Question

The following monthly returns were observed on a security

over 6 consecutive months: 5%, 2%, 6%, -4%, 0% and 3%.

What is the effective annual difference between the 6month geometric return and the 6-month arithmetic return?

Answer 1

Answer-

Monthly returns on observed security

Arithmetic Return

6 month arithmetic return
= ( 5 % + 2 % + 6 % - 4 % + 0 % + 3 % ) / 6

= 12 % / 6
= 2 %

6- month arithmetic return = 2 %

Effective annual return = ( 1+ 0.02)2  - 1
Effective annual return = 1.022  - 1

Effective annual return = 1.0404 - 1
Effective annual return = 0.0404

Effective annual return = 4.04 %

Geometric return

= ​[(1+R1​)×(1+R2​)×(1+R3​)…×(1+Rn​)]1/n  - 1

6 month geometric return = (1.05 x 1.02 x 1.06 x 0.96 x 1.00 x 1.03)1/6 - 1

6 month geometric return = ( 1.122545)1/6   - 1

6 month geometric return = 1.019445 - 1
6 month geometric return = 0.019445

6 month geometric return = 1.9445 %

Effective annual rate = ( 1.019445)2  - 1
Effective annual return = 1.039268 - 1
Effective annual return = 0.039268

Effective annual return = 3.9268 %

The effective annual difference between the 6-month geometric return and the 6-month arithmetic return
= 4.04 % - 3.9268 %
= 0.1132 %