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Risk and Reward

Sports betting is all about taking risks to gain rewards. Intuitively this is perfectly obvious, and most punters are fully aware that one less chance of winning a bigger-priced bet. However, beyond a single bet few actually bother to give much consideration to the relationship between risk and reward, and specifically with regard to our ability to beat the bookmaker over the longer term.

In a previous article reviewing luck versus skill in sports betting I noted that it takes bigger betting at longer odds to demonstrate a statistically significant advantage over the bookmaker for an equivalent profit margin. Conversely, for an equivalent statistical advantage, betting at longer odds would return a bigger yield. This article takes a closer look at why this is the case, and the implications this has for the type of betting one might favour based on chosen risk-reward preferences.

To begin let's just consider a single 50-50 fair money wager, like tossing a coin. We know there is a 50% chance of being correct and a 50% chance of being incorrect; a 50% chance we will show a 100% yield after a win and a 50% chance of a -100% yield after a loss. What happens to these figures if instead we consider a 1-in-6 scenario, for example rolling a 6 in a dice? This time we have a 16.67% chance of a win delivering a 500% yield (5 unit profit from 1 unit stake) and an 83.33% chance of a -100% yield. It's obvious from this that we're taking a bigger risk to make a larger profit. We'll make that profit far less often, but when we do, it's much bigger than for an even-money proposition.

What happens if we have 2 bets of the same proportion? This time there are 4, not 2 possible outcomes: Win, Win; Win, Loss; Loss, Win and Loss, Loss. What happens to our risk-reward numbers? For the even-money coin tossing, each outcome has a 25% probability, with yields of 100%, 0%, 0% and -100% respectively. Thus we have a 50% chance of breaking even, a 25% chance of being 100% up and a 25% chance of being 100% down. By contrast, the probabilities for rolling a 6 twice is just 2.78% for a 500% yield, 27.78% for rolling one 6 (remember there are two ways of doing it, on the first or second roll) for a 200% yield and 69.44% for finishing with no 6s and a -100% yield. We can keep doing this analysis for an ever increasing number of consecutive wagers, but the pattern will be the same: the greater the potential profit, the less likely we will be to realise it.

What, one might now ask, has all this got to do with our ability to beat the bookmaker? Well, recall from luck versus skill in sports betting that I considered a betting record to be showing evidence of something other than chance if the probability that it would arise by chance was less than 1%. So suppose I manage to land 2 consecutive 5-1 winners. The (binomial) analysis above tells me that this could arise 2.78% of the time regardless of whether I had any skill at all in predicting outcomes. What I want to know is what combinations of betting odds and bet numbers I would need to get this figure down to below 1%. For this I can use Excel's BINOMDIST function. The outputs for a sequence of 100 bets are shown below. For example, betting at odds of 4/1 (5.00 in decimal) there is 1% chance that I will have at least 30 winners in 100 bets delivering a yield of at least 50%. By contrast, betting at 1/4 (1.25 in decimal) there is a 1% chance I will show at least 89 winners returning a yield of at least 11.25%.

Why is it the case that for an equivalent probability of occurrence, betting at longer odds will deliver higher yields? This is simply a consequence of the greater uncertainty in the betting propositions. Just remind yourself of the single-bet example discussed above. But if there is just the same chance hitting a 50% yield betting at 4/1 as there is hitting a much smaller 11.25% yield betting at 1/4 surely it makes sense just to forget the 1/4 propositions and bet at the longer prices? Not so. This is because there is a much greater chance that I'll finish with a much bigger loss betting at 4/1 than I would betting at 1/4. Because the population of possible betting returns are binomially distributed, there is more or less the same chance of showing a -50% yield as there is a +50% yield betting at 4/1 prices. Thus whilst, the potential win is greater betting at longer prices, the potential loss is too. Put simply, the more reward you want, the bigger the risk you have to take to get it.

Of course, there is an added disadvantage betting longer prices that is imposed by the bookmakers in the shape of the favourite-longshot bias. Unless one is diligent in seeking out the best market prices, longer odds will generally be of poorer value compared to their shorter counterparts, for reasons discussed in an earlier betting article. And unless one is betting exclusively at the betting exchanges, it is probable that even the best market prices will be weakly inefficient for the longest odds.

The table below compares the betting returns from 250 betting records verified by Sports-Tipsters between 2001 and 2012. With the exception of the very long betting prices yields improve as the odds lengthen. The fact that the over 50s performed so badly is likely to be largely a consequence of a strong market inefficiency (price bias). Nevertheless, in the main, the findings do confirm that provided you can find an edge over the bookmaker, the rewards can be greater for the lower probability betting propositions. Just make sure that you fully understand that if you seek out these greater rewards, there is an implicit greater risk in doing so.