▁the ▁same ▁result ▁but ▁through ▁a ▁different ▁deriv ation . ▁ ▁Remember ▁that ▁ ▁is ▁different ▁from ▁the ▁asset ▁log ▁return ▁. ▁Conf using ▁this ▁is ▁a ▁common ▁mistake ▁made ▁by ▁websites ▁and ▁articles ▁talking ▁about ▁the ▁Kelly ▁C riter ion . ▁ ▁Many ▁assets ▁ ▁Consider ▁a ▁market ▁with ▁ ▁correl ated ▁stock s ▁ ▁with ▁st ochastic ▁returns ▁, ▁ ▁and ▁a ▁risk less ▁bond ▁with ▁return ▁. ▁An ▁invest or ▁puts ▁a ▁fraction ▁ ▁of ▁their ▁capital ▁in ▁ ▁and ▁the ▁rest ▁is ▁inv ested ▁in ▁the ▁bond . ▁Without ▁loss ▁of ▁general ity , ▁assume ▁that ▁invest or ' s ▁starting ▁capital ▁is ▁equal ▁to ▁ 1 . ▁According ▁to ▁the ▁Kelly ▁criter ion ▁one ▁should ▁maxim ize ▁ ▁Exp and ing ▁this ▁with ▁a ▁Taylor ▁series ▁around ▁ ▁we ▁obtain ▁ ▁Thus ▁we ▁reduce ▁the ▁optimization ▁problem ▁to ▁quadratic ▁programming ▁and ▁the ▁un const rained ▁solution ▁is ▁ ▁where ▁ ▁and ▁ ▁are ▁the ▁vector ▁of ▁means ▁and ▁the ▁matrix ▁of ▁second ▁mixed ▁non central ▁moments ▁of ▁the ▁excess ▁returns . ▁ ▁There ▁is ▁also ▁a ▁numerical ▁algorithm ▁for ▁the ▁fraction al ▁Kelly ▁strateg ies ▁and ▁for ▁the ▁optimal ▁solution ▁under ▁no ▁le verage ▁and ▁no ▁short ▁s elling ▁constraints . ▁ ▁Crit ic ism ▁ ▁Although ▁the ▁Kelly ▁strategy ' s ▁promise ▁of ▁doing ▁better ▁than ▁any ▁other ▁strategy ▁in ▁the ▁long ▁run ▁seems ▁comp elling , ▁some ▁econom ists ▁have ▁argued ▁str enu ously ▁against ▁it , ▁mainly ▁because ▁an ▁individual ' s ▁specific ▁invest ing