▁where ▁the ▁fraction ▁devoted ▁to ▁invest ment ▁is ▁based ▁on ▁simple ▁characteristics ▁that ▁can ▁be ▁easily ▁estimated ▁from ▁existing ▁historical ▁data ▁– ▁expected ▁value ▁and ▁variance . ▁This ▁approximation ▁leads ▁to ▁results ▁that ▁are ▁robust ▁and ▁offer ▁similar ▁results ▁as ▁the ▁original ▁criter ion . ▁ ▁Single ▁asset ▁ ▁Consider ing ▁a ▁single ▁asset ▁( stock , ▁index ▁fund , ▁etc .) ▁and ▁a ▁risk - free ▁rate , ▁it ▁is ▁easy ▁to ▁obtain ▁the ▁optimal ▁fraction ▁to ▁invest ▁through ▁geometric ▁Brown ian ▁motion . ▁The ▁value ▁of ▁a ▁log norm ally ▁distributed ▁asset ▁ ▁at ▁time ▁ ▁() ▁is ▁▁▁ ▁from ▁the ▁solution ▁of ▁the ▁geometric ▁Brown ian ▁motion ▁where ▁ ▁is ▁a ▁Wiener ▁process , ▁and ▁ ▁( percent age ▁dr ift ) ▁and ▁ ▁( the ▁percentage ▁vol at ility ) ▁are ▁constants . ▁T aking ▁expect ations ▁of ▁the ▁log arith m : ▁ ▁Then ▁the ▁expected ▁log ▁return ▁ ▁is ▁ ▁For ▁a ▁port folio ▁made ▁of ▁an ▁asset ▁ ▁and ▁a ▁bond ▁pay ing ▁risk - free ▁rate ▁, ▁with ▁fraction ▁ ▁inv ested ▁in ▁ ▁and ▁ ▁in ▁the ▁bond , ▁the ▁expected ▁one - period ▁return ▁is ▁given ▁by ▁ ▁however ▁people ▁seem ▁to ▁deal ▁with ▁the ▁expected ▁log ▁return ▁ ▁for ▁one - period ▁instead ▁in ▁the ▁context ▁of ▁Kelly : ▁ ▁Sol ving ▁ ▁we ▁obtain ▁▁ ▁is ▁the ▁fraction ▁that ▁maxim izes ▁the ▁expected ▁log arith mic ▁return , ▁and ▁so , ▁is ▁the ▁Kelly ▁fraction . ▁ ▁Thor p ▁arrived ▁at