**How high should be a starting bank?
Is 5,000 units enough?**

Well, there is no standard answer to this question. It all depends on the individual strategy.

Image: Sergey Novikov (Shutterstock)

However, what is possible, is to calculate bank fluctuations *(i.e. winning and losing sequences)*.

With the help of knowing the best and worst case scenarios you can determine the ideal starting bank for any betting system of your choice.

At the end of the article you will find a few useful exercises to practise, with the solutions available as a free download to all of you who would like them.

It stands to reason that the smaller the probability of an event occurring *(i.e. higher odds)*, the longer the likely losing streak will be *(in between winning bets)*.

However, the big question is how often and for how long will the losing (and winning) streaks transpire?

It is possible to mathematically calculate many things with statistics, including streaks of luck and bad luck. However, it is important to note that no matter how accurate the results may appear, they are ‘models’ *(a formal representation of a theory)*.

In this article, we are talking about probabilities; what can we ‘predict’ about how things may develop in the future. Please bear in mind that any such hypothesis is always a “could happen” not a “will happen”.

Of course, the larger the sample size *(i.e. number of bets)*, the more likely the prediction is to be correct. But apart from the bookmakers themselves, who else has a betting portfolio comprising thousands of bets every weekend?

The longest expected losing streak *(or winning streak)* can be calculated using the following formula:

**n** = number of trials *(i.e. total number of bets)*

**ln** = natural logarithm^{1}

**P** = probability^{2}

**| .. |** = absolute value or ‘modulus’

^{1}*Suffice to say, explaining what natural logarithm is would be worthy of a series of articles. For the time being, use Excel to calculate this for you.*

^{2}*For winning streak calculations use the positive value (i.e. the probability of winning). For losing streak calculations use the negative probability value. For example, if the probability to win the bet is 33% then the probability that the bet loses (negative probability) is 67%.*

In practice, the formula is best applied to situations where you constantly bet repeatedly on the same probability, for example, on ‘red’ at the roulette wheel: its probability remains exactly the same with every new spin of the wheel.

For football betting the concept is much more difficult to apply as each bet is likely to have a different probability (e.g. one Over 2.5 Goals bet with a 55.3% chance, and the next with a 62.1% chance, etc.).

However, you can group bets in probability clusters – for example, bets with a 55%-60% expected hit rate, bets with a 60%-65% expected hit rate, and so on.

Longest Winning and Losing Streaks, depending on the number of bets (Examples for 50, 500 and 1,000 bets shown)

The tables above show the calculations of the expected maximum number of winning and losing streaks, depending on the expected hit rate *(probability of the bet to win)*.

To read the tables, let’s explain the 70% line *(odds in the region 1.4 and 1.45)*; in other words, bets with a 7 in 10 chance of winning.

The table on the left calculates the expectations of 50 tries *(50 bets in a row, one after the next)*. You can see that the player will experience at least one streak of three lost bets in a row somewhere in the sequence.

On the other hand, he can expect at least one series of 11 winning bets in a row during the same sequence of 50 bets.

In contrast look at the 30% line *(odds in the region of 3.2 to 3.4)*. In a series of 50 bets the bettor must expect at least one sequence of 11 consecutive losing bets, but will probably see only one set of three consecutive winning bets.

To develop a sense of probabilities and sequences, you can experiment with a dice. It has six faces; in other words, a probability of 16.67% (1 in 6 chance) of successfully landing on a chosen number.

Choose a number and count the number of throws until you succeed to roll it. Count also the number of consecutive successful rolls.

Choose two numbers that you do not want to roll (e.g. 5 and 6).

This means you have a 66.67% chance that one of the remaining four numbers is rolled.

In football betting terms, this equates to wagering on something like the full-time ‘Under 3.5 Goals’ market at odds of 1.50. (This experiment is just a little faster than waiting for 50 games to finish!)

Take a pen and paper and record 100 throws of the dice. If one of your four chosen numbers arrives mark a 1 on your paper; if the 5 or 6 are thrown, mark a 0. Count the number of winning and losing streaks you experience.

What is the maximum number of winning and losing streaks you experience in a sample size of 100 throws (bets)?

*Having learned how to calculate the expected length of winning and losing streaks, the next question to ask is: *

**How many bets is it likely to take before I encounter ‘X’ losses in a row?**

This formula is actually very simple:

= **1** divided by **P**, **to the power of** **a**

**P** = probability (expected hit rate or loss rate)

**a** = number of won or lost bets in a row

In the tables below you can see how many attempts (bets) it needs to experience a specific, expected length of luck or bad luck. Again, the assumption is that the bettor bets all the time on the same probability:

Expected time of occurrence of winning and losing streaks, depending on the hit rate

Looking firstly in the right-hand column at the Losing Sequences, if the expected hit rate is 45% *(what you should ‘expect’ at odds of around 2.2)*, then it is likely that you will experience a sequence of three losing bets in a row by the time your sixth bet is settled.

After 20 such bets it is likely that you will have seen a losing streak as long as five bets in a row.

Looking at the Winning Sequences column: you will win three times in a row at some stage during a series of 11 bets.

However, winning five in a row may only be seen once in every 54 bets.

As we mentioned before, in football betting it is extremely difficult, if not impossible, to find bets, all with the same probability of success.

However, you should at least try to understand the theory behind winning and losing streaks, as it will be **easier on your nerves** when you do encounter the inevitable run of bad fortune.

In particular, a thorough understanding of losing streaks is of enormous importance when setting both the size of your starting bank and stakes per bet.

**Example:**

A bettor prefers bets within the odds range of 2.0 to 2.5 with a hit rate between 40% and 50%. He plans to place 50 bets *(e.g. two bets per round on 25 rounds of matches)*.

After looking at the tables, he knows that the maximum losing sequence expected is likely to be as long as six to eight lost bets in a row. Therefore, he knows that there may be at least one sequence of three or four consecutive rounds *(weekends)* when all bets lose.

After every 5^{th} to 8^{th} bet, he is also aware that he is likely to experience a loss of three consecutive lost bets *(e.g. one weekend loses both bets, the following weekend only one loses)*.

He also knows that every 13 to 32 bets there will even be a streak of five losing bets in a row.

The bettor is fully aware that he has to take this into consideration and plan the starting bank accordingly to be able to ‘sit through’ these losing streaks.

Of course, he also knows that winning sequences will arrive too. In his case, with some ‘luck’, he may experience a winning sequence of five bets in a row after 32 bets. Every eight to 16 bets he will have a ‘lucky’ streak of three wins in a row.

This is certainly quite a fluctuation. When these ‘bad luck’ and ‘good luck’ streaks actually happen, nobody knows. However, what we do know is: They will happen!

A starting bank should be approximately five times the maximum expected losing streak. The reason for this is that a losing streak can happen right at the beginning, immediately followed by another bad run of luck. We are talking statistics here!

So if a bettor wants to stake 10 units per bet, the starting bank must be nine times (expected losing streak) the stake of 10 units multiplied by five = 450 units. Then he can risk 2.2% of his bank each time he bets (10 divided by 450). If losing, the stakes will remain constant at 2.2% and, if winning, raised gradually.

**Questions to ask before setting the starting bank:**

- What hit rate is expected
*(probability to win the bets)*? - How many bets are planned for the season?
- How long will the longest losing streak be?
- What is the desired stake per bet?

**Calculation of the starting bank:**

- A bettor pursues a strategy with a win probability of 60% per bet
*(e.g. Under 3.5 Goals)*. He places one bet after the other; in other words, he waits for the outcome of each bet before placing the next. In total he places 50 bets.What is the longest ‘losing streak’ (bad luck) that he can expect? How long is the longest ‘winning streak’ (luck) that can be expected?

- Same example as in (1): A strategy with a probability of 60% per bet; placing one bet after the other.
This time our punter is hoping for a ‘winning streak’ (luck) of 5 consecutive wins. How often does that happen?

- A gambler pursues a strategy with a probability of 20% per bet
*(e.g. ‘betting on the underdog’)*. Again, he places one bet after the other.With a total of 500 bets, how long is the longest ‘losing streak’ that he must expect? After how many bets can he expect the longest ‘winning streak’?

- Same example as in (3): Strategy with a probability of 20% per bet; placing one bet after another
The bettor was hoping for a ‘winning streak’ (good luck) of five consecutive wins. How often does that happen? After which bet number should he expect ‘bad luck’ of five consecutive losses?

- Following the above two strategies
*(one with a 60% chance to win, the other with 20%)*our bettor stakes 10 units per bet.How high should the starting bank be for the 60% strategy, and how much for the 20% strategy?

*Note: The initial bank should be approximately five times the maximum losing streak based on a total of 500 bets placed.*

*Just click on the button above and click on “Proceed to checkout” button in the new tab, then enter your name and e-mail address. Our automatic service will then deliver the file to you via e-mail, free of charge. The size of the PDF file is 320KB.*

The factor **5** used in this article to determine the betting bank is a risk variable for risk-averse bettors. It is also the factor advisable for strategies with a 45% to 55% win probability *(odds between 1.8 and 2.2)*.

Here is another article: **How to Calculate Losing Streaks & Optimal Bankroll** in which we provide a more detailed account of setting the ideal starting bank.

Risk management in sports betting is the foundation stone upon which all of your betting transactions should be built.

Risk management encompasses risk assessment, risk control and capital requirements, all of which cannot be addressed until you understand how winning and losing streaks are likely to impact upon your starting bank.

]]>Bankroll management is one of the most important pillars for success in sports betting.

Image: Alex Roz

A portfolio of sports bets placed over time can be compared to investing in the money markets on a portfolio of stocks and shares.

Indeed, the term ‘bankroll management’ comes from the financial sector and describes the use of the *seed capital* (i.e. in betting terminology, the *initial stake*).

Bankroll is the ‘starting bank’, and the intention is to manage it and increase it at the same time.

Bankroll management therefore deals with how to properly manage your starting bank.

**The good news**: It is actually possible to calculate the required starting bank mathematically.

**The bad news**: The calculations are naturally dependent upon statistics, and the ‘significance’ of the results relies on the amount of data used.

For example, any strategy based on one German Bundesliga team’s home games during a season produces a sample of precisely 17 sets of data, which is a very small number, statistically speaking.

The **Law of Large Numbers** is omnipresent so far as statistical accuracy is concerned: The larger the data sample, the more accurate the final results are likely to be, although a line has to be drawn between sample size and an acceptable level of error.

One way of coping with small data sets is to incorporate a **risk discount** into the equation. More about this later…

On face value, you might assume that calculating the necessary starting bank for a betting strategy can be derived solely from the stake multiplied by the number of bets (n).

With the 17 matches from our example above, and a constant stake of 100 units per bet, the bank would then be: 100 x 17 = 1700 units. But is this maximal amount *really* needed?

Although this may be true where returns from winning bets cannot be immediately re-invested, such a bank can never be optimal because an inordinate amount of capital would be tied-up.

What you should look for is the most cost-effective bankroll where all the money you have at your disposal is working for you as efficiently as possible.

Optimal bankroll is characterized by two things:

- Cash holdings (i.e. money in reserve) is kept as low as possible
**Gambler’s ruin**is avoided

There are five vital criteria you will need to establish:

- What is the size of your stake per bet?
- How many bets does your strategy expect to be placed?
- What is the expected hit rate of your strategy?
- What is its expected longest losing streak?
- Determine the risk variables and incorporate a ‘risk coefficient’.

Okay, we will stick with the German Bundesliga for demonstration purposes and use a system gleaned from its latest full-time 1×2 **HDAFU Simulation Table**.

If you have already bought this table, you can see the full and detailed analysis of **backing the underdog whenever Hamburg plays at home**: This strategy has realised a yield in excess of 58% over the course of five complete seasons from 2010-11 to 2014-15.

In addition, there has been profit produced in every one of those same five seasons.

It’s an ideal candidate for incorporating into a large portfolio of other systems. (When we say ‘large’ we mean a portfolio that will generate at least 500 bets in a season.)

**(1) Size of Stake per Bet:**

This is determined by your own liquidity, and to keep this calculation simple, a **Constant Stake** (CS) of 100 units per bet will be used.

**(2) Number of Bets:**

For this mini portfolio of Hamburg home games, the **Number of Bets** (n) is 17 for the new season.

**(3) Hit Rate:**

The HDAFU Simulation Table reveals that from 85 Hamburg home games over five seasons, 32 underdogs triumphed: a **Hit Rate** of 38%.

The random selection of only 85 matches is a relatively small sample and the possibility of ‘random sample error’ is therefore relatively large.

To compensate, it is worth applying what is known as a ‘risk discount’ to reduce the actual hit rate experienced and to build-in an extra level of security if statistical expectations for the new season are not realised.

Taking a risk discount figure of 5%, the expected hit rate becomes: 38% – 5% = 33%.

[Have a look at **this article** for more information about hit rates].

**(4) Longest Losing Streak Expected (LLSe):**

The longest expected losing streak *(or winning streak)* can be calculated using the following formula:

**n** = number of trials *(i.e. total number of bets)*

**ln** = natural logarithm*

**P** = (negative) probability^{†}

**| .. |** = absolute value or ‘modulus’ *(see Wikipedia if you would like to know more about these mathematical symbols)*

**Suffice to say, explaining what natural logarithm is would be worthy of a series of articles. For the time being, use Excel to calculate this for you: to make life easy, the formulas to use are included in the free spreadsheet download below*.

^{†}*For this calculation, the negative probability or hit rate is used. In this case, having adjusted our hit rate down to 33% using a risk discount, the probability that the bet loses (negative probability) is 67%*.

rounded down to

From a pool of 17 bets, you can therefore statistically expect that a maximum of seven in a row may be lost without winning one in between.

**(5) Risk Coefficient (RC):**

The determination of risk variables depends primarily on your **risk aversion**. Risk-averse bettors choose a high coefficient figure (e.g. 5), whilst gamblers who are happier taking risks choose lower coefficients (e.g. 2).

But why are we including a risk coefficient at all?

We can assume that the longest expected losing streak *(in our example, seven lost bets in a row)*, may already start with the first bet.

Although one bet may win after that, with the gains reimbursing the loss and allowing for reinvestment, there can still be a second stroke of bad luck directly after the first bet that you have won.

Neither winning bets nor losing bets ever line up in a uniform manner; they will always appear in a random pattern, so always better to be safe than sorry.

The formula is:

Our Bundesliga example is an underdog backing system, which by its very nature, is risky. However, as there are only a maximum of 17 bets in this mini system, we will choose a risk coefficient of 1.5: we are happy to take the risks!

It is not very likely that there will be two losing streaks of seven games in a row when betting 17 consecutive times. However, we are aware that it may be quite challenging for the nerves to sit through losing streaks watching the bank balance reduce before your eyes!

The optimal bankroll required to run this system for a season is as follows:

If you remember the sub-optimal bank strategy at the beginning of the article where we touched on a bankroll of 1,700 units (100 units per bet x 17), you can see we have now released 650 units for investing in another strategy elsewhere.

With this **free Excel table download**, you can easily and quickly discover what the longest losing streaks are for your own strategies. Just enter your stake, number of bets, and risk coefficient figures and let it calculate everything for you!

**>>> Excel Workbook – Losing Streaks <<<**

*Click on the above button – in the new tab click on the ‘Continue Checkout’ button. Enter your name and email address to allow our automatic shopping cart to deliver the file by email to you, free of charge. The .xls file size is 93 KB. When you receive your confirmation email, just click on ‘View Purchase Online’ (in the email text) to download the file.*

Many people are able to predict how things will develop in their personal lives over a period of, say, the next 12 months. For example, job prospects or personal events.

Image: Kiselev Andrey Valerevich (Shutterstock)

If you take the time to write down all the expected events planned or envisaged in your life over the next 12 months, you will probably realise in a year’s time that most of them have actually happened *and* more or less as expected.

Perhaps you started with things that are set in stone, the fact that you already have two holidays planned abroad, and more importantly (or not, as the case may be!) that Aunt Emma will be celebrating her 90th Birthday with a small party at your house.

You are kind of expecting a mediocre school report from your son, and you have the gut feeling that next year is finally the one where you are promoted to ‘Head of Department’ at work…

A year on and time for your personal review.

Surprisingly, each of your predictions happened, more or less, but the details turned out to be very, very different from your expectations!

The vacation plans went a bit awry and instead of the usual lazy, beach holidays you went to the mountains.

Aunt Emma made it to her birthday but the party was cancelled, replaced by an even larger and unexpected family gathering at a local hotel.

Your son’s school results were as expected but instead of the usual low maths mark, he made the top half of the class and slipped down the order in sports instead.

You are indeed now Head of Department, exactly as expected, but in a different company.

What we’re trying to say is that although overall trends can be predicted with a fine degree of accuracy, it is almost impossible to make an accurate prediction about a single event which is just one small part of the whole chain.

Another example, from the world of sales.

Once a year, professional companies plan budgets by reviewing their existing client portfolios estimating renewals and/or future sales.

Any professional can predict the increase in the number of customers by ‘X’%. However, it is impossible to say which customers will remain faithful, or how many new clients will be converted.

The only certainty is that after a year there will be ‘X’% more sales. 40% of existing customers will increase their purchases, but which ones will is anyone’s guess.

Of all the budget predictions for the forthcoming year, most will be correct but some will probably have disappeared into thin air to be replaced by positive events that were totally unexpected.

Otherwise, the final overall balance turns out pretty much as predicted.

Before the 2011-12 season our statistics suggested Bayern Munich would have just one or two 0-0 results in their home fixtures in the Bundesliga, but obviously nobody could tell beforehand which games would end goal-less.

Their campaign concluded with just one 0-0 result at home, on 14.04.2012 against Mainz.

The same prediction also applied for one goal matches in Bayern home matches.

There were two such games, 07.08.2011 against Mönchengladbach (0-1) and 19.11.2011 against Dortmund (0-1).

The statistics predicted Bayern would have between two and four home matches with less than 1.5 goals.

During 2011-12 they played three such matches.

]]>*“Companies can go bust. Then the entire share trading system, which is based on historical analyses, becomes totally senseless. On the way to insolvency there is much chaos in a company. Perhaps there are some statistically relevant signs for causes of these failures?*

*The potential of insolvency differentiates football betting from the share market. Football teams rarely disappear.”*

This comment made me really think as I have not viewed football betting from this perspective before.

I write a lot about **value betting and statistical analysis** of football matches, explaining that it is not only possible to analyse football game results but also to identify statistical trends and make profitable value betting decisions.

The main thread of this blog is about how to develop relative reliable prognoses and calculate probabilities to gain a mathematical advantage or ‘edge’ and thereby win money from betting on a long-term basis.

The thought that football betting may be better insulated from bankruptcy than trading in stock market shares had never occurred to me. Thank you, **Ginger Tom**!

Image: Alex Mit (Shutterstock)

Everyone who has had anything to do with share trading since stocks have been traded (and probably everyone else in touch with the news) has certainly seen common headlines such as the following more than once during their lifetimes.

They are always similar, having more or less the same content, with just the country or company names changing from story to story:

- FTSE falls on fears over Spain’s ‘septic’ banks
- Disappointing China GDP sees shares dipping
- Eurozone crisis live: Another Friday the 13th horror story?
- European stocks drop at start of trading
- US STOCKS – Early rally fades as Nasdaq drops sharply

…the above headlines are just a few from last week!

And there are more serious headlines as well:

- Shares in Royal Dutch Shell fell more than 4pc today after an oil ‘sheen’ was spotted
- Rio Tinto Q1 iron ore, copper miss forecasts, shares drop
- Nokia shares dropped over 16 per cent on the Helsinki Stock exchange today

When I read things like this, I am genuinely glad that my funds are tied-up in the much more straight-forward topic of football betting and that I do not burn my fingers in the share markets.

Now, some people may argue that sports betting of any type is just a ‘zero-sum’ game; someone has to lose in order for someone to win.

However, my argument with businesses is that unless shares are issued to finance or float an enterprise (capitalisation), (nearly) every other financial form of trading, especially shares, is also a ‘zero-sum’ game.

Indeed, this ‘paper trading’ has little effect on the day-to-day running of the company and, is itself, no more than pure speculation (i.e. betting).

Furthermore, in the stock market the information available to the public is limited to news which the companies decide to communicate or are legally obliged to reveal.

Companies traded on stock exchanges are usually large enterprises, perhaps internationally active, with a multitude of internal issues, none of which the shareholders know about or have any influence over.

Indeed, when bad news is learned, it is usually too late to act.

]]>In the stock market there is a rule of thumb: Buy shares when the stock chart heads in a direction from bottom left to top right, and don’t burn your fingers when the curve is falling from top left to bottom right.

Whether to buy stock or invest in a fund, an investor usually always employs some sort of technical analysis, in other words, charts, graphs, historical financial statements, profit/losses from recent years, growth forecasts, annual management reports, etc. Indeed, anything that can somehow be expressed in numbers.

Image: wrangler (Shutterstock)

It is probably realistic to assume that the majority of shareholders have never heard the names or know the functions of the individual board members of the companies in which they hold shares.

Investors are not in the least bit interested whether any of these major players is in good or bad health or whether they are enjoying a particularly high level of personal wealth.

No-one cares whether board member ‘A’ lit the fuse and proposed a significant change in the group first, or whether it was actually board member ‘B’, or even the insignificant and always overlooked employee, ‘C’.

The only things of any interest to investors are:

- What effect will particular business decisions have on the future price of the stock?
- Is the share correctly valued?
- Will my portfolio of stocks and shares remain stable?
- Is the share price increasing or dropping?
- When and how much dividend will I be paid?

How many share investors have ever given a thought to the thousands of employees within an organisation, or the state of individual contracts within the businesses they hold stocks from, or the growth and marketing plans, internal documents and policies, accounting procedures, department structures, intrigues and insider plots?

The list of variables is endless!

Businesses traded on the stock market usually have thousands of employees and business partners all around the world; affairs could go wrong a million times, but investors don’t pay much attention to these angles.

In any case, hopefully companies keep their strife, intrigues and internal affairs to themselves and at the end of the day, only the bottom line matters for them and their shareholders, and nothing more.

Large corporations are merely ‘business vehicles’, just the same as football clubs.

I am sorry to all you football fans out there but let’s face it, from Nokia you buy mobile ‘phones, from Manchester United, you get football.

The only difference being that many footballers have a higher salary than a Nokia board member, which is simply due to the fact that a football club business has a relatively high turnover compared to its number of core staff.

As Nokia will not change its business direction overnight, so Manchester United will certainly not become a different team to the one they were last week.

Nokia has been building mobile communications equipment for years and one can probably expect that this will be the same in a year’s time, whilst United have been winning title after title and are the most successful club in English domestic football history.

Nokia builds ‘phones regardless of who is their Chief Executive Officer, and United continue to win matches whether or not Robin van Persie, Wayne Rooney, or David de Gea play.

The most important thing in common is that both organisations have a winning formula/mentality in their respective business sectors.

But when it comes to buying-in to a corporate business or a football club as an investor or fan, in the stock market, technical analysis forms the basis for purchasing decisions.

The **brain of a football supporter**, however, refuses even to accept an analysis!