the point on the lay ladder where the first horse loses). Call this figure W.
2) The proportion of Bismarck lay ladders which reach a successful conclusion within the first two horses.

(The third vital statistic in this respect (the number of Bismarck ladders which are unsuccessful), is already known, I believe, as just a single ladder. Though it would also be useful to know the total number of Bismarck ladders run to date - I only know that it is somewhere between one and two years, whether that represents 300 or closer to 500 ladders would be of interest).

It should also be remembered that all probability calculations are just that - probabilities - and random fluctuations about a mean will always happen. So even if statistics suggest that the probability of a losing Bismarck should only occur once in every 200 days, that doesn’t mean there couldn’t be two losing Bismarck’s within a week! (simply that it’s highly unlikely). “The dice have no memory”, as the phrase goes. There may also be non-random fluctuations, for example horse-racing in February may throw up a higher proportion of losing favourites (due to weather disrupting training, and the new Flat season increasing uncertainty about the true ability of horses). Contrariwise, more settled periods of racing - April to July, say - may lead to a higher proportion of favourites winning. It is also possible that these (well known) variations may be already priced into the market.

The above calculations indicate the great potential profitability in this financial trading model. Given that losing Bismarck’s (i.e. five losing lays on the run) appear to be relatively rare (both theoretically and empirically), then the one final factor that the trader needs to examine is liability.

The whole basis of the Bismarck model of the ‘lay ladder’ is that there must be sufficient capital in the bank (and nerve on behalf of the novice to the system!) to persevere with increasing the bets as the ‘ladder is ascended’. In particular, it is clear that one should not start to ‘climb’ the ladder with a particular level of starting stake, without both the financial resources and the determination to persevere with increasing stakes as the ladder is ascended (and bets could rise quite sharply, if the horses tend to be at the upper end of the prices). The next section will explore liability levels.

Liability Calculations

As with the probability calculations, the model used will simplify the situation by assuming identical odds for the five horses selected to form the lay ladder.

The five races will be numbered 1 to 5, with LN representing the liability after race numbered N (i.e. for race 1, N=1, for race 2, N = 2 etc.).

The letter P is used to represent the profit that the ladder sets out to achieve. For race 1, this will simply represent the level staked, e.g. £20 at a particular set of ‘odds to lay’. If the horse loses, then the layer retains the stake, and the ladder is terminated at race 1. However if the horses wins, the layer has to pay the backer this stake multiplied by the odds that the bet was layed at. By ‘odds’ here are meant decimal odds (as used in Betting exchanges) minus one (as decimal odds, for the person backing the horse, give the total return, including original stake). Algebraically, we will represent these odds by the symbol O. So for decimal odds of 2.7 on Betfair, for example, O = 2.7 - 1 = 1.7.

Now we have defined the key symbols, we can represent the liability in terms of these symbols. The basic Bismarx formula for the liability after race N is deceptively simple:

LN = O ∑(LN-1 + LN-2 + .... L1) + O P

This just means that to work out the liability for a particular race, you add the overall profit (P) you are aiming to achieve with the ladder to the liability for all the previous races on the ladder . This is then multiplied by the ‘odds to lay’, O (decimal odds minus one), to give the liability for the current race.

For the first race on the ladder, there is obviously no previous liability (by definition). So L1 simply equals OP, or the odds for race 1 multiplied by P.

However, applying this simple-looking formula produces quite a complex outcome for race 5 (note difference from earlier versions of this paper, where I made an error in the calculation):

L5 = P(O5 + 4O4 + 6O3 + 4O2 + O)

Two key features of this equation. Firstly, the liability (as common sense suggests) is simply proportional to the original staking level P (which also represents the eventual profit of the lay ladder, at whichever point it is successful). So liability can be halved by simply halving the original stake, for example.

However, far less obvious is the extreme sensitivity of this equation to O, the odds level. Mathematically, this is indicated by the high indices involved (powers from squares right up to a fifth power, for a five leg lay ladder). If O is less than 1 (i.e. decimal odds of less than 2), then this essentially means the liability even after five races has only increased relatively slowly. However, the sting in the tail is that as O becomes greater than 1 (decimal odds greater than 2), then the liability level starts to increase dramatically, the more that O becomes greater than 1. This is indicated in the following table, where the odds this time are given as decimal odds as used on the betting exchanges (as these are what most of us are most familiar with). The liability for the fifth race of the lay ladder is given as a multiple of P, where P is the original staking level (representing the profit aimed at for each lay ladder). Note differences from earlier two versions of this paper, where the calculations were in error.
Decimal Odds
Liability as multiple of profit P for the fifth race of a lay ladder (rounded to nearest whole number)

Here is a slightly fuller table, which shows the liability (to two significant figures) for each stage in the ladder, as a multiple of profit P: 
Decimal Odds
Leg 1
Leg 2
Leg 3
Leg 4
Leg 5

It is obvious that as the odds layed at increase, liability will also increase. What is perhaps not so obvious is how as the decimal odds go above 2 the liability starts to increase quite dramatically, the more they get closer to 3. To put it in practical terms, if the profit aimed at for each Bismarck is £50, then the liability for the fifth race (where the first four horses on the ladder had all won) would be £300 if the horses were all at 1.7. However, if the horses were all layed at 3.0, then the liability would be over £8,000! If the trader had a betting capital of say £30,000, then even this might not be terminal (although the trader might question whether it is worth investing this much for what after all would only be a £50 profit at the end of the ladder). However, for a trader with much less than that available, either there would be insufficient capital to lay the bet at all, or the liability would be such a high proportion of the capital available that the trader could decide to terminate the ladder and accept the loss.

There is an obvious trade-off here, in that horses at higher odds are also significantly less likely to win, so it is much less likely that the fifth race would be reached. However, it does demonstrate the extreme sensitivity of the liability equation to the odds level. Important addendum: the discussion in the first part of this analysis indicated a possible anomaly in the figures (in that having only one losing Bismarck in a period of 400 or so days appears to be inconsistent with the 80% figure quoted for those Bismarck’s losing within the first two legs; according to the assumptions of this model, this figure should perhaps be closer to 90%). A possible explanation for this anomaly was that the variation in the horses selected may be playing a part in producing a greater proportion of winning Bismarck ladders than may have been expected than via the calculations here which assume all the horses are equal. It is a strong possibility if this turns out to be true that those Bismarck selections towards the higher end of the price range do in fact contribute particularly highly to bringing about successful lay ladders. So despite their much greater contribution to the liability, this would also explain why it is in fact advisable to include horses at the upper end of the odds range within the ladder. The original devising of the Bismarck strategy (and in particular, the vital question of choosing the odds limits of 1.7 to 2.7) presumably undertook extensive empirical investigation of the play-off between liability and strike rate. As always in science, empirical observations always take precedence over theoretical calculations! In terms of the Bismarx strategy itself, one might say, ‘if it ain’t broke, don’t fix it’. Its practical success in running speaks for itself.

On the basis of the small number of Bismarck selections I have been exposed to so far, although the actual level of individual suggested lays fluctuated markedly, according to what was available in the betting market that day, the average level of odds was typically somewhere around decimal 2.2 - it should be noted that even at that level, according to this simple model the liability after five races equals twenty-eight times the profit, indicating that the suggested betting bank might not quite be sufficient if the ladder went to the fifth leg. (It would be very interesting to know the number of times the fifth leg of the ladder has actually been reached in practise, over the 400+ Bismarck ladders run so far.)

Though, of course, most of the time the staking levels wouldn’t get anywhere near this point, as the ladder will terminate significantly before the fifth race. And it is important to remember that we are talking about relatively infrequent scenarios here. In addition, the great strength of a ‘lay ladder’ (as opposed to a win ladder) is that every other horse in the race apart from the horse you are laying is running on your behalf. While very long losing sequences when going after ‘winners’ are not that uncommon, the same while laying is very much less likely. So the trader using the Bismarx principle has the odds on their side. But this does indicate why a cut-off of 2.7 is indicated on the ‘strict’ Bismarx selection rules. Even if this is stretched slightly for one or two horses in the ladder, it would be important to balance this by having other horses in the ladder at significantly lower odds, to reduce the risk of the staking getting out of control. This is clearly something which is taken into account in the Bismarx selections I have seen over a number of days.

Apparently 80% of the time, the ‘ladder’ doesn’t get past the second horse; so why are these liability calculations important? Because for 20% of the time, the ladder does go beyond the second horse, and presumably some of the time, even reaches the fifth horse. The essential principle of the ‘Bismarck’ is to persevere with the ladder until there is a loser. If there is a lack of capital to put forward the liability for any race, then it is impossible to stake it on the betting exchanges. So the prospective Bismarck trader needs to have taken this fully on board, and have both the capital and the trust in the system to keep going, even at horse five. In principle, continuing onto horse six, on the very rare occasions when this is necessary. Alternatively, one could simply accept the lost bank capital of twenty points at such a position and restart the bank at a lower level with a new ladder. As was pointed out earlier, the extraordinary mathematics at the heart of the Bismarx lay ladder is that a losing sequence of five lays would only be expected to happen once in some hundreds of days trading this system; however, the bank renews itself every twenty days (assuming a profit level aimed at of 1/20th of the bank with each daily ladder).

The one cautionary point is that according to this simple model (i.e every horse at equal odds), only if average odds are around 2.0 or less for the five horses in the ladder, will a twenty point bank be adequate to fund the fifth leg on the ladder. Quite a few Bismarck ladders are somewhat higher than this, on the basis of the limited sample I have seen. The bold trader could stick to a twenty point bank, accepting the slightly higher risk the bank wouldn’t sustain a 5th leg Bismarck for the sake of greater returns. The more cautious trader could operate with smaller stakes, splitting the same bank into up to sixty points, if one wanted to be reasonably sure the bank could sustain a 5th leg Bismarck even at average odds up to 2.5 (on the basis of the sample I have seen, I expect very few Bismarcks would operate at average odds higher than this). The trade-off for the increased caution, of course, is a rather slower rate of growth of the bank. (It is also intriguing if the figure of only one ‘losing Bismarck’ to date also means that the ‘twenty point bank’ has always been sufficient. That would perhaps be slightly surprising (though pleasing!) in a sequence of 300+ lay ladders, according to the simple model presented here at any rate).


It should be noted that the mathematical models used for these calculations are simplified compared with the real situation. In particular, it is assumed that all horses have the identical odds and prices, which of course is quite different in practise, where there will be quite a range of values - an assumption which may be producing distorting results as far as the probability of an unsuccessful Bismarck is concerned (as discussed in relation to the ‘anomaly’ earlier). However, these calculations should still be indicative of the general trends at work, particularly in terms of liabilities involved..

In conclusion, the profitability of the Bismarck model has been clearly established empirically; this short essay has simply tried to explore some of the mathematics behind it in a little more detail. Clearly, none of this is necessary to operate the system in practise! (Though a good understanding of these principles must have gone into devising the system in the first place, whether or not it was mathematically formalised in the way done here). What these calculations do show are three things: firstly, just how profitable this system is, secondly, the importance of having sufficient capital (and psychological preparedness) to ‘follow through’ on the lay ladder, if the first few horses represent losing lays, and thirdly, the need for extreme caution in backing more than one or two horses in the ladder at levels above 2.7.

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