aking ▁out ▁a ▁loan ▁that ▁requires ▁pay ing ▁interest ▁in ▁order ▁to ▁raise ▁invest ment ▁capital ) ▁decre ases ▁the ▁optimal ▁fraction ▁to ▁be ▁inv ested , ▁as ▁in ▁that ▁case ▁. ▁Obviously , ▁no ▁matter ▁how ▁large ▁the ▁probability ▁of ▁success , ▁, ▁is , ▁if ▁ ▁is ▁sufficiently ▁large , ▁the ▁optimal ▁fraction ▁to ▁invest ▁is ▁zero . ▁Thus , ▁using ▁too ▁much ▁margin ▁is ▁not ▁a ▁good ▁invest ment ▁strategy ▁when ▁the ▁cost ▁of ▁capital ▁is ▁high , ▁even ▁when ▁the ▁opportunity ▁appears ▁prom ising . ▁ ▁Proof ▁ ▁He ur istic ▁proofs ▁of ▁the ▁Kelly ▁criter ion ▁are ▁straightforward . ▁The ▁Kelly ▁criter ion ▁maxim izes ▁the ▁expected ▁value ▁of ▁the ▁log arith m ▁of ▁wealth ▁( the ▁expectation ▁value ▁of ▁a ▁function ▁is ▁given ▁by ▁the ▁sum , ▁over ▁all ▁possible ▁out comes , ▁of ▁the ▁probability ▁of ▁each ▁particular ▁outcome ▁multip lied ▁by ▁the ▁value ▁of ▁the ▁function ▁in ▁the ▁event ▁of ▁that ▁outcome ). ▁We ▁start ▁with ▁ 1 ▁unit ▁of ▁wealth ▁and ▁bet ▁a ▁fraction ▁ ▁of ▁that ▁wealth ▁on ▁an ▁outcome ▁that ▁occurs ▁with ▁probability ▁ ▁and ▁offers ▁odd s ▁of ▁. ▁The ▁probability ▁of ▁winning ▁is ▁, ▁and ▁in ▁that ▁case ▁the ▁resulting ▁wealth ▁is ▁equal ▁to ▁. ▁The ▁probability ▁of ▁losing ▁is ▁, ▁and ▁in ▁that ▁case ▁the ▁resulting ▁wealth ▁is ▁equal ▁to ▁. ▁Therefore , ▁the ▁expected ▁value ▁for ▁log ▁wealth ▁ ▁is ▁given ▁by : ▁▁▁▁ ▁To ▁find ▁the ▁value ▁of ▁ ▁for ▁which ▁the ▁expectation ▁value ▁is ▁maxim ized