Originally posted by wintermute
Everyone read the references posted in Part 0? 
Anyway here goes - for the 2nd time - since the wife accidentally deleted an hour's worth of typing just a few minutes ago.
What Kelly is all about is picking a number in the range of 0 to 1 that finds the best combination of return and risk for your gambling investment. This number between 0 and 1 is the fraction of your capital that you should bet on any individual wager. If the fraction is 0 then you are minimizing risk and return because you are betting nothing at all - if you don't bet you can't lose (or win). If the fraction is 1 then you are betting all your capital every time you wager - maximizing risk and return. Somewhere between 0 and 1 is an optimal fraction that best balances off return and risk. As a bonus it is possible to prove that the optimal fraction not only gives you a good return - it gives you the BEST possible return.
To give a concrete example. Suppose you have a starting capital of 10000 dollars and an optimal Kelly fraction of 5 percent or 0.05 on your first wager. You will wager 500 dollars. If the payoff is even money you will end up with capital of 10500 if you win, 950 dollars if you lose. On the second wager if your optimal Kelly fraction is 10 percent or 0.10 you will wager 1050 dollars if you had won the first wager, 950 dollars if you had lost the first wager. And so on ...
We can simplify the math at no loss in generality if we assume a starting capital of 1 dollar. Let F be the fraction of 1 dollar that we wager on an individual bet and let the payoff be even money as before. If we win the first wager our capital will increase to 1 + F dollars. If we lose it will decrease to 1 - F dollars. If we had two wagers, both at even money, wagered the same fraction F each time and won both wagers our capital would become
( 1 + F ) * ( 1 + F )
where * means multiplication. A win and a loss would lead to a capital of
( 1 + F ) * ( 1 - F )
or
( 1 - F ) * ( 1 + F )
depending on whether or not we had a win followed by a loss or a loss followed by a win. Note that the result is the same whether we win the first game or the second since
( 1 - F ) * ( 1 + F )
equals
( 1 + F ) * ( 1 - F )
to be continued...
Anyway here goes - for the 2nd time - since the wife accidentally deleted an hour's worth of typing just a few minutes ago.
What Kelly is all about is picking a number in the range of 0 to 1 that finds the best combination of return and risk for your gambling investment. This number between 0 and 1 is the fraction of your capital that you should bet on any individual wager. If the fraction is 0 then you are minimizing risk and return because you are betting nothing at all - if you don't bet you can't lose (or win). If the fraction is 1 then you are betting all your capital every time you wager - maximizing risk and return. Somewhere between 0 and 1 is an optimal fraction that best balances off return and risk. As a bonus it is possible to prove that the optimal fraction not only gives you a good return - it gives you the BEST possible return.
To give a concrete example. Suppose you have a starting capital of 10000 dollars and an optimal Kelly fraction of 5 percent or 0.05 on your first wager. You will wager 500 dollars. If the payoff is even money you will end up with capital of 10500 if you win, 950 dollars if you lose. On the second wager if your optimal Kelly fraction is 10 percent or 0.10 you will wager 1050 dollars if you had won the first wager, 950 dollars if you had lost the first wager. And so on ...
We can simplify the math at no loss in generality if we assume a starting capital of 1 dollar. Let F be the fraction of 1 dollar that we wager on an individual bet and let the payoff be even money as before. If we win the first wager our capital will increase to 1 + F dollars. If we lose it will decrease to 1 - F dollars. If we had two wagers, both at even money, wagered the same fraction F each time and won both wagers our capital would become
( 1 + F ) * ( 1 + F )
where * means multiplication. A win and a loss would lead to a capital of
( 1 + F ) * ( 1 - F )
or
( 1 - F ) * ( 1 + F )
depending on whether or not we had a win followed by a loss or a loss followed by a win. Note that the result is the same whether we win the first game or the second since
( 1 - F ) * ( 1 + F )
equals
( 1 + F ) * ( 1 - F )
to be continued...
...and indeed...that is one of the unique aspects of kelly betting...
..it does not matter the ORDERING of wins/losses...one can have 4 wins in a row FOLLOWED BY 3 losses...
...or WLWLWLW...
..at 9.7%...the optimal Kelly Fraction...
THE WIN AMOUNT will be the same for each sequence of W/L
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