Monday, February 23, 2015

quant trading strategies - Questions on continuously compounded return vs long term expected return


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:




  1. Rt1,t=StSt1St1 as the yearly rate of return compounded once per year

  2. Xt1,t=1+Rt1,t=StSt1 as the yearly dollar return

  3. log(Xt1,t)=log(StSt1) as the yearly continuously compounded rate of return

  4. E(Xt1,t)=E(1+Rt1,t) as the expected value of the yearly dollar return and V(Xt1,t) as its variance


He derives that given log(Xt1,t)=log(StSt1)=μ+σN(0,1) (ie log returns follow normal distribution of mean μ and variance σ2), we must have that E(Xt1,t)=exp(μ+σ22).


My questions are:



  1. For the wiki article vs this article, the wiki one says expected log return is Rs=(μσ22)t. This seems similar to the Grandville article in that if you take log(E(Xt1,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?)


  2. In the derivation of the max of the expected value in wiki page. Shouldnt it be G(f)=fE(stock)+(1f)E(bond)=f(μσ22)+(1f)r, but the vol term is different (f2 vs f). Why is that?

  3. 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(Xt1,t)) whereas Rs is the log of the expected returns log(E(Xt1,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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