You calculated the EV of one trial of a 5/4 bet, which is what the bet is in the OP.

The formula is on the wikipedia page, but the better one to use to calculate the Kelly Criterion is the one that is the formula for the Kelly Criterion.

You guys are calculating two different things as a percent of total wealth. Masque is looking at it as an investment of, let's say $100, which will either return $150 or $60. Brain is looking at is a straight bet of $40 which either gets paid the bet+$50 or loses the $40.

Brian's Kelly criteria says the $40 should be 10% of your total wealth to maximize expected wealth growth rate. Since $100 is 2.5 times $40 it's not surprising that Masque's Kelly criteria for investments says the $100 investment should be 25% of total wealth to maximize growth rate.

Also, on the Wikipedia page it says, for betting, the Kelly Criterion boils down to:

Expected Net Winnings / Amount returned if you win

However, I believe your edge, or advantage is:

Expected Net Winnings / Amount Bet

So that's why it's 10% rather than 12.5%. So the rule "Bet your edge" is an approximation for Kelly. Please correct me if I'm wrong.



PairTheBoard

You are definitely correct in the first sentence. There is a section in the wiki page that discusses using Kelly with investments. Not too surprisingly, it gives the same percentage of bankroll to be put at risk as it does for any equivalent odds and payout. Math is math again, which is comforting.

It is fun to think about these things. Excellent job figuring out the discrepancy in language use.

One more correction here the avg ie geometric growth rate factor per trial is ((1+0.25*0.5)*(1-0.25*0.4))^(1/2) ie 1.00623 or 0.623% improvement.


Forgot the root thing because i dropped it in the maximization process that is not affected by it.


So if you did that every day for a year you have a target of 1.00623^365=9.65x your starting position.

BTM you start with the position that allows you to participate ie all of it not the 40% risk you face per event.

How do you calculate the growth in 365 days? Go back to see how you are doing it and that you need to have 25% of your assets used not 10%. So your position grows at avg (1+f*0.5)^(1/2)*(1-f*.4)^(1/2) with f =0.25 not 10%. You are effectively using

(1+0.1*0.5)^(1/2)*(1-0.1*.4)^(1/2)=1.003992

so your result is not consistent with the way one plays the game there because if you use that formula for the growth you should be using mine for the optimization and getting 25% not 10% as well. It is 10% risked effectively per event true because this is how much you stand to lose but the growth then requires to use f=0.25 to use the equation you are using to get your 0.3992% number that comes if you put the wrong for this formula number of 10%.

The 10% is based on money at risk. Using "money at risk" is a fairly standard practice. The 60% is just sitting there effectively doing nothing.

Forgetting to take the root was a silly mistake. These things happen when I only give five minutes of thought to a problem. It doesn't hurt my feelings at all to make a mistake. It would suck if I made such a mistake when planning my trades though!

Also, they have midnight shifts at Stanford? Go to sleep, man!

More on topic, it simply didn't occur to me that you weren't using money at risk. I probably could have spent more effort reading your post in order to ascertain this, but I didn't.

Even more on topic, the reason we use money at risk is because investing bets are closer to "invest $100 to win $6 or lose $4." * Betting only 100% of your bankroll would be silly in this case since margin exists.

*They aren't actually close to that either, but it is closer.

https://www.albionresearch.com/kelly/default.php

Neither you nor Brian is looking at the game as described in the OP. In that game you are only allowed the make the investment if you invest 100% of your total wealth (or equivalently, Bet 40% of your total wealth to win 50% of your total wealth). The growth rate of that investment (or bet) is calculated in the OP as about -5%.

So both you and Brian are making up another game where you are allowed to make the OP investment (or bet) for less than your total wealth and thereby apply Kelly to find the Kelly percentage of your total wealth looking at it as either an investment or a bet. If you look at it as an investment of, say $100 (or equivalently a bet of $40), Kelly tells you what your total wealth should be to optimize avg growth rate. Looked at as a $100 investment Kelly says your wealth should be $400 ( 100 = .25*400). Equivalently, looked at as a $40 bet Kelly says your wealth should be $400 ( 40 = .10*400).

Now the question is, what is the average growth rate of your total wealth starting at $400 and investing 25% (or equivalently betting 10%) at each iteration? Using your equations suitable be the "investment" approach you get about 0.62%. Brian may have incorrectly calculated the avg growth rate of the total wealth (also initially $400) under his "betting" approach. But a correct calculation using the (equivalent) betting approach yields the same avg growth rate of about 0.62% for total wealth (initially $400).

You can easily check this by using this Kelly Calculator I linked to before.

https://www.albionresearch.com/kelly/default.php


Enter
$400 Total Wealth
5 to 4 Odds Offered
50% Probability of Winning
$0.01 Bets must be multiples of
$0.00 Minimum Bet Allowed


PairTheBoard

Some mistakes dont "count" you know among civilized people if they are typos, especially if the person comes back to correct it and if the original idea is crystal clear correct and illuminating on why the formulas are this way. My error was not a mental error only a check your math error.

But your mistake still stands in the growth rate as in if you started with 1000 $ doing this you wont get after a year the result you claim.

Even if you started with your at risk money you get 4.28 vs the actual 9.65


If we did it for 2years every day i get 93x and you get only 18.3x

Clearly you have the wrong result with your 10% growth calculation. You may have the correct idea that you risk only 10% but then you need to use a different growth formula than the one you are using that is derived by the method that gives the 25%.

By all means derive your result mathematically like i did and do not quote equations that do not naturally come from your own assumptions.

Ps: Stanford never sleeps and those that are trying to take down the house of cards its physics department and modern post Standard model physics has built for decades sleep only when the others are lying in their false lectures.

Yes because my intention is to improve on that game that is a losing strategy game that allows one to be bankrupt most of the time.

But Brian is deriving the wrong growth rate for the correct strategy game with his 10% in the formula that only 25% must be inserted. So his growth result is false for the game he is risking 10% each time.

Examine how he gets his growth result after 2 years say and you go ahead and calculate what if we both started with 1000 would be the result after 2 years. What is your answer and what is his answer? Only one can be correct. (see the proof section in wiki and see how he gets his growth result by using the formulae where only 25% must be used not 10%. I do not object 10% is what is "risked". We play the same format game but the growth rate is not calculated that way using 10%.

The point of my post was that both the "investment (25%)" approach and the "betting (10%)" approach are equivalent and both yield the same avg growth rate result if the appropriate respective calculations are done for each.


PairTheBoard

You aren't taking into account the personal importance of being right or wrong. Please adjust your answer accordingly.

The point of my post was that both the "investment (25%)" approach and the "betting (10%)" approach are equivalent and both yield the same avg growth rate result if the appropriate respective calculations are done for each.

I'm giving this point the name "BrianTheMasque" so as to personalize it.

PairTheBoard

Well, in economics two equivalent investments are the same investment*. This is important to economists for some reason. It has something to do with the degree to which a market is "complete."**

I hope our imaginary better doesn't have to "invest"** in multiples of $100. That would be problematic for the calculations. 50% chance he only gets to make one bet before retiring. Kelly can kind of be a ***** sometimes.

*Lack of quotes is important for clarity.

**Quotes necessary for clarity.

***** Not an indication of a footnote.

I'm saying that with an altered version of the OP where the OP game is allowed for amounts less that total wealth, there are two equivalent approaches for Kelly calculations.

You can approach the problem as a $100 investment under altered OP rules, or as a $40 bet at 5-4 odds. Wiki has two different formulas for each approach. Now, you could also consider the $40 bet as an investment and use the Wiki formula for investment, except it will not be under OP rules so you must use different values for the parameters in the Wiki Kelly formulas for investment. You should still get the same result, $400 for optimal initial total wealth, and about 0.62% expected rate of growth under iterations.

The Kelly formula for investment can be used for the $40 5-4 bet but the value of the parameters must be changed because it's a different investment than the $100 under altered OP rules.

So there are 3 approaches to the Kelly calculation which all produce the same result. However, the 3rd approach, considering the $40 5-4 bet as an investment with return ratios of either 2.25 or 0, while equivalent as an approach for Kelly calculations, is not an equivalent investment to the $100 investment with return ratios 1.5 and 0.6. The reason they are not equivalent investments is because the $100 investment ties up an extra $60 for the term of the investment. You would much rather just bet the $40 and use the other $60 for something else. Unfortunately, the OP does not mention the possibility of such a simple bet.


"PairTheBoard"

The OP only mentions one set of terms and a question to which the answer is a simple "no." To get to a "yes" the terms need altering.

Although, I guess the answer to OP could be "yes" if you have some strange jump in utility at $150 and/or have no difference in utility between $60 and $100. This might be true if the ultimate purpose of this bankroll is to purchase a hat.

An alternative to the "no" would be "no, almost regardless of the terms of the bet, if your bankroll is $100 you ought get a job."

I'm not sure if you think I've been disagreeing with or misunderstanding anything you have posted. I've not.

[QUOTE=BrianTheMick2;55402568]The OP only mentions one set of terms and a question to which the answer is a simple "no." [/QUOTE

Right.


PairTheBoard

Do you take into account the personal importance of being right enough to go out and post that your growth rate is wrong and check the wikipedia page and see that the formula you were using was changed in the last 3 days because it was wrong for what it was describing but mine hasnt because i derived it from first principles like we all should if it helps the thread to learn something for anyone new to bankroll management when one has a plus EV or skill edge situation.


Speaking of right or wrong, ego, etc it is facking elementary we all have ego and care not to be wrong but the best of us care to correct things and own it and learn and improve because we have tamed our ego to be more ethical. Also civilized people classify errors in 2 kinds, those that are trivially unimportant and easy to correct at the spot and which all people do because they are universal property of the human brain and those that are deeper mental ones that one has the wrong idea about how things work or logically failing a process. Obviously those are more serious and worthy of studying while the others anyone can correct on the spot and move on as if they never happened.

The problem i have with you is that you are constantly sarcastically after everyone for anything you can pick up to be critical even fake artificial made up things about the worse possible interpretation of what others were doing. This is is massively tilting because it introduces ridiculous aggression and friction to discussions but also because when the time comes for you to fix something wrong of your own there is absolutely no desire to admit or fix anything just make up whatever funny stories about it (like "language"). This is called hypocrisy.

So what is the projected growth outcome if you start with 1k after 2 year of daily application of the process?

The answer to Op is a big yes if you have only a small replenishable net worth and a permanent job you just started. If someone offered you a plus EV for 5k of net worth you can take it if you have more money coming into your future anyway. You earn an avg 250 by doing so.

Can one find an investment that turns 5k in a few minutes to something 5% higher on avg? Not that many things come to mind. Many people would resent it because of volatility but the fact more money is coming in the future and the loss doesnt wipe out all free cash left is still offering some protection.

Or you might want to be more protective of that $5k as a cushion against the vicissitudes of life.


PairTheBoard

But next month another 7k comes in say and your expenses is 4k so that works then doesnt it? Wouldn't you take then since you accumulate 3k every month anyway?

It is interesting to see if it makes sense to take every month for a year actually adding to it all your savings too.

If you get real lucky for example you can have 8-4 result or better that happens 19.5% of the time and suddenly you are looking at >3.3x move or even much more depending on how it happened as you deposit more of your saving every month redefining your total wealth.

If you used the proper fraction once a month you would have in a year 1.0063^12=1.078 which is bs boring lol. If you got the 8/4 result you get >1.68x that is still small compared to >3.3x.

When one is young and bankrupt they must take risk as often as possible if they have steady source of income and are practically printing money every month. Of course they must also stick to a 2k per month permanent savings they do not risk.

Also you only do that until something big happens and then stop because it is futile eventually. When you have little to lose you must go for it if you are covered due to good employment that just started.

At 7% stock market or even 15% if you can beat the market with a little help from derivatives and research, in 30 years that 2k per month will create a massive position like
(annuity)
((1+0.15/12)^360-1)/(.15/12)*2000=13.8 mil with actual deposits of only 720k

Even if it was only the standard market 7% its 2.4 mil.

Inflation adjusted who know how much really but if one could then retire and get some 4% interest rate in some very carefully selected hedged investments they would get
2.4mil*.04-> 8k expenses per month lifestyle financed without working again.

No. I don't and didn't. I'm fairly sure I didn't even use the wiki page to incorrectly calculate the growth rate in the first place.

It is all the same to me whether PairTheBoard or you or me or someone named Charles or Alfred makes a correction to or a criticism of an thought I had. In each case, I suffer no discomfort and get to have a new thought. I'd like to think that I'm smart enough to entirely avoid presenting my thoughts if this weren't the case.

Yes you did use the wiki page or an identical source because you then put that 10% in the formula that had to be 25%. If you do that you get your wrong growth rate. Exactly the same. How else do you get the exact same wrong number? So you used the 10% in the place the 25% should have been. Admit your rate is wrong. The correct rate is 0.62%. But also admit how it got to be wrong. What formula you used from where? We know you used the wrong wikipedia formula to get 10% instead of 25% because you guided me there on that section it was. Then you used the formula for growth that uses by suggested wikipedia bankroll fraction and not risk fraction equation and rate but used 10% there.

When i make a math error i care to find and correct it or good friends will alert me the same way all decent professors in amphitheaters invite colleagues and students in the audience to correct them during a lecture if its some stupid typo or oversight error. But the mental ones are important and must be admitted because we learn from them, one of them being that we are so super confident that we are able to rise from any near term fall or misstep of the mind and offer in exchange an eternity of service to the truth and superior intellect everywhere. People like Trump of course will never accept error or responsibility in anything. Well dont join him.

You get to have a new thought without admitting the other thought was wrong of course or even the fact that i was using the very logic behind Kelly and was not in need of being suggested to use Kelly as you did. Yet you are critical of everyone else and eager to attack all others. So you tried to criticize what i did without understanding it because criticizing me is number one goal every time.

You should care who offers corrections to you. Because some do it because they care for the truth and you and not for scoring points which is the game of some others. When i care for someone and i correct them i dont do it in any sarcastic cynical style ever. I was never here to play facking attitude games with others and you are.

Not exactly right. You can use 10% in the formula for investment. But you then have to change the investment return ratios from the OP's (1.5 or 0.6) to those for the 5-4 bet considered as an investment (2.25 or 0).

PairTheBoard

The 10% is correct. Reread all the posts that PairTheBoard wrote on the subject. You might learn something. You might not.

I didn't bother to read the rest of your post. It is unlikely that there is anything useful to learn on the subjects of the value of thoughts, the value of correction and the value of criticism from someone who finds thoughts being corrected or criticized to be negative.

Even in this case, you may lose your job or suffer injury or illness, disabling you and racking up extraordinary expenses. In case of such bad luck the difference between 3k and 5k may mean the difference between weathering the storm and being out on the street. If you've come from living close to the street and painstakingly saved that 5k you might be more inclined to think that way.

That's not to say you don't have a point.


PairTheBoard