Question

Year	Beginning value	Ending value
1	110	115
2	115	138
3	138	110,4

Find the arithmetic annual mean rate of return and the geometric annual mean rate of return. Which one should be used?

Answer 1

Year	Beginning value	Ending value	Return
1	110	115	=115/110 - 1 = 0.04545454545
2	115	138	=138/115 - 1 = 0.2
3	138	110.4	=110.4/138 - 1 = -0.2

Arithmetic annual mean return = (0.04545454545 + 0.2 - 0.2)/3

Arithmetic annual mean return = 0.01515151515

Arithmetic annual mean return = 1.515151515%

Geometric annual mean return = (Ending value in year 3)/(Beginning value in year 1)^(1/3) - 1

Geometric annual mean return = (110.4/110)^(1/3) - 1

Geometric annual mean return = 1.0036363636^(1/3) - 1

Geometric annual mean return = 1.0012106549 - 1

Geometric annual mean return = 0.0012106549

Geometric annual mean return = 0.12106549%

Geometric annual mean return should be used because it takes compounding into effect, while the arithmetic average return does not.