▁re venue ▁after ▁dedu ction ▁of ▁the ▁track ▁take ▁when ▁- th ▁horse ▁wins . ▁The ▁binary ▁growth ▁exponent ▁is ▁ ▁and ▁the ▁dou bling ▁time ▁is ▁▁▁ ▁This ▁method ▁of ▁selection ▁of ▁optimal ▁bet s ▁may ▁be ▁applied ▁also ▁when ▁prob abilities ▁ ▁are ▁known ▁only ▁for ▁several ▁most ▁prom ising ▁out comes , ▁while ▁the ▁remaining ▁out comes ▁have ▁no ▁chance ▁to ▁win . ▁In ▁this ▁case ▁it ▁must ▁be ▁that ▁▁ ▁and ▁ ▁. ▁ ▁Application ▁to ▁the ▁stock ▁market ▁ ▁In ▁mathematical ▁fin ance , ▁a ▁port folio ▁is ▁called ▁growth ▁optimal ▁if ▁security ▁weights ▁maxim ize ▁the ▁expected ▁geometric ▁growth ▁rate ▁( which ▁is ▁equivalent ▁to ▁maxim izing ▁log ▁wealth ). ▁ ▁Comput ations ▁of ▁growth ▁optimal ▁port fol ios ▁can ▁suffer ▁trem end ous ▁garbage ▁in , ▁garbage ▁out ▁problems . ▁For ▁example , ▁the ▁cases ▁below ▁take ▁as ▁given ▁the ▁expected ▁return ▁and ▁cov ar iance ▁structure ▁of ▁various ▁assets , ▁but ▁these ▁parameters ▁are ▁at ▁best ▁estimated ▁or ▁mode led ▁with ▁significant ▁uncertainty . ▁Ex - post ▁performance ▁of ▁a ▁supposed ▁growth ▁optimal ▁port folio ▁may ▁differ ▁fant ast ically ▁with ▁the ▁ex - ante ▁prediction ▁if ▁port folio ▁weights ▁are ▁largely ▁driven ▁by ▁estimation ▁error . ▁De aling ▁with ▁parameter ▁uncertainty ▁and ▁estimation ▁error ▁is ▁a ▁large ▁topic ▁in ▁port folio ▁theory . ▁ ▁The ▁second - order ▁Taylor ▁polynomial ▁can ▁be ▁used ▁as ▁a ▁good ▁approximation ▁of ▁the ▁main ▁criter ion . ▁Prim arily , ▁it ▁is ▁useful ▁for ▁stock ▁invest ment ,