▁list ▁of ▁arguments ▁that ▁is ▁continuous , ▁strictly ▁increasing ▁in ▁each ▁argument , ▁and ▁symmetric ▁( in variant ▁under ▁permut ation ▁of ▁the ▁arguments ). ▁The ▁average ▁y ▁is ▁then ▁the ▁value ▁that , ▁when ▁replacing ▁each ▁member ▁of ▁the ▁list , ▁results ▁in ▁the ▁same ▁function ▁value : ▁ ▁. ▁ ▁This ▁most ▁general ▁definition ▁still ▁capt ures ▁the ▁important ▁property ▁of ▁all ▁aver ages ▁that ▁the ▁average ▁of ▁a ▁list ▁of ▁identical ▁elements ▁is ▁that ▁element ▁itself . ▁The ▁function ▁▁ ▁provides ▁the ▁arithmetic ▁mean . ▁The ▁function ▁▁ ▁( where ▁the ▁list ▁elements ▁are ▁positive ▁numbers ) ▁provides ▁the ▁geometric ▁mean . ▁The ▁function ▁▁ ▁( where ▁the ▁list ▁elements ▁are ▁positive ▁numbers ) ▁provides ▁the ▁harm onic ▁mean . ▁ ▁A verage ▁percentage ▁return ▁and ▁C AG R ▁ ▁A ▁type ▁of ▁average ▁used ▁in ▁fin ance ▁is ▁the ▁average ▁percentage ▁return . ▁It ▁is ▁an ▁example ▁of ▁a ▁geometric ▁mean . ▁When ▁the ▁returns ▁are ▁annual , ▁it ▁is ▁called ▁the ▁Comp ound ▁Ann ual ▁G row th ▁R ate ▁( CA GR ). ▁For ▁example , ▁if ▁we ▁are ▁considering ▁a ▁period ▁of ▁two ▁years , ▁and ▁the ▁invest ment ▁return ▁in ▁the ▁first ▁year ▁is ▁− 1 0 % ▁and ▁the ▁return ▁in ▁the ▁second ▁year ▁is ▁+ 6 0 %, ▁then ▁the ▁average ▁percentage ▁return ▁or ▁C AG R , ▁R , ▁can ▁be ▁obtained ▁by ▁solving ▁the ▁equation : ▁. ▁The ▁value ▁ ▁of ▁R ▁that ▁makes ▁this ▁equation ▁true ▁is ▁ 0 .