Bankroll Allocation Across Multiple Betting Models

[DeltaLover]

A betting model can be characterized by two fundamental metrics: Its betting frequency (BF) and its ROI.

For simplicity I am assuming we only are betting to win. We can also extend this in dutching over all the other contenders if we need more pool but for now let's only consider one bet...

These two metrics can be combined to a single one which is the expected PNL of the model given a universe of races.

BF and ROI are inversely proportional. The optimal strategy is the one which has the highest PNL which is not necessarily the one with the largest ROI.

This is the reason why models with very low BF presenting very high ROI are inferior to models with lower ROI but much higher BF.

I think this concept is pretty clear and hardly anyone could disagree with it.

What can become a topic of discussion though is the following:


Lets' assume that the best model We can find is one a BF of 35% which presents a 1.20 ROI which can also be expressed as a PNL of $89840.0 for a flat bet of $100.

The event universe consists of 11,098 races out of which 3,893 are the proposed bets...

Lets' also assume that we have another model which out of the remaining 7,205 has the same BF of 35% giving us 2,521 bets but only with 1.05 ROI and let's also assume that its PNL is now only 12,605

We also have a bankroll that allows us based in our risk management plan to escalate our bets up to let's say $2,000 maximum based in the pool size and the price offered. Following such an approach both strategies will present lower ROI although the expected PNL will be reaching its maximum potential.

The questions we need to answer given the described models are the following:

1) What should be the bankroll proportion allocated to each model?

2) How big should we allow our bets to grow in order to achieve maximum composite PNL

Note that for #2 we cannot use KELLY because:
- We do not have a probability for each individual event
- There is a maximum bet imposed by the pool size itself

3) Can we describe it as a linear programming problem?

4) Does the absolute size of the bankroll makes a difference in our approach? Meaning that we should follow a different strategy if our bankroll is smaller that X or larger than Y?

[Robert Fischer]

A betting model can be characterized by two fundamental metrics: Its betting frequency (BF) and its ROI.

For simplicity I am assuming we only are betting to win. We can also extend this in dutching over all the other contenders if we need more pool but for now let's only consider one bet...

These two metrics can be combined to a single one which is the expected PNL of the model given a universe of races.

BF and ROI are inversely proportional. The optimal strategy is the one which has the highest PNL which is not necessarily the one with the largest ROI.

This is the reason why models with very low BF presenting very high ROI are inferior to models with lower ROI but much higher BF.

I think this concept is pretty clear and hardly anyone could disagree with it.

What can become a topic of discussion though is the following:


Lets' assume that the best model We can find is one a BF of 35% which presents a 1.20 ROI which can also be expressed as a PNL of $89840.0 for a flat bet of $100.

The event universe consists of 11,098 races out of which 3,893 are the proposed bets...

Lets' also assume that we have another model which out of the remaining 7,205 has the same BF of 35% giving us 2,521 bets but only with 1.05 ROI and let's also assume that its PNL is now only 12,605

We also have a bankroll that allows us based in our risk management plan to escalate our bets up to let's say $2,000 maximum based in the pool size and the price offered. Following such an approach both strategies will present lower ROI although the expected PNL will be reaching its maximum potential.

The questions we need to answer given the described models are the following:

1) What should be the bankroll proportion allocated to each model?

2) How big should we allow our bets to grow in order to achieve maximum composite PNL

Note that for #2 we cannot use KELLY because:
- We do not have a probability for each individual event
- There is a maximum bet imposed by the pool size itself

3) Can we describe it as a linear programming problem?

4) Does the absolute size of the bankroll makes a difference in our approach? Meaning that we should follow a different strategy if our bankroll is smaller that X or larger than Y?

It sounds like the models in your example are mutually exclusive within the same event universe.
I note/ask this because of the sum of the proposed bets in model1 and model2 is equal to the total races in the event universe.

You may want to eventually "overlap" them in a 2nd example once you master this problem to add complexity.

In a mutually exclusive situation, given the figures in your example, model1 is the bread-winner.
I would start with a focus on model1. Look at win% and bankroll size and risk of ruin to determine the bet size. If your bankroll is sufficiently large (now or as it grows exponentially over time), the pool size will become the ceiling on the bet-size(rather than the bankroll size).

Then you would do the same thing with model2 if you had confidence that the 1.05 ROI was accurate.
A flat bet is going to be less than optimal. If for whatever reason, individual probabilities for model2 can not be obtained, you can also use the odds to form a low estimate.

With mutually exclusive models, you don't really need to do any blending or allocating, other than simply maximizing both models individually.

When you have models that overlap some of the same races, it becomes more complex.

[DeltaLover]

With mutually exclusive models, you don't really need to do any blending or allocating, other than simply maximizing both models individually.

Sure...

But I what I am trying to undrestand is what is the proportion of bankroll that should be assigned to each model to maximize rate of growth while mimimizing risk.

[davew]

You need to consider pool size in your allocation and bet sizing

your 120% ROI can't handle $100 bets at many tracks
and at some tracks $2K is larger than win pool

[Robert Fischer]

Sure...

But I what I am trying to undrestand is what is the proportion of bankroll that should be assigned to each model to maximize rate of growth while mimimizing risk.

Here's what I do -
1. i make a low-estimate of the individual wager's probability of winning.
2. i calculate the max consecutive losses for that probability
3. i use my bankroll size, the max consecutive losses, and an acceptable loss to calculate a bet size.


--
example:

given:
win wager
$1.20 ROI
$6payout
40% individual probability low-estimate


a. 20% loss
bankroll size = $1,000
the max consecutive losses = 18 (over 10,000 proposed bets)
acceptable loss = 20% of bankroll for worst case losing streak
=($1,000*0.2)/18 = $11.50 bet size


b. full bank in play
bankroll size = $1,000
the max consecutive losses = 18 (over 10,000 proposed bets)
acceptable loss = 100% of bankroll for worst case losing streak
=($1,000)/18 = $55.60 bet size

[Robert Fischer]

The above example uses a "fluid" bankroll amount.

It would be re-calculated for each wager (or each significant bankroll change, whatever you choose as the best approach).

And you would simply treat each model separately as if it were the only model in play.

I have to think a little bit about this, and please speak up if you already have the math.

My brain is very lazy/foggy right now, but I am thinking that only danger with multiple models is the possibility of hitting maximum consecutive loss streaks consecutively or in significant overlapping scenario.

With 2 models, such as in your example, It wouldn't be a big problem unless you used a full-bankroll acceptable loss for both, and had a rare worst-case scenario.

The simple solution is to just play both, or only play the bread-winner.

I don't know the math, and the two models are far enough apart in ROI that the exponential gains have to be considered.

here is an example of a problematic caveman approach: multiply ROI*playable races for each model and then allocate to that proportion).
model a = 1.20 ROI * 3,893bets=4671.6
modelb = 1.05 ROI * 2,521bets =2647.05
modela = $638.31/1000bankroll
modelb = $361.69/1000
PITFALL = decreases your bet size.
@$638.31 bankroll for model A
you betsize decreases from $11.50 to $7.09 or $55.60 to $35.40 from the examples in my previous post.

The 'pitfall' is that this ignores the exponential gains to your bankroll.

The $1.20ROI model a , is going to gain exponentially at a much higher rate than the $1.05ROI model b. The bankroll will increase faster --> leading to the bet size increasing faster in a loop.

the caveman ratio approach ignores that. You could also simply use ROI ratios 1.20/1.05 = 4/1 = $800 for model a, $200 for model b.
I can't tell you whether that would work. You could test it over a simulation of plays in a smaller event universe. If it works fine you could use it.

It also begs the "real life" question that model b may not be worth the investment of your time given the fruitfulness of model a. and the opportunity to try to expand model-a into exotics.

There should be some kind of investment formulas out there for investing in several stocks or businesses concurrently.

rambled on a bit here but I don't know an easier way to explain it. :ThmbUp:

[DeltaLover]

No, I really do not now how to analytically resolve the problem...

I've tried to express it as a linear programming problem but I am not satisfied by the result...
The variance of the bankroll size is what makes it more complicated..

Maybe a solution is to just apply some monte carlo simultations and peek an average strategy...

[Robert Fischer]

after thinking about it:

because the growth in bankroll size, and the growth in wager size will increase exponentially in a loop

and
the difference in ROI is so great between the two models

and
the number of playable races is not all that great for model-b ,

Pretty much any reduction in wager-size for model-a (for the purpose of safely including model-b) is going to be a mistake.

Focus on model-a, grow your bankroll and wager-size until the pool size forces a plateau from that growth.

When the pool size forces a plateau from growth(except in the case of a few big tracks, few plays, few big days)

then one of things you can consider is including model-b.

------

of course this only answers your example question - It does not provide a linear formula solution.

What if model-a has an ROI of 1.10 and model-b 1.08 ???
much different

what if the original model-b has 15,000 playable races compared to model-a's 3,893 ???
much different

[DeltaLover]

Focus on model-a, grow your bankroll and wager-size until the pool size forces a plateau from that growth.


I think this is the correct appoach...

Maybe I am over thinking just to find an excuse to have more action which is really bad (but I am affraid true as well!)

For example yesterday out of 30 races I was able to only find a single bet which lost miserably!!!

[Robert Goren]

You are right in trying to expand the numbers bets that show profit. If data cost nothing then you would be right in using only one bankroll for your best bets. you have figure in your outside costs though. The more bets you make the more you spread out those costs.
I have no idea how allocate the two banks rolls. Some math should be able to tell you. I am pretty sure that as one model grows faster than the other one you will be moving funds back forth. I am just sure when.
The more interesting question for me is the bet size. I am sure there is nice equation for it. As your bet size grows your odds decreases and therefore your ROI decreases also for that bet. There is an optimal bet size that maximize your profit the race. It is times like these that I wish I had not skipped the third level calc class for the afternoon bridge game. I am sure that one of the Math guys here should be able to come up some sort of formula using integrals and/or derivatives.
Good luck with your bets and hope somebody come with answer you seek.

[davew]

besides BF and ROI, hit rate should factor in

a 40% cash at $6 is much different than
a 5% cash at $24


what is different about the 2 subsets? other than ROI?
why would you need 2 bankrolls?

[TrifectaMike]

The questions we need to answer given the described models are the following:

1) What should be the bankroll proportion allocated to each model?

2) How big should we allow our bets to grow in order to achieve maximum composite PNL

Note that for #2 we cannot use KELLY because:
- We do not have a probability for each individual event
- There is a maximum bet imposed by the pool size itself

3) Can we describe it as a linear programming problem?

4) Does the absolute size of the bankroll makes a difference in our approach? Meaning that we should follow a different strategy if our bankroll is smaller that X or larger than Y?


Hi Delta,

I believe that using the concept of Gamblers ruin can provide you the answers you require as long as you know the mean and variance of your net gain.

Check your PM's

Mike (Dr Beav)

[DeltaLover]

besides BF and ROI, hit rate should factor in

a 40% cash at $6 is much different than
a 5% cash at $24


what is different about the 2 subsets? other than ROI?
why would you need 2 bankrolls?


How two strategies with same BF + ROI differ based in strike rate? Although there probably exist same differences as far as max loosing streak goes they are pretty much equivalent as far as final PNL goes.

I am not sure that I really need 2 bankrolls.... Maybe having them will simplify housekeeping. But making an upfront decission of what percentage of the (unique) bankroll will be allocated to each strategy will obviously result to different behavior.

The question is what is the optimal way to allocate the bankroll to maximize the PNL based a specific betting universe.

This is not an easy task since we need to calculate slipage and maximum bets based in pool sizes. Of course at one point when the bankroll will exceed a certain level our only concern will be to size the bets so they still remain overlays regardless of the strategy expectation (assuming of course that this is positive). The problem is how to size the bets until you reach this level....

[DeltaLover]

thx doc

[ArlJim78]

Lets' assume that the best model We can find is one a BF of 35% which presents a 1.20 ROI which can also be expressed as a PNL of $89840.0 for a flat bet of $100.


can you show me how this works? I'm trying to understand how the BF of 35% and a 1.2 ROI yields a PNL of $89840 for a $100 wager
thanks

[DeltaLover]

starting with 10,000 while having 3893 bets
we put in action: $389,300
$389,300 * 0.2 = $77,860
so we end up with 87,860

The small difference has to do with the decimals

Actually the real PNL in this examle is $77,860 while 87,860 will be the gross total

[ArlJim78]

gotcha, $77,860 makes sense to me and didn't know where the other figure came from.
thx

[Robert Fischer]

gotcha, $77,860 makes sense to me and didn't know where the other figure came from.
thx

also the "BF" figure stands for his "betting frequency"
he had 3,800 plays out of 11,000 races or whatever , so about 35% of the total races were playable

so the "BF" = 35%

i had to read it a couple times