A new tool

I've recently come across Online LaTeX Editors (see for example here) which make publishing scientific notation on the internet quite easy. So I'm reproducing the previous post on the issue of "Multiple Selections and Sharpe Ratio", using a more readable format for the relevant equations....

Multiple Selections and Sharpe Ratio

Full or fractional Kelly staking has been in the foremost of betting staking plans. Using the available odds, and our estimate of the "fair" probability of an event happening, we can calculate what proportion of our bank should be staked in each bet. However, what happens if there are more than one, say n, concurrent selections? How much should one stake in total on the n selections, and how much on each of the selections?

What if we looked at the concept of expected reward-to-risk ratio? In finance, this is also known as Sharpe Ratio and it measures the excess return of an investment per unit of risk. Let's denote a bet with a letter i. The estimated probability of the bet winning is p_i while the available odds are o_i.

The bet would represent value if > . Now, if we invest one unit in this bet the expected profit becomes:

The standard deviation (a measure of the risk) of the bet's profit can be shown to be:

Denoting the expected profit and its standard deviation as and respectively, the Sharpe ratio for a unit stake in a single bet is given by:



When we have more than one selections, , and we invest units in bet i, the total expected profit is given by:



whereas its standard deviation (because of independence between bets) now becomes:



The Sharpe Ratio is given by:

This means that SR is now simply a function of the available odds, the estimated probabilities and the stake in each bet. We can therefore select those stakes which maximize SR with respect to some constraint e.g .

It turns out that the solution to this optimization problem is simply selecting stakes x_i proportional to or more specifically:



Although this may sound heavy, it's quite simple in practical terms. Consider 4 bets, available at odds of 2.05, 1.50, 5.00 and 2.50. Let the estimated probabilities for these bets be 60%, 68%, 22% and 45% respectively. If you follow the steps above you should get the following figures:

Note that with the calculated allocation we have achieved a reward-to-risk ratio of 0.256, which is higher than the ratio that we would have achieved had we invested all of our stake on Bet 1 (which had a Sharpe ratio of 0.229).

In a future post, I will argue how this approach can be especially useful when the multiple selections of bets include non-independent bets e.g. different bets on the same event.

Multiple Selections and Sharpe Ratio

Full or fractional Kelly staking has been in the foremost of betting staking plans. Using the available odds, and our estimate of the "fair" probability of an event happening, we can calculate what proportion of our bank should be staked in each bet. However, what happens if there are more than one, say n, concurrent selections? How much should one stake in total on the n selections, and how much on each of the selections?

What if we looked at the concept of expected reward-to-risk ratio? In finance, this is also known as Sharpe Ratio and it measures the excess return of an investment per unit of risk. Let's denote a bet with a letter i. The estimated probability of the bet winning is p_i while the available odds are o_i.

The bet would represent value if p_i > 1/o_i. Now, if we invest one unit in this bet the expected profit becomes:

E[Profit_i] = (p_i * o_i - 1)

The standard deviation (a measure of the risk) of the bet's profit can be shown to be:

SD[Profit_i] = o_i * Sqrt[ p_i * (1 - p_i) ]

Denoting the expected profit and its standard deviation as E_i and SD_i respectively, the Sharpe ratio for a unit stake in a single bet is given by:

SR_i = E_i / SD_i

When we have more than one selections, i = 1, 2, ..., n, and we invest x_i units in bet i, the total expected profit is given by:

ExpTotProf = x₁ * E₁ + x₂ * E₂ + ... + x_n * E_n = Sum(x_i * E_i)

whereas its standard deviation (because of independence between bets) now becomes:

SDTotProf = Sqrt ( (x₁ * SD₁)² + (x₂ * SD₂)² + ... + (x_n * SD_n)² )

The Sharpe Ratio is given by:

SR = ExpTotProf / SDTotProf

This means that SR is now simply a function of the available odds, the estimated probabilities and the stake in each bet. We can therefore select those stakes (x₁, x₂, ..., x_n) which maximize SR with respect to some constraint e.g Sum(x_i) = 1.

It turns out that the solution to this optimization problem is simply selecting stakes x_i proportional to E_i / SD_i² or more specifically:

x_{i =} (E_i / SD_i²) / Sum[ E_i / SD_i² ]

Although this may sound heavy, it's quite simple in practical terms. Consider 4 bets, available at odds of 2.05, 1.50, 5.00 and 2.50. Let the estimated probabilities for these bets be 60%, 68%, 22% and 45% respectively. If you follow the steps above you should get the following figures:

Note that with the calculated allocation we have achieved a reward-to-risk ratio of 0.256, which is higher than the ratio that we would have achieved had we invested all of our stake on Bet 1 (which had a Sharpe ratio of 0.229).

Change of course....

It's September already and I've realised that I've left this blog to die a slow and painful death. I'm pretty sure that everyone is wondering what happened to the system (and possibly myself) ... or maybe not! Anyway, work usually expands to fill available time, so there wasn't much time left to keep everything up-to-date.

The system resulted in significant losses (even though the yield was "only" -2.1%) as the following table shows:

I'm not going to post the remaining 200-or-so bets since the last update, but should anyone be interested enough, you can visit my thread at Punters' Lounge which also includes a small end-of-season analysis ...

What does the future hold? Well, I've been focusing on different modeling procedures, so I'll hopefully have some information on this in the not too distant future. Until then, take care!

The sorry state of affairs....

Well, it's been yet another hectic month with absolutely no update to the blog, and this has been a result of a couple of things: on the one hand, it's been a result of actually not getting any decent results (the demoralising factor), whereas it was the inevitable outcome of not having enough time for everything that needed my attention (the lack of time factor). Overall, these factors led to the current state of affairs.

Leaving decelopments in my personal life aside, it has been a yo-yo kind of month with the system fluctuating around the break-even point. In a discussion that I had with a fellow poster on Punters Lounge (where I have been updating my results for monitoring purposes), I accepted that I would have stopped posting the system's results had it not been for a promise I made to myself to follow the system through (on paper) until the end of the season.

Based on the above, and for anyone who keeps monitoring this blog (come back, I'll keep it updated, honest guv) the current state of play is as follows:

As you can see, after almost 900 bets the system is just about breaking even, which more than suggests to me that it's not finding betting value. I already have some ideas on what might be going wrong, and it's something that I will try to correct but this update will most probably take place during the summer. Of course, I'll keep updating it and monitoring its performance, since a promise is actually a promise...

System Selections 20-22/2/2009

Better late than never... The Friday evening matches have just started and in fact the first weekend pick is already going wrong, but a pick is a pick! Another twenty to go!

I am trying to work out what the problem with the system is, as it's failed spectacularly during the past couple of months... But until then, I'll keep monitoring it anyway. Fingers crossed.

System Update

It's been a long three weeks without a post! When you try to take too-many-things-at-the-same-time on, you are bound to fail unless you can cope with e 3-hour sleep every night... Anyway, between work, family and extra-curricular activities one would necessatily attract less attention, so it happened to be my blog. Apologies if anyone has been waiting for predicitions. I can say however, that I've probably saved you some money since the recent results have been hugely disappointing!

Anyway, I am not going to post all the selections up to now, but here is the latest system update:

As always, the record can be checked at Punter's Lounge where I've managed to post the system's selections in time. I'll (hopefully) be back tomorrow with the weekend predictions. Until then, take care!

System Selections 27-28/1/2009 (Results) and 30/1-1/2/2009

Mid-week results were somewhat better but there is still a lot of way to go before the system ends up profitable.

Weekend selections included a pick from Friday night, which however resulted in further loss. It's a good thing therefore that I hadn't posted the selections in time! Anyway, here are the rest of the picks:

System Selections 24/1/2009 (Results) and 27-28/1/2009

Another weekend has come and gone, and the system is registering further losses. In fact, it has returned to an overall loss, wiping out any early-season optisism. Last weekend's results:

Lots of midweek selections which will either dip the system further in trouble or recoup some of the recent losses:

Let's hope for some profit as the system performance is does not make for nice viewing:

System Selections 24/1/2009

Apologies for going missing for more than a week, but things got quite hectic in terms of work and life in general... You'll be relieved to know that you've managed to avoid quite a few losses by the system, though! I'll post a summary update tomorrow, after today's set of selections.

System Selections 13/1/2009 (Results)

Yeovil could not beat Scunthorpe unfortunately so it's another small loss for the system. Not to worry though as it's still in positive figures ... for now!

Until the next update, take care!

System Selections 9-12/1/2009 (Results) and 13/1/2009

Freezing conditions in the UK meant that a number of matches were postponed. As a result, the ammended list of picks follows:

The rescheduled Oldham - Hartlepool match has already been played, and even though I did not post it on the blog on time, I have included it in the results (even though it ended in a loss for the system!). An update of the system's performance will follow tomorrow, after tonight's pick.

System Selections 9-11/1/2009

Twenty selections for the weekend with quite a few longshot picks...

Good luck to anyone follow these picks! Come on ... the underdogs!

New Year's Progress Report

Mid-week games were all called off in the four divisions that the system selections are based on, so this is break presents itself as a good chance to look at a few performance indicators.

There have been a total of 601 selections, 229 of which were successful, resulting in a strike rate of 38.1%. The average strike odds were 2.74 leading to a nice positive yield of +4.2%. These figures are not too dissimilar with the corresponding figures obtained during the validation period (strike rate: 38.0%, average strike odds: 2.79 and yield: 6.0%), something which lends its weight to the assumption that the system actually will produce profit in the long run! Although these figures are very encouraging, one should take into consideration that based on the number of bets so far, the probability of ending in (any kind of) loss in the long run is still high (24.4%). This is a result of the (relatively) long odds that the system selections are placed on, and the subsequent less-than-2-in-5 strike rate. So, I would still urge caution to all those who think that this is the Holy Grail of football betting!

It is furthermore interesting to break the figures down by type of bet. As the tables below show, the system has been hindered by the Home selections which recorded a loss of 87.8 points (or -2.9% yield).

At the same time, Away picks performed admirably well, registering a yield of +8.6% and pulling the system well in profit. This was also helped by a surprisingly high (but probably unsustainable) level of yield from the relatively fewer Draw selections. As anyone who is involved in 1X2 prediction will tell you, prediction of Draws is notoriously difficult in football, and the high positive yield (even after just 51 bets) feels very rewarding especially given the fact that the model tackles the draw probability in a unique way.

It is also of interest to break down the system selections by division. Doing this has been an eye-opener for me: the system has been let down by League One and Two selections, which recorded -3.4% and -5.4% yields respectively, whereas Championship picks have registered only a marginal profit (+1.4% yield).

However, it is obvious that the overwhelming majority of the profit recorded was due to the excellent performance of the system over the Premiership selections. A profit of 356.8 units which translates to an impressive 27.0% yield cannot be easily discarded. These figures, it could be argued, that they are counter-intuitive as one would expect that the higher echelons of the football league structure are better priced in terms of odds rather than the lower leagues and therefore not as profitable. Only time will tell if this result has been a random event or a betting opportunity worth exploiting!

P.S. Watch the system crash in the next couple of weeks, having been given the kiss of death!

System Selections 28-30/12/2008 (Results) and 3/1/2009 (Results)

First of all, let me wish everyone a Happy New Year! Unfortunately, things got a bit busy over the festive period, so I have only just managed to update the blog. This meant that 3 bets over the last weekend were missed but, maybe it was for the better, as they registered small losses. The previous set of picks did not alter the system results in any significant way.

Here are the missed picks (for completeness sake):

Taking into account just over 600 bets, the current state of the system is as follows:

Apart from eating and drinking lots over the Christmas period, I got the chance to research a bit further the system picks. I'll try to put some results up in the next few days. Just a sneak preview, for those backing the picks with real money: the Premiership selections have registered a 27% yield!