I have reading a paper from Oliver Grandville on long term expected return. I am trying to reconcile what I am reading in that paper vs what I see under "Application to Stock Market" in Kelly criterion page on Wikipedia.
http://www.cfapubs.org/doi/abs/10.2469/faj.v54.n6.2227
https://en.wikipedia.org/wiki/Kelly_criterion
In his paper, he define the following:
- Rt−1,t=St−St−1St−1 as the yearly rate of return compounded once per year
- Xt−1,t=1+Rt−1,t=StSt−1 as the yearly dollar return
- log(Xt−1,t)=log(StSt−1) as the yearly continuously compounded rate of return
- E(Xt−1,t)=E(1+Rt−1,t) as the expected value of the yearly dollar return and V(Xt−1,t) as its variance
He derives that given log(Xt−1,t)=log(StSt−1)=μ+σN(0,1) (ie log returns follow normal distribution of mean μ and variance σ2), we must have that E(Xt−1,t)=exp(μ+σ22).
My questions are:
- For the wiki article vs this article, the wiki one says expected log return is Rs=(μ−σ22)t. This seems similar to the Grandville article in that if you take log(E(Xt−1,t))=μ+σ22, but its not exactly the same. Is the because the assumption the wiki article makes is different (ie the stock price moves like Brownian motion?)
- In the derivation of the max of the expected value in wiki page. Shouldnt it be G(f)=fE(stock)+(1−f)E(bond)=f∗(μ−σ22)+(1−f)∗r, but the vol term is different (f2 vs f). Why is that?
- Finally, the last line of the wiki article mentions that "Remember that μ is different from the asset log return Rs. Confusing this is a common mistake made by websites and articles talking about the Kelly Criterion." My understanding is that μ is the expected of the log returns E(log(Xt−1,t)) whereas Rs is the log of the expected returns log(E(Xt−1,t)) and that this distinction is important because E(log(X)) and log(E(X)) are not always equal. Is this understanding correct?
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