11.8 The Expected Return Example ake the following 5 annual returns: 10%, 12%, 8%, 9%, and 11% The arithmetic average is x=∑x=(010+012+098+009+01)=y*050=010 However, the geometric average is ∏I(+x)-1=(1.10*112108*109*1113-1=0099 Thus, the arithmetic average overstates the geometric average The geometric is the actual return that one would have earned The approximation for the geometric return is -a2/2=010-0015811)2=0098 This differs from g as the returns are not normally distributed Options, Futures, and Other Derivatives, 4th edition@ 2000 by John C. Hull Tang Yincai, C 2003, Shanghai Normal University11.8 Options, Futures, and Other Derivatives, 4th edition © 2000 by John C. Hull Tang Yincai, © 2003, Shanghai Normal University The Expected Return Example Take the following 5 annual returns: 10%, 12%, 8%, 9%, and 11% The arithmetic average is However, the geometric average is Thus, the arithmetic average overstates the geometric average. The geometric is the actual return that one would have earned. The approximation for the geometric return is This differs from g as the returns are not normally distributed. (0.10 0.12 0.08 0.09 0.11) 5 * 0.50 0.10 1 1 5 1 1 _ =  = + + + + = = = n i n i x x (1 ) 1 (1.10*1.12*1.08*1.09*1.11) 5 1 0.09991 1 1 1 − = − =         =  + = n n i i g x / 2 0.10 (0.015811 )/ 2 0.09988 2 2 m − = − =