Thanks for your response. I would also like to express my appreciation to you for your work in generalizing the Kelly Criterion concept.
Quote from rvince99:
Equation 2.1 here is correct for determining an optimal fraction, but on the top of page 4, he tries to express it in terms of the sum of natural logs -- the two are NOT equivalent except in what I call the "special case"... .
Please consider the paper by Ralph Vince, page 21, at:
The rhs of Vince Equation (1) is identical in form to the rhs of Zhu equation top of page 4. This form will be referred to as the sum of logs form.
The rhs of Vince Equation (1a) is identical in form to the rhs of Zhu Equation (2.1). This form will be referred to as the product form.
I paraphrase the statement by Vince regarding his Equation (1) and his Equation (1a):
"Rather than taking the sum of the logs of the returns, we can take the product of those returns. Thus, the value for f that maximizes (1) will also maximize (1a)."
Please notice that:
ln(product form) = sum of logs form
exp(sum of logs form) = product form
Because of the strictly increasing and continuity properties of the ln and exp functions, a maximum is attained at a value of f for one of the forms if and only if a maximum is attained at that same value of f for the other form.
In light of this, it can be said that the two forms are equivalent for the purpose of finding values of f which maximize either form. I do not understand what the phrase, “the two are NOT equivalent”, was intended to convey. Any enlightenment would be appreciated.