Monday, 18 April 2011

Value, Kelly and ROI - Blog Post Of The Year Award

Patient Speculation has a blog post of the week 'award' (still awaiting the cheque for my winning entry incidentally), and today on Scott Ferguson's always interesting Sport Is Made For Betting is one of the finest posts I have read for a very long time.

Regular readers of this blog know that I am always on about the necessity of value, and the merits of the Kelly Criterion, and try to give examples to illustrate the points, usually limited to the somewhat tired coin-toss, but this post titled Value Doesn't Pay The Bills - included in full below - offers by far one the clearest and easy to understand explanations of why you need value that I have seen. If this doesn't win Blog Post Of The Week next week, I shall be amazed.

An extremely rare guest post on my blog, from a former bookmaking colleague who has been a pro punter now for several years, Vasu Shan. You can follow him on Twitter @vasman60: “Value doesn’t pay the bills”. I’ve always hated that expression. What does it mean? Surely, it’s simply the ramblings of an unsuccessful punter. Is that old chestnut about finding winners being preferable to finding value rearing its ugly head? Of course, none of us can predict the outcome of an event with any certainty, nothing is “nailed on”, it’s not possible to “buy money”. We’re not in the business of prediction, we’re in the business of probability. A successful punter backs both winners and losers, ensuring that the rewards from the winners outweigh the cost of the losers. Value is what it’s all about, isn’t it?? If you need a winner so badly in order to ensure your electricity doesn’t get cut off, you probably shouldn’t be betting at all. If you need a bit of luck on the punt to pay for dinner tonight, you clearly don’t understand the long-term nature of betting. And if the value simply doesn’t seem to be paying, even in the long-term, well, likely you’re not actually getting value at all!

“Value doesn’t pay the bills”? Nonsense!

So, we’re agreed, we want value from our bets. We want 6/4 about the toss of a coin. We want Even-money about Man Utd at home to some average Premier League side next weekend. We want 33/1 that Tiger Woods wins the Open Championship. Yep, value. Maybe the above are too good to be true, but some kind of value.

And we want as much value as possible. 2/1 about the toss of a coin, even better.

Let’s crunch a few numbers. 6/4 about an Even-money chance represents 25% of value (if I spent my whole career betting Even-money chances at 6/4, my Return on Investment, or ROI, would be 25%). 2/1 about an Even-money chance is better, with Expected ROI on such a bet 50%. Betting 50/1 about a 25/1 chance is a stonking 96% of value. And it gets better, a career betting 1,000,000,000/1 about a 1,000,000/1 chance would yield an Expected ROI of 99,900%. But hold on, back in the real world, are we really saying that betting million-to-1 chances at a billion-to-1 is the gold-paved path to betting glory? After all, even after a consistent outlay, there’s every possibility that you’ll never see a return in your lifetime. Such bets may be astonishing “value”, but in terms of tangible reward are arguably not great bets at all.

Let’s get back to basics. If I offered you the following sets of annual gross profit figures, which would you prefer, as a punter? £200,000 (@ 16%ROI), or £400,000 (@ 4%ROI). Those who would prefer the latter are what economists refer to as “rational”. Winning as much as possible, as quickly as possible, is of course the ultimate betting objective. We want to maximise our Expected Wealth. Which doesn’t mean maximising Expected ROI (aka Value).

Incidentally, when we talk about maximising Expected Wealth, be sure not to confuse that with maximising Expected Profit. If you want to maximise Expected Profit, you should bet your entire bankroll on the next value selection that you come across. A strategy of betting your life savings on an Even-money chance at 11/10 (or any price for that matter) will inevitably end in disaster. Of course, a no-betting strategy will show zero profit. There is, however, an optimal stake between all and nothing...

The Kelly Criterion dictates that you should try to maximise the Expected Rate of Bank Growth rather than trying to maximise the Expected Bankroll itself. Google it for historical details. Dust off those A-Level maths books... in order to ensure Expected Rate of Bank Growth is maximised, our logarithmic utility function is differentiated and set to zero. Google again for details of the boring calculus, but the result is a formula that is able to recommend an optimal stake for each betting opportunity that is a function of the size of your bankroll and the value that you’re getting, with respect to the price that you’re taking.

The uses for Kelly don’t end there. Ever wondered what you should do if you’ve backed a 500/1 Superbowl finalist, with the 50/50 big match now priced at 10/11 pick’em. Though I suspected some economic hedging argument, the absence of a mathematical justification led me to believe that with no remaining value, simply holding the position was the right thing to do. I was wrong. There is an optimal hedge that will see the expected rate of your bank growth maximised, even though you may be accepting negative value on that hedge. Or you’re on Benfica @ 20/1 for the Europa League, and now you make them short favs in the final, and 2/1 is available. How much should you optimally press, if at all? Kelly can help you calculate how best to trade most effectively, based on how much you’ve already risked, the size of your bank and your level of risk aversity. Overstaking, re-assessment, market moves, arbitrage? Ask Kelly. Accept no substitutes.

As the number of bets you make increases, the chance that Kelly betting will beat other systems approaches 100%. The Kelly Criterion is endorsed by no-less-an-investor as Warren Buffett, but it should be noted that while it is economically and mathematically sound, it is not to everyone’s tastes. A common complaint is that it is far too aggressive. Given that we each have a natural level of risk aversity, and knowing that we generally deal with odds estimations with a margin for error (rather than “true” odds), this seems to be a fair grievance. A fraction of Kelly is therefore recommended (30% to 50% seems to be a “happy medium” within the professional punting community), the important thing is that a suitable ratio between your bigger bets and smaller bets remains. Full-Kelly staking should be optimal, but it is certainly volatile. As you reduce your Kelly%, your bank will tend to grow less rapidly, though less volatility will be attached.

The fact is even the best odds compilers in the world would unknowingly be on the road of bankruptcy if they were to mismanage their money, by consistently overstaking. On the other hand, understaking would see a failure to fulfil punting potential (a preferable failure, admittedly).

In an earlier example, we pointed out that a career betting Even-money chances at 6/4 would yield 25% ROI, and a career betting 25/1 chances at 50/1, 96% ROI. However, more significantly, at optimal stakes, the former would on average grow your bank by 2.06% per bet, while the latter would only have an expectation of 0.73% bank growth per bet. And if you’re making a large number of bets, cumulatively, that differential can turn out to be enormous. It should be clear which is the truly productive cash cow and which is merely the efficient red herring.

Those who’ve spent a lifetime maximising ROI, I guess you’ve now realised that in a punting context, those who are able to grow their bank balance more significantly are, by most people’s definition, the more successful.

So, in summary...Return on Investment for show, Rate of Bank Growth for dough. £, not %. As I’ve always said, “Value doesn’t pay the bills”.


DVWooly said...

i agree, its all about how mcuh cash you actually make.

Average Guy said...

Silly question from one who operates on a plane much lower than you guys, can a staking plan a la Kelly be applied to strict trading. Is there a way to calculate the optimum stake size and entry/exit prices for optimum bank growth ? You guys obviously have a much greater understanding of mathematics than I have and I was wondering how to apply it? ps I also posted this query on Scotts blog to maximise chances of a solution, don't be offended.

Sports Betting said...

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Sports Betting said...

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