claim: ( Multiplying .95 by itself 15 times yields 46.3 percent. )”
evidence: For example, assuming reinvestment, the cumulative return for annual returns : 50%, -20%, 30% and -40% is : (  1 + 0.50  ) (  1 − 0.20  ) (  1 + 0.30  ) (  1 − 0.40  ) − 1 = − 0.0640 = − 6.40 % { \displaystyle ( 1+0.50 )( 1-0.20 )( 1+0.30 )( 1-0.40 )-1=-0.0640=-6.40\% } and the geometric average is : (  1 + 0.50  ) (  1 − 0.20  ) (  1 + 0.30  ) (  1 − 0.40  ) 4 − 1 = − 0.0164 = − 1.64 % { \displaystyle { \sqrt[ { 4 } ]{ ( 1+0.50 )( 1-0.20 )( 1+0.30 )( 1-0.40 ) } }-1=-0.0164=-1.64\% } which is equal to the annualized cumulative return : 1 − 0.0640 4 − 1 = − 0.0164 { \displaystyle { \sqrt[ { 4 } ]{ 1-0.0640 } }-1=-0.0164 } In the presence of external flows, such as cash or securities moving into or out of the portfolio, the return should be calculated by compensating for these movements.