Guest post by DOC from Punters Paradise.
Last week DOC began to look at whether you can make more profit from betting on singles or doubles and continues his analysis below.
“It appears that the risk rises fairly dramatically with edge at small values of edge”
It appears that the risk rises fairly dramatically with edge at small values of edge (i.e. from 1% to 3%), reaching a peak at about 5% and then falling off again. To understand why this is happening, we can look at the way in which Kelly stakes are calculated. The typical approximation (which we use here – a future article may look at the implications of this approximation) is that the stake size is given by the formula (edge)/(odds-1).
Our odds are fixed here (for this graph), so that the above reduces to edge/0.4 = (edge)(2.5). This means that our stake size is directly proportional to our edge, so therefore rises very fast at small values of edge – i.e. it doubles as we go from 1% edge to 2% edge, increases by a further 50% as we go to 3% edge etc.
Balanced against that is that as our edge increases we see more winners at any given level of odds. At odds of 1.4, with a 1% edge we expect to win 72%, at 4% edge it is 74%, at 10% edge it is 79% and so on. These two opposing effects give rise to the maximum (relative) risk in the region of 5% edge. We can see similar effects when we plot the equivalent graph for odds of 2.0.
Although this time the noise makes it a little more difficult to see the pattern. Below is the equivalent for odds of10.0:
It shows a similar trend although we don’t appear to reach a peak of maximum risk. The peak is moving in these cases because of the dependence of the stake size on the odds, i.e. stake=edge/(odds-1). Another point worth noting is that the maximum level of risk doesn’t vary that much across the different odds – showing the critical importance of adjusting your stake size for the odds involved.
“A more psychologically important threshold might be when over half of the initial bank is gone.”
Of course most of us with any betting experience wouldn’t be too perturbed if our betting bank dipped to this level – after all we started with 100 units and we still have 80, with time to recover. A more psychologically important threshold might be when over half of the initial bank is gone. I didn’t examine specifically for 50% of the bank in this study but the following graph is for the risk of dipping below 40% of bank.
Again we see the risk peaking for the short-odds betting around an edge of 5% – albeit the risk is low enough at about 7.5% – i.e. over 92% of the time we bet we won’t see our bank dip to these levels. For the odds of 2.0 the peak is at about 9% edge, while for the longer-odds, in this case 5.0, the peak appears to be at very large edge. Similar results can be seen for the risk of total bank loss below.
The results are more ‘noisy’ due to small amount of data – a result of only having 1000 simulations. We can extrapolate however from the totality of the data to say that similar conclusions to earlier may be drawn. The final graph we will look at in this article will compare risk to potential return directly. We can calculate the expected bank growth as follows:
- Total Wagered=Stake size*Number of bets
- Expected Profit=Total Wagered*%Edge
It is possible therefore to achieve the same expected profit level through a range of difference combinations of edge and odds. The following graph shows the risk of losing varying amounts of the bankroll versus expected profit.
“It is possible therefore to achieve the same expected profit level through a range of difference combinations of edge and odds.”
The most important conclusion that can be drawn from this (and the other) graph(s) is that the risk rises very rapidly with expected profit up to the point of bank doubling/trebling, before either levelling out or even falling.
The corollary of this is that the risk is very sensitive to small changes in stake size in this region, allowing risk to be substantially reduced for a small (relatively speaking) change in expected profit. Conversely, getting greedy here really ramps up your risk!
In the next part this article I’ll examine what happens when we use doubles rather than singles as our bet of choice.
DOC is the owner of the Punters Paradise forum. One of best forums online for serious bettors and those looking to learn how to make a profit, a lot of regulars have very strong statistical skills and there is a lot of conversation around these topics. I have had many of my questions answered there.